analysis of imprints on light curves from kerr black holes due to time-dependent accreting...

8/2/2019 Analysis of Imprints on Light Curves from Kerr Black Holes due to Time-dependent Accreting Structures

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Faculty of Physics and AstronomyUniversity of Heidelberg

Diploma thesis

in Physics

submitted by

Adrian Stanislaw Kaminski

born in Opole, Poland.

Year of Submission: 2007



Analysis of Imprints on Light

Curves from Kerr Black Holes dueto Time-dependent Accreting

Structures

This diploma thesis has been carried out by Adrian Kaminski at the

Landessternwarte Konigstuhl

under the supervision of

Prof. Dr. Max Camenzind



Abstract (english)This thesis deals with ray-tracing on the Kerr geometry. As in earlier approaches to thisissue, the radiative transfer from the source to the observer is modelled by a backwardsray-tracing method, that determines the detected intensity distribution generated byradiation, which — emitted in the vicinity of the object investigated — travels alongphoton trajectories to the observer.Motivated by observed quasi-periodic oscillations (QPOs) in spectra of black hole candi-dates this work mainly focuses on time-dependent emitting structures and their imprints

on intensity distributions in time.These structures are represented by regions of enhanced density and emission (hot spots),which due accretion processes are orbiting the compact objects.

Abstract (deutsch)Die hier vorliegende Arbeit beschaftigt sich mit der Strahlungverfolgung in der Kerr

Geometrie. Dazu wird, wie schon bei fruheren Abhandlungen zu diesem Thema, derStrahlungstransport von der Quelle zum Beobachter hin mittels einer ruckwartsgerichteten Strhlungsverfolgung simuliert. Diese Methode bestimmt die gemessene Inten-sitatsverteilung, welche durch die Strahlung hervorgerufen wird, die nach ihrer Emissionin der unmittelbaren Umgebung des betrachteten Systems auf Photon Trajektorien zumBeobachter hin transportiert wird.Angeregt durch quasi-periodische Oszillationen (QPOs), wie sie in Spektren von Ob-

jekten entdeckt wurden, die als Kandidaten fur Schwarze Locher gelten, konzentriertsich diese Arbeit hauptsachlich auf zeitabhangige emittierende Strukturen und ihreAuswirkungen auf zeitliche Intensitatsverteilungen.Solche Strukturen werden durch Regionen erhohter Dichte und Emission, den so genan-

nten Hot Spots reprasentiert, welche aufgrund von Akkretionsprozessen das kompakteObjekt umkreisen.



Contents

List of Figures ix

Notations and Conventions xi

1. Introduction 11.1. Historical Background and Motivation . . . . . . . . . . . . . . . . . . . . 1

1.2. Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Phenomenological and Theoretical Background 5

2.1. Phenomenology of Black Holes . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1. Black Holes’ Masses and Evolution . . . . . . . . . . . . . . . . . . 5

2.1.2. Accretion Theory and Unified Scheme . . . . . . . . . . . . . . . . 8

2.1.3. Chasing Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2. General Relativity and Solutions of Einstein’s Field Equations . . . . . . . 19

2.2.1. Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2. Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3. Kerr Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3. The Time-dependent Ray-Tracer 41

3.1. Structure and Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2. Numerical Integration of the Nullgeodesics . . . . . . . . . . . . . . . . . . 61

3.3. Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4. Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5. Power Density Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4. Data Analysis 77

4.1. Static Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2. Dynamic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.1. Influence of Inclination and Spin on Light Curves . . . . . . . . . . 83

4.2.2. Influence of Hot Spot Radius and Distance on Light Curves . . . . 90

4.2.3. Generating Power Density Spectra . . . . . . . . . . . . . . . . . . 92

5. Conclusion and Outlook 105

A. Curved Spacetimes and Covariant Derivative 107

B. Killing Vectors and Symmetries 111

vii



C. 3+1 Split of Spacetime 113

D. Numerical Integration 115

Bibliography 119

Acknowledgements 123

viii



List of Figures

1.1. Cygnus X-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. spacetime diagram with light cone . . . . . . . . . . . . . . . . . . . . . . 6

2.2. Sagittarius A* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3. orbits of stars in the near of SGR A* . . . . . . . . . . . . . . . . . . . . . 82.4. Unified scheme of AGNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5. Generic spectral profile of AGN . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6. Fluorescence lines in AGN X-ray spectrum . . . . . . . . . . . . . . . . . 13

2.7. Transitions of electrons in atomic inner shells . . . . . . . . . . . . . . . . 14

2.8. Relativistic line broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9. Example for simulated broadened emission lines from . . . . . . . . . . . . 16

2.10. Data from measurements of Kα emission line . . . . . . . . . . . . . . . . 17

2.11. Power spectrum of Microquasar GRS 1915+ 105 . . . . . . . . . . . . . . 18

2.12. Signal propagation through Schwarzschild spacetime . . . . . . . . . . . . 23

2.13. Radial profiles of functions describing Kerr metric in BL coordinates fora = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.14. [Radial profiles of functions describing Kerr metric in BL coordinates fora = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.15. Effective potentials in equatorial plane in Kerr metric . . . . . . . . . . . 29

2.16. Characteristic radii in Kerr metric . . . . . . . . . . . . . . . . . . . . . . 30

2.17. Nullgeodesics on Schwarzschild metric (far view) . . . . . . . . . . . . . . 34

2.18. Nullgeodesics on Schwarzschild metric (close view) . . . . . . . . . . . . . 35

2.19. Nullgeodesics on Kerr metric (perspective along rotation axis) . . . . . . . 36

2.20. Nullgeodesics on Kerr metric (side view) . . . . . . . . . . . . . . . . . . . 37



3.1. Main procedure of ray-tracer . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2. Class hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3. Extent of the implemented accretion disc . . . . . . . . . . . . . . . . . . 54

3.4. Ray-tracing illustrated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1. Line spectra for different inclinations in the Schwarzschild case . . . . . . 78

4.2. Red wing of line spectra influenced by Kerr parameter a . . . . . . . . . . 79

4.3. Spatially resolved image of accretion disc around black hole for a = 0 andi = 30 ° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

ix






4.7. Hot spot orbiting black hole . . . . . . . . . . . . . . . . . . . . . . . . . . 854.8. Light curves over one rotation period for varying inclinations; a = 0 . . . 864.9. Light curves over one rotation period for varying inclinations; a = 0 . . . 864.10. Light curves over one rotation period for varying inclinations; a = 0.3 . . 874.11. Light curves over one rotation period for varying inclinations; a = 0.3 . . 874.12. Light curves over one rotation period for varying inclinations; a = 0.7 . . 884.13. Light curves over one rotation period for varying inclinations; a = 0.7 . . 88

4.14. Light curves over one rotation period for varying inclinations; a = 0.99 . . 894.15. Light curves for varying radial distances of the hot spot. . . . . . . . . . . 914.16. Light curves for varying hot spot radius . . . . . . . . . . . . . . . . . . . 914.17. Long time light curve for hot spot at r = 8M (a = 0) . . . . . . . . . . . . 944.18. PDS for hot spot at r = 8M (a = 0) . . . . . . . . . . . . . . . . . . . . . 944.19. Long time light curve for hot spot at r = 8M (a = 0.3) . . . . . . . . . . . 954.20. PDS for hot spot at r = 8M (a = 0.3) . . . . . . . . . . . . . . . . . . . . 954.21. Long time light curve for hot spot at r = 8M (a = 0.7) . . . . . . . . . . . 964.22. PDS for hot spot at r = 8M (a = 0.7) . . . . . . . . . . . . . . . . . . . . 964.23. Long time light curve for hot spot at r = 8M (a = 0.99) . . . . . . . . . . 974.24. PDS for hot spot at r = 8M (a = 0.99) . . . . . . . . . . . . . . . . . . . 97

4.25. Long time light curve for hot spot at r = ISCO (a = 0) . . . . . . . . . . 984.26. PDS for hot spot at r = ISCO (a = 0) . . . . . . . . . . . . . . . . . . . . 984.27. Long time light curve for hot spot at r = ISCO (a = 0.3) . . . . . . . . . 994.28. PDS for hot spot at r = ISCO (a = 0.3) . . . . . . . . . . . . . . . . . . . 994.29. Long time light curve for hot spot at r = ISCO (a = 0.7 ) . . . . . . . . . 1 0 04.30. PDS for hot spot at r = ISCO (a = 0.7) . . . . . . . . . . . . . . . . . . . 1004.31. Long time light curve for hot spot at r = ISCO (a = 0.99) . . . . . . . . 1014.32. PDS for hot spot at r = ISCO (a = 0.99) . . . . . . . . . . . . . . . . . . 1014.33. Long time light curve for two hot spots (a = 0.5) . . . . . . . . . . . . . . 1024.34. PDS for hot spots at r1 = 8M and r2 = 6M (a = 0.5) . . . . . . . . . . . 1024.35. Dependency of variabilities in light curves on radial coordinate r . . . . . 1 0 3

4.36. Dependency of variabilities in light curve on hot spot radius Rhs . . . . . 1 0 34.37. Dependency of variabilities in light curves on Kerr parameter a . . . . . . 1 0 4

x



Notations and Conventions

• Usually geometrized units will be used throughout the thesis. The gravitationalconstant G, the speed of light c and the Boltzmann constant kB are set to unity.

G = c = kB = 1 (0.1)

• The gravitational radius rg as natural length scale in general relativity is definedby:

rg =GM

c2≡ M (0.2)

• Four vectors are denoted either in component view kν or as bold symbol k

• Tensors are denoted by their indices T µν or by bold symbols T

• Einstein’s summation convention is applied, meaning that summation is performedon any index repeated in a product.

• With tensors, Greek indices µ, ν , κ, λ, . . . cycle the numbers 0 to 3, where Latinindices i, j, k, . . . cycle only the spatial coordinates 1 to 3. The temporal coordinateis always denoted by the letter t or the number 0.

• As not stated differently, the metric signature is (minus, plus, plus, plus)

• The partial derivative on tensors of arbitrary rank will be denoted by ∂ µ, whereas∇µ marks the covariant derivative.

• The solar mass is the typical mass scale in astrophysics

1 M⊙ = 1.989× 1030 kg (0.3)

• The typical length scale in the solar system is the Astronomical Unit (AU).Galactic length scales are given in multiples of the light year (ly) or in multiplesof the parsec (pc).

1 AU = 1.4959787 × 1011 m (0.4)

1 ly = 63240 AU = 9.4605 × 1015 m (0.5)

1 pc = 3.2615 ly = 206264.8 AU = 3.0856 × 1016 m (0.6)

xi



xii



1. Introduction

1.1. Historical Background and Motivation

Even though the term black hole was not used until the late 60’s [Whe1968], the historyof the concept of objects with a mass density sufficiently high to act as a “trap” evenfor light is a long one.

The idea of absolutely dark objects can be derived from Newtonian gravity in connectionwith a finite speed of light. Nothing should be capable of overcoming the object’s gravityfield when its escape velocity equals or exceeds the speed of light.In 1783 John Michell was the first to propose and describe such completely dark objects.Since that kind of thought experiment based on light corpuscles and could not explainthe gravity pull on light in the frame of the wave theory, the existence of these objectswas not substantive enough to become a notable field of interest in science until thebeginning of the last century.With Einstein’s formulation of the special and particularly the general theory of rela-tivity (“SR” and “GR”) in 1905 and 1915 [Ein1905, Ein1915a, Ein1915b, Ein1915c], aframework for further examination of dark objects was created.

As a geometric theory, desribing the link between a mass-energy distribution and thespacetime arising from it, the GR provides tools for the description of motion of anyobjects (fermions/bosons) in curved spacetimes and had a big impact on the scientificsociety.Not even a year later, Karl Schwarzschild derived and presented already the first solutionof Einstein’s field equations.It describes the static external [Sch1916a] and internal [Sch1916b] spacetime metric of a spherically symmetric mass distribution. In this context an event horizon is also pre-dicted. It acts as a boundary in spacetime, beyond which no event can affect the outerworld.Now that a theoretical basis was founded, the issue of compact objects and their space-

times could be investigated further on. Two approaches should be pointed out.On the one hand the solutions of the field equations were examined, leading to resultslike the generalization of the Schwarzschild solution to a spacetime for a spherically sym-metric point charge called the Reissner-Nordstrøm solution [Rei1916, Nor1918], and theBirkhoff theorem [Bir1923], proving the Schwarzschild metric to be the unique spheri-cally symmetric solution of Einstein’s field equations.On the other hand the origins for such metrics and spacetime singularities as a con-sequence thereof were questioned. (Even today the dislike of spacetime singularities,where all known physics fails, results in efforts for alternative solutions.)Examinating the evolution of stars, or massive objects in general, a gravitational collapsto high mass densities turned out to be indispensable for sufficiently massive objects.

1



1. Introduction

Subrahmanyan Chandrasekhar investigated the stability condition for white dwarfs aspossible end configurations for sun-like stars.

He found out that, due to the pressure dependence of the relativistic degenerated elec-trons (P ∝ 4/3), the hydrostatic equilibrium cannot be sustained for a mass exceedingthe critical Chandrasekhar mass of 1.46 M ⊙ [Cha1931a, Cha1931b]. In these cases theobject collapses gravitationally and as a result a neutron star can be formed.Furthermore, in 1939 J.R.Oppenheimer and George Michael Volkoff presented a similarlimiting critical mass for neutron stars [OpVo1939]. Exceeding that one, no stable endconfiguration for the object is possible, and the matter collapses into a singularity, form-ing a black hole.In the decades following, particularly in the 60’s, a lot of research on those extraordinaryobjects was performed. Roger Penrose and Stephen W. Hawking derived the singularity theorems proving the mathematical existence of singularities [Haw1969], and posted the

weak cosmic censorship hypothesis, which conjectures any intrinsic singularities to behidden from observers far away by an event horizon [Pen1969].In the meantime more general solutions of Einstein’s field equations were found.Roy Patrick Kerr developed the Kerr solution in 1963. It describes a stationary andaxially symmetric spacetime arising from rotating mass distributions [Ker1963].This solution was generalized furthermore to the Kerr-Newman solution ; a metric depen-dent on the maximal set of black hole parameters, including its mass, angular momentumand electric charge [New1965].

Lacking in any observational indications, black holes were pure theoretical subjects inthese times and the question, if they were observable at all, was legitimate. Today we

know several common methods for the identification of black hole candidates (BHCs).Although black holes cannot be observed directly, they interact with their environmentand therefore indicators for dark masses are present.The most prominent methods are of kinematical nature, as observing the motion of dy-namical objects in the BHC’s vicinity, and spectro-relativistic such as analyzing spectrafrom matter in the BHC’s environment for influences from GR.In the context of accretion onto black holes, bright X-ray sources and high luminositiesin general, as found at the so called active galaxy nuclei (AGN), are also strong hintsfor massive dark objects.In addition to that, BHC can reveal themselves through gravitational lensing.The first BHC, identified as such one, is a X-ray source observed firstly in 1971 by Tom

Bolton and is called Cyg X-1 [Bol1972]. Today this source is known to be a X-ray bi-nary consisting of a massive O9-B0 supergiant star and a stellar BHC of about ten solarmasses (see Figure 1.1).Being interested in astrophysically observable characteristics of black holes resultingfrom radiation from hot gas accreting around them, many groups examinated the issueof accretion. This resulted in several models which differ from each other in some of their properties. In spite of that, some fundamental features of the accretion geometry,as the presence of a cold accretion disc and a hot corona are widely accepted today.Led by these theoretical examinations, the search for BHCs was dominated by the searchfor bright X-ray sources.

2



1.1. Historical Background and Motivation

Figure 1.1.: Cygnus X-1 in the constellation Cygnus. Identified in 1971 as a BHC,

Cygnus X-1 is a binary consisting of a stellar black hole accreting mat-ter from its companion the supergiant HDE 226868.The distance to the star system is about 2500 pc. [Kal]

The first survey of the sky for such sources was performed by Uhuru 1, followed byprominent satellites like Einstein in 1978, EXOSAT and Tenma in the 80’s and ROSAT in 1990.Today’s major satellites are the RXTE , Chandra , XMM-Newton and the JapaneseAstro-E2 .In fact it is worth mentioning that the first black holes were found as radio sources inthe late 50’s already. Misinterpreted in the first place, the quasars2, known for their

high luminosities, rank among the AGN — galaxies harbouring supermassive accretingblack holes.

The observations performed until today provide data from BHCs throughout a widemass range.The smallest known are the stellar black holes with masses between 1 and 100 M ⊙. Su-permassive black holes (SMBH) with masses between 102 and 106 M ⊙ rank amongst thebiggest.

1“Uhuru” is the Swahili word for “freedom”. The satellite was launched from Kenya.2Term derives from “QUASi-stellAR radio source” and originates from detection circumstances.

3



1. Introduction

Current examinations concentrate on the aim of collecting data on black hole parameters

(particularly mass and angular momentum) in order to provide a link between observa-tions and theory.BHCs can be used as probes to the models describing black holes interacting with mat-ter in their environment, and furthermore for testing the validity of the GR at stronggravitation fields.Motivated by observations of fluorescence lines emitted from the accretion disc, thisthesis deals with the Kerr metric having certain imprints on the observed line profiles.Furthermore accreting dynamic structures breaking the system’s axial symmetry andtheir effect on the total flux are investigated, since flux variabilities, called quasi-periodicoscillations (QPOs) have been observed at many black hole systems.Understanding the origin of such features will contribute to the correct interpretation

of the observation data and can provide a significant approach to physics at stronggravitation fields.

1.2. Thesis Outline

As mentioned in previous section, this thesis’ primary concern is the investigation of dynamic structures in the vicinity of rotating black holes.For that purpose a relativistic ray-tracing method is used to simulate time-dependentintensity distributions originating from such objects.The ray-tracer computes the radiative transfer along null geodesics on the Kerr metricfrom the origin to the observer and provides relativistically deformed line profiles and

total fluxes at different times.In chapter 2 the different kinds of black holes with regard to their evolution and thevariety of detection methods are summarized. The origin of the fluorescence lines isdiscussed in the context of accretion models and the unified scheme of AGN is introduced.Furthermore, properties of the Kerr metric and the derivation of the geodesic equationsis presented.After that, the implementation of the problem is described in chapter 3, taking intoaccount the rendering procedure and the radiative transfer.The results of the simulations performed are presented in chapter 4 then.

4



2. Phenomenological and TheoreticalBackground

2.1. Phenomenology of Black Holes

Since this thesis’ subject is the simulation of black hole observations, an abstract onobjects of this kind is given in this section. Concerning their relevance for astronomyand observation efforts, only objects with masses around the solar mass and above aredescribed here.Black holes as the TeV black holes, being of certain interest in high energy physics, orthe Primordial black holes proposed in cosmology, having been formed in the early uni-verse’s post inflation era, are not regarded here furthermore.At first the formation of compact objects will be outlined. Processes in their environ-ment providing certain possibilities of detection will also be described, leading to theoutline of accretion models developed until today.Consequently the Unified Scheme of AGN , explaining the different AGN classes as man-ifestations of the influence of their orientation to the observer, is presented.

2.1.1. Black Holes’ Masses and Evolution

The term black hole describes a mass configuration with an efficiently high mass densityresulting in a gravitation field that leads zo the formation of an event horizon .That is a null hypersurface acting as a boundary separating spacetime points whichcan be connected to those at infinity1 by a timelike path from those which cannot (seeApp. B). Consequently no information can escape the inner region, although the horizoncan be passed the other way round, and the object should appear comletely dark.In general relativity the existence of an event horizon and its properties can be derivedfrom the metric describing the spacetime resulting from the mass configuration.Despite that, as mentioned in the introduction, the concept of a completely dark objectwas derived already from classical Newtonian gravity. Starting with a spherically sym-metric mass distribution, which gravity potential for points outside the distribution canbe described by that of a point source with the total mass M , the total energy neededby a particle with mass m to escape to infinity from given r (distance to point mass)can be evaluated to:

E =

∞r

F dr =GmM

r. (2.1)

1events sufficiently far away, so the spacetime can be regarded as asymptotically flat

5



2. Phenomenological and Theoretical Background

Figure 2.1.: Spacetime diagram portraying a light cone and space-, null- and timelikepaths. The axes are space (x) and time (t).

Comparing this to the kinetic energy E kin = 1/2 mv2 gained by a particle while fallingfrom infinity to the given r, the escape velocity v

∞can be extracted.

v2∞ =

2GM

r(2.2)

In order to escape the gravitational pull from the source, each particle starting at dis-tance r needs a velocity equal or above v∞ to reach infinity.Combining this to the limiting speed of light c results in a minimum distance rS , de-pending on the source’s total mass M , from which the infinity can be reached at all:

rS =2GM

c2. (2.3)

Comparing that result to the Schwarzschild solution (see Sec. 2.2.2), it should be noticed

that it equals to the position of the event horizon derived there.As this derivation already shows, the mass of the black hole is a crucial parameter.Therefore it is primarily used to distinguish those objects.The black holes of interest in astronomy are usually classified as stellar black holes, mas-sive black holes (MBH) or supermassive black holes (SMBH).

Stellar black holes with masses between 1 and 100 M ⊙ are present relicts of massivestars. During its active state of nuclear burning a star is at hydrostatic equilib-rium. The gravitational pressure pointing inwards is compensated by gas pressure,radiation pressure and by centrifugal forces. When nuclear burning comes to an

6




end, this balanced state cannot be sustained and a gravitational collaps occurs.For low masses this compression, resulting in high mass densities, can be stopped

by fermionic degeneracy pressure of electrons, leading to a white dwarf , or neu-trons at higher masses, leaving a neutron star behind. However, those statesare limited by critical masses — the Chandrasekhar mass of 1.46 M ⊙ for whitedwarfs [Cha1931a, Cha1931b], and the critical mass for neutron stars, which dueto uncertainties in the constitutive equation of matter at densities above thoseof core matter is not determined exactly, but should be located around 1.8 M ⊙[OpVo1939].Since the collapse of massive stars is usually accompanied by a supernova, duringwhich the outer shells are blasted away, the mass of the remnants can drop belowthe critical masses described.For sufficiently massive stars though, there is no known mechanism, which could

prevent the gravitational collapse to a black hole, where its total mass is gatheredin a singularity2.For those reasons, compact dark objects with masses above two solar masses arepassable black hole candidates (BHOs).Objects of that class are usually detected as X-ray binaries (Black Hole X-ray Binary/BHXB), where the black hole evades matter from its companion.

Massive black holes (MBH) , which objects with masses between 100 and 106 M ⊙ arecounted to, could not be confirmed by observations for a long time.First indications for such mid-mass black holes were found with Chandra in theyear 2000 [Kaa2000].Rotational curves from several dwarf galaxies and globular clusters hint for thepossibility of a black hole in the center of these systems.

Supermassive black holes (SMBH) represent the CDOs with masses above 106 up toapproximately 1010 M ⊙. Observations allow the conclusion that almost each galaxyharbours a SMBH in its centre. Quasars of high redshifts are supposed to possessblack holes with highest masses known.As will be described in sec. 2.1.2, the physics of active galactic nuclei require rotat-ing black holes as stimuli in galactic centres in order to explain the various AGNmanifestations as Seyfert galaxies, quasars, blazars and radio galaxies.A very prominent representative of the SMBHs is Sagittarius A* (Sgr A*), a com-pact and bright radio source in the center of the Milky Way , 8 kpc away.

From determinations of the orbit of S2 , a star near to SGR A*, its mass couldbe estimated to approximately 2.6 × 106 M ⊙, being centred within a region of aradius not more than 120 AU [Sch2002]. More recent observations indicate to amass of 3.7 × 106 M ⊙ within a volume with the maximal radius 45 AU [Ghez05](see Figures 2.2 and 2.3).These huge masses in tiny regions can only be explained by a SMBH.

2Although the existence of mathematical singularities is proved [Haw1969], their existence in reality isnot only not clarified, but furthermore is not favourised throughout the scientific community. Alter-native models such as gravastars [Maz2001] and holostars [Pet2003], which dispense with intrinsicsingularities, are developed. As they behave similar to black holes, the search for possible observableindicators favourizing one of the solutions is an important subject.

7




Figure 2.2.: Set of images taken with Chandra.The large picture shows a view of theMilky Way centre with SGR A* la-beled. The smaller images show close-ups of regions with evidences for anX-ray echo reflected at gas clouds inthe environment.(Credit: NASA/CXC/Caltech/M.Muno et al.)

Figure 2.3.: Orbits of starswithin the central1.0 × 1.0 arcseconds of the Milky Way. An-nual average positionsof seven stars andbest fittings of theirorbits are plotted here[UCLA].

2.1.2. Accretion Theory and Unified Scheme

Since isolated black holes could not be detected except by lensing effects due to thespacetime curved by them, common observation methods exploit effects of the interac-tion of CDOs with their environment. Especially the different types of AGN and theradiation emitted by them throughout the whole wavelength spectrum between radioand TeV suggest a complex matter configuration close to black holes.The main feature of black hole-matter systems is the accretion of matter. Due to corre-lations between gravitation and centrifugal forces, ifalling matter forms an accretion discin the equatorial plane, where matter is transported inwards, while angular momentumis carried away outwards. During that process, gravitational energy is transformed intoheat and can be disposed by radiation.With an efficiency factor ε for the transformation of accretion, characterized by theaccretion rate M , to radiation, this process results in the luminosity L:

L = εM . (2.4)

Luminosities and accretion rates of accreting objects can be estimated by utilizing theEddington luminosity LEdd

LEdd =4πGMmP

σT≈ 1.3 · 1046 ergs−1

M

108M ⊙

, (2.5)

8




where M is the mass of the gravitation source, mP the proton mass and σT the Thomsoncross section.

This Eddington limit can be derived from an equilibrium between radiation pressurepointing outwards and compensating the gravitation pressure [Mu2004].From eq. (2.5) it is possible to deduce masses of accreting objects when the luminosityis known.

Standard Accretion Disc

Since accretion is fundamental in black hole physics, the physics of accretion discs areimportant when interpreting observations. The standard accretion disc (SAD) is ananalytical solution to that problem derived from hydrodynamics in 1973/74 [Sha1973,Nov1974]. SADs are considered to be geometrically thin , which means that their half thickness H in vertical direction, in which hydrostatic equilibrium persists, is very smalleverywhere. This fact can be expressed by

H

r≪ 1 , (2.6)

where r is the radial coordinate.The velocity field is dominated by a profile that can be approximated by Keplarian rotation, meaning

Ωφ = 2πν φ ≈ ΩKep =

√M √r3

. (2.7)

For rotating black holes (comp. Sec. 2.2.3), which are characterized by the Kerr param-eter a = J/M (angular momentum/mass), this profile changes to

Ωφ ≈√

M √r3 + a

√M

. (2.8)

The nearly keplerian rotation can only sustain for radii higher than the marginally sta-ble, or innermost stable circular orbit (ISCO) rms. Therefore the SAD describes onlyregions above rms in the equatorial plane3.The orbits in the accretion disc are not purely circular, as a slight radial drift occurs,but still the disc can be regarded as being in hydrodynamical equilibrium.SADs satisfy efficient cooling, meaning that heating from shear effects in the disc is com-pletely radiated away. The temperature T in SADs decreases with the radius T ∝ r−3/4

and since matter at different radii emits black body spectra with maxima at differentwavelengths, the spectrum obtained from SADs is called a multi color black body spec-trum .

Although the SAD solution is self-consistent and used often for simulations, as it isalso in this thesis, it should be mentioned that there are features in black holes’ spectra,that cannot be explained adequately by SADs.

3This results in a gap between the inner edge of the disc and the event horizon. Only for maximal Kerrblack holes, where a = 1 those radii coincide, so the SAD reaces the event horizon.

9




Observations of Seyfert galaxies for example provide indications for truncated discs,which inner radii do not reach the rms.

Consequently further accretion disc solutions as ADAF, RIAF and more can be foundthroughout the literature. A well-organized summary of those can be found in [Mu2004].

Unified Scheme for AGN

As already mentioned, many observations can be explained by AGNs. Due to somevarying features in their spectra they can be subdivided in certain classes. It is widelyaccepted that this classification is not resulting from structural differences, but fromthe object’s orientation to the observer. The unification scheme for AGN (see Fig. 2.4)sketches the environment around a black hole manifesting as an AGN. The AGN are

Figure 2.4.: This sketch shows the matter configuration typical for an AGN, also calledthe AGN paradigm . The most important structures regarding acretion are:the slim accretion disc in the equatorial plane ranging down to a few rg,the hot corona in the central region, and the cold torus feeding the disc[Mu2004].

usually divided into two groups depending on their radio activity. Approximately 90%are rated as radio-quiet , which quasars and Seyferts are counted to. The rest is calledradio-loud , further classified as FR I , with generally low luminosities, a core dominatedradio emission and weak in the optical range, or FR II , with a radio emission being lobedominated, a strong optical emission and higher luminosities at all [Fan1974].

10




The features in the AGN spectra can now be explained by processes in the differentregions around the black hole and by the system’s inclination4 i to the obsever. From

spectral energy distributions from AGNs [Elv1994] a typical radiation profile can beextracted (see Fig. 2.5).

Figure 2.5.: Typical AGN continuum spectrum extracting. Contributions from differentsources in the AGN are distinguished by color. [Mu2004]

The spectrum consists of contributions from different sources:

• Starburst in the galaxy contributes to the lowest photon frequencies in the spec-trum.

• At wavelengths around 10µm the dust torus, located at distances of about 104rg

from the centre, dominates the spectrum. Scattering of hard radiation in the cold(T ≈ 1000 K) dust torus is responsible for the infrared radiation.

• The central region of the spectrum is shaped by the multi color black body spec-trum from the accretion disc.

4For line of sight along the symmetry axis (rotation axis) i = 0° and i = 90° for obsevers in theequatorial plane.

11




• The inner part of the system, where the hot corona is located, is dominant inthe X-ray range, forming a characteristic Compton continuum. This results from

inverse Compton scattering of soft photons at the hot coronal plasma.

• Imposed to the Compton continuum of the corona there is another contribution.Radiation from the hot inner regions hit the accretion disc and get reflected.This is where the fluorescence lines, as the Fe Kα, crucial to this thesis, originatefrom.

An important feature of AGN physics are the relativistic jets along the rotation axis.Jets are directed and highly collimated matter outflows, which can be observed not onlyin AGN, but also in BHXBs and others. They are formed as a result from the accretionprocess. High relativistic velocities in jets are supposed to be driven by twisted magneticfields at rotating black holes.

Observations of those jets served as a matter of excitement, since they indicate velocitieshigher than speed of light. However, this can be explained as a geometric consequencewhen observing low inclined relativistic jets.

Due to the presented matter configuration at AGN, the inclination to the observer isdecisive for interpreting observations of these systems.At high inclinations the dust torus can obstruct the view towards the inner regions.Therefore Seyfert I galaxies, which exhibit narrow and broad emission lines are ex-plained by low inclinations.As broad emission lines are assumed to origin from Doppler shifts at higher velocities,they should be emitted in the inner regions of AGN. A lack in those broad lines, as

observed at Seyfert II galaxies, indicates for high inclinations and the torus obscuringthe centre.

2.1.3. Chasing Black Holes

Supplied by an adequate concept of processes in black holes’ vicinity, indications for suchobjects can be searched for. The main task is then to determine the system’s parameterspace in order to gain an overview of black hole occurrences and to verify the theory.The major parameters are the black hole mass M , its angular momentum, specified bythe Kerr parameter a, the inclination i of the rotational axis to the observer and the

accretion rate˙

M .Several methods are exploited to get those information. Kinematical methods as simplytracing the keplerian orbits of objects in the gravitation field of the source, or reverber-ation mapping techniques measuring time lags between primary radiation and reflectedradiation in broad line regions (BLR), are used for determinations of the central mass.Apart from lensing effects due to the curved spacetime, high luminosities, as obtainedfrom AGNs, strongly hint for black holes.Another important characteristic in the observed spectra is commenly utilized to extractthe parameters. Fluorescence lines from the accretion disc are relativistically broadenedby effects from GR. Depending on underlying models, particularly the inclination andthe spin parameter can be accessed by interpreting those imprints correctly.

12




Fe Kα Fluorescence Line

Fluorescence lines in the keV range can be extracted from the AGN continuum spec-trum by subtracting the contribution from the corona (see Fig. 2.5 in sec. 2.1.2) in thedetected X-ray spectra.Lines from nickel (Ni), iron (Fe), chromium (Cr), calcium (Ca), argon (Ar), sulfur (S),silicon (Si), magnesium (Mg) and neon (Ne) contribute to the X-ray range of the spec-trum. Since they are superposed by the Compton continuum from the corona, whichcan be described by a characteristic power-law, the most dominant line, the Fe Kα, isgenerally observed at Seyferts, quasars and BHXBs.

Figure 2.6.: Fluorescence lines contributing to the X-ray range superposed by the coronalCompton continuum (power-law). Due to its dominance, the Fe Kα line canbe extracted most easily [Rey1996].

It originates from the transition of an electron from the L to the K shell, emitting aphoton with 6.4 keV in the rest frame (see Fig. 2.7), after being lifted by photo-electricabsorbtion of X-ray photons from the corona 5. The threshold value for the absorptionis 7.1 keV for neutral Fe and increases with the level of ionization.

5A significant decay of the detected primary spectrum in the region of about 7 keV results from thatand can be verified in the spectra.

13




Figure 2.7.: Atomic inner shell transitions.

Obviously it can only occur, if the fluorescending matter is not completely ionized. At

least the inner K and L shells should be occupied, which should be the case for thetemperatures in the accretion disc.A reason for the line contributions being relatively low, is another process competingwith the fluorescence. Absorbtion of the photon energy can result in a radiationless statetransition, when another electron is released as an Auger electron .This case is more efficient (66% probability for Fe) and suppresses the fluorescence.For the other elements, listed above, the efficiency ratio for the photon emitting transi-tion is even lower. Therefore the Fe Kα line is the most prominent.

The shape of the fluorescence line, formed by relativistic effects from GR, is now asignificant tool for the determination of the emitting matter’s state of motion and fur-

thermore black hole parameters. The most important imprints on the line’s shape are(see Fig. 2.8):

• The classical Doppler effect results from radiation being blue-shifted when emit-ting source is moving towards the observer, and contrariwisely red-shifted whenmoving in the other direction.The originally monochromatic emission line, originating from the rotating accre-tion disc, is symmetrically broadened and exhibits a characteristic “double-horned”structure.

• Two effects originate from the formulation of SR:

14




The transverse Doppler effect results in a general slight red-shift due to thesource’s motion and relativistic beaming raises the blue wing of the line’s spec-

trum due to collimation of the radiation in the relativistic emitter’s direction of motion.

• The gravitational source determines the curvature of the spacetime in its vicinityand forms the gravitational potential. Photon trajectories (nullgeodesics) frominner to outer regions come with an effort in energy and result in the gravitationalredshift.

Figure 2.8.: A monochromatic emission line from an accretion disc is being formed byvarious effects, resulting in a relativistically broadened line profile [Fab2000].

Since the examination of the Kα line became such a convincing tool, many groups devel-oped techniques in order to simulate lines as they should appear to the observer [Fab1989,Dab1997, Fan1997, Schn05].Those methods can be outlined as followed: Underlying a certain radial emission profile

15




to the accretion disc, contributions from each point on it to the spectrum are summedup. This is performed by implementing the radiative transfer along the photon trajecto-

ries from the emmitter to the observer. This technique is called ray-tracing and providessimulated spectra of relativistically broadened emission lines.The inclination i turned out to be the parameter, the spectra depend mainly on. TheKerr parameter a primarily influences the spectrum’s red wing for accretion discs, whichrange down to the rms. This is due to the fact that for increasing spin parameters theISCO moves towards the event horizon and the gravitational redshift in these regionshas a bigger impact.An example for simulated line profiles for different inclinations, taken from results of [Schn05], is presented in Fig. 2.9, whereas figures 2.10a-d present example data fromextractions of the Kα line.

Figure 2.9.: Spectra of relativistically broadened emission line from an uniformly emit-ting disc, ranging from Rin = rms to Rout = 15 rg. The Kerr parameter isset to a = 0.5 and different inclinations i are evaluated. [Schn05].

16




(a) Seyfert I, MCG-6-30-15; ASCA, [Nan1997] (b) Seyfert I, IC 4329A; ASCA, [Nan1997]

(c) Seyfert I, NGC 3516; ASCA, [Nan1997] (d) Seyfert I, NGC 3227; ASCA, [Nan1997]

Figure 2.10.: Several example data from measurements of the profile of the Kα fluores-cence line at 6.4 keV in emmitter’s rest frame.

17




QPO’s

The X-ray spectra show another interesting property concerning the observed flux distri-bution in time. Nearly periodic features, quasi-periodic oscillations (QPOs), are detectedin X-ray light curves from BHXBs. Two different classes can be distinguished due totheir oscillation frequency. The high frequency quasi-periodic oscillations (HFQPOs) inthe kHz range and the low frequency QPOs (LFQPOs) in the Hz.Due to limits in timing resolution of the observations, those features are usually ex-amined in Fourier space of time-depending light curves. The resulting power density spectra (PDS) then exhibit peaks with characteristic amplitudes at certain frequencies(see Fig. 2.11 as example).Several approaches are analyzed in order to explain and to simulate these observations.Dynamic structures, regions of higher emission co-rotating within the accretion disc (hot spots), should imprint a characteristic signature to light curves and are implemented inray-tracing applications [Bao1992, Schn05], as it is performed within this thesis (seechap. 3).Another concept is often favorized to explain the variabilities described. Accretion discs,slightly inclined with respect to the black hole’s rotation axis, should perform a precession(Lense-Thirring precession) due to effects from GR. This interaction between gyroscopes(rotating masses) is known as gravitomagnetic spin-spin interaction 6.

Figure 2.11.: Average power spectrum from observations of GRS 1915+ 105 with theRossi X-Ray Timing Explorer in the 13-27 keV energy band featuring aQPO at 40 Hz. The solid curve represents the best fitting [Str2001].

6The term originates from the analogy to classical electrodynamics, where moving electrical chargesproduce magnetic fields.

18



2.2. General Relativity and Solutions of Einstein’s Field Equations

2.2. General Relativity and Solutions of Einstein’s FieldEquations

In contradiction to classical formulations, in general relativity gravitation arises fromthe geometric structure of the generally not flat but curved spacetime. The concepts of GR describe, how the spacetime, which can be defined by its metric, is formed due topresent mass (energy) distributions and contrariwisely how it affects those. Equations of motion for particles arise directly from the metric, since they are moving along geodesics.Therefore the procedure of exploiting a given system’s properties usually demands thederivation of the corresponding metric from Einstein’s field equation (2.13). That canbe thought of as a set of second-order differential equations for the metric tensor gµν .Due to its structure and its nonlinear nature, resulting from the fact that in GR thegraviatational field couples to itself, solving the field equation is a non-trivial task. Even

in vacuum cases simplifying assumptions, as isometries in the solution, are usually nec-essary.

In this part of the thesis, the field equation and its major solutions are introduced,following the concepts presented in [Carr, Fli, MTW].

2.2.1. Field Equations

The metric of a general Riemann spacetime is defined by the line element

ds2 = gµν (x)dxµdxν , (2.9)

where gµν (x) is the coordinate depending metric tensor. The dependency of its compo-nents on the coordinates is a manifestation of the spacetime curvature.This is why coordinate transformations to a flat Minkowski metric ds2 = ηµν dx′µdx′ν

are only possible locally, yielding the relation

gµν = ηαβ ∂x′α

∂xµ∂x′β

∂xν (2.10)

between gµν and the metric tensor η = diag(−1, 1, 1, 1) of the minkowski metric.Since the metric describes the spacetime curvature, which is formed by masses, or energyin general, the sources of gravity in a given system must be taken into account.

They are summed up by the divergence-free7 and symmetric stress-energy tensor T µν .Now the ansatz for the field equation can be stated:

G ∝ T (2.11)

On the righthand side the sources are placed, whereas on the lefthand side Gµν is ageometric object characterizing the gravity. As such one, some constraints to G can beposted:

• In flat spacetimes G vanishes.

7This condition expresses the conservation of momentum and energy.

19




• It is constructed from and only from the Riemann curvature tensor R (see App. A)and the metric components, with

– being linear in R

– being symmetric and of second rank

– having a vanishing divergence ∇ ·G ≡ 0

• For weak fields (gµν ≈ ηµν ) the field equation should reduce to the Newtonian caseR00 = 4π

.

It can be proved that apart from a multiplicative constant, only

Gµν = Rµν − 1

2gµν R (2.12)

satisfies those requirements, where Rµν is the Ricci tensor and R is the curvature scalar .Those can be derived from contractions of the Riemann curvature tensor by Rµν =gκλRκµλν and R = Rµ

µ = gµν Rνµ .Exploiting the weak field approximation, the proportionality in (2.11) can be determined,leading to the final Einstein field equation

Gµν = Rµν − R

2gµν = 8πT µν . (2.13)

Gµν is often referred to as the Einstein curvature tensor 8.A contraction of this one yields R = −8πT . Plugging this into (2.13) results in analternate form

Rµν = 8π

T µν − T

2gµν

. (2.14)

When interested in vacuum solutions, T µν is set to zero and the vacuum Einsteinequation for those cases is then given by:

Rµν = 0 . (2.15)

2.2.2. Schwarzschild Metric

The Schwarzschild solution9 describes the metric outside a spherically symmetric mass

distribution. To derive such one, it is necessary to exploit the Einstein field equation forthe vacuum case (2.15).Introducing polar coordinates xµ = (t,r,θ,φ) and imposing spherical symmetry also forthe solution, the ansatz for the metric searched for can be written as

ds2 = −e2α(r) dt2 + e2β (r) dr2 + e2γ (r)r2dΩ2 , (2.16)

where dΩ2 = dθ2 + sin2 θ dφ2 is the metric on a unit two-sphere, and α(r), β (r), γ (r)are functions of the radial coordinate r.

8Due to geometrized units, the factor G/c4, often found in literature, is ignored in (2.13).9More precisely the exterior Schwarzschild solution.

20




Choosing a new radial coordinate r′ = eγ (r)r, the factor e2γ (r) can be disposed, and themetric can be denoted as

ds2 = −e2α(r) dt2 + e2β (r) dr2 + r2dΩ2 .10 (2.17)

To determine α(r) and β (r) from (2.15), the components of the Ricci tensor must becalculated. The nonvanishing components are:

Rtt = e2(α−β )

∂ 2rα + (∂ rα)2 − ∂ rα∂ rβ +2

r∂ rα

(2.18a)

Rrr = −∂ 2rα− (∂ rα)2 + ∂ rα∂ rβ +2

r∂ rβ (2.18b)

Rθθ = e−2β [r(∂ rβ − ∂ rα)− 1] + 1 (2.18c)

Rφφ = sin2 θ Rθθ (2.18d)

By exploiting the relation Rtt = Rrr = 0 and rescaling the time coordinate, the resultα = −β can be derived. Rθθ = 0 yields ∂ r(re2α) = 1, which can be solved to obtain

e2α = 1 − c

r, (2.19)

where c is a constant to be determined.This can be achieved by comparing the derived metric to the metric around a pointmass in the weak-field limit, where the gtt component satisfies gtt = − 1− 2GM

c2r

=

− 1− rSr

. Since for r →∞ the schwarzschild solution should reduce to the weak-field

case, the missing constant can be identified as c = rS , called the Schwarzschild radius.

The final form of the Schwarzschild metric reads then:

ds2 = −

1− rS r

dt2 +

1− rs

r

−1dr2 + r2dΩ2 . (2.20)

Properties of the Schwarzschild Metric

The Schwarzschild solution describes the metric outside spherically symmetric masses.The fact that for r →∞ the metric becomes flat (gµν ≈ ηµν ) is called asymptotical flat-ness. The temporal coordinate t can be interpreted as the time (proper time), measuredby an observer at infinity.For some objects this metric is not applicable, since the Schwarzschild radius rS wouldlie within the mass distribution. Assuming a homogenous mass distribution with the

density within the radius R, a criterion for can be generated.With the condition that rs = 2GM/c2 > R, where M = V is the central mass, a criticalmass density crit can be approximated to

crit =3M

4πr3S

=3c6

32πG3M 2. (2.21)

For the Earth for example it would mean that its mass had to be compressed to thesphere with the radius r ≈ 2.7cm.

10Precisely instead of r, r′ should be used here, but this is only a matter of labeling and will not beconsidered.

21




So, for objects not exceeding crit this exterior Schwarzschild solution needs to be ex-panded.

Nevertheless, for non-rotating black holes the Schwarzschild metric is applicable.

When examining the metric (Eq. 2.20), two regions arise as a matter of concern.For r → 0 and r → rS the metric coefficients gtt respectively grr diverge. Since the metriccoefficients are coordinate-dependent, such singularities can result from the breakdownof the coordinate system chosen, and do not need to be meaningful for the spacetimegeometry. In fact, an adequate coordinate system can be found in this case that thesingularity at r = rS vanishes (Eddington-Finkelstein coordinates).Singularities, originating from the choice of the coordinate system are therefore calledcoordinate singularities.Intrinsic singularities, arising from the metric curvature, can be detected by investigat-

ing scalars, constructed from the curvature tensor. Such scalars can be the curvature(Ricci) scalar R = gµν Rµν , or any higher-order scalars as Rµν Rµν , RµνρσRµνρσ, and soon. A singularity of the metric’s curvature occurs, whenever one of those scalars divergesat a certain point. For the schwarzschild metric it can be shown that r = 0 implies sucha singularity due to

RµνρσRµνρσ =48G2M 2

r6. (2.22)

Since the metric coefficient gtt changes its sign at r = rS , the surface, described by thatcondition, is still interesting.Considering radial null curves, for which θ and φ are constant and ds2 = 0, yields

dt

dr= ±

1− rS

r

−1. (2.23)

When approaching rS , the quantity dt/dr diverges. Interpreting t as the proper timeof an observer far away, it implies that from his point of view, light rays would neverreach that surface, or the other way round would never reach infinity when originatingfrom there. This time lapse is also the reason for photons being gravitationally redshifted(blueshifted respectively), when their frequency is measured at different radial distances.The contribution from the gravitational redshift of photons measured at different radialdistances rA, rB can be denoted as:

z = λBλA

− 1 =

gtt(rB)gtt(rA)

1/2

− 1 . (2.24)

Those time lapses are illustrated in Fig. 2.12, which shows the propagation of signalsthrough spacetime when emitted at different r.The surface defined by r = rS is the event horizon of the Schwarzschild metric. Itseparates points that are connected to infinity by timelike paths from those that are not.This property manifests itself in the Killing vector K µ = (∂ t)

µ (see App. B) changingfrom being timelike to spacelike at r = rS .It is important to mention that the event horizon, being that interesting for observersat infinity, derives from the coordinate choice (the temporal coordinate t particularly).

22




Figure 2.12.: Spacetime diagram showing signals being emitted at intervals of constantproper time ∆τ 1 and being detected by an observer at fixed r with longertime intervals ∆τ 2. [Carr]

For objects moving along geodesics the event horizon is not a boundary and can be

traversed. Therefore free falling particles, moving along geodesic tra jectories, reach theevent horizon and even the centre at finite proper times.The curvature of the Schwarzschild metric around black holes can be visualized by thegeodesic flux around the event horizon (see Fig. 2.17, 2.18).It can be proved that the Schwarzschild metric is the unique static and sphericallysymmetric solution to the Einstein’s field equation in vacuum [Bir1923].

2.2.3. Kerr Metric

Since the Schwarzschild solution, presented in the previous section, is a spherically sym-metric solution and for rotating black holes this symmetry breaks down, it is not appli-cable to those objects. The most general solution, the Kerr-Newman metric [New1965],was found in 1965 and is characterized by the maximum set of black hole’s parameters,namely the mass, the angular momentum and electrical charges.Due to the argument that any electric charge from the black hole should be compensatedby electric currents in its vicinity, not the general Kerr-Newman metric but the Kerrsolution is considered commenly. This one is parametrized only by the central mass M and the Kerr (spin) parameter a = J/M .Due to conservation of angular momentum during collapses, the assumption of rotatingcompact objects seems sensible.

23




Formulation

Before introducing the Kerr metric in Boyer-Lindquist form , which is widely spreadthroughout the literature, a general formulation for axially symmetric and stationaryspacetimes is presented. The Papapetrou line element applies to both, the vacuum andthe non-vacuum case [Pap1966]11

ds2 = e2Φdt2 − e2Ψ(dφ− ωdt)2 − e2µ2(dx2)2 − e2µ3(dx3)2 (2.25)

Here the set of coordinates is (t,φ,x2, x3) and the functions Φ, Ψ, ω, µ2 and µ3 are onlydependent of the spatial coordinates x2 and x3. The two obvious symmetries imply theexistence of two conserved quantities, which turn out to be the total energy E and thetotal angular momentum J .Applying the vacuum Einstein equation to the ansatz and exploiting the gauge freedomfor the functions µ2 and µ3, the Kerr metric can be derived.In cartesian coordinates it can be written as [Cha1983]:

ds2 = dt2 − dx2 − dy2 − dz2

− 2M r3

r4 + a2z2

dt− 1

r2 + a2[r(x dx + y dy) + a(x dy − y dx)]− z

rdz

2 (2.26)

Since the function y depends only on x, y, z, and a, the only parameters, the metricdepends on, are as stated the mass M and the specific angular momentum, also calledKerr parameter a. In geometrized units a ∈ [−M, M ] and with M normalized to 1,

a ∈ [−1, 1].Since the formulation (2.26) is quite unhandy, the Kerr solution is often denoted inthe pseudo spherical Boyer-Lindquist coordinates (t,r,θ,φ) [Boy1967], where the lineelement takes the form

ds2 = −α2dt2 + 2(dφ− ωdt)2 +ρ2

∆dr2 + ρ2dθ2 , (2.27)

yielding the metric coefficients gµν

gµν =−α2 + ω2 2 0 0 −ω

2

0 ρ2/∆ 0 0

0 0 ρ2 0−ω

2 0 0 2

(2.28)

and the components of the inverse metric tensor gµν

gµν =

−1/α2 0 0 −ω/α2

0 ∆/ρ2 0 00 0 1/ρ2 0

−ω/α2 0 0 1/

2 − ω2/α2

. (2.29)

11Here, as well as in Eq. 2.26, the metric signature (plus, minus, minus, minus) is used.

24




The functions α, ∆, ρ, ω, are functions of the two spatial coordinates r, θ and thetwo parameters a and M :

∆ = r2 − 2M r + a2 (2.30a)

ρ2 = r2 + a2 cos2 θ (2.30b)

Σ2 = (r2 + a2)2 − a2∆sin2 θ (2.30c)

α2 =ρ2∆

Σ(2.30d)

ω =2M ra

Σ2(2.30e)

=Σ

ρsin θ (2.30f)

They can be interpreted as follows.

• The function α is often referred to as the lapse function (see App. C) describingthe general relativistic time dilatation.

• ω is called the frame-dragging frequency and, being defined as ω = −gtφ/gφφ, itarises from the cross term gtφ.Since the rotation of the metric is described by this function, it vanishes for theSchwarzschild case when a = 0 and the metric becomes diagonal.

• As · 2π equals the circumference of cylinders with radius r and centred at therotation axis, it is also called cylindrical radius.

• ∆ and ρ are geometric functions. ρ is related to the radial coordinate r by degen-erating to that for θ = π/2 in the equatorial plane. From ∆ = 0, the positions of the event horizons can be derived.

The radial profiles of those functions in the equatorial plane for the extreme Kerr casea = 1 and the Schwarzschild case a = 0 are illustrated in Fig. 2.14 and Fig. 2.13respectively. The profiles are visualized for radii above the (outer) event horizon, whichlies at r = M for the extreme Kerr case and at r = rS = 2M for the Schwarzschild case.Interesting to point out is the behaviour of the lapse function α, which in both casesbecomes null at the event horizon, implying an infinite redshift, and converges to α = 1for r

→∞. That describes the asymptotical flatness of the Kerr metric.

For low radii, the frame-dragging frequency ω increases when a = 0. Consequently thedragging of the spacetime, evoking from the rotating central mass, primarily has animpact at small distances, whereas it becomes null for large r.Evaluating these quantities for a = 0 makes clear that the Kerr solution (2.27) reducesto the Schwarzschild metric as it should, due to the spheric symmetry being restored.The uniqueness of the Kerr metric as a stationary axially symmetric solution of theEinstein’s field equation in vacuum is stated by the Robinson theorem [Rob1975].

25




2 4 6 8 1 0

f

u

n

c

t

o

n

v

a

u

e

s

[

g

e

o

m

t

r

z

e

d

u

n

t

s

]

r a d i u s r [ M ]

a = 0

Figure 2.13.: Radial profiles for the Boyer-Lindquist functions in the equatorial planefor the Schwarzschild case a = 0. Only radial distances above the eventhorizon rS are considered.

2 4 6 8 1 0

f

u

n

c

t

o

n

v

a

u

e

s

[

g

e

o

m

t

r

z

e

d

u

n

t

s

]

r a d i u s r [ M ]

a = 1

Figure 2.14.: Radial profiles for the Boyer-Lindquist functions in the equatorial plane forthe extreme Kerr case a = 1. Only radial distances above the (outer) event

horizon are considered.

26




Properties

Looking for regions, where the lapse function α becomes null for arbitrary values of the

Kerr parameter a, the condition ∆ = 0 yields

r±H = M ±

M 2 − a2 . (2.31)

It is interesting that for a = 0 there are two horizons, namely the outer horizonr+H = M +

√M 2 − a2 and the inner horizon12 at r−H = M +

√M 2 − a2.

For a = 0 the two horizons coincide at r = rS . Since the outer horizon is a boundaryfor any information being able to reach infinity, the inner horizon is not considered herefurthermore.Actually, both horizons are independent of the poloidal coordinate θ, resulting in spher-ical symmetry.

The result in (2.31) also implies that the absolute value of a must not exceed M , whichwould lead to the breakdown of the horizons and to a naked singularity.

A divergence of metric components at r±H implies singularities there. Since those can beavoided by a transformation to an adequate coordinate system, they can be consideredas coordinate singularities.In analogy to the Schwarzschild metric, which exhibits an intrinsic singularity at r = 0,there is also an intrinsic singularity at r = a in the equatorial plane. In contrast to theSchwarzschild solution, this unavoidable singularity is ring-shaped.

Another interesting feature arises from the fact that the metric component gtt changes

signs outside the event horizon. Solving gtt = 0 gives

rerg = M +

M 2 − a2 cos2 θ . (2.32)

The surface, defined by (2.32), is called the ergosphere and due to its θ-dependency hasan oblate structure, coinciding with the outer event horizon at the poles at θ = 0 andθ = π.Given the angular frequency of an observer in the Kerr metric Ω = U φ/U t13 and thecondition for a globally time-like velocity field gtt + 2Ωgtφ + Ω2gφφ > 0, the angularfrequency is limited to:

Ω− ≤

Ω≤

Ω+

, with Ω±

= ω±

α

(2.33)

Depending on the frame-dragging frequency ω this limitations become important forsmall radii. In (2.33) the upper limit Ω+ is the limit for prograde rotating observers,whereas Ω− stands for the limit of retrograde rotation.Evaluating that at the ergosphere leads to the conclusion that for radii lower than rerg

no static14 observers can exist, as the lower limit Ω− becomes positive. Therefore thesurface, defined by rerg(θ), is also called the static limit .

12Also called Cauchy horizon .13U µ denotes the four-velocity.14Static in relation to the coordinate system.

27




At the outer event horizon, where α = 0, the metric and anything else rotates with theangular frequency of the event horizon

ΩH = ω(r+H) = a

(r+H)2 + a2

.15 (2.34)

The region between the ergosphere and the outer event horizon, where frame-draggingof the metric forces anything to co-rotate with the central object is called the ergoregion .

Discussing the effective potentials in Kerr metric, the so called marginally stable or-bit or innermost stable circular orbit ISCO can be found. Regarding movement in theequatorial plane, it is the minimal radius, where stable rotation is possible. In the ef-fective potential curves (see Fig. 2.15), which are parametrized by the specific angularmomentum λ, the minima correspond to stable circular and the maxima to unstable

circular orbits. The marginally stable orbit rms is marked by a saddle point in the po-tential curve of the minimal λ ≈ 3.464.The ISCO is given by

rms = M

3 + Z 2 ∓

(3− Z 1)(3 + Z 1 + 2Z 2)

, (2.35)

with the functions Z 1 and Z 2

Z 1 = 1 +

1− a2

M 2

1/31 +

a

M

1/3+

1− a

M

1/3

(2.36a)

Z 2 =

3a2

M 2 + Z 21 , (2.36b)

where the upper signs correspond to prograde and the lower signs to retrograde orbits.It is worth noticing that for the extreme Kerr case (a = 0) the ISCO lies at the outerevent horizon, so stable circular orbits occur in its vicinity.Those characteristic radii ascribed above are visualized for the equatorial plane inFig. 2.16 for the whole range of possible values of the spin parameter a.The radii coincide for the extreme Kerr case at the event horizon rH = 1rg = 1M .With a = 0, specifying the Schwarzschild case, the ring singularity becomes point-likeand the event horizon lies at rH = rS = 2rg = 2M .It is remarkable that unstable circular photon orbits exist in the Kerr metric.

The radius for those orbits depends on a and lies between the ergosphere and themarginally stable orbit rms.

15Notice that for the Schwarzschild case, no rotation is allowed at the event horizon.

28




Figure 2.15.: The effective potential curves are parametrized by the specific angular mo-mentum. The minima mark stable circular orbits. The saddle point of thepotential with the limiting λ = 3.464 corresponds to the marginally stablecircular orbit rms. [Cam1997].

29




- 1 , 0 - 0 , 5 0 , 0 0 , 5 1 , 0

r

a

d

u

s

r

[

M

]

K e r r p a r a m e t e r a [ M ]

r i n g

Figure 2.16.: The characteristic radii of the Kerr spacetime in the equatorial plane fordifferent values of a, where a < 0 corresponds to retrograde rotation and

a > 0 for prograde rotation.

30




Geodesics in Kerr Metric

In order to examine the radiative transfer in the Kerr metric, the trajectories, alongwhich photons are moving, need to be determined. Usually for this purpose, the geodesicequation (A.8) needs to be solved.Alternatively the equations of motion can be derived from the Hamilton-Jacobi formal-ism adapted to the GR. This was done by B. Carter in 1968 [Car1968] and resulted infinding the fourth conserved quantity (Carter’ constant C) and an elegant extraction of the integrals of motion.

Applying the Hamilton-Jacobi formalism for geodesics parametrized by the affine pa-rameter λ on a metric, defined by the metric components gµν , the Hamiltonian is givenby:

H =

1

2 gµν

pµ pν =

1

2 gµν ∂S

∂xµ∂S

∂xν , (2.37)

where pµ are the covariant momentum components given by the partial derivatives of the action S .Evaluating the hamiltonian for the Kerr metric, one can see that it is not explicitelydependent on the variables t and φ. Taking into account the Hamilton function

d pµdλ

= − ∂ H∂xµ

, (2.38)

this fact implies the two quantities pt and pφ to be conserved along the geodesic.Those can be associated to the energy E of the particle, as it is measured at infinity,

and its axial component of angular momentum Lz.The third conserved quantity is the particle’s rest mass, given by the norm of the four-momentum µ =

√−gκσ pκ pσ.The Hamilton-Jacobi equation

∂S

∂λ+H

xµ,

∂S

∂xµ

= 0 (2.39)

evaluated using (2.37) yields a differential equation for the action function S :

2∂S

∂λ=

Σ2

ρ2∆ ∂S

∂t 2

+4aM r

ρ2∆

∂S

∂t

∂S

∂φ

− ∆− a2 sin2 θ

ρ2∆sin2 θ

∂S

∂φ

2

− ∆

ρ2

∂S

∂r

2

− 1

ρ2

∂S

∂θ

2 (2.40)

To solve this, a separation of S in all variables proves to be adequate:

S =1

2µ2λ− Et + Lzφ + S r(r) + S θ(θ) (2.41)

Here the first three terms are fixed by the known conserved quantities ∂S/∂λ = −H =µ2/2, pr = ∂S/∂t = −E and pφ = ∂S/∂φ = Lz. The functions S r(r) and S θ(θ) arefunctions of only r respectively θ.

31




Plugging this ansatz into (2.40) and arranging the terms according to their variablesgives

µ2r2 − 1

∆

(r2 + a2)E − aLz

2+ (Lz − aE )2 + ∆

∂S r∂r

2

+

a2µ2 + (L2

z csc2 θ − a2E 2)

cos2 θ +

∂S

∂θ

2

= 0 .

(2.42)

Obviously the upper term here depends only on r and the lower term only on θ. Conse-quently the both terms need to be constant independently from each other. Exploitingthis fact, (2.42) can be split by imposing the constant C, which is the fourth conservedquantity:

∂S θ∂θ

2

= C − (L2z csc2 θ − a2E 2 + a2µ2)cos2 θ = Θ (2.43a)

∆

∂S r∂r

2

=1

∆

(r2 + a2)E − aLz

2 − C + (Lz − aE )2 + µ2r2

=R

∆(2.43b)

with the abbreviations

P = (r2 + a2)E − aLz , (2.44a)

R = P 2 −∆C + (Lz − aE )2 + µ2r2

, (2.44b)

Θ = C −

a2(µ2 − E 2) + L2z csc2 θ

cos2 θ . (2.44c)

Applying (2.43a,b) to (2.41) gives now

S =1

2µ2λ−Et + Lzφ +

√R

∆dr +

√Θdθ . (2.45)

The equations of motion for the Kerr metric can now be derived by evaluating xµ =gµν pν = gµν (∂S/∂xν ). Since we are interested in radiative transfer, only nullgeodesics(µ2 = 0) are considered:

t =1

ρ2∆(Σ2E − 2aMrLz) (2.46a)

r2

=

R

ρ4 (2.46b)

θ2 =Θ

ρ4(2.46c)

φ =1

ρ2∆

2aMrE + (ρ2 − 2M r)Lz csc2 θ

(2.46d)

The algorithm, implemented to solve these, is presented in the next chapter.Sometimes in literature another quantity can be found in this context, namely the con-stant K = C + (Lz − aE )2.With this relation the abbreviations R and Θ become:

32




R =

(r

2

+ a

2

)E − aLz2 −∆K (2.47a)

Θ = −(aE sin θ − Lz csc θ)2 − a2µ2 cos2 θ +K (2.47b)

To obtain a better idea of the metric, it is useful to take a look on the geodesic flux.Figures 2.17-2.22 visualize the null-geodesics around a central mass. Those photon tra-

jectories are computed by numerical integration of (2.46a-d) on the Kerr metric, wherethe initial conditions are fixed at the position of a virtual observer far away. At thispoint in spacetime the geodesics coincide.Applying Boyer-Lindquist coordinates with the central mass in the origin and its rota-

tion axis marking θ = 0, the observer’s position is defined by the radial coordinate rand the poloidal coordinate θ. Due to stationarity and axial symmetry, t and φ can beneglected here.For the visualizations only θ, the Kerr parameter a and the perspective were varied.The radial distance of the observer was set to r = 800M . Two different accuracies forthe intagration of the geodesics are indicated by color, where the red ray serves as areference with a precision 100 times higher than the white ray. The yellow circle marksthe event horizon, whereas the cyan structure represents the ergosphere.The Schwarzschild case (a = 0) can be examined at the figures 2.17 and 2.18, whichdiffer in scale.The nullgeodesics for the extreme Kerr case (a = 1) can be found at figures 2.19-2.22.

Unlike in the Schwarzschild metric, the spherical symmetry is disturbed here, and there-fore the image’s perspective and the poloidal coordinate of the virtual observer (initialcondition for geodesic tracing) need to be specified.The illustrations show the influence of frame-dragging on the geodesics close to thecentral mass, where the trajectories are wound up.

33




Figure 2.17.: Far view on nullgeodesics in the Schwarzschild case; rH = rS = 2M .

34




Figure 2.18.: Close view on nullgeodesics in the Schwarzschild case.

35




Figure 2.19.: View on nullgeodesics in the extreme Kerr case as viewed along the axis of symmetry.

36




Figure 2.20.: View on nullgeodesics in the extreme Kerr case as viewed from the equa-torial plane. The geodesics’ start point is fixed by θ = 90°.

37





38





39




Zero Angular Momentum Observer

In some cases it becomes useful to dismiss the global coordinate system and concentrate

on small scales. In spite of spacetimes being generally curved, on scales sufficiently smallthe local metric appears flat. Therefore it is allowed to define a locally flat, orthonormalcoordinate basis (tetrad).Such a system given, the corresponding transformation matrix αµ

ν and its inverse can beused to change between the coordinate systems.

In the case of the Kerr metric this can be further exploited. An observer, consideredorbiting the central mass with the frame-dragging frequency ω, has a vanishing four-momentum component in the eφ direction ( pφ = 0). Consequently such an observer iscalled a Zero Angular Momentum Observer (ZAMO).The ZAMO tetrad eµ in Boyer-Lindquist coordinates is given by:

et =1

αet +

ω

αeφ (2.48a)

er =

√∆

ρer (2.48b)

eθ =1

ρeθ (2.48c)

eφ =1

eφ (2.48d)

The transformation matrix for the change of basis is then:

αµµ =

1/α 0 0 ω/α

0

∆/ρ2 0 00 0 1/ρ 00 0 0 1/

(2.49)

40



3. The Time-dependent Ray-Tracer

As already outlined in the introduction, simulated spectra of monochromatic radiationfrom the surroundings of rotating black holes are examined in this thesis. In order toachieve that, a ray-tracing method is applied. Trajectories of photons, being detected

by a fixed observer far away, are back-traced through spacetime, until they either endup at the event horizon, or reach regions of no concern. The numerical integration of the ordinary differential equations of first order (2.46 a-d) is performed by an adaptiveFehlberg algorithm (see Sec. 3.2 and App. D). With those geodesics given, the spectracan be simulated then by applying a certain matter distribution in the central mass’environment.This matter contributes by radiation to the flux being transported along the geodesicsand being detected at the obsever in the end. Throughout the simulations presented inchapter 4, only monochromatic radiation was considered to contribute to the spectra.Doing that, contributions from fluorescence lines like the Kα can be studied.Similar methods were successfully used in the past to obtain simulated line profiles from

accretion discs [Fab1989, Dab1997, Fan1997, Mu2000].In this thesis a further feature is analyzed. Regions of higher density, formed withinthe accretion disc, the so called hot spots (see page 54 in Sec. 3.1), are considered toinfluence the line spectra and the intensity distribution in time.That dynamic feature derives from the hot spots corotating with the accretion disc andgeneral relativistic effects influencing the spectra differently, depending on the positionof the radiating matter.So, when simulating the detected radiation for equal intervals during a certain period of time, light curves can be obtained.This concept is applied to the volume ray-tracing application implemented by BurkhardZink in 2002 [Zin2002]. Volume ray-tracing means that matter distributions are consid-

ered as three dimensional objects and can be intersected by the calculated geodesics.While resulting in higher processing times than those of planar ray-tracing applications,this concept allows absorption being taken into account.

In this chapter the implementation of the dynamic ray-tracing concept, performed bymeans of the object oriented programming language C++, will be presented.After outlining the general functionality and process sequence and discussing the classes,that were implemented for this purpose (Sec. 3.1), the numerical integration of thegeodesics (Sec. 3.2) and the radiative transfer along them will be described (Sec. 3.3).Before introducing the discrete Fourier transform (DFT) method, used to generate powerdensity spectra (Sec. 3.5), the code of the application’s main objects is revealed (Sec. 3.4).

41




3.1. Structure and Functionality

To produce time-dependent light curves, it is inevitable to simulate detected spectrafor each timestep separately. Since the integration of the geodesics demands for a notnegligible processing time, but the geodesic flux remains static in time, it only needs tobe performed once for each pixel on the detector’s screen.Intesity contributions, arising from the dynamic matter distribution along the photontrajectories, are evaluated for each timestep then. Being monochromatic in the emit-ter’s rest frame, those contributions get red- or blue-shifted due to relativistic effectsfrom GR. That frequency shift depends on the emitter’s position and state of motion.Summing up all intensity contributions along a geodesic results in obtaining a spectrumI (ν ).Spatially resolved pictures of the environment of the central mass can be obtained by

color-encoding these spectra for each pixel.Usually the spatial resolution of detector’s being used today is not sufficiently high toget spatially resolved pictures of the objects concerned here.Therefore it is more reasonable to sum over each pixel on the detector’s screen.A total spectrum can be simulated by doing that.A further integration

I (ν )dν yields the total energy detected. Varying in time, these

can be used to obtain time depending light curves.

The major class implemented, that controls the described procedure is named Raytracer ,derived from the class camera . Initialized with its position in spacetime, the metric ap-plied for the simulation1 and the camera’s aperture, which fixes the initial conditions

for the numerical integration (see below), it sets up the application’s environment. Sup-plied with certain parameters, defined in two separate parameter files parameter.h andparhotspot.h , an instance of the object Matterfield is initialized at first when the class’method PerformSimulation(. . . ) is called. Matterfield encapsulates the matter objectsconsidered for the simulation2.After that, instances of the classes Spectrum and image are initialized. The class Spec-trum represents the simulated spectra, storing values of the specific intensity for differentfrequencies ν and provides useful operations on them, whereas Image handles the data,which can be used to generate the spatially resolved images.Consequently each timestep demands its own instances of these.Since spectra are evaluated for all pixels independently and need to be summed up then

to the time dependent total spectra, the double amount, namely twice the timesteps, of the spectrum class’ instances are required. The one half, called the working spectra , isused during the evaluation of spectra, differing in time, for each pixel successively.The other half is exploited for summing up those and storing the total spectra.Having set up the basic conditions, the computation is started by looping over each pixelin the detector frame. The equations of motion for photons are numerically integratedhere, and the intensity contribution at each integration step is taken into account for

1The Kerr metric only is used for this thesis, where the Schwarzschild case can be achieved by settingthe Kerr parameter a to null.

2An accretion disc and hot spots were used for this thesis, but further matter objects (dust torus,corona etc.) could be introduced here easily.

42




each timestep separately. By that procedure, the radiation transport along the geodesicregarded is evaluated time-dependently. The resulting working spectra are then on the

one hand added to the total ones, and on the other hand used to generate a RGB-colorrepresentation for the given pixel.Having performed that for each pixel, the aspired information is available. By normaliz-ing the images to the brightest contribution and by integration of the total spectra, thedata is prepared for output.Usually spatially resolved color-coded time dependent pictures of the virtually observedobjects are generated and can be used to obtain animations. In addition to that, thetime depentent spectra, which can be used for examining the line broadening, are storedin adequate data files.Finally, the data, representing the total energy flux, is stored and can be used to visu-alize time dependent light curves.

This main routine, performed by the class method PerformSimulation(string* simname)of the class Raytracer is illustrated in Fig.3.1. The classes of the time-depending ray-

Figure 3.1.: Illustration of the main sequence performed by the class method Perform-Simulation(string* simname) of the major class Raytracer .

tracer designed for the purpose of this thesis are structured due to an underlying hierar-chy, as it is generally intended in object oriented programming. This hierarchy dependson the classes’ meaning and their functionality and should help to prevent confusion tothe software engineer himself and any user as well.

43




The classes and their methods concerning their function are described next, whereasparts of the cource code will be revealed in Sec. 3.4. Only public attributes of the classes

Figure 3.2.: Class hierarchy of the ray-tracing application.

are listed, since the private ones serve solely for inner operations.Figure 3.2 visualizes the class hierarchy of the following classes.

Object is pure abstract acting as the base class for all other classes, and thereforeprovides only a constructor and destructor.

FourTupel represents a four-tuple of real numbers. It can be initialized by:

• FourTupel():Standard initialization, where the components are set to null.

• FourTupel(float8 a0, float8 a1, float8 a2, float8 a3)3:Initialization with defined components.

• FourTupel(const FourVector& vector):Here the components are extracted from the Fourvector (see below) provided.

The four components are stored as public attributes and can be accessed directly.

FourVector stores the four components of a vector and provides basic operations onthem:

• FourVector():Standard constructor, where the components are set to null.

3The type float8 is a double precision float number defined in the file ntypes.h .

44




• FourVector(float8 x0, float8 x1, float8 x2, float8 x3):Initialization with defined components.

• FourVector(const FourTupel& tupel):The components are extracted from a provided FourTupel during this initial-ization.

• FourVector operator+(const FourVector& add):This defines the operator “+” for the class as simple addition of the compo-nents.

• FourVector operator*(const float8& smul):The multiplication of the vector with a scalar is defined as the operation ”*”.

• void MakeLightlike():The three spatial components are normalized according to the temporal, so

that the vector becomes light-like (the Minkowski metric is applied here).• void NormalizeTimelike():

The vector V µ is normalized by the condition V µV µ = ηµν V ν V µ = 1, where ηis the Minkowski metric tensor.

• bool IsZero():Returns true if the vector’s components are null.

As at the class FourTupel , the components of the vector are stored as public at-tributes.

Chart is also an abstract class. Derivations from that one represent a coordinate chart

corresponding to a special coordinate system. Supplied with a certain four-tuple,those classes should be able to check, if such one is valid for the given chart.Furthermore it provides the method int GetIdentifier();, by which each derivedclass should be identified with regard to its integer identifier defined in the filecharttypes.h .

BoyerLindquistChart models a Boyer-Lindquist coordinate representation in the Kerrmetric and is initialized with the Kerr parameter a and the central mass M 4. Itprovides the following methods:

• BoyerLindquistChart(float8 kerr a, float8 kerr M):The constructor must be supplied with the parameters a and M .

• float8 GetRms():Delivers the radial distance of the marginally stable circular orbit (ISCO) rms

in the equatorial plane.

• void GetSafeSpot(FourTupel& result):Simply delivers a point in spacetime outside the ergosphere.

• bool IsValid(const FourTupel& tupel):Delivers true if the position in spacetime provided is outside the event hori-zon. The region inside the horizon is considered non-valid.

4Usually the mass M will be set to 1.

45




• float8 GetEventHorizon():Yields the radial coordinate of the event horizon.

• float8 GetErgosphereBorder(float8 theta):The radial coordinate of the ergosphere is delivered according to the poloidalangle provided.

• int GetIdentifier():The integer identifier of the charttype is returned.

• float8 Evaluate g(const Event& event,const FourVector& v1,constFourVector& v2):Evaluates and delivers the coordinate dependent result of vµuµ = gµν vν uµ,where the two fourvectors and the point in spacetime, where it shall be eval-uated, are provided by the parameters.

Event is an object defining an event in spacetime. It is represented by its coordinateson a special chart.The chart, which is used for its coordinate representation and the coordinatesthemself are public. The initialization of this class’ instances can be achieved by:

• Event(const Chart* chart):The coordinates are set to null.

• Event(const Chart* chart, const FourTupel& tupel):The coordinates are provided by an instance of the class FourTupel .

where the chart needs to be provided.

The chart used is can be required by the method: Chart* GetChart().

ColorRGB simply stores three real numbers for the purpose of a RGB-color representa-tion.Instances are initialized by the constructor ColorRGB(float8 r, float8 g,float8 b), wherein the components must be defined.Public access is allowed then to those member attributes.

Spectrum is an important object for the ray-tracing application in this thesis. Storinga specified amount of values for a defined frequency domain it models a spectraldistribution, that can represent intensity, opacity or emissivity distributions. For

this purpose it provides also several useful methods:

• Spectrum():The constructor can be called without any parameters, but for a successfulinitialization the parameters defining the domain must be provided withinthe parameter file parameter.h . Those parametersPR SPECTRUM NU MIN ,PR SPECTRUM NU MAX and PR SPECTRUM RES define the domain’sminimum, maximum and resolution.

• void CreateShiftRGB(float8& ref, ColorRGB& result):This method generates a redshift-color representation of the spectral distribu-tion in relation to a provided reference frequency (float8& ref). This method

46




is used by the class Raytracer to generate spatially resolved images of thematter distribution.

• bool AddDot(float8 frequency, float8 intensity):Value, specified by float8 intensity , is added to the spectrum at the frequencyprovided by float8 frequency . Should the given frequency be out of range, themethod returns false.

• void Clear():The spectrum is cleared, which means that all values are set to null.

• float8 GetIntensity(float8 frequency):The value for a given frequency is delivered.

• void GlobalBrighten(float8 factor):The whole spectrum is multiplied by a provided factor, that needs to be higher

than null. Otherwise nothing is done here.• void AddSpectrum(const Spectrum& spectrum),

void AddSpectrum(const Spectrum* spectrum):Those methods perform a spectrum addition. They only differ in the pa-rameter providing the spectrum to be added. At the upper it needs to be areference, whereas at the lower a pointer is accepted.

• void MultiplySpectrum(const Spectrum& spectrum):A multiplication of spectra is performed5.

• void AddWeightedSpectrum(const Spectrum& spectrum, const float8&weight):

The provided spectrum, weighted by the factor float8 weight , is added.• void AddWeightedConstantSpectrumLinear(const float8& intensity,

const float8& weight):This method adds a linear weighted spectrum of the form I ·w · ν , where I isa constant value weighted by w and ν denotes the frequency.Within its method UpdateIntensity Abs(. . . ) the class matterfield uses thismethod during each step of geodesic integration in order to add intensitycontributions from matter objects.

• void CumulateOpacity(const Spectrum& tau):Opacity contributions are cumulated acoording to the algorithm presented insection 3.3 on page 67.

• void FtoI():The spectrum is weighted by ν 3 in order to transform from the relativisticinvariant radiation flux F to the local intensity I = ν 3 · F .

• void AddConst(const float8& c):The constant provided is added to spectrum.

• float8 GetTotIntensity():In order to obtain the total intensity detected

I (ν )dν , a summation over

5Note that this method as well as AddSpectrum(. . . ) reasonably should only be used with instances of the same size (same frequency domain).

47




the spectrum’s values is performed, which can be done since the frequencyintervals are equal.

• void CreatePlot(Image& result) const:Spectrum data is written to the provided instance of the class Image (seebelow), so that the spectrum can be plotted then.

• void CreateDataFile(const string& filename):Spectrum data is exported to a data file. The resulting file consists of tworows: (ν ,I (ν ))

The intensity values corresponding to different frequencies are stored in a publicSTL6-container (vector) and therefore can be accessed by any objects.

Image encapsulates the RGB-color data of an image with a certain resolution as it is

defined within initialization:

• Image(uint sizeX, uint sizeY):Constructor with well defined image resolution given by the parameters uint sizeX and uint sizeY 7.

• Image():Constructor to use without additional parameters. The resolution here mustbe provided by the parameter PR OUTPUT PIXELS within the file param-eter.h .

As an image representing object, this class provides methods, by which its data

can be manipulated and exported:

• void CreateTGA(const string& filename, uint multiply=1):With a provided filename this method exports a TGA-file from the image.The resolution of the output file can be defined by the scale parameter uint multiply .

• void SetPixel(uint x, uint y, float8 r, float8 g, float8 b) ,void SetPixel(uint x, uint y, const ColorRGB& colorRGB):The RGB-color value of a specific pixel can be set by those methods. The pixelis identified by the parameters x and y , whereas the color data is provided bythree separated components float8 r , float8 g , float8 b at the upper method

or by an instance of the class ColorRGB at the lower.

• void GetPixel(uint x, uint y, float8& r, float8& g, float8& b):Delivers the RGB values of a specified pixel.

• void Clear():Image data is cleared (set to null).

• uint GetSizeX():Image resolution in x-direction is returned.

6Standard Template Library is a software library included in the C++ Standard Library.7The type uint is simply an unsigned int defined in the file ntypes.h .

48




• uint GetSizeY():Image resolution in y-direction is returned.

• float8 GetHighestBrightness():Delivers the brightest picture component.

• void Normalize():The brightness is normalized, so that the brightest color component has thevalue 1.

• void Normalize(float8 brightness):This method normalizes the brightness to the brightness provided by the pa-rameter float8 brightness. After having determined the brightest component,this method can be used to normalize multiple images to the same value.

Frame is an object modelling a local flat (Minkowski) frame. In order to provide coordi-nate transformations on objects between the frame basis and the global coordinatebasis, it needs to be initialized with a Chart , used for the coordinate representa-tion. Furthermore a FourTupel defining the event on the given chart, frame basisvectors e0, e1, e2, e3 and coordinate basis vectors f 0, f 1, f 2, f 3 must be providedwithin the initialization. Public access is allowed to the basis vectors.After the initialization, following methods are available:

• void TransformIntoFrameBasis(const FourVector& vector, FourVec-tor& result):This method transforms the provided vector representation from the coordi-nate basis to the frame basis.

• void TransformIntoCoordinateBasis(const FourVector& vector, FourVec-tor& result):Transformation of a vector representation from the frame basis to the coor-dinate basis.

• Event GetEvent():The position in spacetime of the frame is returned.

ZAMO is derived from the class Frame and represents a ZAMO frame (see Sec. 2.2.3)in the Kerr metric. As such one, only the chart for coordinate representation andthe position in spacetime need to be provided for the initialization:

• ZAMO(const Chart* chart, const FourTupel& tupel).

As a derived class, it provides additionally the same methods as its parent classFrame.The ZAMO is not defined along the axis of symmetry (θ = 0 and θ = π/2).

ZAMOCameraFrame is derived from the class ZAMO and designed to represent a framelooking towards the centre. For this purpose the orientation of er is changed rela-tively to that in instances of the ZAMO class. It is initialized by the constructor:

• ZAMOCameraFrame(const Chart* chart, const FourTupel& tupel).

49




Instances of the class Raytracer (see below), which mimic a virtual detector ob-serving the whole scene, are generally initialized with instances of ZAMOCamer-

aFrame.

Camera is a pure abstract class modelling a generally far away observer, who the null-geodesics are backtraced from. By the constructor

• Camera(const Frame& frame, float8 aperture),

which needs to be provided with the camera’s frame, where its position is alsospecified, the observer is defined entirely. The parameter float8 aperture suppliesthe object with the camera’s viewing angle (in degrees) in direction to the centre.This angle γ must lie within the range [0 °, 180 °] and is used by the Raytracer class’ method PerformSimulation(.. . ) (see below) to fix the initial conditions of

geodesics being traced.

Raytracer is the main class of the ray-tracing application in this thesis. It is derivedfrom the class Camera and therefore the observer is completely specified by theinitialization:

• Raytracer(const Frame& frame, float8 aperture, string* name=0),

where additionally a name of the simulation can be provided.The following methods cover all procedures, that have to be performed in order toobtain spectra from and/or spatially resolved pictures of the system considered:

• bool PerformSimulation(string* simname):This is the main procedure executed when spectra from the source are de-

sired. It controls the numerical integration of the nullgeodesics and callsmethods, which handle the cumulation of the radiation flux along them. Fin-ished with evaluation, the output data is formatted.This process was already described above and is illustrated by Fig. 3.1 onpage 43.

• bool RayTestImage(string* simname):This method is called to generate ray images of the geodesic flux. Theamount of geodesics that are to be plotted is specified by the parametersPR RAY IMAGE NR X and PR RAY IMAGE NR Y in the file parame-ter.h , where x and y denote coordinates in the observer’s frame (screen).Supplied with those, the nullgeodesics, which differ in their initial momentum(direction) are integrated one after another. This is performed by calling themethod:

• void CreateRayImage(const Event& event, const FourVector& k,float8 max coord, bool top, Image* result), which needs to be suppliedwith the geodesic’s initial position (Event event), its initial four-momentum(FourVector k) and further parameters specifying the maximum coordinate rat which the tracing is stopped (float8 max coord), defining the perspectiveof the ray image (bool top), and the object, which the image data is written

50




to (Image* result).

Finished with all geodesics, the image data is written to a TGA-file. Suchray image files were introduced in the previous chapter (see Fig. 2.17-2.22).

• void Measurement(const FourVector& basedirection, const Event&basepos, Spectrum* &specs, MatterField& matter, const float8&timeincr):

This method is called from within the method PerformSimulation(. . . ) foreach pixel in the detector’s frame.It is responsible for the geodesic evaluation and the radiation transfer as de-scribed in Sec. 3.3. For this purpose it must be supplied with the geodesic’sinitial position and momentum, the matterfield, which contributes to theradiation detected, and the spectra objects, where the evaluated flux contri-

butions are added to.Since the matter distribution around the central mass is a function of timeM(t) if hot spots are considered, the matter’s contribution to the radiationflux at a given position must be considered for different times i separately.Therefore the parameter float8 timeincr is provided. It specifies the incrementof time for consecutive evaluations. Given that, at each geodesic integrationstep M(t, xµ) is evaluated for t = timeincr · i−∆t, where ∆t is the propaga-tion time of the photon from the detector to the position considered. It canbe extracted from the temporal coordinate x0 evaluated by integration. Thisseparation in time allows the generation of time-dependent spectra.

• void Measurement planar(const FourVector& basedirection, constEvent& basepos, Spectrum* &specs, MatterField& matter, constfloat8& timeincr):This method has the same meaning for ray-tracing as the previous one, butis used for planar ray-tracing, which means that the accretion disc cannot beintersected by the traced geodesics and therefore acts as a boundary, wherethe numerical integration is stopped. The parameter PR TRACE PLANARin the parameter file decides if planar or volume ray-tracing is to be performed.Note that no absorption can be considered in the planar case8.

Geodesic acts as a parent class for further special derivations and represents generalgeodesics in spacetime.

KerrGeodesic is derived from the class Geodesic and is designed exclusively for null-geodesics on the Kerr metric. It provides an adaptive ray propagation algorithm(see Sec. 3.2) by which the equations of motion (2.46 a-d) can be solved. Adaptivemeans that the step size used for integration is adjusted so that the computationalerror lies within a given range defined in the parameter file.

8For the simulations of this thesis, only volume ray-tracing was performed

51




The initialization of this object’s instances

• KerrGeodesic(const FourVector& dir in, const Event& pos in)

can be supplied with initial conditions for the ray propagation, namely the actualposition in spacetime Event pos in and the actual four-momentum FourVector dir in , which specifies the ray direction. The geodesic integration is performedwithin the methods:

• bool Fehlberg4 5(const FourVector& dir in, const Event& pos in,float8& lambda, float8& lambda used, FourVector& dir out, Event&pos out):Here a one step integration is performed. Therefore, for a complete integra-tion of the nullgeodesics this method needs to be called consistently, until the

boundary conditions are satisfied. The main boundary is that ray propagationstops at the event horizon. Additionally to that, the integration of geodesicsis halted throughout the simulations in this thesis once the ray reaches theradial distance of the detector, where no matter is supposed to be presentand consequently cannot contribute to the radiation flux.This method needs to be supplied by the ray’s initial position and four-momentum as well as with the actual step size used. This one is adjustedduring the integration if the error turns out to be too high or to low. Thenew step size and the ray’s new position and four-momentum are written tothe parameters float8 lambda , Event pos out , Fourvector dir out .

• bool Fehlberg4 5 lowerr(const FourVector& dir in, const Event&

pos in, float8& lambda, float8& lambda used, FourVector& dir out,Event& pos out):This method is similar to the previous one, but the geodesic integration isperformed with a higher accuracy. The factor specifying it, with regard to theerror boundaries given for the previous method, is defined by the parameterPR RAY IMAGE REF PRECISION .That operation is called while generating ray images by the method RayTes-tImage(...) of the class Raytracer . Doing that, reference rays with higheraccuracy can be visualized simultaneously.

Matterobject acts as a base class for objects representing any mass distributions around

the central mass.

MatterItem is derived from Matterobject and serves itself as a parent class for classesmodelling specific matter objects. As a pure abstract class, it provides methods,that need to be defined within all derived classes. Those methods are consid-ered to specify the matter objects’ density distributions, their state of motion andfurthermore the emissivity and the opacity essential for the radiation transport.

52




AccDisc represents a standard accretion disc as described in section 2.1.2.The initialization of instances of this class

• AccDisc()

must be supplied with several parameters defined in the parameter file. Those pa-rameters are firstly the Kerr metric specific parameters M , a, but also parametersspecifying the disc properties as the inner and outer edge and the disc matter’sopacity.The following methods are available within this class and define the disc’s ap-pearence.

• float8 GetInnerRadius():Delivers the disc’s inner edge, which is defined by

PR ACCRETIONDISK INNER RADIUS in the parameter file. The inneredge’s minimum radial distance to the central mass is the marginally stableradius rms.

• float8 GetOuterRadius():Delivers the disc’s outer edge defined byPR ACCRETIONDISK OUTER RADIUS in the parameter file.

• void GetFourVelocity(const Event& event, FourVector& u):Delivers the four-velocity of the disc matter in the coordinate frame, whichdepends on the position in spacetime. Since only planar Keplerian orbitsare considered, V φ is the only non vanishing spatial coordinate. It can beextracted from the angular frequency Ωφ

Ωφ(r) = 2πν φ(r) ≈√

M √r3 + a

√M

. (3.1)

by

V φ =Ω− ω

α(3.2)

• float8 GetDensity(const Event& pos, const float8& time):Delivers the source rest frame density (xµ). Generally a homogeneous den-sity distribution is assumed and consequently the normalized value = 1 isdelivered if the position lies within the accretion disc. The extent of the ac-

cretion disc, as it is implemented, is shown in Fig. 3.3. With those parameterschosen, the case of a slim disc is still satisfied.Since the density controls the emissivity within this implementation, a radialemissivity law of the form ε(r) = const. for rin ≤ r ≤ rout is represented bythis density distribution.In addition to that, the density distribution can be chosen to satisfy a single( (r) ∝ r−α) or a double ( (r) ∝ r−α for r < rbreak and (r) ∝ r−β else)power law as it is proposed in [Dab1997, Mu2000].Which distribution is chosen, depends on the parameterPR ACCDISC DENSITY , whereas for the power laws the constants α, β must be specified.

53




Figure 3.3.: The image illustrates the accretion disc for the extreme Kerr case, whererms = rH and therefore the disc’s inner edge touches the event horizon. Forlower values of the Kerr parameter a the radial distance of the inner edgeincreases.

• void GetEmissivity(const float8& g, Spectrum& result, const Event&event):This method delivers the source’s rest frame emissivity. Since the emission

is weighted by the density distribution, a constant value can be deliveredhere if monochromatic emission is considered9. This contribution is shiftedin frequency by the redshift factor g and added to the spectrum providedby method parameter Spectrum result . The redshift factor g depends on therelation between the source’s four-velocity and the geodesic’s four-momentumat the given position and is evaluated by the method Evaluate g(.. . ) of theclass BoyerLindquistChart (see above).

• void GetOpacity(const float8& g, Spectrum& result, const Event&event):Delivers the accretion disc matter’s opacity that is specified in the parameterfile and stored as a private attribute of the class.

When absorption is considered, this method is called from within the methodUpdateIntensity Abs(. . . ) of the class MatterField , which handles the radia-tion flux contributions from different matter items.

HotSpot is also a class derived from MatterItem and represents regions of higher den-sity and emission within the accretion disc.

9When simulating other emissivity distributions, for example black body radiation, this profile needsto be adjusted.

54




Instances can be initialized by

•

HotSpot(int index),where an index parameter must be provided. This index acts as an identifier if multiple hot spots are considered. The hot spot properties, which the radius, theinitial position, the density and opacity count to, are specified in the file param-eter.h when only one hot spot is used, or in the file parhotspot.h for multiple hotspots.The following methods are provided in HotSpot :

• void Setomega(const float8& r):By that method the angular frequency of the hot spot is evaluated and storedto a private attribute. According to Eq.(3.1), which holds also for hot spots

corotating within the accretion disc, the angular frequency depends on theradial distance, that needs to be provided.

• float8 GetOmega():Returns the hot spot’s angular frequency.

• float8 Get r():Delivers the hot spot’s radial distance to the centre.

• float8 Get phi():The hot spot’s initial azimuthal coordinate φ is returned.

• float8 GetRadius():The radius Rhs of the hot spot is returned.

• float8 Get density():The density within the hot spots is described by a spherical symmetric Gaus-sian distribution.When this method is called, the peak density ˆ in the hot spot’s centre isreturned.

• float8 Distance(const Event& pos, const float8& time, float8& dis-tance):This method evaluates and delivers the spatial distance of a given point inspacetime Event pos to the centre of the hot spot. Pseudo-Cartesian coordi-nates are used for this evaluation, so that the curvature of the spacetime is

neglected.This deviation should not have a great impact for the distances d evaluatedhere (d < 4Rhs).

• void GetFourVelocity(const Event& event, FourVector& u):The four-velocity for the hot spot at a given point in spacetime is delivered.Since the hot spots are assumed to co-rotate within the accretion disc, thismethod delivers the same result as the corresponding one within the classAccDisc (see above).

• float8 GetDensity(const Event& pos, const float8& time):The hot spots represent a density distribution, that is super-imposed to that

55




of the accretion disc.For this application, it is described by a spherical symmetric Gaussian profile

in local Cartesian space

hs = ˆ exp(− d2

2R2hs

) , (3.3)

where Rhs is a specified radius and d denotes the distance to the hot spotcentre. Therefore, with a given event on spacetime, the time depending dis-tance to the hot spot’s centre can be evaluated and the density hs, weightedas described by Eq. (3.3), can be delivered. For distances higher than 4Rhs,the rest frame density = 0 is returned, since the contribution at this pointalready drops to ˆ exp(−8) ≈ 3.4 · 10−4 ˆ .The peak density is specified in the parameter files10.Note that all points within the hot spot are assumed to co-rotate with itscentre. Therefore no shearing effects can be simulated and the hot spots’form remains constant. This should be taken into account, when simulatinglight curves of long time periods (many rotation periods). Generally shearingwithin the accretion disc should deform the hot spots. Such arched structureswere considered for ray-tracing in [Schn05].

• bool CriticalEvent(const Event& pos):This method is only implemented to decrease processing time. As alreadydescribed, matter contributions to the radiation flux need to be evaluated ateach considered position for each timestep separately. This time consuming

computation affects the whole application’s performance. Now, since the hotspots’ radius is known and only their azimuthal position changes, there arewell defined regions on the manifold, where never contributions from thoseobjects can impact. Due to that, this method checks, if the provided positionlies within a region, which can be passed by any defined hot spots. If thatshould not be the case, then only static radiation contributions need to beconsidered and so evaluated only once for the specific geodesic integrationstep.

• void GetEmissivity(const float8& g, Spectrum& result, const Event&event),

• float8 GetOpacity():Those two methods do not differ from the corresponing ones in the classAccDisc (see above). The opacity for the hot spots must be specified withinthe parameter files.

MatterField is also a class derived from Matterobject and is the main object handlingthe communication of the application with all matter objects considered.

10Note that a radial dependency is super-imposed to the value of the peak density if a single or doublepower law for the emissivity is chosen in the parameter file.

56




• void UpdateIntensity(MatterItem* item, const float8& rho, constEvent& position, const FourVector& k, float8& stepsize, Spectrum*

result, const uint& specnumber),

• void AddIntensity(const Event& position, const FourVector& k,float8 stepsize, const float8& time, Spectrum* result, const uint&specnumber):Those two methods correspond to the methods void UpdateIntensity Abs(. . . )and void AddIntensity Abs respectively. They are called instead of the upperif no absorption is considered.

• void AddIntensity planar(const Event& position, const FourVec-tor& k, const float8& time, Spectrum* result, const uint& spec-number):When planar ray-tracing is performed, the procedures of the methods void UpdateIntensity(. . . ) and void AddIntensity(. . . ) are combined within thismethod. Obviously no absorption is considered in that case.

• float8 GetMaximalR():This method delivers the maximum radial distance from the centre, whereany matter can be found. Consequently it delivers either the outer edge of the accretion disc or the radial coordinate of the hot spot being farthest.

As already mentioned, the essential parameters for the ray-tracing application are storedin the parameter files parameter.h and parhotspot.h . Those parameters are listed anddescribed here.Global application parameters are:

• PR TRACE PLANAR: Defines if planar or volume ray-tracing is to be per-formed. Possible values are: 1 = planar; 0 = volume.

• PR ABSORPTION: Defines if absorption is to be considered. Possible valuesare: 1 = yes; 0 = no.

• PR NUMBER TIMESTEPS: Integer defining the number of timesteps evalu-ated.

• PR NUMBER OF PERIODS: Defines the time span for the time depending

ray-tracing in units of the rotation period T = 2π/ω of the hot spot with the lowestradial coordinate r.

• PR SPACETIME A: The Kerr parameter a; (0 ≤ a ≤ 1).

• PR SPACETIME M: The central mass M . (Generally set to 1.)

Parameters defining the virtual observer (detector):

• PR CAMERA T: Detector’s initial temporal coordinate.

• PR CAMERA R: Detector’s radial distance.

58




• PR CAMERA THETA: Detector’s poloidal coordinate. This parameter de-fines the inclination to the system observed.

• PR CAMERA PHI: Detector’s azimuthal coordinate.

• PR CAMERA APERTURE: Dector’s aperture.

Parameters concerning the spatially resolved images of the scene:

• PR IMAGES: Defines if images are to be created. 1 = yes; 0 = no.

• PR OUTPUT PIXELS: Defines the image resolution in both directions. Con-sequently this parameter defines the number of pixels and so geodesics that arebacktraced. (number of geodesics = (PR OUTPUT PIXELS)2)

• PR REFERENCE FREQ: The reference frequency for the method void Cre-ateShiftRGB(...), which defines the color representation.

Parameters concerning the simulated spectra:

• PR SPECTRUM NU MIN: Minimum spectrum frequency.

• PR SPECTRUM NU MAX: Maximum spectrum frequency.

• PR SPECTRUM RES: Spectral resolution.

• PR SPECTRUM PLOT: Defines if spectra should be plotted additionally.1 = yes; 0 = no.

• PR SPECTRUM ERRORLINE: Defines if error message should be showedwhen a contribution is to be added to a frequency, that is out of spectrum’s range.1 = yes; 0 = no.

Parameters concerning ray images:

• PR GENERATE RAY IMAGE: If set to 1, ray image is generated but nospectra are simulated.

• PR RAY IMAGE NR X: Number of rays to be shown (in screen’s x-direction).

• PR RAY IMAGE NR Y: Number of rays to be shown (in screen’s y-direction).

• PR RAY IMAGE VIEW TOP: Defines the perspective of the ray image rel-ative to rotation axis. 1 = along axis; 0 = perpendicular to axis.

• PR RAY IMAGE MAX R: Scale for the images. The highest radial coordi-nate to be plotted.

• PR RAY IMAGE REF PRECISION: Accuracy factor for reference rays.

Parameters concerning the integration of the geodesics (ray propagation):

• PR RAY MAX STEPS: Maximum number of intergration steps per geodesic.

59




• PR RAY STOP AT CAMERA RADIUS: If set to 1, ray propagation isstopped, when radial coordinate reaches the radial distance of the detector.

• PR RAY LOG TRACED STEPS: Number of integration steps is shown onscreen.

• PR RAY NEAR ERROR MIN: Minimum error boundary for regions withinr < 6M .

• PR RAY NEAR ERROR MAX: Maximum error boundary for regions withinr < 6M .

• PR RAY MED ERROR MIN: Minimum error boundary for regions within6M < r < 200M .

• PR RAY MED ERROR MAX: Maximum error boundary for regions within6M < r < 200M .

• PR RAY FAR ERROR MIN: Minimum error boundary for regions withinr > 200M .

• PR RAY FAR ERROR MAX: Maximum error boundary for regions withinr > 6M .

Parameters concerning accretion disc:

• PR ACCRETIONDISK USE: If set to 1, the accretion disc is considered.

• PR ACCRETIONDISK INNER RADIUS: Disc’s inner edge. If set to null,the ISCO is used.

• PR ACCRETIONDISK OUTER RADIUS: Disc’s outer edge.

• PR ACCDISC DENSITY: Defines the radial emissivity profile for the discmatter. Possible values: 1 = constant, 2 = single power law, 3 = double power law.

• PR ACCDISC ALPHA S: Constant to be defined when single power law ischosen for the radial emissivity profile.

• PR ACCDISC R BREAK: Breaking radius to define for double power law.

• PR ACCDISC ALPHA D: Constant to be defined when double power law ischosen for the radial emissivity profile.

• PR ACCDISC BETA D: Constant to be defined when double power law ischosen for the radial emissivity profile.

• PR ACCRETIONDISK OPACITY: Opacity constant.

Parameters concerning hot spots:

• PR HOTSPOT USE: If set to 1, one or multiple hot spots are considered.

60




The starting point when studying covariant radiation transport is to find an adequatequantity representing the radiation flux along the geodesics, since the classical specific

intensity I ν is not a local scalar. The phase space density of particle number F turnsout to fulfil those requirements. In a local rest frame and for photons ( p0 = ν ) it can beassociated to the specific intensity I ν by

F =n

ν 2dνdΩ=

I ν ν 3

, (3.6)

where n denotes the photon number density.By the means of that relation, the specific intensity as detected in the camera frame canbe evaluated from the relativistic invariant transport quantity F .The general relativistic Boltzmann equation

σ · dF =

dF dλ

, (3.7)

where σ generates a godesic flow field on the given manifold, describes the change of F while being transported along geodesics. The only non-vanishing term here derives fromsources along the geodesics.Therefore to obtain the total F , this source term needs to be evaluated for each photontrajectory reaching the detector’s position in spacetime.The backtracing method applied for this purpose is illustrated in Fig. 3.4.

Figure 3.4.: The principle of backwards ray-tracing is illustrated here. The figure showsthe camera’s position, which acts as an initial boundary for the ray prop-agation, the geodesics (on generally curved spacetime), and sources alongthem, where the contributions to the relativistic invariant transport quantityF originate from.

The camera’s screen with the sufrace area A is assumed to be an array of N × N pixels of the partial areas AXY . It is positioned curtly in front of the position, wherethe nullgeodesics are supposed to coincide and backtraced from. Consequently to beexact, the screen should have to be treated as a sphere section, but in reality for small

62



3.3. Radiative Transfer

apertures γ it can be approximated by a flat screen representation with a regular grid.Therefore the different AXY can be treated uniformally.

Such a screen discretization leads to the following expression of the total F C

, where C implies the camera position:

F C,total =

A

F C (ν, Ω)dΩ

=N −1X=0

N −1Y =0

||AXY ||F C (ν, ΩXY )

∝N −1X=0

N −1Y =0

F C (ν, ΩXY ) , (3.8)

where with suitable¯ΩXY , assumed in the center of the pixel areas AXY , and for a regularangular distance for the pixels, the are measures ||AXY || are pixel independent.

To obtain F C for a given pixel, we need to integrate over sources along the correspondinggeodesic:

F C (ν, ΩXY ) =

λCλ0

dF (x(λ), k(λ))

dλ

src

dλ . (3.9)

In this equation λ parametrizes the photon trajectory (x(λ), k(λ)) with the boundaryconditions xα(λC ) = xαC and kαrest(λC ) = (ν C ,−ν C Ω) in the camera frame12.If scattering is neglected, the source term can be split up into two fractions

dF

dλsrc

=dF

dλem

+dF

dλabs

(3.10)

representing emission and absorption processes.

The part containing the emission can be described bydF

dλ

em

= J (x(λ), k(λ), F (x(λ))) · n(x(λ)) , (3.11)

where n(x(λ)) is the rest frame number density of the source field. J is called theinvariant emissivity and is a function of the spacetime event, the direction of emissionand of the incoming F .

Defining the length element ds = || k||, where k denotes the spatial components of kand exploiting kµkµ = 0, the equation (3.11) can be expressed by classical quantities byintegrating over a step dλ and transforming into the emitter’s rest frame:

dF =1

ν 3dI ν = J (x, k, F (x))n(x)dλ

= J (x, k, F (x))n(x)ds

ν

=1

4πν 3ρ(x)ǫν ds (3.12)

12Since the photons are viewed as incoming, the mirrored unit vector − Ω is used here.

63




For the last step, the classical radiation transport equation dI ν /ds = ρǫν /4π − ρκν I ν was inserted.

Consequently the invariant emissivity can be expressed by classical quantities (rest framedensity ρ, rest frame emissivity ǫν ):

J (x, k, F (x)) =1

4πν 2ρ(x)

n(x)ǫν (3.13)

In analogy to this procedure, the absorption term in (3.10) can be examined.The invariant opacity K can be defined by:

dF

dλ

abs

= −K (x(λ), k(λ)) · n(x(λ)) · F (x(λ), k(λ)) . (3.14)

Integration over dλ and transformation into the absorber’s local rest frame gives:

dF =1

ν 3dI ν = −K (x, k)n(x)F (x, k)dλ

= −K (x, k)n(x)I ν ν 3

ds

ν

= − 1

ν 3ρ(x)κν I ν ds , (3.15)

where κν denotes the classical opacity.Like in the emission case, K can now be described by its classical analogon:

J (x, k) = ν

ρ(x)

n(x) κν . (3.16)

Given that, equation (3.9) takes the following form:

F C (ν, ΩXY ) =

λCλ0

n(x(λ))

J (x(λ), k(λ), F (x(λ)))

−K (x(λ), k(λ))F (x(λ), k(λ))

dλ .

(3.17)

Introducing the covariant optical depth ω by

ω = λ1

λ0

n(λ)K (λ)dλ (3.18)

and the effective source function Ξ by

Ξ(λ, F ) =J (λ, F )

K (λ)(3.19)

the transport equation can be expressed by

dF

dω= Ξ(ω, F )− F (ω) .13 (3.20)

13Due to readability, the full dependencies of quantities are not denoted any further.

64




After defining the quantities J and S

J = F (ω)e

ω

(3.21)S = Ξ(ω, F )eω , (3.22)

an integration of J over ω can be performed:

J (ω) = J (ω = 0) +

ω0S (ω)dω . (3.23)

To obtain this, the derivative

dJ dω

= eω

F +dF

dω

= eωΞ = S (3.24)

was used.

Exploiting the definition of J , the integral (3.23) can be expressed by thequantity F :

F (ω) = F (0)e−ω +

ω0

e(ω′−ω)Ξ(ω)dω . (3.25)

When resubstituting ω and Ξ by the means of (3.18) and (3.19), and taking the boundaryF (0) = 0 into account, the equation (3.25) takes the form:

F C (ν, ΩXY ) = λC

λ0

n(λ)J (λ, F )

·e−

R λCλ n(λ′)K (λ′)dλ′dλ . (3.26)

This is now the rendering equation for the transport of F along the nullgeodesics. Inorder to be used within the ray-tracing application, it needs to be discretized.The integral is split into small steps λi ∈ [λ0, λC ], i = (0, 1, 2, . . . , n) with λ0 = 0 andλn = λC .This procedure yields:

F C (ν, ΩXY ) =n−1i=0

λi+1λi

n(λ)J (λ, F ) · e−R λCλ

n(λ′)K (λ′)dλ′dλ . (3.27)

This can be transformed to

F C (ν, ΩXY ) =n−1i=0

n−1 j=i+1

ψ j

λi+1

λi

n(λ)J (λ, F ) · e−R λi+1λ

n(λ′)K (λ′)dλ′dλ , (3.28)

where the exponent could be split up by the means of the covariant step absorption coefficients ψi given by

ψi = e− R λi+1

λin(λ)K (λ)dλ

. (3.29)

65




This expression can further be transformed by the use of the mean value theorem ,which states that there exists a suitable λi ∈ [λi, λi+1] so that

λi+1λi

f (λ)dλ = (λi+1 − λi)f (λi) (3.30)

is satisfied.That relation is applied twice to (3.28), once to the base and once to the exponent,which yields

F C (ν, ΩXY ) =n−1i=0

n−1 j=i+1

ψ j

(λi+1 − λi)n(λi)J (λi, F )e−(λi+1−λi)n(λ′i)K (λ′i) (3.31)

for an adequate λi ∈ [λi, λi+1] and λ′i ∈ [λi, λi+1].The same principle can be utilized on the covariant step absorbtion coefficients.So for a suitable λ′′i ∈ [λi, λi+1] they can be described by:

ψi = e−

R λi+1λi

n(λ)K (λ)dλ= e−(λi+1−λi)n(λ′′i )K (λ′′i ) . (3.32)

The discretization is now performed explicitly with the assumptions

λi ≈ λi+1 + λi2

= λi +1

2∆λi , (3.33a)

λ′i≈

λi+1 + λi

2

= λi +3

4

∆λi , (3.33b)

λ′′i ≈λi+1 + λi

2= λi +

1

2∆λi where ∆λi = λi+1 − λi (3.33c)

Plugging those preparations into (3.31) and expressing the covariant quantities J andK by the classical ones (ρ, ǫν and κν ) leads to the discretized general relativistic volumeray-tracing equation:

F C (ν, ΩXY ) =1

4π

n−1i=0

n−1 j=i+1

ψ j

∆λi

· ρ(λi +

1

2 ∆λi)

1

ν 2i ǫν i(λi +

1

2 ∆λi)

· e−1

2∆λi ρ(λi+

1

2∆λi) ν i κνi(λi+

1

2∆λi) ,

(3.34)

withψi = e−∆λi ρ(λi+

1

2∆λi) ν i κνi(λi+

1

2∆λi) . (3.35)

In order to decrease evaluations, it is assumed that the quantites ρ, ǫν , κν are equal at(λi + 1

2∆λi) and (λi + 34∆λi).

66




The ray-tracing algorithm is implemented in the methods UpdateIntensity Abs(. . . )and UpdateIntensity(. . . ) of the class MatterField , whereas the second one does not

take absorption into account. The invariant spectral distribution F and the cumulatedabsorption Ψ =

n−1 j=i+1 ψ j are being transported along the photon trajectories obtained

by the stepwise numerical integration.Starting with the pixel geodesics at the camera, normalized by k0

C = ν C = 1 in thecamera rest frame, two arrays F [ν ] and Ψ[ν ] are filled at each integration step14.So at each position λ with the actual stepsize ∆λ the following procedure needs to beconsistently performed:

• The local density ρ(λ), emissivity ǫ(λ) and opacity κ(λ) are determined.Generally the emissivity and opacity are functions of the frequency ν , but con-sidering monochromatic emission and a frequency independent opacity, κ(ν ) canbe handled as a constant and ǫ(ν ) as a distribution with a specific value (set to1 due to simplicity) at ν i = giν C , where gi is the local red-shift factor and ν thefrequency as measured in the camera’s frame (normalized to 1)15.

• The array τ [ν ] is set up temporarily by evaluating for each ν : τ [ν ] = ∆λ ρ(λ) g ν κ(λ).For the monochromatic case the array τ [ν ] reduces to only one contribution atν = g−1 yielding τ ν = ∆λ ρ(λ) κ(λ).

• The array F [ν ] is increased by the source contributionS [ν ] = Ψ[ν ] ∆λ ρ(λ) ǫ(λ)[ν ] exp(−τ [ν ]/2)/(gν )2.For monochromatic sources this reduces to S [ν ] = Ψ[g−1] ∆λ ρ(λ) exp(−τ [ν ]/2).

• The transported opacity is cumulated by Ψn

ew[ν ] = Ψ[ν ]·

exp(−

τ [ν ]).

• The next evaluation point x(λ′) and step size ∆λ′ are generated by integration of the geodesic equations.

This routine is performed until a termination condition is satisfied. Usually this is thecase, when backtraced geodesics reach the event horizon or leave the region of interest.In the end the intensity I ν as measured at the camera is obtained by I [ν ] = F [ν ] ν 3.

14The arrays are represented by instances of the class Spectrum .15The red-shift factor is evaluated by the method Evaluate g(. . . ) of the class BoyerLindquistChart (see

at page 46).

67




3.4. Source Code

Since the structure of the time depending volume ray-tracer and the procedures, whichare performed during the evaluations, already were described in the previous sections,only the C++ source code of the main methods controlling the sequence of the applica-tion are presented here.Within the main routine an instance of the class Raytracer is initialized and during thatsupplied with the most important parameters, namely the metric specifying Kerr pa-rameter a and the central mass M and additionally the camera’s position in spacetime.In order to perform time dependent volume ray tracing, its method PerformSimula-tion(...) (see page 50) is called. The following code is executed then:

Listing 3.1: Source code of Raytracer::PerformSimulation(. . . )

1 bool R a ytr a cer : : P er f o r m S i m u l a ti o n ( s tr i n g ∗ simname)2 3 M a t t e r F i e l d m a t te r ; // d is c and h o t s p o ts a r e i n i t i a l i z e d h er e4

5

6 // wo rk in g s p ec t ra a re i n i t i a l i z e d 7 Spectrum∗ s p e c t i m e=new Sp ec tr um [ PR NUMBER TIMESTEPS] ;8 i f ( ! s p e c t i m e ) return 0 ; // c he ck a l l o c a t i o n 9 // t o t a l s p ec t ra a re i n i t i a l i z e d 10 Spectrum∗ s p e c t o t =new Sp ec tr um [ PR NUMBER TIMESTEPS] ;11 i f ( ! s p e c t o t ) return 0 ; // c he ck a l l o c a t i o n

12 #i f (PR IMAGES)13 // The I m ag es w he re c o l o r d a t a i s w r i t t e n t o

14 Image ∗ images=new Ima ge [PR NUMBER TIMESTEPS ] ;15

i f ( ! i m a g es ) return 0 ; // c he ck a l l o c a t i o n 16 #endif

17 f l o a t 8 t i me i n cr e me n t ; // v a r i a b l e s t o r i n g t im e i n cr e me n t 18 f l o a t 8 o meg a ma x = 0; // v a r i a b l e f o r a ng u la r f r e qu e nc y o f n e ar e s t h ot

s p o t 19 f o r ( i n t k = 0 ; k<PR HOTSPOT NUMBER; k++)20 21 i f (( matter . matterit ems [ k ] )−>GetOmega()>omega max)22 omega max=(matter . matteritems [k ])−>GetOmega() ; / / om eg a ma x is

d e t e rm in e d 23 24 / / s e t t i n g t i m e in c r e me n t ( # P e r i o d s∗2PI/( omega max ∗ # t i m e s t e p s ) )

25 i f (omega max!=0)26 ti m ei n cr em en t= 2∗PI∗PR NUMBER OF PERIODS/( omega max ∗( f l o a t 8 )

PR NUMBER TIMESTEPS) ;27 e l s e ti m ei n cr em en t = 1 . 0 ; // i f no h ot s p o t s p r es e n t 28 #i f (PR IMAGES)29 ColorRGB c o l o r ( 0 . 0 , 0 . 0 , 0 . 0 ) ; // v a r i a b l e f o r s t o r i n g RGB −d a t a 30 #endif

31 f l o a t 8 r e f f r e q=PR REFERENCE FREQ; // r e f e r e n c e f r e q u en c y ( u s u a l l y 1 )32

33 // r en de ri ng f o r eac h p i x e l :34 // C o or di n at e s o f t a r g e t p i x e l on p r o j e c t i on p la ne35 f l o a t 8 ty , t z ;36 // d e pe n di n g on t h e a p e r t ur e ( v i e w in g a n g l e ) t h o s e a r e t h e m ax imal y , z

( a t x =1)37 f l o a t 8 m ax ty = t a n ( m Ap er tu re / 3 6 0 . 0∗ PI) ;

68



3.4. Source Code

38 f l o a t 8 m ax t z = m ax t y ; // when r e s o l u t i o n i n b ot h d i r e c t i o n s t h esame

39 // V iew d i r e c t i o n i n f ram e b a s i s40 F o ur V ec t o r v i e w f ;41 // V iew d i r e c t i o n i n c o or d i na t e b a s i s42 F o ur V ec t o r v i e w c ;43 // Get t h e b as e e v en t 44 const Event eve nt = mFrame. GetEvent () ;45

46 f o r ( u i n t y= 0; y<PR OUTPUT PIXELS ; y++) // l i n e s ( y )47 48 f o r ( u i n t x= 0; x<PR OUTPUT PIXELS ; x++) / / r ow s ( x )49 50 // T ra ns fo rm p i x e l c o o r d in a t e on p l an e i n t o ( x , y , z )−s p a c e

51 t y = m a x t y ∗ ( 1 . 0 − 2 . 0 ∗ ( f l o a t 8 )x /( f l o a t 8 ) (PR OUTPUT PIXELS−1)) ;

52 t z = m a x t z ∗ ( 1 . 0 − 2 . 0 ∗ ( f l o a t 8 )y /( f l o a t 8 ) (PR OUTPUT PIXELS−1)) ;

53

54 v i e w f = F o u r V e c t o r ( 1 . 0 ,−1 . 0 ,− ty ,− tz ) ;55 vi e w f . Ma keL i g h tl i ke () ; / / i n i t i a l momentum s e t 56 // Tran sf orm i n t o l o c a l c o or d i na t e b a s i s57 mFrame. T r a n s f o r m In to C o o r d i n a teB a s i s ( vi ew f , vi ew c ) ;58 cout<<” R e nd e ri ng L i ne ”<<y+1<<” Row ”<<x+1<<” . . . . . . ”<<100∗((

f l o a t 8 ) ( y∗PR OUTPUT PIXELS+x) / ( f l o a t 8 ) (PR OUTPUT PIXELS∗PR OUTPUT PIXELS) )<<” % f i n i s h e d . ”<<endl ;

59 // i n t e g r a t i o n+r a d i at i o n t r a n s f e r i s done h er e :60 #i f ( ! PR TRACE PLANAR)61 M ea su re me nt ( v i e w c , e v e nt , s p e c t i m e , m at t er , t i m e i n c re m e n t

) ;62 #endif

63 #i f (PR TRACE PLANAR)64 M e a su r e me n t p l an a r ( v i e w c , e v e nt , s p e c t i m e , m a tt e r ,

t i m ei n cr em en t ) ;65 #endif

66

67 // s p e ct ra f o r a l l t im es t e ps f o r g iv en p i x e l f i n is h e d 68 f o r ( i n t j = 0 ; j<PR NUMBER TIMESTEPS; j ++)69 70 // s p e c t r a a r e a d de d t o t o t a l s p ec t ru m s .71 ( s p e c t o t + j )−>AddSpectrum(∗ ( spe c tim e+j ) ) ;72 #i f (PR IMAGES)73 // e v a l u a t io n o f c o l o r r e p r e s en t a t i o n a nd s t o r i n g t o i mag e

74 ( s p ec ti m e+ j )−>C re a te S hi f tR GB ( r e f f r e q , c o l o r ) ;75 ( images+j )−>S e t P i x e l ( x + 1 , y + 1 , c o l o r ) ;76 #endif

77 78 79 80 #i f (PR IMAGES)81 // i ma ge s n ee d t o b e n o r ma li ze d t o h i g h e s t i n t e n s i t y c o n t r i b u t i on 82 f l o a t 8 h i g h e s t b r i g h t n e s s = 0 .0 ; // v a r i a b l e f o r b r i g h t e s t c om pon en t 83 f l o a t 8 b r i g h t n e s s c h e c k = 0 .0 ;84 cout<<” P e r f o r mi n g i ma ge n o r m a l i z a t i o n . ”<<endl ;85 f o r ( i n t i =0; i<PR NUMBER TIMESTEPS; i ++)86

69




87 b r i g h t n e s s c h e c k = (i m ag e s+ i )−>G e t H i g h e s t B r i g h t n e s s ( ) ;88 i f ( b r i g h t n e s s c h e c k > h i g h e s t b r i g h t n e s s ) h i g h e s t b r i g h t n e s s =

b r i g h t n e s s c h e c k ;89 90 f o r ( i n t i =0; i<PR NUMBER TIMESTEPS; i ++)91 ( images+i )−>N o rm a li z e ( h i g h e s t b r i g h t n e s s ) ; // n o r m a li z i n g t o

h i g h e s t v a lu e92 #endif

93

94 s t r i n g i n t d i s t ri b u t io n n a m e =∗simname+” in tt o t . dat” ;95 s t r i n g t ga n a me ;96 s t r i n g s pe c n am e ;97 f l o a t 8 I t [PR NUMBER TIMESTEPS ] ;98 f l o a t 8 I me an = 0 .0 ;99 f l o a t 8 I ma x = 0. 0 ;100 o f s t r ea m i n t t o t ( i n t d i s t r i b u t i o n n a m e . c s t r ( ) ) ;

101 // i ma ge s and t o t a l s p e ct r a f o r e ac h t i m e s te p f i n i s h e d 102 / / c r e a t e I m ag e Pl o ts , s p ec t ru m d a t a f i l e s an d 103 // t im ed ep en de nt i n t e n s i t y d i s t r i b u t i o n d a t a f i l e s104 f o r ( i n t i =0; i<PR NUMBER TIMESTEPS; i ++)105 106 o s t r i n g s t r e a m temp ;107 i f ( i <10 )108 temp<<” 0 ”<< i ;109 e ls e i f ( i >9)110 temp << i ;111 tga name=∗simname+temp. s t r ( )+” . tga ” ;112 spec name=∗simname+temp. s t r ( )+” . dat ” ;113 #i f (PR IMAGES)114 cout<<” Ima ge ”<< i+1<<” o f ”<<PR NUMBER TIMESTEPS<<” i s w ri tt en . ”<<

endl ;115 / / im ag e d a t a w r i t t e n t o TGA− f i l e

116 ( images+i )−>CreateTGA( tga name , ( ui nt ) (8 00 /PR OUTPUT PIXELS) ) ;117 #endif

118

119 ( s p e c t o t + i )−>C r ea teD a ta F i l e ( s p ec n a m e ) ; // s pe ct ru m d a t a i s w r i t t e n 120 i f (PR SPECTRUM PLOT==1)121 // s pe ct ru m i s p l o t t e d i f s e t i n ” p ar am et er . h ”122 spec name=∗simname+temp . s t r ( )+” spe c . tg a” ;123 Ima ge s p ec P l o t (8 0 0 , 8 0 0 ) ;124 ( s p e c t o t + i )−>C r ea teP l o t ( s p ec P l o t ) ;125 spe cPl ot .CreateTGA(spec name ,1) ;126

127 I t [ i ] = ( s p ec to t+i )−>G e t T o t I n t e n s i t y ( ) ;128 i f ( I t [ i ]> I m a x ) I m a x= I t [ i ] ; 129 I mean+=I t [ i ] ;130 // w r i t in g d at a f o r l i g h t c ur ve s

131 i n t t o t<<ti m ei n cr em en t ∗( f l o a t 8 ) i<<” ”<<I t [ i ]<<en d l ;132 133 // some s t a t i s t i c s134 f l o a t 8 rms ;135 f l o a t 8 v a r i an c e ;136 f l o a t 8 s t d d e v i a t i o n ;137 f l o a t 8 v ar = 0 . 0;138 I me an=I me an / ( ( f l o a t 8 )PR NUMBER TIMESTEPS) ;139 f o r ( i n t i =0; i<PR NUMBER TIMESTEPS; i ++)

70



3.4. Source Code

140 141 var+=(I t [ i ]− I m e a n ) ∗( I t [ i ]− I m ean ) ;142 143 v a ri an c e=var / ( ( f l o a t 8 )PR NUMBER TIMESTEPS) ;144 s t d d e v i a t i o n =s q r t ( v a r i a n c e ) ;145 rms=sqr t ( I mean∗ I m e an+s t d d e v i a t i o n ∗ s t d d e v i a t i o n ) ;146

147 i n t t o t . c l o s e ( ) ; / / c l e a n i n g u m 148 d elete [ ] s p e c t i m e ;149 d elete [ ] s p e c t o t ;150 #i f (PR IMAGES)151 d elete [ ] i m a g es ;152 #endif

153 // p r e p a ri n g and w r i t i n g i n f o rm a t io n f i l e

154 s t r i n g t e x t=∗simname+” . tx t ” ;155 o f s t r e a m p a r ( t e x t . c s t r ( ) ) ;

156 i f (PR TRACE PLANAR)157 pa r<<” P l a n a r Ray−T r a c in g p e r f or m e d . ”<<en d l ;158 e l s e 159 i f (PR ABSORPTION) pa r<<”Volume Ray−T r ac i ng p e rf o rm e d c o n s i d e r i n g

a b s o r p ti o n . ”<<endl ;160 e l s e pa r<<”Volume Ray−T r ac i ng p e rf o rm ed w i th o ut c o n s i d e r i n g a b s o r p t i o n

. ”<<end l ;161 pa r<<” K er r p a r a m eter a=”<<PR SPACETIME A<<en d l ;162 pa r<<”Black Hole mass M=”<<PR SPACETIME M<<en d l ;163 pa r<<” Came ra p o s i t i o n s : ”<<endl ;164 pa r<<” r : ”<<PR CAMERA R<<endl ;165 pa r<<” t h e t a : ”<<PR CAMERA THETA<<endl ;166 pa r<<” p h i : ”<<PR CAMERA PHI<<en d l ;167 pa r<<” a p e r t ur e : ”<<PR CAMERA APERTURE<<en d l ;168 pa r<<endl ;169 pa r<<”Ray−Modu lati ons : rms=”<<rms<<endl ;170 pa r<<”Mean Flu x=”<<I mean<<endl ;171 pa r<<” H i g h es t A m pl i tu d e = ”<<I max<<endl ;172 pa r<<” The s t a n d a rd d e v i a t i o n =”<<s t d d e v i a t i o n <<endl ;173 pa r<<” H i g h e st M od ul at io n i s : ”<<100.0∗(I m ax−I m ea n ) / I m ea n<<”% of mean

f l u x . ( = ”<<100.0∗(I m ax−I m ea n )/ r ms<<” % r ms a nd ”<<(I max−I m e an ) /s t d d e v i a t i o n <<” t i m e s t h e s t a nd a r d d e v i a t i o n . ) ”<<endl ;

174 pa r<<endl ;175 i f (PR ACCRETIONDISK USE)176 177 pa r<<” A c c r e t i o n d i s c : ”<<en d l ;178 pa r<<” I n n er r a d i us : ”<<matter . dis c−>GetInnerRadius ()<<endl ;

179 pa r<<” O ut er r a d i u s : ”<<matter . dis c−>GetOuterRadius ()<<endl ;180 pa r<<” D e n si t y : ”<<PR ACCDISC DENSITY<<en d l ;181 pa r<<” O p a ci t y : ”<<PR ACCRETIONDISK OPACITY<<endl ;182 pa r<<en d l ;183 184 e l s e 185 pa r<<” no A c c re t i on d i s c u se d ”<<endl ;186 pa r<<en d l ;187 188 pa r<<” Number o f H ot S p o ts : ”<<PR HOTSPOT NUMBER<<endl ;189 f o r ( i n t j = 0 ; j<PR HOTSPOT NUMBER; j ++)190 191 pa r<<” ho t s po t ”<< j+1<<” : ”<<en d l ;

71




192 pa r<<” R a d i us = ”<<matter . matt erit ems [ j ]−>GetRadius ()<<endl ;193 pa r<<” r a t s t ar t = ”<<matter . matt erite ms [ j ]−>G e t r ( )<<endl ;194 pa r<<” ph i a t s t a r t = ”<<matter . matt erite ms [ j ]−>Get p h i ()<<en d l ;195 pa r<<” C i r c u l a r f r e q u e n c y Omega= ”<<matter . matt erite ms [ j ]−>GetOmega()

<<en d l ;196 pa r<<”−−> nu=Omega/(2 Pi )= ”<<matter . matt erite ms [ j ]−>GetOmega( ) / (2 ∗PI )

<<en d l ;197 pa r<<” d e n s i t y p ea k= ”<<matter . matt erite ms [ j ]−>G e t d e n s i t y ( )<<en d l ;198 pa r<<” O v e r b r i g ht n e s s : ”<<matter . matt erit ems [ j ]−>G e t d e n s i t y ( ) ∗100.0<<

”%”<<en d l ;199 pa r<<en d l ;200 201 pa r<<endl ;202 pa r<<endl ;203 pa r<<” Number o f t i m e s t e p s : ”<<PR NUMBER TIMESTEPS<<en d l ;204 pa r<<” Number o f o b s e r v e d p e r i o d s : ”<<PR NUMBER OF PERIODS<<endl ;

205 pa r<<endl ;206 pa r<<”Sp ectrums : ”<<en d l ;207 pa r<<” l o w e st f r e q . : ”<<PR SPECTRUM NU MIN<<en d l ;208 pa r<<” h i g h es t f r e q . : ”<<PR SPECTRUM NU MAX<<en d l ;209 pa r<<” r e s o l u t i o n : ”<<PR SPECTRUM RES<<endl ;210 pa r<<” r e f e r e n c e f r e q . : ”<<PR REFERENCE FREQ<<endl ;211 p a r . c l o s e () ;212

213 retu rn tru e ; / / f i n i s h e d 214

As mentioned before, the nullgeodesics are traced for each pixel on the screen separately.This pixel iteration begins in code line 46.After setting up the initial conditions for the integration of the geodesic equations,the method Measurement(. . . ) (see page 51) of the same instance is called in line 61.This method controls the sequence of the stepwise integration and the cumulation of the relativistic invariant transport quantity F and the cumulated opacity Ψ along thegeodesics:

Listing 3.2: Source code of Raytracer::Measurement(. . . )

1 void Ray tra cer : : Measurement( const F o u r V e c t o r & b a s e d i r e c t i o n ,2 const Event& basepos , Spectrum∗ & s p e c s ,3 Ma tter F i el d & m a tter , const f l o a t 8 & t i m e in c r )

4 5 // w o rk i ng s p e c t r a a re c l e a r e d 6 f o r ( i n t i = 0 ; i<PR NUMBER TIMESTEPS; i ++)7 8 ( spec s+i )−>Clea r () ;9 10

11 // t he o b je c t r e p re se n t i ng g eo de si c i s i n i t i a l i z e d 12 K e r r Ge o d e s ic r a y ( b a s e d i r e c t i o n , b a s e p o s ) ;13 Spectrum ∗ Psi ;14 i f (PR ABSORPTION==1)15 // c u mu la te d o p a c i t y f o r e ac h t i m e s t e p16 P s i = new Sp ec tr um [ PR NUMBER TIMESTEPS ] ;

72



3.4. Source Code

17 f l o a t 8 o p a c i ty c o n s t = 1 .0 ;18 f o r ( i n t j = 0 ; j<PR NUMBER TIMESTEPS; j ++)19 20 ( Psi+j )−>AddConst( opac it yc on st ) ; / / S e t i n i t i a l l y t o 121 22 23 // I n t e g r a t e and e v a l ua t e r e nd e ri n g e q ua t io n f o r a l l t i me s a t e ac h

i n t e gr a t i on s t ep24 S p ectru m a ct s p ectr u m ;25 f l o a t 8 s t e p s iz e = −0 . 1 ; // i n i t i a l s t e p s i z e ( u se d f o r p r op ag a ti o n )26 f l o a t 8 s t e ps i z e u s e d ; // v a r i a bl e f o r s t o r in g a c tu a l s t ep s i z e27 u i n t s tepn u m ber = 0 ;28 E ve nt a c t p o s i t i o n =b a se p o s ;29 E ve nt n e w p o s i t i o n =b a s e p o s ;30 F ou rV ec to r a c t d i r e c t i o n =b a s e d i r e c t i o n ;31 F ou rV ec to r n e w d i r e c t i o n ;

32 while (stepnumber<=PR RAY MAX STEPS)33 34 // c h e c k i n g f o r t e r m in a t i on c o n d i t i o n s35 i f ( ( PR RAY STOP AT CAMERA RADIUS==1) && ( ac t p o s i t i o n . mTupel . a1>

PR CAMERA R) ) break ;36 i f ( ! r ay . F e hl b er g 4 5 ( a c t d i r e c t i o n , a c t p o s i t i o n , s t e p s i z e ,

s t e p s i z e u s e d , n e w d i r e c t io n , n e w p o s i t i o n ) ) break ;37

38 // d et er mi ne s ou r ce c o n t r i b u t i o n s t o F 39 // i f s t a t i c t r ac i n g i s pe rf or me d −> e v al u at i on i s same f o r a l l

t i m e s t e p s40 i f ( ! matter .CheckTimeDep( new pos iti on ) )41 42 i f (PR ABSORPTION==0)43 / /no a b s o r p t i o n 44 m a t te r . A d d I n t e n s i t y ( n e w p o s i t i o n , n e w d i r e c t i o n , − s t e p s i z e u s e d

, 0 , s pe cs , PR NUMBER TIMESTEPS) ;45 e l s e

46 // w i t h a b s o r p t i on 47 m a t te r . A d d I n t en s i t y A b s ( n e w p o s i t i o n , n e w d i r e c t i o n , −

s t e p si z e u se d , 0 , Psi , sp ec s , PR NUMBER TIMESTEPS) ;48 49 e l s e

50 51 // e v a lu a te f o r a l l t i me s te p s s e pp a ra t e ly 52 i f (PR ABSORPTION==0)53

54 f o r ( i n t i = 0 ; i<PR NUMBER TIMESTEPS; i ++)55 m a t te r . A d d I n t e n s i t y ( n e w p o s i t i o n , n e w d i r e c t i o n , −

s t e p s i z e u s e d , ( t i m e in c r ∗( f l o a t 8 ) i−f a b s ( n e w p o s i t i o n .mTupel. a0 ) ) , ( spe cs+i ) ,1 ) ; // e v a l u a t e w i t ho u t a b s o rb t i o n

56 57 e l s e

58 59 f o r ( i n t i = 0 ; i<PR NUMBER TIMESTEPS; i ++)60 m a t te r . A d d I n t e ns i t y A b s ( n e w p o s i t i o n , n e w d i r e c t i o n , −

s t e p s i z e u s e d , ( t i m e in c r ∗( f l o a t 8 ) i−f a b s ( n e w p o s i t i o n .mTupel. a0) ) , ( Psi+i ) , ( spec s+i ) ,1) ; // w i t h a b s o r b t i o n

61 62

73




63 // u pd at e o f i n i t i a l b ou nd ar ie s f o r n ex t i n t e gr a t i on s t ep

64 a c t p o s i t i o n =n e w p o s i t i o n ;65 a c t d i r e c t i o n =n e w d i r e c t io n ;66 stepnumber=stepnumber+1;67 68 i f (PR ABSORPTION==1) // c l e a n i n g up69 d elete [ ] P s i ;70 f o r ( i n t k = 0 ; k<PR NUMBER TIMESTEPS; k++)71 ( spe cs+k)−>FtoI () ; // F −>I 72

73

3.5. Power Density Spectra

As mentioned in Sec. 2.1.3, variabilities in light curves obtained from observations areusually analyzed in the frequency space. Therefore the time depending signal is decom-posed to a sum of periodic oscillations with specific frequencies ω and amplitudes.The power density spectra, arising from this procedure, are well suited to examine theperiodic structure of the signal distribution in time.Consequently, in order to simulate such power density spectra, the time depending inten-sity distributions I (t), as obtained from the ray-tracing application, must be transformedto frequency space. This can be achieved by applying a discrete Fourier transform tothe finite-domain discretized light curves.Generally the Fourier series of a time depending signal S (t), defined in the domaint ∈ [0, T ], with the period T is given by

S (t) =∞

n=−∞cn exp(2πi

nt

T ) (3.36)

with the Fourier coefficients

cn =1

T

T 0

S (t) exp(−2πint

T ) . (3.37)

By the means of the trapezoidal rule this integral can be discretized and approximated.This procedure yields the corresponding discrete Fourier transform

F (n∆ω) =

N −1m=0 S (m∆t) exp(−i n ∆ω m∆t)

=N −1m=0

S (m∆t) exp(−2πim n

N ) , (3.38)

where ∆ω = 2π/T and N ∆t = T .So given the discretized light curve I (n∆t) with n = (0, 1, . . . , N −1), the discrete fouriertransform can be obtained by

I (n∆ω) =N −1m=0

I (m∆t)exp(−2πimn

N ) . (3.39)

74



3.5. Power Density Spectra

From this equation one could suppose that it should be possible now to obtain N inde-pendent coefficients I n. Unfortunately this is not the case, since I (n∆t) ∈ R for all n.

The representative in frequency space of a pure real signal in the time space is hermitian ,meaning I N −n = I n. As the Nyquist-Shannon sampling theorem states, with a samplingangular frequency ωs = 2πN/T = N ∆ω, where ∆ω is the resolution in frequency spaceand T the time interval on which the signal is given, only frequencies lower than theNyquist frequency ωny = ωs/2 = N ∆ω/2 can be reconstructed.Consequently transforming the given light curves I (n∆t) by a DFT to the frequencyspace, a resolution ∆ω = 2π/T is obtained, and the highest accessible angular frequencyis ωmax = ((N/2)− 1)∆ω.The discrete Fourier transform is implemented in a separate application. The transformis being performed by the function void dft(...), which, supplied with a data file of the format (t, S (t)) and the number N of timesteps, determines the measuring interval

T = tmax − tmin, performs the evaluations and stores the results to a file of the format(ω, S (ω)).This procedure is performed by:

Listing 3.3: Implementation of the DFT

1 void d f t ( char ∗ name , i n t p u n kta n za h l )2 3 i f s t r e a m d a t e i ;4 da te i . open( name) ;5 double t ;6 double i n t e n s t ;7

8

d a t e i>>t>>i n t e n s t ;9 double t min=t ;10 double t max=t ;11 while ( ! d a te i . eo f () )12 d a t e i>>t>>i n t e n s t ;13 i f ( t<t m i n ) t min=t ; // d e t e rm i n in g t h e t im e i n t e r v a l T=t ma x −t m i n 14 i f ( t>t m a x ) t max=t ;15 16 d a t e i . c l o s e ( ) ;17

18 o f st r ea m r e s u l t a t ( ” d f t r e s u l t . d at ” ) ;19 i n t N=punkta nzah l ; // number o f g i v e n t i m e s t e p s

20 double d el ta o m eg a = (2 . 0∗P I ) / ( t m a x−t m i n ) ;21 complex<double> g ( 0 . 0 , 0 . 0 ) ;

22

23 f o r ( i n t n=0;n<N; n++)24 i f s t r e a m i n t e n s ( name ) ;25 g = ( 0 . 0 , 0 . 0 ) ;26 cout<<”DFT e v a l u a t e s s t e p ”<<n+1<<” o f ”<<N<<endl ;27

28 f o r ( i n t m=0;m<N;m++)29 i n t e n s >>t>>i n t e n s t ; // g e t ti n g t he i n t e n s i t y I n 30 g=g+ i n t en s t ∗ p o l a r (1 . 0 , − ( ( 2 . 0∗ PI∗m∗n) /N) ) ; / / e v a l u a t i o n 31 32 // w r i t i n g d at a t o f i l e33 i f ( n<(N/2) ) 34 r e s u l t a t <<n∗deltaomega<<” ”<<a b s ( g )<<endl ;

75




35 e ls e i f ( n==(N/2 ) ) 36 e ls e i f ( n>(N/2) ) 37 / / v a l u e s f o r omega >n y qu i st f re qu en cy a re s h i f t e d t o n e ga t iv e

f r e q u e n c i e s38 r e s u l t a t <<(n−N) ∗deltaomega<<” ”<<a b s ( g )<<endl ;39 i n t e n s . c l o s e ( ) ;40 41 r e s u l t a t . c l o s e ( ) ;42 cout<<” d el ta Om eg a i s ”<<deltaomega<<” = 2∗P I / ( ”<<t max<<”−”<<t m i n<<”

) ”<<en d l ;43 cout<<” H i gh e st d et e rm in ed f r e qu e n cy i s ”<<deltaomega ∗ ((N/2)−1)<<endl ;44 cout<<”Nyquist−f re q ue n cy i s ”<<deltaomega ∗(N/2)<<en d l ;45

76



4. Data Analysis

In this chapter the results obtained from different kind of simulations performed withthe time depending volume ray-tracing application are presented.For testing purpose static simulations were performed and are presented in Sec. 4.1.The imprints on light curves from the hot spots co-rotating within the accretion disc areexamined thereafter in Sec. 4.2.

Before turning to the data obtained, let us take a look on some global parameters, whichare specified in the application’s parameter file, and that were kept constant throughoutall evaluations.The metric, the ray-tracing is performed on, is exactly defined by the Kerr parameter aand the central mass M , which was continuously set to the value M = 1.Volume ray-tracing, considering absorption, was applied to all simulations. For thatpurpose partially opaque accretion discs were mimicked by setting the opacity in theemitter’s rest frame to κ = 1. The same was done for the hot spot matter.The accretion disc was generally assumed to extend from the marginal stable orbit tothe radius r = 30M , and a constant radial profile of the emissivity was used.

The camera was positioned at the radius rc = 1800M and the aperture was chosen tobe γ = 2. With the size of the accretion disc those set ups provide an adequate view onthe given scene.The camera’s screen resolution was set to 400 in both directions, which results in 160 , 000geodesics to be propagated. Espacially in connection with the evaluation of multipletimesteps and the number of maximum integration steps, set to 1500, this parameteressentially affects the processing time. Consequently due to time constraints a higherresolution was rejected.The error boundaries for the adaptive ray propagation, which control the step size ateach integration step were defined diversely for three different regions. Comparing theevaluated geodesics to those computed with a higher accuracy, the following boundaries

turned out to approximate the nullgeodesics sufficiently accurate in the scopes of inter-est:For regions specified by r < 6M the local error was approved to be d ∈ [3 ·10−6, 3 ·10−5],for 6M ≤ r ≤ 200M the interval [3 · 10−5, 3 · 10−4] was applied, and for r > 200M theboundaries were given by [r/106, r/105].Finally the instances of the class Spectrum , which during the evaluations store the fre-quency dependent intensity contributions, need to be specified. A spectral resolution of 2, 000 within the range given by ν min = 0.0001 and ν max = 3 was chosen. Note that thefrequency is given in relation to the rest frame frequency of the radiation emitted. Thusit is positioned at ν = 1.As the spectra of the broadened lines will show, this spectral range is suitable.

77



4.1. Static Simulations

Comparing the line profiles of the Schwarzschild case to those of the maximum Kerrcase (a = 1), the Kerr parameter turns out to have only small influence on the line

broadening. As figure 4.2 visualizes, only the red wing is extended to lower frequencies,when a = 1 is applied. This results from the fact that for increasing Kerr parameterthe accretion disc ranges down to lower radial coordinates, at which the lapse functionα decreases, and consequently the radiation originating from those regions experiencesa higher red-shift from the general relativistic time dilatation.

0 , 5 1 , 0

1 0 0 0

r

e

a

t

v

e

f

r

e

q

u

e

n

c

y

[

a

r

b

t

r

a

r

y

u

n

t

s

]

f r e q u e n c y /

a = 0

a = 1

Figure 4.2.: Logarithmic plotting of relative intensity of a single line emission at ν = 1for the inclination i = 30 °. Solely the red wing is noticeably influenced bythe Kerr parameter a.

79



4. Data Analysis

The spatially resolved scene is displayed by the figures 4.3-4.6 for the two inclinationsi = 30 °, i = 60 ° applying the Schwarzschild case as well as the maximum Kerr case.

The local red-shift is color-coded so that red means red-shift and blue color implies ablue-shift. Additionally the relative intensity is implied by brightness1.Those images approve, as the line spectra already had shown, that for higher inlinationsthe Doppler effect has a remarkable impact. Consequently the regions moving in direc-tion of the observer are blue-shifted.The images, representing the Schwarzschild case, nicely show the accretion disc’s inneredge at rin = rms = 6M . A close look to the inner region of the images reveals a slightshift (to the right) when comparing the two cases a = 0 and a = 1. This is due to theframe dragging forming the curvature and consequently the nullgeodesic flow (compareto Fig. 2.19).A further feature, that derives from the volume character of the ray-tracing applica-

tion, and cannot be generated by planar tracing when ray propagation is stopped in theequatorial plane, is the manifestation of a secondary structure in the direct vicinity of the event horizon. It results from lensing effects, as near the horizon nullgeodesics arebeing “back-curved” to the observer’s direction. Consequently the form of the horizonis mirrored by the inner edge of those secondary images [Zin2002].

1This pixel color-encoding is performed by the method void CreateShiftRGB(. . . ) of the class Spectrum .

80



4.1. Static Simulations

Figure 4.3.: Spatially resolved image of accretion disc around black hole for a = 0 andi = 30 °. The red-shift is color-coded and the brightness is weighted by therelative intensity contribution.

Figure 4.4.: Spatially resolved, color-coded image of accretion disc around black hole fora = 1 and i = 30 °.

81



4. Data Analysis

Figure 4.5.: Spatially resolved, color-coded image of accretion disc around black hole fora = 0 and i = 60 °.

Figure 4.6.: Spatially resolved, color-coded image of accretion disc around black hole for

a = 1 and i = 60 °.

82



4. Data Analysis

value. As the diagrams show, the opposite applies to the light curves of the highestinclination possible i = 90 °. Furthermore a shift in time can be noticed between the

peak at maximum inclination to those at lower viewing angles.When the spin parameter is increased, the ISCO and consequently the hot spot’s radialdistance decrease to lower r. The velocities therefore become higher and result in theamplfication of the relativistic beaming. This explains the general increase of the vari-abilities at medium inclinations. In the near maximum Kerr case a = 0.99 peaks of allinclanations get damped, as the gravitational red-shift becomes more effective at smallhot spot distances to the event horizon.Furthermore at lower radial distances the geodesic bending by the spacetime curvatureis more effective, and the peak in the detected intensity moves to earlier times, whenthe hot spot is still located behind the black hole. This does not apply to the cases of maximum inclination, due to the circumstance that for those the peak in the intensity

occurs at times when the hot spot is positioned on the black hole’s far side anyway.In those cases the inner region of the scene is obscured by the disc itself and mainly theopposite regions, that are visible due to the curvature of the spacetime, contribute tothe variabilities detected.

Analyzing the light curves, it should be noticed that mainly for low values of a thepeak in intensity is followed by a noticeable decline.This occurs at times when the hot spot is moving away from the observer and conse-quently the radiation experiences a red-shift. The absence of this feature at high Kerrparameters again can be explained by extreme frame dragging in the vicinity of the eventhorizon.

As the figure 2.19 on page 36 clearly shows, nullgeodesics, originating from small dis-tances to the horizon, are orbiting the central mass repeatedly with slightly increasingradial coordinate r before they are able to escape to infinity.Due to that, for hot spots at low distances to the centre, there should be a backgroundcontribution to the relativistic beaming that impacts for all spot’s angular phases.Consequently at high values of a, where the ISCO is located at such distances, no de-crease in the intensity can be observed and the light curves appear more symmetric.

84



4.2. Dynamic Simulations

(a) Hot spot at t = 0. (b) Hot spot at t = 1/20T 0.

(c) Hot spot at t = 2/20T 0. (d) Hot spot at t = 3/20T 0.

(e) Hot spot at t = 4/20T 0. (f) Hot spot at t = 5/20T 0.

Figure 4.7.: A series of a hot spot with the radius Rhs = 0.5M orbiting a black hole with

a = 0.7 at the radial distance r = 7M . The rest frame density of the spotis specified to ˆ = 1.5 and the inclination i = 60 ° is applied.T 0 denotes the hot spot’s rotation period. 85



4. Data Analysis

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 5

1 , 0 0 0 0

1 , 0 0 0 5

1 , 0 0 1 0

1 , 0 0 1 5

1 , 0 0 2 0

1 , 0 0 2 5

1 , 0 0 3 0

1 , 0 0 3 5

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

i = 0 . 0 0 1

i = 3 0

i = 6 0

i = 9 0

t i m e t / T

Figure 4.8.: Frequency-integrated light curves of a hot spot orbiting at the ISCO definedby the Kerr parameter a = 0. The intensities are normalized to their meanvalues, and different inclinations are implied by color.

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 8

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

i = 0 . 0 0 1

i = 3 0

i = 6 0

Figure 4.9.: Frequency-integrated light curves of a hot spot orbiting at the ISCO definedby the Kerr parameter a = 0. The intensities are normalized to their mean

values, and different inclinations are implied by color.

86




0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 5

1 , 0 0 0 0

1 , 0 0 0 5

1 , 0 0 1 0

1 , 0 0 1 5

1 , 0 0 2 0

1 , 0 0 2 5

1 , 0 0 3 0

i = 0 . 0 0 1

i = 3 0

i = 6 0

i = 9 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

Figure 4.10.: Frequency-integrated light curves of a hot spot orbiting at the ISCO definedby the Kerr parameter a = 0.3. The intensities are normalized to theirmean values, and different inclinations are implied by color.

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 8

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

1 , 0 0 0 5 i = 0 . 0 0 1

i = 3 0

i = 6 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T


87



4. Data Analysis

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

1 , 0 0 0 0

1 , 0 0 0 5

1 , 0 0 1 0

1 , 0 0 1 5

1 , 0 0 2 0

1 , 0 0 2 5

i = 0 . 0 0 1

i = 3 0

i = 6 0

i = 9 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T


0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 8

1 , 0 0 0 0

1 , 0 0 0 2

1 , 0 0 0 4

1 , 0 0 0 6

1 , 0 0 0 8

i = 0 . 0 0 1

i = 3 0

i = 6 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T


88




0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 8

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

1 , 0 0 0 5

1 , 0 0 0 6

i = 0 . 0 0 1

i = 3 0

i = 6 0

i = 9 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

Figure 4.14.: Frequency-integrated light curves of a hot spot orbiting at the ISCO defined

by the Kerr parameter a = 0.99. The intensities are normalized to theirmean values, and different inclinations are implied by color.

89



4. Data Analysis

4.2.2. Influence of Hot Spot Radius and Distance on Light Curves

In order to examine how the light curves are affected by the hot spot’s radius and radial

distance, the Kerr parameter and the inclination were kept constant for the simulationspresented next.A moderate inclination of i = 60 ° was chosen, whereas a was set to a = 0.7.Figures 4.15 and 4.16 visualize the light curves obtained while the radius Rhs and theradial distance to the mass centre were varied.As expected, they do not reveal any specific influences on the curves’ form, but solelythe size of the peaks is affected.It is higher for increasing radii, and also increases at lower radial distances r. This isdue to the applied Keppler rotation and therefore increasing velocities at lower r.Note that in Fig. 4.15, where the light curves for different distances are presented, theangular phases of the hot spots do not coincide. This results from the fact that thepropagation time of light from positions at different radial coordinates to the observeris unequal. When now for the different simulations the initial azimuthal position of thespots is set to the same value, consequently this time lag leads to the spots appearingat different azimuthal angles at the first timestep evaluated. This is why, compared toeach other, the light curves are shifted in time2.In spite of that, the major effect from differing distances r can be extracted.

2Naturally the angular frequencies differ for the different radial coordinates. This does not need to betaken into account, since the time is given in units of the rotation period T .

90




0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 9 8

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

1 , 0 0 0 5

1 , 0 0 0 6

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

r = 6 M

r = 1 0 M

r = 1 2 M

r = 1 5 M

Figure 4.15.: Frequency-integrated light curves of a hot spot orbiting at different radialdistances, implied by color. The inclination is i = 60 ° and the Kerr pa-rameter is a = 0.7. The intensities are normalized to their mean values.The angular phases of the spots, which the curves are obtained from, do

not coincide.

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 9 9 8

1 , 0 0 0

1 , 0 0 2

1 , 0 0 4

1 , 0 0 6

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

= 0 . 3 M

= 0 . 5 M

= 0 . 8 M

= 1 . 0 M

= 1 . 5 M

Figure 4.16.: Frequency-integrated light curves of a hot spot orbiting at the distance

r = 6M . The inclination is i = 60 ° and the Kerr parameter is a = 0.7.The intensities are normalized to their mean values, and different hot spotradii are implied by color. 91



4. Data Analysis

4.2.3. Generating Power Density Spectra

As the qualitative results from the last section show, light curves, obtained from ob-

servations of black hole systems, could provide an approach to the investigation of thesystems’ parameters.Unfortunately with current observational capabilities it is not possible to get a strongenough X-ray signal over individual periods in order to be able to differentiate betweensuch light curves.This is why variabilities, the QPOs, in the spectra are usually examined in the frequencyspace (see Sec. 2.1.3).In this section, the way how it should be possible to simulate such PDSs, and what kindof conclusions about the system should be able to be derived from them, is outlined.For this purpose simulations over multiple rotation periods of hot spots were performed.Due to what the light curves have tought us so far, only medium inclinations were ap-plied. At low inclinations the variabilities are very faint, and high inclinations werediscarded because a present dust torus, as assumed in theory but not implemented tothe ray-tracer application of this thesis, should obscure the inner region and generallyshould be considered when simulating observations at low viewing angles.Another aspect, not considered by the application, must be taken into account. Due toshearing in the accretion disc, the hot spots exhibit a finite life time. In the course of time, the regions of higher emission should be deformed into an arc-like structure anddissolve thereafter.Due to this fact the light curves for this part were generated by evaluating the inte-grated total intensity over a moderate number of rotation periods. The “observation”time T = 4.5T 0 was usually applied when not stated differently.This value is important when transforming the light curves to the frequency space, sinceit defines the resolution in frequency to

∆ω = 2π∆ν =2π

T =

ω0

4.5, (4.1)

where ω0 = 2π/T 0 denotes the hot spot’s angular frequency. The highest accessiblefrequency ωmax in the generated PDS is specified by (compare to Sec. 3.5)

ωmax =

N

2− 1

∆ω , (4.2)

where N is the number of evaluated timesteps, and was set to N = 40.The diagrams 4.17-4.32 show the light curves with varying Kerr parameter and hot spotdistance to the centre and the corresponding power density spectra obtained by themethod introduced in Sec. 3.5.The non-sinusoidal shape of the light curves results in declining power in the higherharmonic frequencies at nω0 (n ∈ N).Despite that, except for the near maximum Kerr case, the PDS reveal power contribu-tions up to n = 4.

92




A χ2-Lorentzian fit of the form

f (ω) = aoffset +A

π

HWHM

(ω − ω0)2 + HWHM2 (4.3)

was applied to the base peaks in order to reversely verify the hot spot’s angularfrequency ω0

3.As the method shows, from sufficiently resolved power density spectra it is possible toobtain the circular frequencies of high emission regions orbiting Kerr black holes. Thisaspect gains in importance, when analyzing QPOs, deriving from multiple hot spotsorbiting the compact object.Considering the assumption that the minimum radial coordinate of hot spots withinthe accretion disc equals the marginally stable orbit (ISCO), the Kerr parameter a of arotating black hole can be estimated from the highest base frequency found in the PDS.

Such a light curve for two hot spots is presented by Fig. 4.33. The spots were situatedat the radial coordinates r1 = 8M , r2 = 6M , and 60 time steps were evaluated over 6.5times the rotation period of the inner hot spot.The corresponding PDS reveals power at both angular frequencies and their higher har-monics. Particularly the contributions for n = 2 are clearly distinguishable at ω ≈ 0.095and ω ≈ 0.14.

The figures 4.35-4.36 once again visualize the already discussed dependency of the X-rayvariabilities on the radial coordinate r, the hot spot’s radius Rhs and the spinparameter a.The highest modulation in the light curves I mod = (I max

−I mean)/I mean was therefore

evaluated in units of the root mean square rms of the intensity distributions in time fordifferent values of the concerning parameters.In order to compare the variabilities at different orbit radii (Fig. 4.35), an adjustmentin the hot spot radii Rhs had to be performed. Since the ray-tracing application doesnot consider any intensity contributions from matter located below the ISCO, only theouter hemisphere of the hot spots, centered at rms, is taken into account.Therefore to compare the contribution from such high emission regions at the ISCO tothose at higher radial distances, the radius Rhs(ISCO) was multiplied by a factor of 2.

3The fit was performed by the scientific graphing and analysis software OriginPro 7.5 SR0 developedby the OriginLab Corporation .

93



4. Data Analysis

0 1 2 3 4

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

a = 0

i = 6 0

= 1 3 . 2 0 x 1 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

Figure 4.17.: Light curve of hot spot with radius Rhs = 0.5M orbiting at r = 8M . Theangular frequency is ω0 = 0.04419.

0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0

0 , 0 0

0 , 0 5

0 , 1 0

0 , 1 5

0 , 2 0

0 , 2 5

= 0 . 0 1 0 1

= 0 . 1 9 1

F i t r e s u l t :

= 0 . 0 4 5 5 ± 0 . 0 0 0 8

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)

f r e q u e n c y

Figure 4.18.: Power density spectrum obtained from upper light curve. A Lorentzianfit is applied to the base peak to reversely obtain the hot spot’s angular

frequency ω0. The fit was performed with χ2

/DoF = 0.22 × 10−3

, whereDoF denotes the degrees of freedom.

94



4. Data Analysis

0 1 2 3 4

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

a = 0 . 7

i = 6 0

= 1 2 . 1 x 1 0

t i m e t / T

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n


0 , 0 0 0 , 0 2 0 , 0 4 0 , 0 6 0 , 0 8 0 , 1 0 0 , 1 2 0 , 1 4 0 , 1 6 0 , 1 8 0 , 2 0

0 , 0 0

0 , 0 2

0 , 0 4

0 , 0 6

0 , 0 8

0 , 1 0

0 , 1 2

0 , 1 4

0 , 1 6

0 , 1 8

0 , 2 0

0 , 2 2

= 0 . 0 0 9 8

= 0 . 1 8 6

F i t r e s u l t :

= 0 . 0 4 3 7 ± 0 . 0 0 0 1

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)


frequency ω0. The fit was performed with χ2/DoF = 1.69× 10−3.

96




0 1 2 3 4

0 , 9 9 9 9

1 , 0 0 0 0

1 , 0 0 0 1

1 , 0 0 0 2

1 , 0 0 0 3

1 , 0 0 0 4

a = 0 . 9 9

i = 6 0

= 1 1 . 4 x 1 0

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T


0 , 0 0 0 , 0 2 0 , 0 4 0 , 0 6 0 , 0 8 0 , 1 0 0 , 1 2 0 , 1 4 0 , 1 6 0 , 1 8 0 , 2 0

0 , 0 0

0 , 0 5

0 , 1 0

0 , 1 5

0 , 2 0

= 0 . 0 0 9 7

= 0 . 1 8 3

F i t r e s u l t :

= 0 . 0 4 3 6 ± 0 . 0 0 0 6

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)


frequency ω0. The fit was performed with χ2/DoF = 0.73× 10−3.

97



4. Data Analysis

0 1 2 3 4

0 , 9 9 9 0

0 , 9 9 9 5

1 , 0 0 0 0

1 , 0 0 0 5

1 , 0 0 1 0

1 , 0 0 1 5

1 , 0 0 2 0

1 , 0 0 2 5

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

a = 0

i = 6 0

= 8 . 4 1 x 1 0

Figure 4.25.: Light curve of hot spot with radius Rhs = 1M orbiting at r = ISCO. Theangular frequency is ω0 = 0.06804.

0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 5 0 , 3 0

= 0 . 0 1 5 5

= 0 . 2 9 5

F i t r e s u l t :

= 0 . 0 7 0 3 ± 0 . 0 0 1 4

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)



/DoF = 5.26× 10−2

.

98




0 1 2 3 4

0 , 9 9 9 0

0 , 9 9 9 5

1 , 0 0 0 0

1 , 0 0 0 5

1 , 0 0 1 0

1 , 0 0 1 5

1 , 0 0 2 0

1 , 0 0 2 5

1 , 0 0 3 0

1 , 0 0 3 5

a = 0 . 3

i = 6 0

= 9 . 8 9 x 1 0

t i m e t / T

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n


0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 5 0 , 3 0 0 , 3 5 0 , 4 0

= 0 . 0 1 2 0

= 0 . 3 8 0

F i t r e s u l t :

= 0 . 0 9 0 3 ± 0 . 0 0 1 4

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)



/DoF = 7.69× 10−2

.

99



4. Data Analysis

0 1 2 3 4

0 , 9 9 9

1 , 0 0 0

1 , 0 0 1

1 , 0 0 2

1 , 0 0 3

1 , 0 0 4

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

t i m e t / T

a = 0 . 7

i = 6 0

= 1 2 . 4 7 x 1 0


0 , 0 0 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 7

= 0 . 0 3 2 8

= 0 . 6 2 3

F i t r e s u l t :

= 0 . 1 4 7 2 ± 0 . 0 0 3 1

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)



/DoF = 1.86× 10−1

.

100




0 1 2 3 4

0 , 9 9 9 0

0 , 9 9 9 5

1 , 0 0 0 0

1 , 0 0 0 5

1 , 0 0 1 0

1 , 0 0 1 5

1 , 0 0 2 0

1 , 0 0 2 5

a = 0 . 9 9

i = 6 0

= 9 . 2 0 x 1 0

t i m e t / T

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n


0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0 1 , 2 1 , 4 1 , 6

= 0 . 0 8 3 1

= 1 . 5 7 8

F i t r e s u l t :

= 0 . 3 7 5 8 ± 0 . 0 0 6 0

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)

Figure 4.32.: Power density spectrum obtained from upper light curve. A Lorentzian

fit is applied to the base peak to reversely obtain the hot spot’s angularfrequency ω0. The fit was performed with χ2/DoF = 0.33× 10−1.

101



4. Data Analysis

0 1 2 3 4 5 6 7

0 , 9 9 9 6

0 , 9 9 9 8

1 , 0 0 0 0

1 , 0 0 0 2

1 , 0 0 0 4

1 , 0 0 0 6

1 , 0 0 0 8

a = 0 . 5

i = 6 0

= 2 . 2 2 x 1 0

t i m e t / T

0 ( m i n )

t

o

t

a

n

t

e

n

s

t

y

I

/

I

m

e

a

n

Figure 4.33.: Light curve of two hot spots with radius Rhs = 0.5M orbiting at r1 = 8M and r2 = 6M . The corresponding angular frequencies are ω1 = 0.0432 andω2 = 0.0658.

0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 5 0 , 3 0

= 0 . 0 1 0 3

= 0 . 2 9 8 6

F i t r e s u l t :

= 0 . 0 4 5 9 ± 0 . 0 0 3 5

= 0 . 0 6 6 4 ± 0 . 0 0 1 9

f r e q u e n c y

a

m

p

t

u

d

e

(

a

r

b

t

r

a

r

y

u

n

t

s

)

Figure 4.34.: Power density spectrum obtained from upper light curve. A multi-peak

Lorentzian fit is applied to the base peaks to reversely obtain the hot spots’angular frequencies. The fit was performed with χ2/DoF = 1.19× 10−3.

102




2 4 6 8 1 0 1 2 1 4 1 6

0 , 0 0

0 , 0 5

0 , 1 0

0 , 1 5

0 , 2 0

0 , 2 5

0 , 3 0

0 , 3 5

H

g

h

e

s

t

M

o

d

u

a

t

o

n

[

r

m

s

x

1

0

0

]

r a d i a l d i s t a n c e r / M

a = 0 . 7

i = 6 0

Figure 4.35.: This diagram shows the highest modulations (in units of 100×rms) in lightcurves with different hot spot radial coordinate.

0 , 2 0 , 4 0 , 6 0 , 8 1 , 0 1 , 2 1 , 4 1 , 6

H

g

h

e

s

t

M

o

d

u

a

t

o

n

[

r

m

s

x

1

0

0

]

H o t S p o t R a d i u s R / M

a = 0 . 7

i = 6 0

= 6 M

Figure 4.36.: Highest modulations in light curves against the hot spot radius Rhs.

103



4. Data Analysis

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0

0 , 0 5

0 , 1 0

0 , 1 5

0 , 2 0

0 , 2 5

0 , 3 0

0 , 3 5

K e r r p a r a m e t e r a

H

g

h

e

s

t

M

o

d

u

a

t

o

n

[

r

m

s

x

1

0

0

]

= 1 . 0 M

= I S C O

i = 3 0

i = 6 0

Figure 4.37.: Highest modulations in light curves at different values of spin parametera. The spots were located at the ISCO, which moves towards the eventhorizon when a is increased. Relativistic beaming is then more effective

due to higher velocities and the variabilities are amplified. For a ≈ 1, thegravitational red-shift damps the modulations.

104



5. Conclusion and Outlook

106



A. Curved Spacetimes and CovariantDerivative

The Riemann space

The line element

ds2 =

N i,k=1

gik(x1, . . . , xN )dxidxk = gik(x)dxidxk (A.1)

defines the metric of the N -dimensional Riemann space.Important properties of the metric tensor gik are:

• Its components are differentiable, implying the possibility to approximate the met-ric locally by a quadratic form with constant coefficients. Due to that it can thenbe locally described by a flat metric.

• The tensor is symmetric gik = gki.

• The determinant is not vanishing (det(gik) = 0) and therefore an inverse matrixcan be defined by gipg pk = δki .

• Since the metric tensor’s components are generally coordinate dependent, gik trans-forms to g′ pm by gik = α pi αmk g′ pm, where α is the coordinate dependent transforma-

tion matrix αik(x) = ∂x′i/∂xk.1

The coordinate differentials are also transformed by the transformation matrix:dx′i = αik(x)dxk. Tensors, denoted by their co- and contravariant indices, transform bycomponents as the coordinate differentials.

Covariant Derivative

Resulting from the transformation matrix’ components being coordinate depenent, thetotal differential of a vector (tensor) field is not trivial, since the quantities transformdifferently at different locations in space. Therefore, in order to construct the differential,objects at same locations need to be compared. For this purpose an additional term,arising from a parallel shift (along a geodesic) of the quantity being regarded must betaken into account.

1The line element remains form invariant under general coordinate transformations x′i =x′i(x1, . . . , xN ).

107



A. Curved Spacetimes and Covariant Derivative

Following that concept, a covariant derivative of vector fields on curved spacetimes canbe constructed:

∇µV ν

= ∂ µV ν

+ Γν µλV

λ

(A.2)This operation now is coordinate independent, since that contribution is included bythe term with the connection coefficients Γν µλ. Those coefficients are often referred to asChristoffel symbols and are described by the components of the metric tensor:

Γσµν =1

2gσρ(∂ µgνρ + ∂ ν gρµ − ∂ ρgµν ) (A.3)

For scalars the covariant derivative reduces to the partial.The general expression for the covariant derivative for tensors of arbitrary rank can benoted as:

∇σT µ1µ2···µkν 1ν 2···ν l = ∂ σT µ1µ2···µkν 1ν 2···ν l+ Γµ1σλT λµ2···µkν 1ν 2···ν l + Γµ2σλT µ1λ···µkν 1ν 2···ν l + · · ·− Γλσν 1T µ1µ2···µkλν 2···ν l − Γλσν 2T µ1µ2···µkν 1λ···ν l − · · ·

(A.4)

Riemann Curvature Tensor

The Riemann curvature tensor R provides a local description of the space’s curvatureat each point. As a linear transformation on a vector, it can be regarded as a descriptionof its transformation by parallel transport along an infinitesimal loop. It can be denotedas:

Rρσµν = ∂ µΓ

ρνσ − ∂ ν Γ

ρµσ + Γ

ρµλΓ

λνσ − Γ

ρνλΓ

λµσ (A.5)

Geodesics

Given a curve xµ(λ), parametrized by λ, parallel transport of a tensor T along that pathcan be defined as the requirement that the covariant derivative of T along the pathvanishes:

dxσ

dλ∇σT µ1µ2···µkν 1λ···ν l = 0 (A.6)

With an alternative definition for geodesics, paths of shortest distance between twopoints in space, namely that a geodesic is a path, along which its tangent vector is par-

allel transported, the geodesic equation can be derived.

108



With the tangent vector dxµ/dλ to the path xµ, the condition for it being paralleltransported can be written as

dx

µ

dλ ∇µdx

µ

dλ = 0 (A.7)

or asd2xµ

dλ2+ Γµρσ

dxρ

dλ

dxσ

dλ= 0 , (A.8)

called the geodesic equation.When λ is an appropriate affine2 parameter along a null geodesic, it can be normalizedsuch that dxµ/dλ equals the momentum four-vector:

pµ =dxµ

dλ. (A.9)

For timelike paths the four-momentum is

pµ = mU µ = mdxµ

dτ , (A.10)

where U µ denotes the four-velocity.The geodesic equation (A.8) can be expressed by terms of pµ:

pλ∇λ pµ = 0 (A.11)

2Related to the proper time τ by τ → λ = aτ + b with some constants a, b.

109



A. Curved Spacetimes and Covariant Derivative

110



B. Killing Vectors and Symmetries

Isometries of a given metric describing a manifold M can be understood as transforma-tions, maps M→ M, under which the geometry of the manifold is invariant.Such a symmetry is present, when the metric coefficients gµν are independent of a certaincoordinate xσ∗. Then the translation along this coordinate xσ∗ → xσ∗ +aσ∗ is consideredas a symmetry. Using the relation pλ∂ λ pµ = mdxλ

dτ ∂ λ pµ = mdpµdτ

1, the geodesic equation(A.11) can be expanded to

mdpµdτ =

1

2 (∂ µgνλ) pλ pν . (B.1)

From this it can be deduced that with gµν being independent of xσ∗ the correspndingmomentum component pσ∗ is a conserved quantity of motion.Those isometries can be described by the formalism of Killing fields. The vector

K µ = (∂ σ∗)µ = δµσ∗ , (B.2)

describing the symmetric transformation, is called a killing vector , if the gµν are inde-pendent of xσ∗ .From the constancy of pσ∗ = K ν p

ν , which is equivalent to the statement that its direc-tional derivative along the geodesic vanishes pµ∇µ(K ν p

ν ) = 0, the Killing equation

∇µK ν +∇ν K µ = 0 (B.3)

can be derived.Each Killing vector satisfying (B.3) implies the existence of a conserved quantity whenmoved along geodesics.

There is a relation between event horizons in spacetimes and certain Killing vectorfields.If a Killing vector field χµ is null along a null2 hypersurface Σ, that surface is calledKilling horizon of χµ, and:

• In stationary, asymptotically flat spacetimes each event horizon is a Killing horizon

for some Killing vector field χµ.

• In those spacetimes which are static, such a vector field is that one representingtime translations at infinity K µ = (∂ t)

µ.

• For stationary but not static spacetimes the event horizon is axially symmetricand a Killing horizon for the linear combination χµ = K µ + Ω0Rµ, where Rµ isthe rotational Killing vector field Rµ = (∂ φ)µ.

1This argument only holds for timelike paths, but can be developed analogously for null geodesics withaffine parameter λ.

2Null here means light-like.

111



B. Killing Vectors and Symmetries

112



C. 3+1 Split of Spacetime

The four-dimensional spacetime can be decomposed into three-dimensional pure spatialhypersurfaces with t = const. The metric for this situation reads then

ds2 = −α2dt2 + hik(dxi + β idt)(dxk + β kdt) , (C.1)

where i, k = 1, 2, 3.α = dτ/dt is called then the lapse function , describing the time lapse between the proper

time τ at the surface regarded and the time t at infinity.The hik denote the metrics of the hypersurfaces with t = const, and the β i = dxi/dt,which are normal to these, are called shift functions.

113



C. 3+1 Split of Spacetime

114



D. Numerical Integration

In order to solve an ordinary differential equation of the form

y′(x) = f (x, y(x)) (D.1)

with a given boundaryy(x0) = y0 , (D.2)

it is helpful to discretize the problem and by means of that to approximate the solutionby evaluating a specific number of points (xk, yk).By decreasing the number of evaluations of f , usually an error is introduced due to thediscretization.In general, algorithms propagating the function f from the initial boundary on step bystep are called one-step integrators and can be described by the calculation rule

yk+1 = yk + hΦ(xk, yk, yk+1, h) , (D.3)

where h = xk+1 − xk is the step size.In order to estimate the accuracy of such methods, the local discretization error dk atxk

can be defined by

dk+1def = y(xk+1)− y(xk)− hΦ(xk, y(xk), y(xk+1), h) , (D.4)

with the exactly integrated value y(xk+1).The algorithms, specified by (D.3), can be furthermore classified by the one-step in-tegration order p ∈ R, which is defined if the following inequality is satisfied by thecorresponding local discretization error

maxk |dk| ≤ D = const · h p+1 = O(h p+1) . (D.5)

The most simple representation of such one-step integrators is the Euler method , which

uses the slope at xk that can be evaluatd directly from (D.1) to compute the next valueat xk + h. This method is described by the computation rule

yk+1 = yk + hf (xk, yk) , (k = 0, 1, 2, . . .) . (D.6)

115




Runge-Kutta algorithms

Rewriting (D.1) in its integral form

y(xk+1) = y(xk) +

xk+1xk

f (x, y(x))dx (D.7)

leads to an approximation ansatz for the so called Runge-Kutta integrators. The integral(D.7) is discretized arbitrarily, yielding

yk+1 = yk + hi

cif (ξi, y(ξi)) (D.8)

with the sampling points ξ ∈ [xk, xk+1] and the integration weight factors ci.

Since the values f (ξi, y(ξi)) are not known, the equation is not directly solvable.So the main task is to find approximations for y(ξi), called predictors y∗i , that can be usedto evaluate (D.8). This principle is consequently called a predictor-corrector method .From parametrizing ξi and the predictors y∗i by

ξi = xk + aih (D.9a)

y∗i = yk +i−1 j=1

hbijf (ξ j , y∗ j ) (D.9b)

with a1 = 0, an algorithm for the evaluation of the yk+1 can be derived:

ki = f (ξi, y∗i ) = f (ξi, yk + hi−1 j=1

bijk j) , (D.10a)

yk+1 = yk + hi

ciki . (D.10b)

Now the parametrs ai, bij and the integration weights ci need to be specified. This can beachieved by demanding the predictors y∗i to be exact for the specific differential equationy′ = 1 and by special demands to the order p of the algorithm (for demonstrations see[Sch1997]).Finding an adequate step size h for the integration algorithm is very essential. A wellworking method for adjusting this parameter is to approximate the local error d

k+1of the

used algorithm by an integrator of higher order p. If both algorithms base on evaluationof the same ki, this can be achieved with a tolerable amount of effort. This is calledembedding an algorithm of lower into one of higher order.The evaluated local error can be used then to adjust the step size for each integrationstep. Algorithms exploiting such a dynamic control are called adaptive.

116



The adaptive Fehlberg algorithm

The algorithm for the integration of the nullgeodesics in this thesis is the adaptiveFehlberg(4,5) [Zin2002], which is a Runge-Kutta integrator of the order p = 4 using fourevaluations of the function f to determine yk+1. It is embedded in an algorithm of theorder p = 5 using six evaluations of f . As described above, the higher-order integratoris used to approximate the local error. With the step size h, the computation rule of this method is given by [Sch1997, Zin2002]:

k1 = f (xk, yk) , (D.11a)

k2 = f

xk +

2

9h, yk +

2

9hk1

, (D.11b)

k3 = f xk +1

3h, yk +

1

12hk1 +

1

4hk2 , (D.11c)

k4 = f

xk +

3

4h, yk +

69

128hk1 − 243

128hk2 +

135

64hk3

, (D.11d)

k5 = f

xk + h, yk − 17

12hk1 +

27

4hk2 − 27

5hk3 +

16

5hk4

, (D.11e)

yk+1 = yk + h

1

9k1 +

9

20k3 +

16

45k4 +

1

12k5

, (D.11f)

k6 = f

xk +

5

6h, yk +

65

432hk1 − 5

16hk2 +

13

16hk3 +

4

27hk4 +

5

144hk5

, (D.11g)

dk+1 ≈h

300 (−2k1 + 9k3 − 64k4 − 15k5 + 72k6) . (D.11h)

117




118



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122



Acknowledgements

• Zu aller erst mochte ich mich bei Prof. Dr. Max Camenzind dafur bedanken, dasser mir die Moglichkeit gegeben hat, mich mit diesem Thema zu beschaftigen.

• Weiterhin danke ich Dr. Christian Fendt dafur, dass er sich noch so kurzfristigdazu bereit erklart hat, das zweite Gutachten fur diese Arbeit zu erstellen.

• Der großte Dank gebuhrt meinen Eltern, die mich immer unterstutzt haben undmir das Studium der Physik ermoglicht haben.Sie haben mich in allen meinen Vorhaben bestarkt und mich gelehrt, dass es immerlohnenswert ist, sich fur seine Ziele einzusetzen, und dass es wichtig ist, sich nichtvon moglichen Misserfolgen einschuchtern zu lassen.

• Meiner Freundin Sabrina mochte ich fur Ihr uneingeschranktes Vertauen in meineFahigkeiten danken, und dafur, dass sie es verstand, mit meinen in letzter Zeithaufigen Launen umzugehen.

• Zu guter Letzt will ich noch einen Dank an die Studienkollegen loswerden, die mich

seit dem ersten Semester bis heute in Heidelberg begleitet haben. Die lustigenGrillabende, das gemeinsame Aufgaben Rechnen, lange Diskussionen, abendlicheAltstadttouren, UB-Prufungslernereien und vieles mehr. . . das sind alles Dinge, diemich diese tolle Zeit in Heidelberg niemals werden vergessen lassen.

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Bibliography

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Erklarung

Ich versichere, dass ich diese Arbeit selbststandig verfasst und keine anderen als dieangegebenen Quellen und Hilfsmittel benutzt habe.

Heidelberg, den . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

analysis of imprints on light curves from kerr black holes due to time-dependent accreting...

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