Artificial black hole singularitiesM. Cadoni,

University of CagliariCOSLAB 2005, Smolenice, August 31 2005

Based on M.C. , Class. and Quant. Grav. 22 (2005) 409

M.C and S. Mignemi, gr-qc/0503059, gr-qc/0504143

Abstract

We look for acoustic analogues of a spherical symmetric black hole with a pointlike source. We show that the gravitational system has a dynamical counterpart in the constrained, steady motion of a fluid with a planar source. The equations governing the dynamics of the gravitational system can be exactly mapped in those governing the motion of the fluid. The different meaning that singularities and sources have in fluid dynamics and in general relativity is also discussed.

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IntroductionIntroduction Since the seventies of the last century black hole Since the seventies of the last century black hole

physics plays a crucial role for fundamental physics plays a crucial role for fundamental theoretical physicstheoretical physics: :

Quantum mechanics and spacetime structureQuantum mechanics and spacetime structure: Hawking : Hawking radiation, information loss for black holes, meaning of radiation, information loss for black holes, meaning of singularitiessingularities

Thermodynamics and statistical mechanicsThermodynamics and statistical mechanics:: Microscopic Microscopic derivation of the Bekenstein-Hawking area law derivation of the Bekenstein-Hawking area law S=A/4 S=A/4

Non perturbative solutions of string theoryNon perturbative solutions of string theory: Branes, : Branes, AdS/CFT correspondenceAdS/CFT correspondence

Holographic principleHolographic principle : Fundamental or Emergent : Fundamental or Emergent??

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• Problem: Present (and near-future) astrophysical observations can not test most of theoretical predictions

• Recently has become increasingly clear that condensed matter system (e.g. fluids) can be used to mimic various kinematical aspects of general relativity

• Condensed matter analogues have been used to mimic Black holes and event horizons Cosmological solutions Field theory in curved spacetime Hawking radiation

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Condensed matter system are experimentally testable in Condensed matter system are experimentally testable in laboratory laboratory in the near future we could have a “ in the near future we could have a “ Black Black hole phenomenologyhole phenomenology” based not on astrophysical but ” based not on astrophysical but condensed matter experimentscondensed matter experiments

Can we also describe in this way spacetime Can we also describe in this way spacetime singularities?singularities?

Main obstructionMain obstruction: usual approach works for kinematical : usual approach works for kinematical but not dynamical aspects of gravitationbut not dynamical aspects of gravitation

To extend the analogy at a dynamical level is a very difficult task: Gravitational systems:Gravitational systems:

1)1) Huge redundancy of gauge degrees of freedom Huge redundancy of gauge degrees of freedom

2)2) Separation between gravitational field and sources Separation between gravitational field and sources

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Fluids:

1) Few physical parameters (pressure, velocity, density)

2) Do not seem to allow for a source-field description

We will show that an analogy gravitational dynamics (constrained) fluid dynamics can be found at least for spherically symmetric black holes with ( or without) sources with gauge degrees of freedom completely fixed

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SummarySummary1.1. Gravitational dynamics and fluid dynamics Gravitational dynamics and fluid dynamics

A.A. Gravitational dynamics without sourcesGravitational dynamics without sources

B.B. Fluid dynamicsFluid dynamics

C.C. Gravitational dynamics with pointlike sourcesGravitational dynamics with pointlike sources

2.2. Black hole thermodynamics and fluid dynamicsBlack hole thermodynamics and fluid dynamics

3.3. Solutions of the constrained fluid dynamics Solutions of the constrained fluid dynamics

4.4. ConclusionsConclusions

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1. Gravitational dynamics and fluid dynamics

dt

dx

dt

dxgdtmLgxdRgxdA jiijMF

G 44

16

1

• We only consider spherically symmetric solutions (Topological theory, no propagating gravitational degrees of freedom). They can be described by an effective 2D (dilaton) gravity model obtained retaining only the radial modes of Einstein gravity ( G=-2)

ds2(4)= ds2

(2)+(2/2)d2(2)

• We consider Einstein gravity coupled to matter fields and We consider Einstein gravity coupled to matter fields and a pointlike source of mass a pointlike source of mass mm

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• After a After a Weyl rescaling of the metricWeyl rescaling of the metric we get a 2D model with we get a 2D model with potential potential V(V()) and coupling function and coupling function W(W())

dt

dx

dt

dxgdtWmVRgxdA jiij )())((

2

1 22)2(

• For static solutions in the Schwarzschild gauge For static solutions in the Schwarzschild gauge (ds(ds22=Ud=Ud22++ UU--

11dxdx22)) and source at rest in the origin the field equations are and source at rest in the origin the field equations are

)(22

)(2

22

2

2

22

2

rmWVdr

d

dr

dU

dr

dU

Vdr

d

dr

dU

rd

dWm

d

dV

dr

Ud

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A. Gravitational dynamics without sourcesA. Gravitational dynamics without sources

))()((2

)( 220 dVF

rM

• 2D dilaton gravity without matter allows for the definition of a scalar mass function, which on shell is constant (F0 is a normalization constant)

• The gravitational field equation take the simple form

0)(

,2 dr

rdMV

dr

d

dr

dU

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• where J=Vd, the event horizon is located at r=rh, with J(rh)=2M/, in all physically interesting situations the curvature singularity is located at r=0

ds2 J() 2M

d 2 J()

2M

1

dr2, r

• The general solutions of the field equations, describing BH are

B. Fluid dynamics

• Fluid dynamics is a classical field theory which can be Fluid dynamics is a classical field theory which can be completely understood in terms of Newtonian physics. The completely understood in terms of Newtonian physics. The fundamental equations are the equation of continuity and Euler fundamental equations are the equation of continuity and Euler equationequation

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, v and P are the fluid density, velocity and pressure, is the potential for external forces. We will consider zero inviscid, irrotational and barotropic fluids

• In particular this means that P is a function of only so that we can define a specific entalpy h and an equation of state and that v can be derived from a potential

dP dh, P P(), v

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• The notion of acoustic horizon (and acoustic black hole) arises by considering perturbations (sound waves) around some fixed background at the linearized level

101010 , , PPP

• Let us consider the steady flow of a fluid whose motion is essentially one-dimensional transverse velocities (along y,z) are small with respect to that along x

• The flux tube has a slab geometry with v , and the flux tube profile A depending only on x

• In this case the fluid dynamics equations become ( F is the flux of matter fluid)

0)v( ,0v

v Adx

d

dx

d

dx

d

dx

dP

dx

d F

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• Unruh has shown that the equation of motion for the sound wave is that of a scalar massless field propagating in a Lorentzian manifold,

1

g ( gg1) 0

Where gis the acoustic metric

ds2 g dxdx

0

c (c 2 v0

2)dt 2 2v0dxdt dx2

Where c is the local speed of sound

c dP

d0

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• As long as the propagation of sound waves is concerned we can use the usual (kinematical) spacetime notions of general relativity

Ergo-regions: K=(1,0) , K2= gtt= -(c) (c2-v2). Ergo-regions are regions where K2>0. Supersonic regions, v2>c2 are ergo-regions

Trapped surface: region where the normal component of v is always greater then c

Apparent Horizon: the boundary of a trapped region

Event horizon: the boundary of a region from where phonons cannot escape

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• At the dynamical level the gravity/fluid correspondence can be realized defining the new variables

r dx0 , t dxv0

c2 v02

J 2M

0

cc2 v0

2

• The causal structure of the gravitational black hole can be put in correspondence with that of the acoustic black hole using a Painlevè Gullstrand-like coordinate transformation

X 0

cc2 v0

2 , Y = 0c, F = lnc

0

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• They can be put in the gravitational form by the identification

0 ,22X 2 XYYA

dr

d

dr

de

dr

dFX

dr

dY

dr

d F

2

2 )( ,

F

rM

XYYAUX

• So that the fluid equations become

and imposing the constraint

dr

dV

dr

de

dr

dFX

dr

dY F 222

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• Dynamical equivalence of gauge fixed spherical symmetric gravity with a constrained fluid dynamics

• If the external parameters for the fluid (A(x),(x)) are given the constraint is equivalent to an equation of state for the fluid

• If the equation of state for the fluid is given (usual physical situation) the constraint determines the external parameters (,A) for the fluid.

• The conserved quantity of fluid dynamics ( the flux F) is identified with the conserved quantity of the gravitational dynamics ( the black hole mass M)

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C. Gravitational dynamics with pointlike sources

• In order to give an acoustic meaning to the curvature In order to give an acoustic meaning to the curvature singularity the discussion has to be generalized to the singularity the discussion has to be generalized to the case case mm0 0

• In the presence of the pointlike source the gravitational In the presence of the pointlike source the gravitational equations can be written asequations can be written as

)()(

,2

2 rdr

dW

m

dr

rdMV

dr

d

dr

dU

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• Everything works as in the previous case, the only Everything works as in the previous case, the only difference is that now the continuity equation for the difference is that now the continuity equation for the fluid acquires a source term proportional to fluid acquires a source term proportional to mm

)()v( 200 r

dr

dW

mA

dx

d

dx

d

F

• The acoustic analogue of the gravitational curvature singularity is a The acoustic analogue of the gravitational curvature singularity is a planar source for the fluid fluxplanar source for the fluid flux

• In the acoustic description the delta function singularity is not associated In the acoustic description the delta function singularity is not associated with a singularity of the dynamics ( the Euler equation is not singularwith a singularity of the dynamics ( the Euler equation is not singular ) )

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• Explicit solutions of the dynamical equations can be Explicit solutions of the dynamical equations can be found taking the solutions as function of found taking the solutions as function of |r| |r| ( to ( to generate the delta function singularity at generate the delta function singularity at r=0r=0))

22

2

2m ),()( ),(

2)(

, , )(

d

dWmVW

mJ

rJ

XU

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2. 2. Black hole thermodynamics and fluid dynamicsBlack hole thermodynamics and fluid dynamics

• Considering the acceleration of fiducial observers or the periodicity of the Euclidean section at the horizon one can define also for acoustic black holes the notion of Surface gravity and Hawking temperature:

TH 1

2nc v

• However, usually it is very difficult to find the acoustic counterpart of the other BH thermodynamical parameters mass M and entropy S • Our approach allows us to give a natural acoustic meaning to M and S

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• In fact it is well known that black hole thermodynamics follows directly from the gravitational field equations

• Having found a dynamical correspondence gravity/fluid we just need to express the BH mass and entropy in terms of the fluid parameters

• We find acoustic analogues Ma ,Sa satisfying automatically the first principle dMa =Ta dSa

• The explicit expression are simple and transparent in the case of a flux tube of constant section ( M(rh) is the fluid mass inside the horizon)

)(

2)(2 ),()()(

0

0 h

r

r

ahhha rA

xdxSrYrArMMh

M

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• The entropy has a simple physical meaning is proportional to the total mass of the fluid outside the horizon, whereas the first principle gives

h

h

a

aa dr

rd

dS

dMT

)(

2

12

F

• The temperature measures the rate of change of the flux of fluid mass when the horizon position is changed

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3. 3. Solutions of the constrained fluid dynamicsSolutions of the constrained fluid dynamics • Let us now solve the equations of the constrained fluid Let us now solve the equations of the constrained fluid

dynamics dynamics find the form of v0 , 0, c, A,

• Let us consider a generic power law equation of state Let us consider a generic power law equation of state for the fluid (for the fluid (aa arbitrary real constant), arbitrary real constant),

n

n

aP 0

24

• This Equation describes almost all physically interesting fluids: perfect fluid (n=1), Bose-Einstein condensates (n=2), Chaplygin gas (n= - 1)

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• To solve completely the dynamics we have to consider separately two cases

1)1) Flux tube with constant section Flux tube with constant section

2)2) Flux tube with non-constant section and Flux tube with non-constant section and homogeneous external potential homogeneous external potential

• In terms of the new variables the equation of state reads

4)1(3 aeY Fnn

• Using this equation. one has

1

1

1

2

1

2

1

2

0)1(2

3

1

2

0 , ,v

n

n

nnnn

n

n YacYaXYYa

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Flux tube with constant section

• In this case we can use the continuity equation to solve for X=X(Y). The constraints determines the form of the external potential. The solution can be given as function of Y.

Where is the (constant) flux of mass in the tube.

• The acoustic horizon is located at Y=Yh= (X=0), the subsonic region Y> (X>0), the black hole singularity at Y=Ys= [(A2+1)1/2-A] .

)1

1

2(

, , ,v

1

12

1

421

4

1

1

1

2

1

2

1

2

0

1n

2

0

n

n

nn

n

n

nnn

Yn

Ya

YacYaY

a

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• The external potential becomes extremal on the horizon (null force condition)

• All the fluid parameters remain finite both at the horizon and the singularity

Behaviour of the fluid Behaviour of the fluid parameters ( External parameters ( External potential bold black line, potential bold black line, velocity black line, density velocity black line, density dashed line, speed of sound dashed line, speed of sound grey line) normalized at their grey line) normalized at their horizon values as a function horizon values as a function of of YY, for , for n=2n=2, , =1, =1, M=(3/4)M=(3/4). . The acoustic The acoustic horizon is located at horizon is located at Y=1Y=1, the , the singularity at singularity at Y=1/2Y=1/2

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Behaviour of the fluid Behaviour of the fluid parameters ( External parameters ( External potential bold black line, potential bold black line, velocity black line, density velocity black line, density dashed line, speed of sound dashed line, speed of sound grey line) normalized at their grey line) normalized at their horizon values as a function horizon values as a function of the of the YY, for , for n=1/2n=1/2, , =1, =1, M=(3/4)M=(3/4). . The acoustic The acoustic horizon is located at horizon is located at Y=1Y=1, , the singularity at the singularity at Y=1/2Y=1/2

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Flux tube with non-constant section

• In this case we take the external potential constant whereas A changes along the flux tube.

• We can use the continuity equation to find A(X,Y)

)(

A2

2

XYY

• The constraint can be now easily solved to give ( integration constant)

) ( 1-n

1nX 1

32

n

n

YY

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• Again the fluid parameters can be given (for n generic in implicit form) as function of the variable Y

• The acoustic horizon is located at Y=Yh= (n+1)/(n-1), the

black hole singularity at Y=Ys , the supersonic region at

Ys<Y<Yh

• All the fluid parameters remain finite both at the horizon and the singularity

• The horizon must forms at a minimum of the section A

the flux tube has the form of a Laval nozzle

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• This means that the constraint, which is necessary to have a correspondence between gravitational and fluid dynamics takes the form of a “geometrical constraint” on the form of the flux tube.

• This “geometrical constraint” forces the fluid to develop an acoustic horizon

• This is an well-known fact from hydrodynamics: the acoustic horizon must form at the narrowest cross-section of the nozzle

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Behaviour of the fluid Behaviour of the fluid parameters ( Tube cross-parameters ( Tube cross-section bold black line, section bold black line, velocity black line, density velocity black line, density dashed line, speed of sound dashed line, speed of sound grey line) normalized at their grey line) normalized at their horizon values as a function horizon values as a function of of YY, for , for n=2n=2, , =1, =1, M=(9/16)M=(9/16). . The acoustic The acoustic horizon is located at horizon is located at Y=1Y=1, , the singularity at the singularity at Y=1/8. Y=1/8. The range of the coordinate The range of the coordinate YY is is 1/8<Y<(27/8)1/8<Y<(27/8)1/21/2. For . For this value of this value of YY the tube the tube cross-section diverges.cross-section diverges.

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Behaviour of the fluid Behaviour of the fluid parameters ( Tube cross-parameters ( Tube cross-section bold black line, section bold black line, velocity black line, density velocity black line, density dashed line, speed of sound dashed line, speed of sound grey line) normalized at grey line) normalized at their horizon values as a their horizon values as a function of the function of the YY, for , for n=1/2n=1/2, , =1, M=(9/16)=1, M=(9/16). . The The acoustic horizon is located acoustic horizon is located at at Y=1Y=1, the singularity at , the singularity at Y=1/8. Y=1/8. The range of the The range of the coordinate coordinate YY is is 1/8<Y<(8/3)1/8<Y<(8/3)3/23/2. For this . For this value of Y the tube cross-value of Y the tube cross-section diverges.section diverges.

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4. Conclusions

Main resultsMain results: : The gravity/fluid analogy can be promoted from The gravity/fluid analogy can be promoted from

the pure kinematical to a full dynamical levelthe pure kinematical to a full dynamical level It is possible to find a dynamical equivalence It is possible to find a dynamical equivalence

between high symmetric gauge-fixed gravitational between high symmetric gauge-fixed gravitational systems and a fluidsystems and a fluid

We can construct artificial BH singularities and We can construct artificial BH singularities and give a meaning to artificial BH thermodynamicsgive a meaning to artificial BH thermodynamics

Singularities have different meaning in general Singularities have different meaning in general relativity and fluid dynamicsrelativity and fluid dynamics

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In the former case they are related to true singularities In the former case they are related to true singularities of the dynamics in the latter they appear as mere source of the dynamics in the latter they appear as mere source terms for the matter (cusp terms in the fiels)terms for the matter (cusp terms in the fiels)

Open questionsOpen questions

How far reaching is the analogy? How far reaching is the analogy?

It seems to hold for gravitational systems with high It seems to hold for gravitational systems with high symmetry and with no propagating gravitational degrees symmetry and with no propagating gravitational degrees of freedom (topological gravity). It is therefore likely of freedom (topological gravity). It is therefore likely that it can be extended to cosmological solutions that it can be extended to cosmological solutions ((Cosmological singularityCosmological singularity) but extremely unlikely that it ) but extremely unlikely that it can be used to describe gravitational systems with can be used to describe gravitational systems with propagating degrees of freedompropagating degrees of freedom (solutions depending (solutions depending on both the timelike and spacelike coordinate, on both the timelike and spacelike coordinate, gravitational waves), gravitational waves), backreactionbackreaction etc etc

Role of the Weyl transformation Role of the Weyl transformation

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