BLACK HOLESIN GENERAL RELATIVITY

Leonardo Gualtieri & Valeria Ferrari

Dipartimento di Fisica, Universita degli studi di Roma “Sapienza”

(2011)

Contents

1 The Schwarzschild solution 31.1 Line element and general properties . . . . . . . . . . 31.2 Horizon . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Spacetime extension . . . . . . . . . . . . . . 61.3.2 Kruskal and Eddington-Finkelstein coordinates 111.3.3 The Finkelstein diagram . . . . . . . . . . . . 13

1.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The far field limit of an isolated, stationary object 202.1 The case of a weakly gravitating source . . . . . . . . 212.2 The case of a general source . . . . . . . . . . . . . . 282.3 Mass and angular momentum of an isolated object . 34

3 The Kerr solution 393.1 The Kerr metric in Boyer-Lindquist coordinates . . . 403.2 Symmetries of the metric . . . . . . . . . . . . . . . . 433.3 Frame dragging and ZAMO . . . . . . . . . . . . . . 443.4 Horizon structure of the Kerr metric . . . . . . . . . 45

3.4.1 Removal of the singularity at ! = 0 . . . . . . 453.4.2 The horizon . . . . . . . . . . . . . . . . . . . 50

3.5 The infinite redshift surface and the ergosphere . . . 523.6 The singularity of the Kerr metric . . . . . . . . . . . 55

3.6.1 The Kerr-Schild coordinates . . . . . . . . . . 563.6.2 The metric in Kerr-Schild coordinates . . . . . 583.6.3 Some strange features of the inner region of

the Kerr metric . . . . . . . . . . . . . . . . . 603.7 General black hole solutions . . . . . . . . . . . . . . 65

1

4 Geodesics of the Kerr metric 674.1 Equatorial geodesics . . . . . . . . . . . . . . . . . . 69

4.1.1 Null geodesics . . . . . . . . . . . . . . . . . . 734.1.2 Is E < 0 possible? . . . . . . . . . . . . . . . . 764.1.3 The Penrose process . . . . . . . . . . . . . . 794.1.4 The innermost stable circular orbit for time-

like geodesics . . . . . . . . . . . . . . . . . . 814.1.5 The 3rd Kepler law . . . . . . . . . . . . . . . 82

4.2 General geodesic motion: the Carter constant . . . . 84

5 Black hole thermodynamics 905.1 Limits for energy extraction from a Kerr black hole . 905.2 The laws of black hole thermodynamics . . . . . . . . 945.3 The generalized IInd law of thermodynamics . . . . . 965.4 The Hawking radiation . . . . . . . . . . . . . . . . . 101

5.4.1 Quantum fields in Minkowski spacetime . . . 1025.4.2 Quantum fields in a general spacetime . . . . 1065.4.3 Particle creation in Schwarzschild spacetime . 109

These notes have been written for a graduate course on blackholes in general relativity; it is assumed that the students have al-ready followed a basic course on general relativity. The notes arebased on the extensive literature existing on the subject, in partic-ular on R.W. Wald, General relativity, University of Chicago Press,Chicago, 1984; S.W. Hawking, G.F.R. Ellis, The large scale struc-ture of spacetime, Cambridge University Press, Cambridge, 1973;and on the original articles cited in these books. For more basictopics on general relativity I refer the reader to C.W. Misner, K.S.Thorne, J.A. Wheeler, Gravitation, W.H. Freeman and co., S. Fran-cisco, 1973; S. Weinberg, Gravitation and cosmology, Wiley, NewYork, 1972; B.F. Schutz, A first course in general relativity, Cam-bridge University Press, Cambridge, 1985; H. Stephani, Relativity,Cambridge University Press, Cambridge, 2004; N. Straumann, Gen-eral Relativity, Springer Verlag, Berlin, 2004; and on the notes of theundergraduate course “General Relativity” held by Valeria Ferrari.

I use the geometric units G = c = 1.

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Chapter 1

The Schwarzschild solution

In this chapter we discuss some features of the Schwarzschild so-lution; some of them are already covered by fundamental generalrelativity courses, and will be only mentioned; some others will bediscussed in greater detail.

1.1 Line element and general properties

The Schwarzschild solution can be written as

ds2 = !!1! 2M

r

"dt2 +

1

1! 2Mr

dr2 + r2(d!2 + sin2 !d"2) (1.1)

in the coordinates (t, r, !,"). This solution is asymptotically flat,since as r " # it tends to the Minkowski metric in polar coordi-nates.

The spacetime metric (1.1) depends on the positive constant M ,which can be interpreted as the BH mass. A reason for this inter-pretation can be the following: if a body moves in the Schwarzschildspacetime, and if r if enough large to assume, with good accuracy,the weak-field approximation, then the motion of the body is de-scribed by Newton’s laws, with gravitational potential

" = !M

r(1.2)

which coincides with the gravitational potential of a central bodywith mass M . As we will see later on, there are other reasons togive to M the interpretation of BH mass.

3

The Schwarzschild solution is spherically symmetric; furthermore,it is static, i.e., it admits a timelike Killing vector (#µ = (1, 0, 0, 0))hypersurface orthogonal (g0i = 0 if i $= 0). As a consequence, theSchwarzschild metric is invariant for temporal translations (it is sta-tionary) and for time reversal t % !t. The Schwarzschild solutionadmits four independent Killing vectors: one timelike, and threespacelike, corresponding to the symmetries of the rotation groupSO(3).

It is possible to show that any asymptotically flat and spheri-cally symmetric solution of Einstein’s equations in vacuum is alsostatic, and then it is the Schwarzschild solution. This result, namedthe Birkho! theorem, is very important; it implies, for instance,that the exterior of spherically symmetric stars is described by theSchwarzschild solution, and that spherically symmetric objects can-not be sources of gravitational radiation.

1.2 Horizon

The metric, in these coordinates, is singular at r = 0 and at r = 2M .However, the singularity r = 2M is only an artifact of the coordinatechoice, and can be removed by changing coordinates (and then it iscalled “coordinate singularity”); the singularity r = 0, instead, is atrue singularity of the metric (and is called “curvature singularity”)1.

Before discussing the removal of the coordinate singularity, letus consider in greater detail the surface r = 2M , and the otherhypersurfaces r = constant.

The normal of a hypersurface whose equation is # & r!constant =0 is

nµ = #,µ = (0, 1, 0, 0) . (1.3)

Let us consider a generic hypersurface. At any point of such hyper-surface we can introduce a locally inertial frame, and rotate it insuch a way that the components of the normal vector are

n! = (n0, n1, 0, 0) and n!n! = (n1)2 ! (n0)2 . (1.4)

1Actually, there are other coordinate singularities: " = 0, # and $ = 0, 2#; the removalof such singularities is trivial: it is su!cient to define new angular variables by performing arotation; therefore, in the following we will not consider these singularities.

4

Consider a vector t! tangent to the surface at the same point. t!

must be orthogonal to n% :

n!t! = !n0t0 + n1t1 = 0 ' t0

t1=

n1

n0(1.5)

thus

t! = $(n1, n0, a, b) with a, b e $ constant and arbitrary.(1.6)

Consequently the norm of the tangent vector is

t!t! = $2[!(n1)2+(n0)2+(a2+b2)] = $2[!n!n

!+(a2+b2)] . (1.7)

We have that

• If nµnµ < 0, the hypersurface is called spacelike, and tµ isnecessarily a spacelike vector.

• If nµnµ > 0, the hypersurface is called timelike, and tµ can betimelike, spacelike or null.

• If nµnµ = 0, the hypersurface is called null, and tµ can bespacelike or null.

Let us consider a point P on a surface # = 0. If the surface isspacelike, the tangent vectors of the surface lie all outside the light-cone in P . Therefore, a particle passing through P , whose velocityvector lies inside the cone, can cross the surface only in one direction.If the surface is null, the situation is nearly the same: the tangentvectors to the surface lie inside to the light-cone in P , or are tangentto it, thus a particle can cross the surface in one direction only. Itthe surface is timelike, some tangent vectors of the surface are insidethe cone, some others are outside, i.e. the surface cuts the cone,and then a particle passing through P can cross the surface in bothdirections.

From (1.3), (1.1), in the case of an r = constant surface

nµn&gµ& = grr = 1! 2M

r(1.8)

thus the surfaces r = constant are spacelike if r < 2M , null ifr = 2M , timelike if r > 2M .

The null hypersurface r = 2M , then, separates regions of space-time where r = const are timelike hypersurfaces from regions where

5

r = const are spacelike hypersurfaces; therefore, an object crossinga null hypersurface r = const can never come back; for this rea-son, the null hypersurface r = 2M is called horizon. As we will seewhen studying other BH solutions, this is a general property of nullhypersurfaces. The horizon r = 2M separates the spacetime in:

• the region with r > 2M , where the r = const. hypersurfacesare timelike; the r " # limit, where the metric becomes flat,is in this region, so we can consider this region as the exteriorof the black hole;

• the region with r < 2M , where the r = const. hypersurfacesare spacelike; an object which falls inside the horizon and enterin this region can only continue falling to decreasing values ofr, until it reaches the curvature singularity r = 0; this region isthen considered the interior of the BH; the name BH is due tothe fact that nothing, neither objects nor signals of any kind,can escape from this region.

1.3 Singularities

The fact that r = 0 is a curvature singularity can be shown bycomputing the curvature scalar

R!%'(R!%'( = 48

M2

r6(1.9)

which diverges at r = 0. The fact that r = 2M is a coordinatesingularity is less easy to prove: the finiteness of the polynomialsin the curvature tensor does not exclude, in principle, that thereis a curvature singularity there. Therefore we need to study thissingularity, finding the coordinate change which allows to remove it.

1.3.1 Spacetime extension

To study the structure of singularities, we must first define rigor-ously the concept of singularity in general relativity. This is notobvious, since a singularity does not belong to the spacetime man-ifold; changing coordinate frame, a singularity can be mapped toinfinity, so that at a first look the metric does not appear singular(there is nothing strange in a singular behaviour at infinity).

6

The key notion is the length of geodesics. In Schwarzschild space-time, for instance, an observer falling into the BH reaches the sin-gularity r = 0 in a finite amount of proper time, thus its (timelike)geodesic has a finite length and cannot be extended; this means thatr = 0 is a singularity: whatever coordinate frame we choose, thatgeodesic has a finite length.

Therefore, to characterize a singular behaviour we need the no-tion of geodesic completeness: a spacetime is geodesically complete ifevery timelike and null geodesic can be extended to values arbitrar-ily large of the a%ne parameter. If, instead, the spacetime admitsat least one incomplete (i.e., which cannot be extended) timelikeor null geodesic, we say that it is geodesically incomplete, and thismeans that there is a singularity (either a true curvature singular-ity or a coordinate singularity). We only consider timelike or nullgeodesics because they represent worldlines of massive or masslessparticles, and then they represent observers or signals, while space-like geodesics do not correspond to worldlines of physical objects.

Let us consider the Schwarzschild metric in the coordinates (t, r, !,"),with line element (1.1)

ds2 = !!1! 2M

r

"dt2 +

1

1! 2Mr

dr2 + r2(d!2 + sin2 !d"2) . (1.10)

To be rigorous, the spacetime manifold described by (1.10) is notdefined at r = 0 and at r = 2M . We have then two disconnectedmanifolds, M1 with 0 < r < 2M (the interior of the BH) and M2

with r > 2M (the exterior of the BH). A geodesic correspondingto an observer falling into the BH cannot be extended across thehypersurface r = 2M , since it does not belong to M1(M2, and thegeodesic terminates at a finite value of the a%ne parameter (noticethat the observer arrives at r = 2M with t = +#). On the otherhand, since this singularity is not a true curvature singularity, it canbe removed with the following two steps:

• we change the coordinate frame, for instance to the Kruskalframe (U, V, !,"), for which (as we shall show in Section 1.3.2)

ds2 = !32M3

re!

r2M dUdV + r2(d!2 + sin2 !d"2) (1.11)

(note that the r in (1.11) should be considered as a function ofthe coordinates, r(U, V )).

7

U

r=0

V

r > 2M

r < 2M

Figure 1.1: Interior and exterior of a Schwarzschild black hole in Kruskal coor-dinates.

In these coordinates (see Fig. 1.1), the BH exterior r > 2Mcorresponds to (U < 0, V > 0) and the BH interior r < 2Mcorresponds to (U > 0, V > 0 with UV < 1); the singularityr = 2M (and t = +#) corresponds to the semiaxis (U = 0,V > 0); the curvature singularity r = 0 corresponds to theupper branch of the hyperbole UV = 1.

• In the new frame the manifold can be extended across the semi-axis (U = 0, V > 0), separating M1 and M2, since the lineelement (1.11) is not singular there; this means that we considera new manifold,

M ) M1 (M2 , (1.12)

defined byV > 0 , UV < 1 . (1.13)

Therefore, in order to eliminate a coordinate singularity we have toextend the spacetime manifold.

Notice that when we think of a Schwarzschild BH, we are im-plicitly considering the extended manifold M: indeed, we assumethat an object falling inside the BH crosses the horizon r = 2M .Even the discussion of the previous section about the hypersurfacesr = constant (and, among them, r = 2M), assumes that the man-ifold is M; we stress that the manifold M is not covered by the

8

U

r=0

V

r=0

I

II

IV

III

t =

r = 2M

r = 2M

t = +

Figure 1.2: Maximal extension of Schwarzschild spacetime in Kruskal coordi-nates. The dashed line represents the worldline of an observer falling into theblack hole.

coordinates (t, r, !,") and the line element (1.10).The manifold M should still be extended: timelike and null

geodesics from V = 0 cannot be extended backwards, unless weextend the manifold to V * 0. By considering (!# < U < +#,!# < V < +# with UV < 1) we have the maximal extension(i.e., which cannot be further extended) of the Schwarzschild space-time. This is the usual Kruskal construction, shown in Fig. 1.2; thedashed line represents the worldline of an observer falling into theblack hole, and the wave-like curves represent the curvature singu-larity r = 0.

In the Kruskal coordinates (1.11) the null worldlines with !,"constant, are straight lines at 45o, i.e. U = const. and V = const.;this can be seen easily from the line element (1.11): assuming !,"constant, if either dU = 0 or dV = 0, then ds = 0 and the infinites-imal tangent vector is null (to say in di&erent words: any worldlinewith tangent vector either (1, 0, 0, 0) or (0, 1, 0, 0), is null). There-

9

fore, the light-cone can be drawn as in Minkowski spacetime, andthen it is easy to describe the causal connections among events (seefor instance, in Fig. 1.2, the dashed line representing the possi-ble worldline of a massive particle falling inside the black hole). Inparticular, we see that Sector I can send signals (and matter andenergy) to sector II only, and receive signals from sector III; fur-thermore, there is another copy of sector I, i.e. sector IV, which iscausally disconnected from I, but can receive signals from III andsend signals to II.

The incomplete (timelike and null) geodesics of the maximallyextended manifold correspond to a true singularity: in this case, tor = 0, i.e., in Kruskal coordinates, to UV = 1. There are geodesicsreaching the singularity with finite a%ne parameter, and cannot beextended through it; for instance, an observer that falls inside theBH reaches the singularity in a finite amount of proper time.

It should be stressed that the maximal extension of Schwarzschildspacetime has no meaning if we consider a BH as an astrophysicalobject. Therefore, we do not have to worry about the meaning ofthe sector IV, and leave the discussion of other universes to science-fiction writers. Indeed, the above construction describes an eternalblack hole, whereas astrophysical BHs result from stellar collapse,which happen at finite values of t. In particular, the region IIIcannot exist for an astrophysical BH, because the semiaxis (U < 0,V = 0) corresponds to t = !#.

A comment on the r = 0 singularity. The fact that we cannotsay what happens to the observer as it reaches the singularity, con-stitutes a problem for the theory; on the other hand, such problemis not severe from an operational point of view, since no signal fromthe observer reaching the singularity can be sent outside the blackhole: the consistency of the theory, in a certain sense, is preservedby the existence of the horizon. Roger Penrose has conjectured thatthere is a fundamental principle, the cosmic censorship hypothesis,stating that all singularities in the universe (with the exception of apossible initial singularity) are covered by horizons. In other words,there is no naked singularity. There is no definitive proof of thisconjecture, but there are indications supporting it. Therefore, it iscommonly believed that the cosmic censorship hypothesis is likelyto be correct; it is worth noticing that many mathematical proper-ties of spacetime would be challenged by the existence of a naked

10

singularity.

1.3.2 Kruskal and Eddington-Finkelstein coordinates

The Kruskal coordinates are defined as follows. Let us consider ther > 2M manifold, and define

u & t! r" , v & t+ r" (1.14)

withr" & r + 2M ln

# r

2M! 1

$(1.15)

tortoise coordinate, which tends to !# as r " 2M . Notice thatr > 2M corresponds to

!# < u < +# !# < v < +# . (1.16)

and the limit r " 2M , t " +# corresponds to u " +#, withfinite v. Since

dr"dr

=1

1! 2Mr

, (1.17)

the metric is

ds2 = !!1! 2M

r

"(dt2 ! dr2") + r2(d!2 + sin2 !d"2)

= !!1! 2M

r

"dudv + r2(d!2 + sin2 !d"2) . (1.18)

We have

r" ! r

2M= ln

# r

2M! 1

$' 1!2M

r=

2M

re

r!"r2M =

2M

re

"r2M e

v"u4M .

(1.19)Then, defining

U & !e!u

4M , V & ev

4M (1.20)

we have the Kruskal metric:

ds2 = !32M3

re!

r2M dUdV + r2(d!2 + sin2 !d"2) (1.21)

with U < 0, V > 0.Let us now consider the manifold 0 < r < 2M , and define u, v as

in (1.14),u & t! r" , v & t+ r" (1.22)

11

but with a new tortoise coordinate

r" & r + 2M ln#1! r

2M

$(1.23)

which is always negative, tends to zero as r " 0, and to !# asr " 2M . Di&erentiating this new tortoise coordinate we get thesame expression as (1.17),

dr"dr

=1

1! 2Mr

, (1.24)

then the metric in u, v is still given by

ds2 = !!1! 2M

r

"dudv + r2(d!2 + sin2 !d"2) (1.25)

but

1! 2M

r= !2M

re

"r2M e

v"u4M (1.26)

thus, definingU & +e!

u4M , V & e

v4M , (1.27)

we have the same metric of eq. (1.21),

ds2 = !32M3

re!

r2M dUdV + r2(d!2 + sin2 !d"2) , (1.28)

but with U > 0, V > 0.This metric, with V > 0 and extended to !# < U < +#,

describes then the exterior and the interior of the BH, as anticipated.Actually, there is a simpler coordinate system that covers the

region I and II: the Eddington-Finkelstein (EF) coordinates

(v, r, !,") !# < v < +# 0 < r < +# . (1.29)

The two definitions (1.15), (1.23) can be put together as

r" & r + 2M ln%%%r

2M! 1

%%% (1.30)

and v = t + r". Then, since

dt2!dr2" = dv2!2dvdr" = dv2!2dr"dr

dvdr = dv2!2dvdr

1! 2Mr

, (1.31)

12

the metric in the EF coordinates is

ds2 = !!1! 2M

r

"dv2 + 2dvdr + r2(d!2 + sin2 !d"2) . (1.32)

This metric covers both the interior and the exterior of the BH, i.e.the sectors I and II of the Kruskal construction, and is not singularat the horizon, which is simply r = 2M . Notice that on the hori-zon v is finite because t " +# and r" " !#, while u is insteaddivergent. All the computations and derivations involving the inte-rior and the exterior of the BH, like for instance the study of ther = constant surfaces of Section 1.2, and the study of what happensto a spaceship falling inside the BH, can be rigorously performed inthe EF coordinates.

In principle, one could also build the maximal extension of theSchwarzschild metric by patching together the EF chart, which cov-ers sectors I,II, with other three (similar) charts, one covering sectorsII,III, one covering sectors III,IV, and one covering sectors IV,II;fortunately, we do not need to discuss this complicate construction,since there exists a single coordinate frame, the Kruskal frame, whichcovers all the four sectors of the maximal extension. But for otherBH solutions there is no equivalent to the Kruskal coordinates, andthe only way to build the maximal extension is to patch togetherdi&erent coordinate frames (analogue to the EF frame), which elim-inate the coordinate singularities but do not provide, alone, themaximal extension.

1.3.3 The Finkelstein diagram

A useful way to visualize the Schwarzschild spacetime is the Finkel-stein diagram, in which the axes are (t, r) where

t & v ! r = t+ 2M ln%%%r

2M! 1

%%% . (1.33)

In the Finkelstein diagram the null lines v = const., correspond-ing to ingoing massless particles, are straight lines at 45o; the nulllines u = const., corresponding to outgoing massless particles, arehyperbolic curves. These two sets of curves define the light conescentered in any point of the spacetime, and then allow to establishthe causal relationship among di&erent events. Di&erently from the(t, r) diagram, the light-cones behave regularly at the horizon.

13

~t

r

r = 2M

u = co

nst.

r = 0

v = const.

u = const.

t = co

nst.

Figure 1.3: Eddington-Finkelstein diagram of a Schwarzschild black hole.

Since t + t for r , 2M and r - 2M , the coordinate t coincideswith t far away from the horizon, but they are very di&erent close tothe horizon (where, anyway, the coordinate t has not a clear physicalmeaning). Still, inside the horizon t cannot be considered a “time”,because the vector $/$t is spacelike; this is clear from Fig. 1.3,where $/$t (directed vertically) falls outside the null cones in theinterior of the BH.

Notice that the r = constant lines are vertical lines, while thet = constant lines are hyperbolic curves (dashed lines in Fig. 1.3);the t-axis represents the singularity, and for this reason it is drawnwave-like.

As we mentioned above, astrophysical BHs are the product ofgravitational collapse, and then are not eternal black holes. A qual-itative view of the spacetime of a realistic black hole is shown in theFinkelstein diagram in Fig. 1.4, where the shadowed area representsthe interior of the star. The r = 0 axis is a curvature singularity fort . t0, i.e. after the singularity formation; t < t0, the r = 0 axisis simply the (trivial) coordinate singularity at the origin of polarcoordinates. Also the horizon, represented by the dashed line, isformed during the collapse.

It is important to stress that, although we have discussed theentire Schwarzschild solution, only the r > 2M region is directly

14

~t

t0~

r

r = 2Mv = const.

u = const.

fluid

r = 0

Figure 1.4: Eddington-Finkelstein diagram of a stellar collapse originating aSchwarzschild black hole. The shadowed area represents the fluid interior of thestar. The curvature singularity (wave-like line) is formed at t = t0. The horizonis represented by the dashed line.

relevant for astrophysical observations: no signal can come from theinterior of the BH. Therefore, the most physically relevant propertiesof black holes are the properties of the exterior region. On theother hand, it is impossible to have a general understanding of thephysics of black holes (and then of the behaviour of astrophysicalblack holes) without having at least a general idea of what’s goingon inside; for this reason we have briefly discussed the features ofthe entire solution.

1.4 Geodesics

We remind that the geodesic equations are equivalent to the Euler-Lagrange equations with Lagrangian L = 1/2gµ& xµx& where a dotindicates di&erentiation with respect to the a%ne parameter % (wecall the four-velocity of the particle xµ or, equivalently, uµ). For theSchwarzschild metric, the Lagrangian is

L =1

2

&!!1! 2M

r

"t2 +

r2'1! 2M

r

( + r2!2 + r2 sin2 !"2

). (1.34)

15

The equations of motion for t, " and ! are:

• Equation for t:

$L$t

! d

d%

$L$(t)

= 0 " d

d%

*!1! 2M

r

"2t

+= 0 (1.35)

i.e.

t =const'1! 2M

r

( =E'

1! 2Mr

( . (1.36)

In the case of a massive particle, the constant E can be inter-preted as the energy of the particle per unit mass at infinity,since at r " #, where the spacetime becomes flat and specialrelativity applies, the energy of the particle is P 0 = mu0 = mE.

It should be reminded that, since the Schwarzschild metric ad-mits a timelike Killing vector )

)t " #!(t) = (1, 0, 0, 0), then

g!%#!(t)u

% = const " !!1! 2M

r

"t = const = !E (1.37)

thus E is the constant of motion associated with invariance fortime translations.

• Equation for ": in a similar way it is easy to show that

" =const

r2 sin2 !=

L

r2 sin2 !. (1.38)

Since there is a spacelike Killing vector ))$ " #!($) = (0, 0, 0, 1),

such that

g!%#!($)u

% = const " r2 sin2 !" = const " L = const,(1.39)

thus L is the constant of motion associated with invariance for"-rotations. It can be interpreted as the angular momentum ofthe particle per unit mass.

• Equation for !:

$L$!

! d

d%

$L$(!)

= 0 " d

d%(r2!) = r2 sin ! cos !"2 . (1.40)

Therefore the equation for ! is

! = !2

rr! + sin ! cos !"2 . (1.41)

16

Given the spherical symmetry, polar axes can be chosen in ar-bitrary way. We choose them such that, for a given value of thea%ne parameter, say % = 0, the particle is on the equatorialplane ! = #

2 and its three-velocity (r, !, ") lays on the same

plane, i.e. !(% = 0) = 0. Thus, we have to solve the followingCauchy problem

! = !2

rr! + sin ! cos !"2

!(% = 0) = 0

!(% = 0) =&

2(1.42)

which admits an unique solution. Since

!(%) & &

2(1.43)

satisfies the di&erential equation and the initial conditions, itmust be the solution. Thus the orbit is plane and to hereafterwe shall assume ! = #

2 and ! = 0.

• Equation for r: it is convenient to derive this equation fromthe condition u!u! = !1, or u!u! = 0, respectively valid formassive and massless particles.

A) massive particles:

g!%u!u% = !

!1! 2M

r

"t2+

r2'1! 2M

r

(+r2!2+r2 sin2 !"2 = !1

(1.44)which becomes, by substituting the equations for t and "

r2 +

!1! 2M

r

"!1 +

L2

r2

"= E2 (1.45)

B) massless particles:

g!%u!u% = !

!1! 2m

r

"t2 +

r2'1! 2M

r

( + r2!2 + r2 sin2 !"2 = 0

(1.46)which becomes

r2 +L2

r2

!1! 2M

r

"= E2 . (1.47)

17

Summarizing, the geodesic equations are:

! =&

2

t =E'

1! 2Mr

(

" =L

r2

r2 = E2 ! V (r) (1.48)

with

V (r) =

!1! 2M

r

"!1 +

L2

r2

"(1.49)

for massive particles and

V (r) =L2

r2

!1! 2M

r

"(1.50)

for massless particles; it is worth noticing that the potential of mass-less particles depends on L as a multiplication constant, and thenthe location of maxima and minima of V (and, more generally, theshape of V ) does not depend on L, while for massive particles thedependence on L is non trivial.

We also point out that for massive particles (for which uµuµ = !1and the a%ne parameter is the proper time) E,L are, as we saidabove, the energy (at infinity) and the angular momentum per unitmass, but this is not the case for massless particles. If the particleis massless, there is an arbitrariness in the definition of the a%neparameter (% " a% preserve the normalization uµuµ = 0), and E,Lscale with %; then, we can normalize % so that E,L are the energy(at infinity) and the angular momentum of the massless particle.

The radial equations

r2 = E2 ! V (r) (1.51)

allow a qualitative study of the motion analogue to the study per-formed in Newtonian mechanics. Some results of this qualitativestudy are the following:

• geodesics of massless particles describe either open orbits, cor-responding to deflection of the ingoing particle, or unstablecircular orbits with r = 3M ;

18

• geodesic of massive particles describe either open orbits, cor-responding to deflection of the ingoing particle, or closed or-bits, with r ranging between a minimum and a maximum value(these are not really closed curves due to periastron advance);

• circular orbits of massive particles are allowed only if r . 3M ;if 3M * r < 6M they are unstable orbits; if r . 6M theyare stable orbits; therefore, the innermost stable circular orbit(ISCO) has r = 6M .

It is important to stress that the classical tests of general rela-tivity are based on the properties of Schwarzschild’s solution: thegravitational redshift, the perihelion advance, the deflection of lightsignals. But, as we will discuss later on, astrophysical observationsare starting to test more complex spacetimes, like the Kerr metricwhich describes a rotating BH.

19

Chapter 2

The far field limit of anisolated, stationary object

In this chapter we study the gravitational field generated by anisolated, stationary object.

We require that the source is an isolated object, thus we areassuming that the spacetime outside the source is vacuum (Tµ& = 0).Therefore, it is reasonable to assume that the spacetime, far awayfrom the source, tends to the Minkowski spacetime; this conditionis called asymptotic flatness.

If the spacetime is asymptotically flat, we can define, in an ap-propriate coordinate frame, a space coordinate r such that1

limr#$

gµ& = 'µ& . (2.1)

We call far field limit the region of spacetime in which r - R, whereR is a lengthscale of the source. In the far field limit,

gµ& = 'µ& +O

!1

r

". (2.2)

We also assume that the source is stationary, i.e. it does not dependon time. In this case, it can be shown that Einstein’s equations implythat the spacetime is also stationary.

We want to determine the metric in the far field limit of an iso-lated, stationary object. In other words, we want to determine the

1Actually a complete, coordinate independent definition of asymptotic flatness would re-quire the introduction of the concept of conformal infinity, which is beyond the scope of theselectures. However, for the derivation of the far field limit of isolated stationary objects thedefinition (2.1) is su!cient.

20

lowest order terms in 1/r in the expansion (2.2). We will show thatthe spacetime metric in the far field limit is

ds2 = !!1! 2M

r

"dt2 +

!1 +

2M

r

"dr2

+r2(d!2 + sin2 !d"2)! 4J

rsin2 !dtd"

+ higher order terms in 1/r . (2.3)

with M mass and J angular momentum of the central object.Far away from the source the expansion (2.2) applies; it can also

be written in the form

gµ& = 'µ& + hµ& (2.4)

with |hµ& | , 1. The perturbation hµ& is a solution of the equationsof linearized gravity gµ& = 'µ& + hµ& ,

!hµ& = 0 (2.5)

hµ&,µ = 0 (2.6)

where we have defined

hµ& & hµ& !1

2'µ&h

!! (2.7)

and we neglect terms O(|h|2) (which are also higher order terms inthe 1/r expansion). We are assuming that the spacetime is station-ary, therefore (2.5), (2.6) become

/2hµ& = 0 (2.8)

hi&,i = 0 . (2.9)

We stress that (2.8), (2.9) hold only in the far field limit r - R; weare not making any assumption on the nature of the source.

Nevertheless, to have more insight on the physical meaning of(2.3), we will start deriving it in the particular case where the grav-itational field on the source is weak. Subsequently, we will considerthe general case.

2.1 The case of a weakly gravitating source

If the gravitational field on the source is weak, it is possible to lin-earize the metric on the source, and the linearized Einstein’s equa-

21

tions give

!hµ& = !16&Tµ&

hµ&,µ = 0 , (2.10)

whose solution is

hµ&(t,x) = 4

,

V

Tµ&(t! |x-x%|,x% )

|x-x% | d3x% (2.11)

where V is the three-volume of the source. We choose V such thatits boundary falls outside the source, i.e. Tµ& = 0 on the boundary$V .

In the standard derivation of the quadrupole formula for gravi-tational radiation, one considers the oscillating part of the integral(2.11), neglecting the terms constant in time. Here, on the contrary,we are considering a stationary source Tµ& = Tµ&(x%); the integral(2.11) becomes

hµ&(x) = 4

,

V

Tµ& (x%)

|x! x%|d3x% . (2.12)

We denote by Latin letters i, j the space indexes 1, 2, 3; notice that,in linearized gravity, the indexes of hµ& are raised by using theMinkowski metric, thus hiµ = hi

µ.Let us consider the Taylor expansion of 1/|x!x%|; x is the position

vector of the point where we compute the gravitational perturbationhµ& (with radial coordinate r = |x|); x% is the position vector of ageneric point inside the source; then, |x%| * R , r = |x|. Such anexpansion is commonly named multipolar expansion; we have

1

|x! x%| =1

r+

-i x

ix%i

r3+O

!1

r3

". (2.13)

Substituting in (2.12) we find

hµ&(x) =4

r

,

V

Tµ&d3x% +

4-

i xi

r3

,

V

Tµ&x%id3x% . (2.14)

The first integral in (2.14) gives,

V

Tµ&d3x% . (2.15)

22

Being T00, in the limit of weak gravitational field, the mass-energydensity of the system, by definition

,

V

T00d3x% = M . (2.16)

To compute the components µi of the integral (2.15), we first noticethat the stress-energy tensor conservation, for a stationary source,reduces to

T µ&,& = T µ0

,0 + T µi,i = T µi

,i = 0 . (2.17)

Using (2.17), and the property )xi

)xj = (ij , we find,

V

T µid3x% =

,

V

T µk(ikd3x% =

,

V

T µk $x%i

$x%k d3x%

= !,

V

!$T µk

$x%k

"x%id3x% = 0 (2.18)

where we have integrated by parts; the surface terms do not con-tribute, because, on the boundary of V , Tµ& = 0. Therefore, thecomponents µi of the integral (2.15) vanish.

Let us now compute the second integral in (2.14),,

V

Tµ&x%id3x% . (2.19)

The 00 component gives the position of the center of mass, whichcan be set to zero by a proper choice of the origin of the coordinates:

,

V

T00x%id3x% = Mx%i

cdm = 0 . (2.20)

To compute the µi components of the integral (2.19), we start byproving that it is antisymmetric in the last two indexes:

,

V

T µix%jd3x% = !,

V

T µjx%id3x% . (2.21)

Indeed,,

V

'T µix%j + T µjx%i( d3x% =

,

V

T µk

!$x%i

$x%k x%j +

$x%j

$x%k x%i"d3x%

=

,

V

T µk $

$x%k

'x%ix%j( d3x% = !

,

V

x%ix%jT µk,kd

3x% = 0 (2.22)

23

where we have integrated by parts, using the fact that, on the bound-ary of V , Tµ& = 0.

Let us consider now the components µ) = jk of the integral(2.19) ,

V

T jkx%id3x% . (2.23)

It is symmetric in the first two indexes, but, from (2.21), it followsthat it is also antisymmetric in the last two indexes. Therefore, itis easy to show that it is the opposite of itself, and then it vanishes:

,

V

T kix%jd3x% = !,

V

T kjx%id3x% = !,

V

T jkx%id3x%

=

,

V

T jix%kd3x% =

,

V

T ijx%kd3x% (2.24)

= !,

V

T ikx%jd3x% = 0 . (2.25)

The last integral we have to compute is,

V

T 0ix%jd3x% = !,

V

T 0jx%id3x% . (2.26)

To express it di&erently we introduce the tensor *ijk, which is anti-symmetric in any exchange of its indexes, and has *123 = 1. Then,if we define

J i & !,

V

*ijkT 0jx%kd3x% , (2.27)

we can write ,

V

T 0ix%jd3x% = !1

2*ijkJk (2.28)

It is easy to prove that (2.27) implies (2.28), using the very usefulproperty2

*ijk*klm = (il(jm ! (im(jl . (2.29)

Then, given an antisymmetric tensor Bij = !Bji, we have

J i = *ijkBjk 0' Bij =1

2*ijkJk : (2.30)

2The prove of (2.29) is the following. *ijk &= 0 only if its three indexes are all di"erent,thus i &= k and j &= k; the same for *lmk. Therefore *ijk*lmk &= 0 only if the indexes ij and lmare the same, with any ordering. If they have the same order, i.e. ij = lm, then *ijk*lmk = 1,while if they have the opposite order, i.e. ij = ml, then *ijk*lmk = !1. Thus (2.29).

24

1

2*ijkJk =

1

2*ijk*klmBlm =

1

2

'(il(jm ! (im(jl

(Blm = Bij . (2.31)

Finally, replacing

Bjk = !,

V

T 0jx%kd3x% , (2.32)

we find that (2.27) implies (2.28).The physical meaning of Jk is simple. The tensor T 0i represents

the momentum density of the source,

T 0i = P i (2.33)

i.e. an element of the matter source in the volume d3x% has momen-tum Pd3x%. Writing (2.27) as a vector product,

J =

,x% 1 Pd3x% (2.34)

since (u1 v)i = *ijkujvk.We remind that the angular momentum of a point particle is the

vector product between its momentum and its position vector. Theangular momentum of an element of matter in the source, which hasmomentum Pd3x% and position x%, is then x% 1 Pd3x%. Therefore, Jis the total angular momentum of the source.

Summarizing, the multipolar expansion (2.14) gives

h00 =4M

r+O

!1

r3

"

h0i =2

r3*ijkxjJk +O

!1

r3

"

hij = O

!1

r3

"(2.35)

because,

V

T00d3x% = M

4

r3xj

,

V

T0ix%jd3x% = ! 4

r3xj

,

V

T 0ix%jd3x% =2

r3*ijkxjJk .

(2.36)

Notice that in flat space h00 = h00, h0i = !h0i, hij = hij .

25

In terms of hµ& , given by 3

hµ& = hµ& !1

2'µ&h

!! , (2.37)

we have4

h00 =2M

r+O

!1

r3

"

h0i =2

r3*ijkx

jJk +O

!1

r3

"

hij =2M

r(ij +O

!1

r3

". (2.38)

Let us transform the solution (2.38) in polar coordinates

x1 = r sin ! cos"

x2 = r sin ! sin"

x3 = r cos ! . (2.39)

We have .

i

(dxi)2 = dr2 + r2d!2 + r2 sin2 !d"2 , (2.40)

thus

hijdxidxj =

2M

r

'dr2 + r2d!2 + r2 sin2 !d"2

(. (2.41)

The transformation of h0idx0dxi is less trivial. If we choose theorientation of our polar coordinates such that

J = (0, 0, J) , (2.42)

then

h0idx0dxi =

!2

r3dx0

"*ijkx

jJkdxi = !!

2

r3dx0

"J(x1dx2 ! x2dx1)

= !!

2

r3dx0

"Jr2 sin2 !d" = !2J

rsin2 !dtd" (2.43)

3To invert (2.7) we take the trace of (2.7), finding h!! = !h!

!, and replace this relationinto (2.7).

4Notice that xj

r3is an O

!1r2

"term, because in the far field limit xj ' r.

26

where the property x1dx2 ! x2dx1 = r2 sin2 !d" can be proven bydi&erentiation of (2.39).

The corresponding line element is

ds2 = !!1! 2M

r+O

!1

r3

""dt2

+

!1 +

2M

r+O

!1

r3

""/dr2 + r2(d!2 + sin2 !d"2)

0

+

!!4J

rsin2 ! +O

!1

r2

""dtd" . (2.44)

This is the solution of linearized Einstein’s equations. If we considerthe complete Einstein’s equations

Rµ& = 0 (2.45)

we have terms O(|hµ& |2), which produce terms quadratic in M andJ in the metric (and also higher order terms). Therefore, there areterms 2 M2/r2, 2 J2/r2 in the metric. Then, we cannot say, forinstance, that

g00 = !1 +2M

r+O

!1

r3

"(2.46)

because there is also a term 2 M2/r2 arising from nonlinear termsin Einstein’s equations. If we don’t want to compute these terms,we truncate our expansion at O(1/r2):

ds2 = !!1! 2M

r+O

!1

r2

""dt2

+

!1 +

2M

r+O

!1

r2

""/dr2 + r2(d!2 + sin2 !d"2)

0

+

!!4J

r+O

!1

r2

""sin2 !dtd" . (2.47)

Finally, we make the following redefinition of the radial coordinate:

r " r !M . (2.48)

Making this change in (2.47), and neglecting contributions O(1/r2),the only term which produces a change in the metric is

!1 +

2M

r

"r2(d!2 + sin2 !d"2) "

!1 +

2M

r

"(r !M)2(d!2 + sin2 !d"2)

27

= r2!1 +O

!1

r2

""(d!2 + sin2 !d"2) . (2.49)

With this coordinate redefinition, we find the expression (2.3):

ds2 = !!1! 2M

r

"dt2 +

!1 +

2M

r

"dr2

+r2(d!2 + sin2 !d"2)! 4J

rsin2 !dtd"

+ higher order terms in 1/r . (2.50)

Notice that in the case of a spherically symmetric spacetime (inwhich J = 0), the line element (2.50) corresponds to the lineariza-tion of the Schwarzschild metric in the standard coordinates (t, r, !,").In the same limit, (2.38) corresponds to the linearization of theSchwarzschild metric in isotropic coordinates.

2.2 The case of a general source

Let us now consider a general source. The gravitational field can bestrong near the source, whereas far away, where we want to solveEinstein’s equations, it is weak. Thus, far away from the source, wecan consider linearized gravity, neglecting terms O(|hµ&|2), and wecan neglect terms with a large power of 1/r, where r is the distancefrom the source.

Therefore, we expand the metric far away from the source as

gµ& = 'µ& + hµ& (2.51)

with

hµ& =aµ&(!,")

r+

bµ&(!,")

r2+O

!1

r3

". (2.52)

The coe%cients aµ& , bµ& do not depend by r, but they depend bythe angular variables !,", so that they remain finite for r " #. Wecan express the angular variables in an alternative way, using thedirector cosines ni defined as

ni =xi

r= (sin ! cos ", sin ! sin", cos !) (nini = 1). (2.53)

Thus, we can say that aµ& , bµ& are functions of the ni.

28

The metric perturbation, which we assume to be stationary, sat-isfies equation (2.8),

/2hµ& = 0 . (2.54)

The Laplace operator in spherical coordinates has the form5

/2 =1

r2$rr

2$r +IL

r2(2.55)

where IL is an operator acting on the angular variables only:

IL & $2" + cot !$" + sin!2 !$2

$ . (2.56)

Substituting (2.52) in (2.55) we easily find

ILaµ&(!,") = 0 (2.57)

ILbµ&(!,") = !2bµ&(!,") . (2.58)

The eigenfunctions of the operator IL are the spherical harmonicsYlm(!,"), with l = 0, 1, . . . and m = !l,!l + 1, . . . , l ! 1, l. Theyare defined by the property

ILYlm = !l(l + 1)Ylm . (2.59)

Equation (2.58) tells us that bµ& is a linear combination of the spher-ical harmonics with l = 1, which are

Y11 = !1

3

8&sin !ei$

Y10 =

13

4&cos !

Y1!1 =

13

8&sin !e!i$ . (2.60)

This is equivalent to say that bµ& is a linear combination of ni = xi/r,because

n1 =x1

r= sin ! cos" =

18&

3

!Y11 + Y1!1

2

n2 =x2

r= sin ! sin" =

18&

3

!Y11 ! Y1!1

2i

n3 =x3

r= cos ! =

14&

3Y10 . (2.61)

5The theory of Laplace equation and the properties of spherical harmonics are extensivelydiscussed in the literature. See for instance Jackson’s book Electromagnetism, Chapter 3.

29

Therefore, aµ& does not depend on the angular variables ni, whilebµ& , being a combination of the three harmonics Y1m, is linear in ni,and can be written as

bµ&(ni) = bµ&i ni . (2.62)

Expansion (2.52) can then be written as

hµ& =aµ&

r+

bµ&i xi

r3+O

!1

r3

", (2.63)

with aµ& , bµ&i constant coe%cients.We now impose on (2.63) the gauge condition (2.6)

hµ&,& = 0 (2.64)

which in the case of stationary perturbations becomes

hµi,i = 0 . (2.65)

We get (remember that in linearized gravity it is irrelevant if a spaceindex i is high or low)

hµj,j = !aµjxj

r3+

bµji ((ijr2 ! 3xixj)

r5= 0 (2.66)

which has to be satisfied for all (large) values of r and for all valuesof ni = xi/r. Thus,

aµj = 0'(ij ! 3ninj

(bµij = 0 . (2.67)

These equations do not involve a00, b00i, which are then in generalnonvanishing free constants; to simplify the notation, we rewritethem as

a & a00bi & b00i . (2.68)

The first of the conditions (2.67) states that all the constants aµ&di&erent from a00 vanish. The second one is more complicate toanalyze. It can be rewritten as

H ijb0ij = 0 (2.69)

H ijbkij = 0 (2.70)

30

where we have defined

H ij & (ij ! 3ninj . (2.71)

The general solution of (2.69), (2.70) is

b0ij = b(ij + ck*ijk (2.72)

bkij = dk(ij + di(kj ! dj(ki (2.73)

where b, ck, dk are constants. A rigorous proof of (2.72), (2.73) wouldrequire an use of the structures of Group Theory which goes beyondthe scope of these lectures. We only give here an intuitive, non-rigorous proof of the first solution, (2.72).

Equation (2.69) must be satisfied for every value of ni, i.e. forevery value of the angular variables !,". But, while H ij depends onthe angles, bµij cannot depend on the angles. Thus, equation (2.69)can be satisfied only because of the symmetry properties of H ij: itis symmetric and traceless

H ij = Hji (ijHij = 0 . (2.74)

All the quantities considered here are tensors in the euclidean three-dimensional space. The only constant tensors which vanish whencontracted with H ij are the Kronecker delta (ij and the completelyantisymmetric tensor *ijk: the former vanishes because H ij is trace-less, the latter vanishes because H ij is symmetric. The solution b0ijmust be a combination of them, thus we have (2.72).

Summarizing, by imposing the gauge condition (2.6) on the ex-pansion (2.63) we get

h00 =a

r+

bixi

r3+O

!1

r3

"

h0i =bxi

r3+ *ijk

xjckr3

+O

!1

r3

"

hij =1

r3'!(ijdkxk + dix

j + djxi(+O

!1

r3

"(2.75)

depending on the constants a, bi, b, ck, dk.The constants b, dk can be eliminated by a (position dependent)

infinitesimal di&eomorphism xµ " xµ + #µ with parameter

#µ =

!! b

r,!di

r

"(2.76)

31

(it is infinitesimal in the sense that r is large and # 2 1/r).The change in the metric is

gµ& = 'µ& + hµ& " gµ& + gµ!#!,& + g&!#

!,µ + gµ&,!#

!

= 'µ& + hµ& + gµ!#!,& + g&!#

!,µ + gµ&,!#

!

(2.77)

therefore the change in the perturbation (hµ& is

(hµ& = gµ!#!,& + g&!#

!,µ + gµ&,!#

! (2.78)

and, being #µ = O(|hµ&|), we have that, neglecting terms quadraticin hµ& ,

(hµ& = 'µ!#!,& + '&!#

!,µ . (2.79)

The change in the variable

hµ& = hµ& !1

2'µ&'

!%h!% (2.80)

is

(hµ& = (hµ& !1

2'µ&'

!%(h!%

= 'µ!#!,& + '&!#

!,µ ! 'µ&#

!,! . (2.81)

We have

#µ,0 = 0

#0,i =bxi

r3

#k,i =dkxi

r3(2.82)

then

(h00 = !'00#k,k +O

!1

r3

"=

dkxk

r3+O

!1

r3

"

(h0i = '00#0,i +O

!1

r3

"= !bxi

r3+O

!1

r3

"

(hij = 'ik#k,j + 'jk#

k,i ! 'ij#

k,k =

1

r3/dixj + djxj ! 'ijd

kxk0

(2.83)

32

thus, after the di&eomorphism,

h00 =a

r+

bixi

r3+O

!1

r3

"

h0i = *ijkxjckr3

+O

!1

r3

"

hij = O

!1

r3

"(2.84)

where we have definedbi & bi + di . (2.85)

Furthermore, we can get rid of bi by making the (rigid) translation

xi " xi +bia

(2.86)

which produces the following change in the a/r term:

a

r= a

'(xi)2

(!1/2 " a

2

34xi +

bia

526

7!1/2

= a

4r2

41 + 2

bixi

r2a

55!1/2

+O

!1

r3

"

=a

r

41! bixi

r2a

5+O

!1

r3

"=

a

r! bixi

r3+O

!1

r3

".

(2.87)

Therefore,

h00 =a

r+O

!1

r3

"

h0i = *ijkxjckr3

+O

!1

r3

"

hij = O

!1

r3

". (2.88)

Finally, we compute

hµ& = hµ& !1

2'µ&'

!%h!% . (2.89)

33

We have1

2'!%h!% = ! a

2r(2.90)

therefore

h00 =a

2r+O

!1

r3

"

h0i = *ijkxjckr3

+O

!1

r3

"

hij = (ija

2r+O

!1

r3

". (2.91)

With the identifications

a = 4M ck = 2Jk (2.92)

the solution (2.91) coincides with the solution (2.38), which we havederived in the case of a weak field source, and that we have alreadyshown to be equivalent to the solution (2.3).

2.3 Mass and angular momentum of an isolatedobject

As we have seen, the metric

ds2 = !!1! 2M

r

"dt2 +

!1 +

2M

r

"dr2

+r2(d!2 + sin2 !d"2)! 4J

rsin2 !dtd"

+ higher order terms in 1/r . (2.93)

describes the far field limit of an isolated, stationary source. Ifthe source is weakly gravitating, we can apply the definitions ofNewtonian physics on the source, and in this case we have seen thatM and J have a simple interpretation: they are the mass and theangular momentum of the source, respectively.

In the case of a source which is not weakly gravitating, M andJ are simply integration constants of the general far field solution(2.93). We could ask: which is their physical interpretation in thiscase?

34

One answer which is often given to this question is the follow-ing. The motion of a test body in the metric (2.3), far away froma strongly gravitating source, cannot be distinguished from the mo-tion it would have if the source were weakly gravitating, with massM and angular momentum J . Thus, we can give an operationaldefinition of the mass and angular momentum of the strongly gravi-tating source, by looking to the motion of test bodies far away fromthe source. The mass will be defined by looking to Kepler’s thirdlaw, and the angular momentum by looking to the precession ofgyroscopes orbiting around the source.

A di&erent answer is based on the stress-energy pseudotensor tµ& ,which describes the energy and momentum carried by the gravita-tional field, and satisfies, together with the stress-energy tensor Tµ& ,a conservation law:

[(!g)(T µ& + tµ&)],& = 0 . (2.94)

It can be expressed as a divergence:

(!g)(T µ& + tµ&) =$+µ&!

$x!(2.95)

where

+µ&! =1

16&

$

$x%

/(!g)(gµ&g!% ! gµ!g&%)

0. (2.96)

We can also write the quantity +µ&! in a di&erent way, which willbe useful later:

+µ&! =$%µ&!%

$x%(2.97)

with

%µ&!% & 1

16&

/(!g)(gµ&g!% ! gµ!g&%)

0. (2.98)

Furthermore, in a stationary spacetime, gµ&,0 = 0 therefore

+µ&! =$%µ&!i

$xi. (2.99)

Let us now consider a spherical three-dimensional volume V centeredon the source, extending up to a radius r far away from the source.Notice that it is di&erent from the volume V considered in Section2.1, which was only covering the source.

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The total four-momentum enclosed in the volume V is due bothto the source and to the gravitational field, and is

P µ =

,

V

d3x(!g)(T 0µ + t0µ) . (2.100)

As explained in Section 2.1, if the three-momentum in the volumeelement d3x located at the space point with coordinates xi is

P id3x , (2.101)

then the angular momentum in the same volume element is

*ijkPjxkd3x . (2.102)

Therefore, the total angular momentum is

J i = *ijk,

V

d3x(!g)(T 0j + t0j)xk . (2.103)

Substituting (2.95) in (2.100) and imposing the stationarity of thespacetime,

P µ =

,

V

d3x$+0µ!

$x!=

,

V

d3x$+0µk

$xk. (2.104)

Using Gauss’ theorem we can express this integral as an integral onthe spherical surface S, orthogonal, at each point, to the vector n,which is the unit vector normal to the surface S:

P µ =

,

S

dS+0µknk . (2.105)

Substituting (2.95) in (2.103) we find that the total angular momen-tum enclosed in the volume V is

J i = *ijk

,

V

d3x$+0jl

$xlxk = *ijk

,

V

d3x

*$(+0jlxk)

$xl! +0jl

$xk

$xl

+

= *ijk

,

V

d3x

*$(+0jlxk)

$xl! +0jk

+= *ijk

,

V

d3x$

$xl

'+0jlxk ! %0jkl

(

= *ijk

,

S

dS'+0jlxk ! %0jkl

(nl . (2.106)

Substituting (2.96) and (2.98) in (2.105), (2.106) one finds

P µ = (M, 0, 0, 0) , J i = (0, 0, J) . (2.107)

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We show this explicitly in the case of M . We have that

P 0 =

,

S

dS+00ini (2.108)

and dS = r2d'. The metric perturbation is

h00 =2M

r+O

!1

r2

"

h0i =2

r3*ijkx

jJk +O

!1

r3

"

hij =2M

r(ij +O

!1

r2

". (2.109)

Being gµ& = 'µ&+hµ&+O(|hµ&|2), we have that the property gµ&g&+ =(µ+ implies

gµ& = 'µ& ! hµ& +O(|hµ& |2) (2.110)

where the indexes of hµ& have been raised with the Minkowski met-ric. Indeed,

('µ& ! hµ&)('&+ + h&+) = (µ+ +O(|hµ& |2) . (2.111)

Therefore,

g00 = !1! h00 +O(|hµ&|2) = !1! 2M

r+O(|hµ& |2) (2.112)

gij = (ij ! hij +O(|hµ&|2) =!1! 2M

r

"(ij +O(|hµ&|2) .

(2.113)

Finally, the determinant of gµ& is

g = (!1 + h00)(1 + hii) = !1! 4M

r+O(|hµ& |2) . (2.114)

Putting together (2.112), (2.113), (2.114), and neglecting the termsO(|hµ&|2) (like the terms 2 M2, 2 J2), we find

+00ini =1

16&ni$j

/(!g)

'g00gij ! g0ig0j

(0

=1

16&ni$j

*!1 +

4M

r

"!!1! 2M

r

"!1! 2M

r

"(ij

++O(|hµ& |2)

= ! 1

16&ni$i

4M

r+O(|hµ&|2) =

1

4&

M

r2nini =

1

4&

M

r2(2.115)

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thus

P 0 =

,

S

+00inir2d' = M . (2.116)

The case of angular momentum is conceptually similar, but the com-putation is more lengthy and we don’t repeat it here.

We can conclude that the integration constants M and J ap-pearing in the far field limit metric of an isolated source (2.3) canbe correctly interpreted as the mass and the angular momentum, re-spectively, of the system. In the case of a weakly gravitating source,the system is the source alone, whereas if the source has a stronggravitational field, the field itself is part of the system and con-tributes to the total mass M and angular momentum J appearingin the line element (2.3).

We stress again that, since the source is isolated, the metric ap-proaches the Minkowski metric far away from the source; this allowsus to assume, for r su%ciently large, that gµ& = 'µ& + hµ& with hµ&

small. Furthermore, the hµ& terms can be expanded in powers of1/r. The dominant contribution in this expansion tells us which arethe total mass and angular momentum of the system.

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