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Mass and Angular Momentum in General Relativity 121

5.3 Black Hole Extremality: a Mass–Angular Momentum

Inequality

Subextremal Kerr black holes are characterized by presenting angular momenta

bounded by their total masses. Keeping axisymmetry, it has been recently shown

[27,28,31](seealso[32]) that the inequality

jM

ADM

; (75)

holds also for vacuum, maximal (K = 0), axisymmetric Euclidean data. Moreover,

equality only holds for slices of extremal Kerr. Inequality (75) provides a nontrivial

relation for black hole spacetimes between precisely the two physical quantities we

are focusing on in this review. It is natural to explore if some analogous inequality

holds when moving away from axisymmetry and when considering only the local

region around the black hole. Attempts have been done in this sense, but they all

must face the ambiguities resulting from the absence of canonical expressions for

quasi-local masses and angular momenta. In order to illustrate the caveats to keep

in mind when undertaking this kind of discussion, one can consider the case in

which Komar quantities are used for constructing a truly quasi-local analogue of

expression (75) for axisymmetric stationary data: initial data have been constructed

[2] where the quotient jJ

j=M

on the black hole horizon can become arbitrarily

large. Interestingly, these studies have led to the formulation [3] in axisymmetry

of the related conjecture 8jJ

jA, only involving intrinsic quantities on the

horizon. This inequality has been proved to hold in the stationary axisymmetric

case [59], as well as in a generalization including the electromagnetic ﬁeld and the

associated electric charge on the horizon [60].

6 Conclusions

The problem of characterizing the energy–momentum and angular momentum as-

sociated with the gravitational ﬁeld in General Relativity has been present since the

birth of the theory and controversies have plagued its already long-standing history.

Understanding that no local density of energy–momentum can be identiﬁed for the

gravitational ﬁeld has challenged the validity of the mass and angular momentum

cherished notions from nongravitational physics, when trying to perform a straight-

forward extension of these concepts to the gravitational ﬁeld in a general relativistic

setting.

The study of speciﬁc problems suggests concrete and/or partial solutions. In this

spirit, at low velocities and weak self-gravities post-Newtonian approaches handle

consistent notions of mass and angular momentum and the same holds in per-

turbative approaches around exact solutions, for which physical quantities can be

identiﬁed unambiguously. In the same line, a (quasi-local) notion of the energy car-

ried by a gravitational wave can be introduced as an average along the wavelength,

122 J.L. Jaramillo and E. Gourgoulhon

proving to provide a useful notion in practical applications. A particular setting of

singular conceptual importance is that of isolated systems in General Relativity,

speciﬁcally through their characterization as asymptotically ﬂat spacetimes. The

notions of total ADM and Bondi–Sachs energy–momentum provide well-deﬁned

quantities that, on the one hand, have clariﬁed important conceptual issues such as

the capability of gravitational waves to actually carry energy away from a system

and, on the other hand, they also represent inestimable tools in practical applications

due to their intrinsic/geometric character. Positivity theorems for the total mass rep-

resent without any doubt some of the most important and profound results in General

Relativity. The combination of the success in isolated systems, together with the

absence of a gravitational local energy–momentum density, has led to the consider-

ation that the whole effort for the search of a local expression for the gravitational

energy represents an ill-posed or pseudo-problem (see, e.g., [73]).Butatthesame

time, and motivated by practical needs and/or fundamental physical reasons (cf. in

this sense [48] for a related discussion on quasi-local issues regarding observables in

Quantum Field Theory), important efforts have been devoted to the introduction of

quasi-local notions of gravitational energy–momentumassociated with extended but

ﬁnite regions of the spacetime. In this respect, signiﬁcant insights into the structure

of the gravitational ﬁeld have been achieved, with applications in diverse conceptual

and practical contexts. But it must be acknowledged, as it is referred in [85], that

the status of the quasi-local mass studies is in a kind of post-modern situation in

which the devoted intensive efforts have resulted in a plethora of proposals with no

obvious deﬁnitive and entirely satisfying candidate.

A moderate (intermediate) position that avoids radical skepticism against the

quasi-local approach would consist in assuming that mass and (in a more restricted

sense) angular momentum can be unambiguously deﬁned only as global quantities

for isolated systems. But accepting, at the same time, that quasi-local expressions

provide meaningful and insightful quantities that are inextricably subject to the need

of making systematically explicit the speciﬁc setting in which they are deﬁned (one

can make the analogy with the notion of effective mass in solid state physics, where

different masses can be simultaneously employed for the same particle as long as

their speciﬁc purposes are clearly stated

). The moral of the whole discussion in this

article is that the formulation of meaningful global or quasi-local mass and angu-

lar momentum notions in General Relativity always needs the introduction of some

additional structure in the form of symmetries, quasi-symmetries, or some other

background structure. This point must be explicitly kept in mind whenever employ-

ing the so-deﬁned physical quantities, specially when extrapolating or performing

compared analysis.

Acknowledgements The authors wish to thank the organizers of the Orl´eans School on Mass for

their kind invitation and encouragement.

We thank B. Carter for his many comments and insights in this discussion, and in particular for

bringing us to the solid state analogy for gravitational masses.

Mass and Angular Momentum in General Relativity 123

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Post-Newtonian Theory and the Two-Body

Problem

Luc Blanchet

Abstract Reliable predictions of general relativity theory are extracted using

approximation methods. Among these, the powerful post-Newtonian approximation

provides us with our best insights into the problems of motion and gravitational radi-

ation of systems of compact objects. This approximation has reached an impressive

mature status, because of important progress regarding its theoretical foundations,

and the successful construction of templates of gravitational waves emitted by

inspiralling compact binaries. The post-Newtonian predictions are routinely used

for searching and analyzing the very weak signals of gravitational waves in current

generations of detectors. High-accuracy comparisons with the results of numerical

simulations for the merger and ring-down of binary black holes are going on. In this

article we give an overview on the general formulation of the post-Newtonian

approximation and present up-to-date results for the templates of compact binary

inspiral.

1 Introduction

Although relativists admire the mathematical coherence – and therefore beauty – of

Einstein’s general relativity, this theory is not easy to manage when drawing ﬁrm

predictions for the outcome of laboratory experiments and astronomical observa-

tions. Indeed only few exact solutions of the Einstein ﬁeld equations are known,

and one is obliged in most cases to resort to approximation methods. It is beyond

question that approximation methods in general relativity do work, and in some

cases with some incredible precision. Many of the great successes of general rel-

ativity were in fact obtained using approximation methods. However, because of

the complexity of the ﬁeld equations, such methods become awfully intricate at

high approximation orders. On the other hand it is difﬁcult to set up a formalism

in which the approximation method would be perfectly well deﬁned and based on

L. Blanchet (



)

G R"CO, Institut d’Astrophysique de Paris, C.N.R.S. & Universit´e Pierre et Marie Curie,

bis

boulevard Arago, 75014 Paris, France

e-mail: blanchet@iap.fr

L. Blanchet, A. Spallicci, and B. Whiting (eds.), Mass and Motion in General Relativity,

Fundamental Theories of Physics 162, DOI 10.1007/978-90-481-3015-3

 Springer Science+Business Media B.V. 2011

125

126 L. Blanchet

clear premises. Sometimes it is impossible to relate the approximation method to

the exact framework of the theory. In this case the only thing one can do is to rely

on the approximation method as the only representation of real phenomena, and to

discover “empirically” that the approximation works well.

The most important approximation scheme in general relativity is the post-

Newtonian expansion, which can be viewed as an expansion when the speed of

light c tends to inﬁnity, and is physically valid under the assumptions of weak grav-

itational ﬁeld inside the source and of slow internal motion. The post-Newtonian

approximation makes sense only in the near zone of an isolated matter source,

deﬁned as r ,where cT is the wavelength of the emitted gravitational

radiation, with T being a characteristic time scale of variation of the source. The ap-

proximation has been formalized in the early days of general relativity by Einstein

[51], de Sitter [47, 48], and Lorentz and Droste [71]. It was subsequently devel-

oped notably by Einstein, Infeld and Hoffmann [52], Fock [55, 56], Plebanski and

Bazanski [80], Chandrasekhar and collaborators [32–34], Ehlers and his school

[50,61,62], and Papapetrou and coworkers [74,76].

Several long-standing problems with the post-Newtonian approximation for gen-

eral isolated slowly-moving systems have hindered progress untill recently. At

high post-Newtonian orders some divergent Poisson-type integrals appear, casting

a doubt on the physical soundness of the approximation. Linked to that, the domain

of validity of the post-Newtonian approximation is limited to the near zone of the

source, making it a priori difﬁcult to incorporate into the scheme the condition of

no-incoming radiation, to be imposed at past null inﬁnity from an isolated source.

In addition, from a mathematical point of view, we do not know what the “reliabil-

ity” of the post-Newtonian series is, that is, if it comes from the Taylor expansion

of a family of exact solutions.

The post-Newtonian approximation gives wonderful answers to the problems of

motion and gravitational radiation, two of general relativity’s corner stones. Three

crucial applications are:

1. The motion of N point-like objects at the ﬁrst post-Newtonian level [52], that is,

1PN,

is taken into account to describe the Solar System dynamics (motion of

the centers of mass of planets).

2. The gravitational radiation-reaction force, which appears in the equations of

motion at the 2.5PN order [39–42], has been experimentally veriﬁed by the

observation of the secular acceleration of the orbital motion of the Hulse–Taylor

binary pulsar PSR 1913C16 [90–92].

3. The analysis of gravitational waves emitted by inspiralling compact binaries –

two neutron stars or black holes driven into coalescence by emission of gravita-

tional radiation – necessitate the prior knowledge of the equations of motion and

radiation ﬁeld up to high post-Newtonian order.

As usual, we refer to nPN as the order equivalent to terms (v=c)

in the equations of motion be-

yond the Newtonian acceleration, and in the asymptotic waveform beyond the Einstein quadrupole

formula, where v denotes the binary’s orbital velocity and c is the speed of light.

Post-Newtonian Theory and the Two-Body Problem 127

Strategies to detect and analyze the very weak signals from inspiralling compact

binaries involve matched ﬁltering of a set of accurate theoretical template wave-

forms against the output of the detectors. Measurement-accuracy analyses have

shown that in order to get sufﬁciently accurate theoretical templates, one must at

least include conservative post-Newtonian effects up to the 3PN level, and radiation-

reaction effects up to 5.5PN order, that is 3PN beyond the leading order of radiation

reaction, which is 2.5PN [36–38,54,81,86].

The appropriate description of inspiralling compact binaries is by two structure-

less point-particles, characterized solely by their masses m

or m

(and possibly

their spins). Indeed, most of the nongravitational effects usually plaguing the

dynamics of binary star systems, such as the effects of a magnetic ﬁeld, of an

interstellar medium, the inﬂuence of the internal structure of the compact bodies,

are dominated by purely gravitational effects. Inspiralling compact binaries are very

clean systems which can essentially be described in pure general relativity.

Although point-particles are ill deﬁned in the exact theory, they are admissible

in post-Newtonian approximations. Furthermore the model of point particles can

be pushed to high post-Newtonian order, where an a priori more realistic model

involving the internal structure of compact bodies would fail through becoming

intractable. However there is an important worry: a process of regularization to deal

with the inﬁnite self-ﬁeld of point particles is crucially needed. The regularization

should be carefully deﬁned to be implemented at high orders. It should hopefully be

followed by a renormalization.

The orbit of inspiralling compact binaries can be considered to be circular, apart

from the gradual inspiral, with an excellent approximation. Indeed, gravitational

radiation-reaction forces tend to circularize rapidly the orbital motion. At each in-

stant during the gradual inspiral, the eccentricity e of the orbit is related to the

instantaneous frequency ! 2=P by [79]

' const !

19=9

(for e  1): (1)

For instance one can check that the eccentricity of a system like the binary pul-

sar PSR 1913+16 will be e '510

6

when the gravitational waves become visible

by the detectors, that is, when the signal frequency after its long chirp reaches

f != '10 Hz. Only systems formed very late, near their ﬁnal coalescence,

could have a non-negligible residual eccentricity.

The intrinsic rotations or spins of the compact bodies could play an impor-

tant role, yielding some relativistic spin–orbit and spin–spin couplings, both in the

binary’s equations of motion and radiation ﬁeld. The spin of a rotating body is of the

order of S mav

spin

,wherem and a denote the mass and typical size of the body,

and where v

spin

represents the velocity of the body’s surface. In the case of compact

bodies we have a Gm=c

, and for maximal rotation v

spin

c; for such objects the

magnitude of the spin is roughly S Gm

=c. It is thus customary to introduce a

dimensionless spin parameter, generally denoted by ,deﬁnedby

S D

: (2)

128 L. Blanchet

We have   1 for black holes, and  . 0:63  0:74 for neutron stars (depending

on the equation of state of nuclear matter inside the neutron star). For binary pulsars

such as PSR 1913 C16, we have  . 510

3

. Considering models of evolution of

observed binary pulsar systems when they become close to the coalescence we ex-

pect that the spins will make a negligible contribution to the accumulated phase in

this case. However, astrophysical observations suggest that black holes can have

non-negligible spins, due to spin up driven by accretion from a companion during

some earlier phase of the binary evolution. For a few black holes surrounded by mat-

ter, observations indicate a signiﬁcant intrinsic angular momentum and the spin may

even be close to its maximal value. However, very little is known about the black-

hole spin magnitudes in binary systems. For black holes rotating near the maximal

value the templates of gravitational waves need to take into account the effects of

spins, both for a successful detection and an accurate parameter estimation.

We devote Section 2 of this article, including Sections 2.1–2.6, to a general

overview of the formulation of the post-Newtonian approximation for isolated

sources. We emphasize that the approximation can be carried out up to any post-

Newtonian order, without the aforesaid problem of divergences. The main technique

used is the matching of asymptotic expansions, which permits to obtain a complete

post-Newtonian solution incorporating the correct boundary conditions at inﬁnity.

Then Section 3, that is, Sections 3.1–3.6, will deal with the application to systems

composed of compact objects. The subtle issues linked with the self-ﬁeld regular-

ization of point-particles are discussed in some details. The results for the two-body

equations of motion and radiation ﬁeld at the state-of-the-art 3PN level are presented

in the case of circular orbits appropriate to inspiralling compact binaries. We put the

accent on the description of spinning particles and particularly on the spin–orbit

coupling effect on the binary’s internal energy and gravitational-wave ﬂux.

2 Post-Newtonian Formalism

2.1 Einstein Field Equations

General relativity is based on two independent tenets, the ﬁrst one concerned with

the dynamics of the gravitational ﬁeld, the second one dealing with the coupling of

all the matter ﬁelds with the gravitational ﬁeld. Accordingly the action of general

relativity is made of two terms,

S D

16G

gRC S

matter



; g

˛ˇ



: (3)

The ﬁrst term represents the kinetic Einstein–Hilbert action for gravity and tells that

the gravitational ﬁeld g

˛ˇ

propagates like a pure spin-2 ﬁeld. Here R is the Ricci

scalar and g D det.g

˛ˇ

/ is the determinant of the metric. The second term expresses

the fact that all matter ﬁelds (collectively denoted by  ) are minimally coupled to

Post-Newtonian Theory and the Two-Body Problem 129

the metric g

˛ˇ

, which deﬁnes the physical lengths and times as measured in local

laboratory experiments. The ﬁeld equations are obtained by varying the action with

respect to the metric (such that ıg

˛ˇ

D 0 when jx



j!1) and form a system of

ten differential equations of second order,

˛ˇ

Œg; @g; @

g D

8G

˛ˇ

Œg: (4)

The Einstein tensor G

˛ˇ

 R

˛ˇ



˛ˇ

is generated by the matter stress–energy

tensor T

˛ˇ

 .2=

g/.ıS

matter

=ıg

˛ˇ

/. Four equations give, via the contracted

Bianchi identity, the conservation equation for the matter system as



˛

 0 H) r



˛

D 0; (5)

which must be solved conjointly with the Einstein ﬁeld equations for the gravita-

tional ﬁeld. The matter Eq. 5 reads also





gT





gT



: (6)

Let us introduce an asymptotically Minkowskian coordinate system such that the

gravitational-wave amplitude, deﬁned by h

˛ˇ



gg

˛ˇ

 

˛ˇ

is divergence-

less, that is, satisﬁes the de Donder or harmonic gauge condition



˛

D 0: (7)

With this coordinate choice the Einstein ﬁeld equations can be recast into the

d’Alembertian equation

h

˛ˇ

16G



˛ˇ

: (8)

Here  D 







is the usual (ﬂat-spacetime) d’Alembertian operator. The source

term 

˛ˇ

can rightly be interpreted as the “effective” stress–energy distribution of

the matter and gravitational ﬁelds in harmonic coordinates; note that 

˛ˇ

is not a

tensor and we shall often call it a pseudo-tensor. It is conserved in the sense that





˛

D 0: (9)

This is equivalent to the condition of harmonic coordinates (7) and to the covariant

conservation (5) of the matter tensor. The pseudo-tensor is made of the contribution

of matter ﬁelds described by T

˛ˇ

, and of the gravitational contribution 

˛ˇ

that is a

Here, g

˛ˇ

denotes the contravariant metric, inverse of the covariant metric g

˛ˇ

,and

˛ˇ

repre-

sents an auxiliary Minkowski metric. We assume that our spatial coordinates are Cartesian so that

.

˛ˇ

/ D .

˛ˇ

/ D diag.1; 1; 1; 1/.

130 L. Blanchet

complicated function of the gravitational ﬁeld variable h



and its ﬁrst and second

derivatives. Thus,



˛ˇ

DjgjT

˛ˇ

16G



˛ˇ

Œh; @h; @

h: (10)

The point is that 

˛ˇ

is at least quadratic in the ﬁeld strength h



, so the ﬁeld equa-

tions (8) are naturally amenable to a perturbative nonlinear treatment. The general

expression is



˛ˇ

Dh







˛ˇ

C @



˛



ˇ

˛ˇ











 g

˛





ˇ





 g

ˇ





˛





C g







˛



ˇ



˛

ˇ

 g

˛ˇ



























: (11)

To select a physically sensible solution of the ﬁeld equations in the case of a

bounded matter system, we must impose a boundary condition at inﬁnity, namely,

the famous no-incoming radiation condition, which ensures that the system is truly

isolated from other bodies in the Universe. In principle the no-incoming radiation

condition is to be formulated at past null inﬁnity I



. Here, we can simplify the

formulation by taking advantage of the presence of the Minkowski background 

˛ˇ

to deﬁne the no-incoming radiation condition with respect to Minkowskian past null

inﬁnity say I





. Within approximate methods this is legitimate as we can view the

gravitational ﬁeld as propagating on the ﬂat background 

˛ˇ

; indeed 

˛ˇ

does exist

at any ﬁnite order of approximation.

The no-incoming radiation condition should be such that it suppresses any ho-

mogeneous (regular in R

) solution of the d’Alembertian equation h

hom

D0 in a

neighborhood of I





. We have at our disposal the Kirchhoff formula that expresses

hom

at some ﬁeld point .x

/ in terms of its values and its derivatives on a sphere

centered on x

with radius  jx

 xj and at retarded time t t

 =c,

hom

/ D

d˝

4



@

.h

hom

/ C

.h

hom



.x;t/; (12)

where d˝ is the solid angle spanned by the unit direction N  .x  x

/=.From

this formula we deduce the no-incoming radiation condition as the following limit

at I





,thatis,whenr jxj!C1with t C r=c D const,

lim





.rh

˛ˇ

/ C

.rh

˛ˇ



D 0: (13)

In fact we obtain also the auxiliary condition that r@



˛ˇ

should be bounded near I





.This

comes from the fact that  differs from r and we have  D r  x

n C O.1=r / with n D x=r.