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Introduction to Gravitational Self-Force 261
7 How Should Gravitational Self-Force Be Derived?
Although there is a general consensus that Eqs. 11 and 12 (or the Detweiler–Whiting
version of Eq. 12) should provide a good description of the self-force corrections to
the motion of a sufficiently small body, it is important that these equations be put on
a firmer foundation both to clarify their range of validity and to potentially enable
the systematic calculation of higher order corrections. It is clear that in order to
obtain a precise and rigorous derivation of gravitational self-force, it will be nec-
essary to take some kind of “point particle limit,” wherein the size, R, of the body
goes to 0. However, to avoid difficulties associated with the nonexistence of point
particles in general relativity, it is essential that one lets M go to 0 as well. If M goes
to 0 more slowly than R, the body should collapse to a black hole before the limit
R ! 0 is achieved. On the other hand, one could consider limits where M goes to
0 more rapidly than R, but finite size effects would then dominate over self-force
effects as R ! 0. This suggests that we consider a one-parameter family of solu-
tions to Einstein’s equation, g
ab
./, for which the body scales to 0 size and mass in
an asymptotically self-similar way as ! 0, so that the ratio R=M approaches a
constant in the limit.
Recently, Gralla and I [7] have considered such one-parameter families of bodies
(or black holes). In the limit as ! 0 – where the body shrinks down to a worldline
and “disappears” – we proved that must be a geodesic. Self-force and finite
size effects then arise as perturbative corrections to . To first order in ,these
corrections are described by a deviation vector Z
i
along .In[7], Gralla and I
proved that, in the Lorenz gauge, this deviation vector satisfies
d
2
Z
i
dt
2
D
1
2M
S
kl
R
kl0
i
R
0j 0
i
Z
j
h
tail
i
0;0
1
2
h
tail
00
;i
: (15)
The first term in this equation corresponds to the usual “spin force” [12], that is, the
leading order finite size correction to the motion. The second term is the usual right
side of the geodesic deviation equation. (This term must appear since the corrections
to motion must allow for the possibility of a perturbation to a nearby geodesic.) The
last term corresponds to the self-force term appearing on right side of Eq. 12.It
should be emphasized that Eq. 15 arises as the perturbative correction to geodesic
motion for any one-parameterfamily satisfying our assumptions, and holds for black
holes as well as ordinary bodies.
Although the self-force term in Eq. 15 corresponds to the right side of Eq. 12,
these equations have different meanings. Equation 15 is a first order perturbative
correction to geodesic motion, and no Lorenz gauge relaxation is involved in this
equation since h
ab
is sourced by a geodesic . By contrast, Eq. 12 is supposed to
hold even when the cumulative deviations from geodesic motion are large, and
Lorenz gauge relaxation is thereby essential. Given that Eq. 15 holds rigorously
as a perturbative result, what is the status of the MiSaTaQuWa equations (11),
(12)? In [7], we argued that the MiSaTaQuWa equations arise as “self-consistent
262 R.M. Wald
perturbative equations” associated with the perturbative result (15). If the deviations
from geodesic motion are locally small – even though cumulative effects may yield
large deviations from any individual geodesic over long periods of time – then the
MiSaTaQuWa equations should provide an accurate description of motion.
Acknowledgements This research was supported in part by NSF grant PHY04-56619 to the
University of Chicago.
References
1. S. Detweiler, B.F. Whiting, Phys. Rev. D 67, 024025 (2003)
2. B.S. DeWitt, R.W. Brehme, Ann. Phys. (N.Y.) 9, 220 (1960)
3. P.A.M. Dirac, Proc. R. Soc. Lond. A 167, 148 (1938)
4. P.R. Garabedian, Partial Differential Equations (Wiley, New York, 1964)
5. D. Garfinkle, Class. Q. Grav. 16, 4101 (1999)
6. R. Geroch, J. Traschen, Phys. Rev. D 36, 1017 (1987)
7. S.E. Gralla, R.M. Wald, Class. Q. Grav. 25, 205009 (2008)
8. L. Hormander, The Analysis of Linear Partial Differential Operators, I (Springer, Berlin, 1985)
9. L. Hormander, The Analysis of Linear Partial Differential Operators, IV (Springer, Berlin,
1985)
10. W. Israel, Nuovo Cimento B 44, 1 (1966); Lettere alla Redazione, ibidem 48, 463 (1967)
11. Y. Mino, M. Sasaki, T. Tanaka, Phys. Rev. D 55, 3457 (1997)
12. A. Papapetrou, Proc. R. Soc. Lond. A 209, 248 (1951)
13. E. Poisson, Living Rev. Rel. 7, URL (cited on 23 December 2008): http://www.livingreviews.
org/lrr-2004-6
14. T.C. Quinn, R.M. Wald, Phys. Rev. D 56, 3381 (1997)
Derivation of Gravitational Self-Force
Samuel E. Gralla and Robert M. Wald
Abstract We analyze the issue of “particle motion” in general relativity in a
systematic and rigorous way by considering a one-parameter family of metrics cor-
responding to having a body (or black hole) that is “scaled down” to 0 size and
mass in an appropriate manner. We prove that the limiting worldline of such a
one-parameter family must be a geodesic of the background metric and obtain the
leading order perturbative corrections, which include gravitational self-force, spin
force, and geodesic deviation effects. The status of the MiSaTaQuWa equation is
explained as a candidate “self-consistent perturbative equation” associated with our
rigorous perturbative result.
1 Difficulties with Usual Derivations
It is of considerable interest to determine the motion of a body in general relativity
in the limit of small size, taking into account the deviations from geodesic motion
arising from gravitational self-force effects. There is a general consensus that the
gravitational self-force is given by the “MiSaTaQuWa equations”: In the absence of
incoming radiation, the motion of a particle of mass M is given by
u
r
u
D
1
2
.g
C u
u
/.2r
h
tail
r
h
tail
/
ˇ
ˇ
z./
u
u
; (1)
where g
is the background metric, u
is the four-velocity of the worldline z
,and
h
tail
.x/ D M
Z
1
G
C
0
0
1
2
g
G
C
0
0
x; z.
0
/
u
0
u
0
d
0
; (2)
S.E. Gralla (
)andR.M.Wald(
)
Enrico Fermi Institute and Department of Physics, University of Chicago,
5640 S. Ellis Avenue, Chicago, IL 60637, USA
e-mail: rmwa@uchicago.edu; sgralla@uchicago.edu
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
9,
c
Springer Science+Business Media B.V. 2011
263
264 S.E. Gralla and R.M. Wald
in which G
C
is the retarded Green’s function for the wave operator r
˛
r
˛
Q
h
2R
˛
ˇ
Q
h
˛ˇ
. (Note that the
limit of integration indicates that only the part of G
C
interior to the light cone contributes to h
tail
.) However, all derivations contain some
unsatisfactory features. This is not surprising in view of the fact that, as noted in [6],
“point particles” do not make sense in nonlinear theories like general relativity!
Derivations that treat the body as a point particle require unjustified “regulariza-
tions.”
Derivations using matched asymptotic expansions [3, 4] make a number of ad
hoc and/or unjustified assumptions.
The axioms of the Quinn–Wald axiomatic approach [5] have not been shown to
follow from Einstein’s equation.
All of the above derivations employ at some stage a “phoney” version of the lin-
earized Einstein equation with a point-particle source, wherein the Lorenz gauge
version of the linearized Einstein equation is written down, but the Lorenz gauge
condition is not imposed.
2 Rigorous Derivation Requirements
How should gravitational self-force be rigorously derived? A precise formula for
gravitational self-force can hold only in a limit where the size, R, of the body goes
to 0. Since “point-particles” do not make sense in general relativity – collapse to a
black hole would occur before a point-particle limit could be taken – the mass, M ,of
the body must also go to 0 as R ! 0. In the limit as R;M ! 0, the worldtube of the
body should approach a curve, , which should be a geodesic of the “background
metric.” The self-force should arise as the lowest order in M correction to .In
the following, we shall describe an approach that we have recently taken to derive
gravitational self-force in this manner. Details can be found in [1].
The discussion above suggests that we consider a one-parameter family of solu-
tions to Einstein’s equation, .g
./; T
.//, with R./ ! 0 and M./ ! 0 as
! 0. But, what conditions should be imposed on .g
./; T
.// to ensure that
it corresponds to a body that is shrinking down to 0 size, without undergoing wild
oscillations, drastically changing its shape, or doing other crazy things as it does so?
3 Limits of Spacetimes
As a very simple, explicit example of the kind of one-parameter family we seek,
consider the Schwarzschild–de Sitter metrics with M D ,
ds
2
./ D
1
2
r
Cr
2
dt
2
C
1
2
r
Cr
2
1
dr
2
C r
2
d˝
2
: (3)
Derivation of Gravitational Self-Force 265
If we take the limit as ! 0 at fixed coordinates .t;r;;/with r>0, it is easily
seen that we obtain the de Sitter metric – with the de Sitter spacetime worldline
defined by r D 0 corresponding to the location of the black hole “before it dis-
appeared.” However, there is also another limit that can be taken. At each time t
0
,
introduce rescaled coordinates Nr D r=,
N
t D .t t
0
/=. We then correspondingly
“blow up” the metric g
./ by multiplying it by
2
, that is, we define
Ng
./
2
g
./: (4)
We then have
d Ns
2
./ D.1 2=Nr
2
C Nr
2
/d
N
t
2
C .1 2=Nr
2
C Nr
2
/
1
d Nr
2
CNr
2
d˝
2
: (5)
In the limit as ! 0 (at fixed .
N
t; Nr;;/) the “de Sitter background” becomes
irrelevant. The limiting metric is simply the Schwarzschild metric of unit mass. The
fact that the limit as ! 0 exists can be attributed to the fact that the Schwarzschild
black hole is shrinking to zero in a manner where, in essence, nothing changes
except the overall scale.
4 Our basic Assumptions
The simultaneous existence of both of the above types of limits characterizes the
type of one-parameter family of spacetimes g
./ that we wish to consider. More
precisely, we wish to consider a one parameter family of solutions g
./ satisfying
the following properties:
(i) Existence of the “ordinary limit”: There exist coordinates x
˛
such that the
metric g
.; x
˛
/ is jointly smooth in .; x
˛
/,atleastforr>
N
R for some
constant
N
R,wherer
p
P
.x
i
/
2
(i D 1; 2; 3). For all and for r>
N
R, g
./
is a vacuum solution of Einstein’s equation. Furthermore, g
. D 0; x
˛
/ is
smooth in x
˛
, including at r D 0, and, for D 0,thecurve defined by r D 0
is timelike.
(ii) Existence of the “scaled limit”: For each t
0
,wedefine
N
t .t t
0
/=, Nx
i
x
i
=. Then the metric Ng
NN
.It
0
INx
˛
/
2
g
NN
.It
0
INx
˛
/ is jointly smooth in
.; t
0
INx
˛
/ for Nr r= >
N
R.
4.1 Additional Uniformity Requirement
The above two conditions must be supplemented by an additional “uniformity
requirement,” which can be explained as follows. From the definitions of Ng
NN
and Nx
, we can relate coordinate components of the barred metric in barred
266 S.E. Gralla and R.M. Wald
coordinates to coordinate components of the unbarred metric in corresponding
unbarred coordinates,
Ng
NN
.It
0
I
N
t; Nx
i
/ D g
.It
0
C
N
t;Nx
i
/: (6)
Now introduce new variables ˛ r and ˇ =r D 1=Nr, and view the metric
components g
./ as functions of .˛;ˇ;t;;/,where and are defined in
terms of x
i
by the usual formula for spherical polar angles. We have
Ng
NN
.˛ˇ; t
0
I
N
t;1=ˇ;;/ D g
.˛ˇ; t D t
0
C
N
tI˛; ; / : (7)
Then, by assumption (ii) we see that for 0<ˇ<1=
N
R, g
is smooth in .˛; ˇ/ for
all ˛ including ˛ D 0. By assumption (i), we see that for all ˛>0, g
is smooth
in .˛; ˇ/ for ˇ<1=
N
R, including ˇ D 0. Furthermore, for ˇ D 0, g
is smooth in
˛, including ˛ D 0.
We now impose the additional uniformity requirement on our one-parameter
family of spacetimes:
(iii) g
is jointly smooth in .˛; ˇ/ at .0; 0/.
We already know from our previous assumptions that g
.It
0
;r;;/ and its
derivatives with respect to x
˛
approach a limit if we let ! 0 at fixed r and then
let r ! 0. The uniformity requirement implies that the same limits are attained
whenever and r both go to 0 in any way such that =r goes to 0.
It has recently been proven in [2] that an analog of the uniformity requirement
holds for electromagnetism in Minkowski spacetime in the following sense: Con-
sider a one-parameter family of charge-current sources of the form J
.;t;x
i
/ D
Q
J
.;t;x
i
=/ where
Q
J
is a smooth function of its arguments and x
i
D 0 defines a
timelike worldline. Then the retarded solution, F
.; x
/, is a smooth function of
the variables .˛;ˇ;t;;/in a neighborhood of .˛; ˇ/ D .0; 0/. In the gravitational
case, we do not have a simple relationship between the metric and the stress–energy
source, and in the nonlinear regime, it would not make sense to formulate the uni-
formity condition in terms of the behavior of the stress–energy. Consequently, we
have formulated this condition in terms of the behavior of the metric itself.
5GeodesicMotion
The uniformity requirementimplies that the metric components can be approximated
near .˛; ˇ/ D .0; 0/ with a finite Taylor series in ˛ and ˇ,
g
.It;r;;/ D
N
X
nD0
M
X
mD0
r
n
r
m
.a
/
nm
.t;;/; (8)
Derivation of Gravitational Self-Force 267
where remainder terms have been dropped. This gives a far zone expansion.
Equivalently, we have
Ng
NN
.It
0
I
N
t; Nr;;/ D
N
X
nD0
M
X
mD0
.Nr/
n
1
Nr
m
.a
/
nm
.t
0
C
N
t;;/: (9)
Further Taylor expanding this formula with respect to the time variable yields a near
zone expansion. Note that since we can express Ng
NN
at D 0 as a series in 1= Nr as
Nr !1and since Ng
NN
at D 0 does not depend on
N
t, we see that Ng
N N
. D 0/ is a
stationary, asymptotically flat spacetime.
The curve to which our body shrinks as ! 0 (see condition (i) above) can
now be proven to be a geodesic of the metric g
. D 0/ as follows: Choose the
coordinates x
˛
so that at D 0 they correspond to Fermi normal coordinates about
the worldline . In particular, we have g
D
on at D 0. It follows from
(8) that near (i.e., for small r) the metric g
must take the form
g
D
C O.r/ C
C
.t;;/
r
C O.1/
C O.
2
/: (10)
Now, for r>0, the coefficient of , namely,
h
D
C
r
C O.1/; (11)
must satisfy the vacuum linearized Einstein equation off the background spacetime
g
. D 0/. However, since each component of h
is a locally L
1
function, it
follows immediately that h
is well defined as a distribution. It is not difficult to
show that, as a distribution, h
satisfies the linearized Einstein equation with source
of the form N
.t/ı
.3/
.x
i
/,whereN
is given by a formula involving the limit as
r ! 0 of the angular average of C
and its first derivative. The linearized Bianchi
identity then immediately implies that N
is of the form Mu
u
with M constant,
and that is a geodesic for M ¤ 0.
6 Corrections to Motion
Our main interest, however, is not to rederive geodesic motion but to find the lead-
ing order corrections to geodesic motion that arise from finite mass and finite size
effects. To define these corrections, we need to have a notion of the “location” of
the body to first order in . This can be defined as follows: Since Ng
N N
. D 0/ is an
asymptotically flat spacetime, its mass dipole moment can be set to 0 (at all t
0
)asa
gauge condition on the coordinates Nx
i
. These coordinates then have the interpreta-
tion of being “center of mass coordinates” for the spacetime Ng
NN
. D 0/.Interms
of our original coordinates x
˛
, the transformation to center of mass coordinates at
all t
0
corresponds to a coordinate transformation x
˛
!Ox
˛
of the form
Ox
˛
.t/ D x
˛
A
˛
.t; x
i
/ C O.
2
/: (12)
268 S.E. Gralla and R.M. Wald
To first order in , the worldline defined by Ox
i
D 0 should correspond to the
“position” of the body. The first-order displacement from in the original coor-
dinates is then given simply by
Z
i
.t/ A
i
.t; x
j
D 0/: (13)
The quantity Z
i
is most naturally interpreted as a “deviation vector field” defined
on . Our goal is to derive relations (if any) that hold for Z
i
that are independent of
the choice of one-parameter family satisfying our assumptions.
6.1 Calculation of the Perturbed Motion
We now choose the x
˛
coordinates – previously chosen to agree with Fermi normal
coordinates on at D 0 – to correspond to the Lorenz/harmonic gauge to first
order in .Toorder
2
, the leading order in r terms in g
˛ˇ
are
g
˛ˇ
.It;x
i
/ D
˛ˇ
C B
˛iˇj
.t/x
i
x
j
C O.r
3
/
C
2M
r
ı
˛ˇ
C h
tail
˛ˇ
.t; 0/ C h
tail
˛ˇ i
.t; 0/x
i
C M R
˛ˇ
.t/ C O.r
2
/
C
2
M
2
r
2
2t
˛
t
ˇ
C 3n
˛
n
ˇ
C
2
r
2
P
i
.t/n
i
ı
˛ˇ
C
1
r
2
t
.˛
S
ˇ/j
.t/n
j
C
1
r
K
˛ˇ
.t;;/C H
˛ˇ
.t;;/C O.r/
C O.
3
/: (14)
Here B and R are expressions involving the curvature of g
. D 0/ and we
have introduced the “unknown” tensors K and H . The quantities P
i
and S
˛ˇ
turn out to be the mass dipole and spin of the “near-zone” background spacetime
Ng
NN
. D 0/. For simplicity, we have assumed no “incoming radiation.” Hadamard
expansion techniques and the second-order perturbation theory were used to derive
this expression.
Using the coordinate shift x
! x
A
to cancel the mass dipole term, the
above expression translates into the following expression for the scaled metric
Ng
N˛
N
ˇ
.
O
t
0
/ D
˛ˇ
C
2M
Nr
ı
˛ˇ
C
M
2
Nr
2
2t
˛
t
ˇ
C 3n
˛
n
ˇ
C
1
Nr
2
t
.˛
S
ˇ/j
n
j
C O
1
Nr
3
C
h
tail
˛ˇ
C 2A
.˛;ˇ/
C
1
r
K
˛ˇ
C
N
t
Nr
2
t
.˛
P
S
ˇ/j
n
j
C O
1
Nr
2
C
N
tO
1
Nr
3
C
2
B
˛iˇj
Nx
i
Nx
j
Ch
tail
˛ˇ;
Nx
CM R
˛ˇ
. Nx
i
/ C 2B
˛iˇj
A
i
Nx
j
C 2A
.˛;ˇ/
Nx
C H
˛ˇ
C
N
t
Nr
P
K
˛ˇ
C
N
t
2
Nr
2
t
.˛
R
S
ˇ/j
n
j
C O
1
Nr
C
N
tO
1
Nr
2
C
N
t
2
O
1
Nr
3
C O.
3
/;
(15)
Derivation of Gravitational Self-Force 269
where quantities on the right side are evaluated at t D t
0
, and the overdots denote
derivatives with respect to t.
The terms that are first order in in this equation satisfy the linearized vacuum
Einstein equation about the background “near zone” metric (i.e., the terms that are
0th order in ). From this equation, we find that dS
ij
=dt D 0, that is, to lowest
order, spin is parallely propagated along .
The terms that are second order in in this equation satisfy the linearized
Einstein equation about the background “near zone” metric with source given by
the second-order Einstein tensor of the first order terms. Extracting the ` D 1, elec-
tric parity, even-under-time-reversal part of this equation at O.1=Nr
2
/ and O.
N
t=Nr
3
/,
we obtain (after considerable algebra!)
Z
i;00
D
1
2M
S
kl
R
kl0i
R
0j 0 i
Z
j
h
tail
i0;0
1
2
h
tail
00;i
: (16)
In other words, in the Lorenz gauge, the deviation vector field, Z
a
,on that de-
scribes the first order perturbation to the motion satisfies
u
c
r
c
.u
b
r
b
Z
a
/ D
1
2M
R
bcd
a
S
bc
u
d
R
bcd
a
u
b
u
d
Z
c
.g
ab
Cu
a
u
b
/
r
d
h
tail
bc
1
2
r
b
h
tail
cd
u
c
u
d
: (17)
7 Interpretation of Results
Equation 17 gives the desired leading order corrections to motion along the geodesic
. The first term on the right side of this equation is the Papapetrou “spin force,”
which is the leading order “finite size” correction. The second term is just the usual
right hand side of the geodesic deviation equation; it is not a correction to geodesic
motion but rather allows for the possibility that the perturbation may make the body
move along a different geodesic. Finally, the last term describes the gravitational
self-force that we had sought to obtain, that is, the corrections to the motion caused
by the body’s self-field. Equation 17 gives the correct description of motion when
the metric perturbation is in the Lorenz gauge. When the metric perturbation is
expressed in a different gauge, the force may be different [1].
8 Self-Consistent Equations
Although we have now obtained the perturbative correction to geodesic motion due
to spin and self-force effects, at late times the small corrections due to self-force
effects should accumulate (e.g., during an inspiral), and eventually the orbit should
deviate significantly from the original, unperturbed geodesic . When this happens,
270 S.E. Gralla and R.M. Wald
it is clear that our perturbative description in terms of a deviation vector defined on
will not be accurate. Clearly, going to any (finite) higher order in perturbation the-
ory will not help (much). However, if the mass and size of the body are sufficiently
small, we expect that its motion is well described locally as a small perturbation of
some geodesic. Therefore, one should obtain a good description of the motion by
making up (!) a “self-consistent perturbative equation” that satisfies the following
criteria: (1) It has a well-posed initial value formulation. (2) It has the same “number
of degrees of freedom” as the original system. (3) Its solutions correspond closely
to the solutions of the original perturbation equation over a time interval where the
perturbation remains small. In some sense, such a self-consistent perturbative equa-
tion would take into account the important (“secular”) higher order perturbative
effects (to all orders), but ignore other higher order corrections. Such equations are
commonly considered in physics. The MiSaTaQuWa equations appear to be a good
candidate for a self-consistent perturbative equation associated with our perturbative
result.
9 Summary
In summary, we have analyzed the motion of a small body or black hole in general
relativity, assuming only the existence of a one-parameter family of solutions satis-
fying assumptions (i), (ii), and (iii) above. We showed that at lowest (“0th”) order,
the motion of a “small” body is described by a geodesic, , of the “background”
spacetime. We then derived a formula for the first order deviation of the “center of
mass” worldline of the body from . The MiSaTaQuWa equations then arise as (can-
didate) “self-consistent perturbative equations” based on our first order perturbative
result. Note that it is only at this stage that “phoney” linearized Einstein equations
come into play.
We have recently applied this basic approach to the derivation of self-force
in electromagnetism [2], and have argued that the reduced order form of the
Abraham–Lorentz–Dirac equation provides an appropriate self-consistent perturba-
tive equation associated with our first order perturbative result (whereas the original
Abraham–Lorentz–Dirac equation is excluded). It should be possible to use this
formalism to take higher order corrections to the motion into account in a system-
atic way in both the gravitational and electromagnetic cases.
References
1. S. Gralla, R. Wald, Class. Q. Grav. 25, 205009 (2008)
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