Derivation of Gravitational SelfForce
Abstract
We analyze the issue of “particle motion” in general relativity in a systematic and rigorous way by considering a oneparameter family of metrics corresponding to having a body (or black hole) that is “scaled down” to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a oneparameter family must be a geodesic of the background metric and obtain the leading order perturbative corrections, which include gravitational selfforce, spin force, and geodesic deviation effects. The status the MiSaTaQuWa equation is explained as a candidate “selfconsistent perturbative equation” associated with our rigorous perturbative result.
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 selfforce effects. There is a general consensus that the gravitational selfforce is given by the “MiSaTaQuWa equations”: In the absence of incoming radiation, the motion is given by
(1) 
(2) 
where is the retarded Green’s function for the wave operator . (Note that the limit of integration indicates that only the part of interior to the light cone contributes to .) However, all derivations contain some unsatisfactory features. This is not surprising in view of the fact that, as noted in thisvolume , “point particles” do not make sense in nonlinear theories like general relativity!

Derivations that treat the body as a point particle require unjustified “regularizations”.

Derivations using matched asymptotic expansions matchedexpansions make a number of ad hoc and/or unjustified assumptions.

The axioms of the QuinnWald axiomatic approach quinnwald have not been shown to follow from Einstein’s equation.

All of the above derivations employ at some stage a “phoney” version of the linearized 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.
How should gravitational selfforce be rigorously derived? A precise formula for gravitational selfforce can hold only in a limit where the size, , of the body goes to zero. Since “pointparticles” do not make sense in general relativity—collapse to a black hole would occur before a pointparticle limit could be taken—the mass, , of the body must also go to zero as . In the limit as , the worldtube of the body should approach a curve, , which should be a geodesic of the “background metric”. The selfforce should arise as the lowest order in correction to . In the following, we shall describe an approach that we have recently taken to derive gravitational selfforce in this manner. Details can be found in grallawald .
The discussion above suggests that we consider a oneparameter family of solutions to Einstein’s equation, , with and as . But, what conditions should be imposed on to ensure that it corresponds to a body that is shrinking down to zero size, but is not undergoing wild oscillations, drastically changing its shape, or doing other crazy things as it does so?
As a very simple, explicit example of the kind of oneparameter family we seek, consider the SchwarzschilddeSitter metrics with ,
(3) 
If we take the limit as at fixed coordinates with , it is easily seen that we obtain the deSitter metric—with the deSitter spacetime worldline defined by corresponding to the location of the black hole “before it disappeared”. However, there is also another limit that can be taken. At each time , one can “blow up” the metric by multiplying it by , i.e., define
(4) 
We correspondingly rescale the coordinates by defining , . Then
(5) 
In the limit as (at fixed ) the “deSitter background” becomes irrelevant. The limiting metric is simply the Schwarzschild metric of unit mass. The fact that the limit as 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.
The simultaneous existence of both of the above types of limits charaterizes the type of oneparameter family of spacetimes that we wish to consider. More precisely, we wish to consider a one parameter family of solutions satisfying the following properties:

(i) Existence of the “ordinary limit”: There exist coordinates such that is jointly smooth in , at least for for some constant , where (). For all and for , is a vacuum solution of Einstein’s equation. Furthermore, is smooth in , including at , and, for , the curve defined by is timelike.

(ii) Existence of the “scaled limit”: For each , we define , . Then the metric is jointly smooth in for .
The above two conditions must be supplemented by an additional “uniformity requirement”, which can be explained as follows. From the definitions of and , we can relate coordinate components of the barred metric in barred coordinates to coordinate components of the unbarred metric in corresponding unbarred coordinates,
(6) 
Now introduce new variables and , and view the metric components as functions of , where and are defined in terms of by the usual formula for spherical polar angles. We have
(7) 
Then, by assumption (ii) we see that for , is smooth in for all including . By assumption (i), we see that for all , is smooth in for , including . Furthermore, for , is smooth in , including .
We now impose the additional uniformity requirement on our oneparameter family of spacetimes:

(iii) is jointly smooth in at .
We already know from our previous assumptions that and its derivatives with respect to approach a limit if we let at fixed and then let . The uniformity requirement implies that the same limits are attained whenever and both go to zero in any way such that goes to zero.
It has recently been proven in grallahartewald that an analog of the uniformity requirement holds for electromagnetism in Minkowski spacetime in the following sense: Consider a oneparameter family of chargecurrent sources of the form where is a smooth function of its arguments and defines a timelike worldline. Then the retarded solution, , is a smooth function the variables in a neigborhood of . In the gravitational case, we do not have a simple relationship between the metric and the stressenergy source, and in the nonlinear regime, it would not make sense to formulate the uniformity condition in terms of the behavior of the stressenergy. Consequently, we have formulated this condition in terms of the behavior of the metric itself.
The uniformity requirement implies that the metric components can be approximated near with a finite Taylor series in and ,
(8) 
where remainder terms have been dropped. This gives a far zone expansion. Equivalently, we have
(9) 
Further Taylor expanding this formula with respect to the time variable yields a near zone expansion. Note that since we can express at as a series in as and since at does not depend on , we see that is a stationary, asymptotically flat spacetime.
The curve to which our body shrinks as (see condition (i) above) can now be proven to be a geodesic of the metric as follows: Choose the coordinates so that at they correspond to Fermi normal coordinates about the worldline . In particular, we have on at . It follows from (8) that near (i.e., for small ) the metric must take the form
(10) 
Now, for , the coefficient of , namely
(11) 
must satisfy the vacuum linearized Einstein equation off of the background spacetime . However, since each component of is a locally function, it follows immediately that is well defined as a distribution. It is not difficult to show that, as a distribution, satisfies the linearized Einstein equation with source of the form , where is given by a formula involving the limit as of the angular average of and its first derivative. The linearized Bianchi identity then immediately implies that is of the form with constant, and that is a geodesic for .
Our main interest, however, is not to rederive geodesic motion but to find the leading 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 is an asymptotically flat spacetime, its mass dipole moment can be set to zero (at all ) as a gauge condition on the coordinates . The new coordinates then have the interpretation of being “center of mass coordinates” for the spacetime . In terms of our original coordinates , the transformation to center of mass coordinates at all corresponds to a coordinate transformation of the form
(12) 
To first order in , the world line defined by should correspond to the “position” of the body. The firstorder displacement from in the original coordinates is then given simply by
(13) 
The quantity is most naturally interpreted as a “deviation vector field” defined on . Our goal is to derive relations (if any) that hold for that are independent of the choice of oneparameter family satisfying our assumptions.
We now choose the coordinates—previously chosen to agree with Fermi normal coordinates on at —to correspond to the Lorenz/harmonic gauge to first order in . To order , the leading order in terms in are,
(14) 
Here and are expressions involving the curvature of and we have introduced the “unknown” tensors and . The quantities and turn out to be the mass dipole and spin of the “nearzone” background spacetime . For simplicity, we have assumed no “incoming radiation”. Hadamard expansion techniques and 2nd order perturbation theory were used to derive this expression.
Using the coordinate shift to cancel the mass dipole term, the above expression translates into the following expression for the scaled metric
(15) 
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 th order in ). From this equation, we find that , i.e., to lowest order, spin is parallelly 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 , electric parity, evenundertimereversal part of this equation at and , we obtain (after considerable algebra!)
(16) 
In other words, in the Lorenz gauge, the deviation vector field, , on that describes the first order perturbation to the motion satisfies
(17) 
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 selfforce that we had sought to obtain, i.e., the corrections to the motion caused by the body’s selffield. 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 will be different grallawald .
Although we have now obtained the perturbative correction to geodesic motion due to spin and selfforce effects, at late times the small corrections due to selfforce effects should accumulate (e.g., during an inspiral), and eventually the orbit should deviate significantly from the original, unperturbed geodesic . When this happens, it is clear our perturbative description in terms of a deviation vector defined on will not be accurate. Clearly, going to any (finite) higher order in perturbation theory 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 “selfconsistent 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 of the original perturbation equation over a time interval where the perturbation remains small. In some sense, such a selfconsistent perturbative equation 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 selfconsistent perturbative equation associated with our perturbative result.
In summary, we have analyzed the motion of a small body or black hole in general relativity, assuming only the existence of a oneparameter family of solutions satisfying assumptions (i), (ii), and (iii) above. We showed that at lowest (“zeroth”) order, the motion of a “small” body is described by a geodesic, , of the “background” spaceetime. We then derived is a formula for the first order deviation of the “center of mass” worldline of the body from . The MiSaTaQuWa equations then arise as (candidate) “selfconsistent 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 selfforce in electromagnetism grallahartewald , and have argued that the reduced order form of the AbrahamLorentzDirac equation provides an appropriate selfconsistent perturbative equation associated with our first order perturbative result (whereas the original AbrahamLorentzDirac equation is excluded). It should be possible to use this formalism to take higher order corrections to the motion into account in a systematic way in both the gravitational and electromagnetic cases.
Acknowledgments
This research was supported in part by NSF grants PHY0456619 and PHY0854807 to the University of Chicago.
References
 (1) R.M. Wald, “Introduction to Gravitational SelfForce” (contribution to this volume).

(2)
Y. Mino, M. Sasaki, and T. Tanaka, Phys. Rev. D, 55,
34573476, (1997);
E. Poisson, Liv. Rev. Rel. 7 6 (2004).  (3) T.C. Quinn and R.M. Wald, Phys. Rev. D, 56, 33813394, (1997).
 (4) S. Gralla and R. Wald, Class. Quantum Grav. 25 205009 (2008).
 (5) S. Gralla, A. Harte, and R. Wald, arXiv:0905.2391 (2009).