Concluding remarks

For each of the three cases (scalar, electromagnetic, and gravitational) I have presented two different derivations of the equations of motion. The first derivation is based on a spatial averaging of the retarded field, and the second is based on a decomposition of the retarded field into singular and radiative fields. In the gravitational case, a third derivation, based on matched asymptotic expansions, was also presented. These derivations will be reviewed below, but I want first to explain why I have omitted to present a fourth derivation, based on energy-momentum conservation, in spite of the fact that historically, it is one of the most important.

Conservation of energy-momentum was used by Dirac [25

] to derive the equations of motion of a point electric charge in flat spacetime, and the same method was adopted by DeWitt and Brehme [24

] in their generalization of Dirac’s work to curved spacetimes. This method was also one of the starting points of Mino, Sasaki, and Tanaka [39

] in their calculation of the gravitational self-force. I have not discussed this method for two reasons. First, it is technically more difficult to implement than the methods presented in this review (considerably longer computations are involved). Second, it is difficult to endow this method with an adequate level of rigour, to the point that it is perhaps less convincing than the methods presented in this review. While the level of rigour achieved in flat spacetime is now quite satisfactory [56

], I do not believe the same can be said of the generalization to curved spacetimes. (But it should be possible to improve on this matter.)

The method is based on the conservation equation $Tαβ;β = 0$ , where the stress-energy tensor $Tαβ$ includes a contribution from the particle and a contribution from the field; the particle’s contribution is a Dirac functional on the world line, and the field’s contribution diverges as $4 1∕r$ near the world line. (I am using retarded coordinates in this discussion.) While in flat spacetime the differential statement of energy-momentum conservation can immediately be turned into an integral statement, the same is not true in a curved spacetime (unless the spacetime possesses at least one Killing vector). To proceed it is necessary to rewrite the conservation equation as

where $μ gα (z,x)$ is a parallel propagator from $x$ to an arbitrary point $z$ on the world line. Integrating this equation over the interior of a world-tube segment that consists of a “wall” of constant $r$ and two “caps” of constant $u$ , we obtain

where $dΣ β$ is a three-dimensional surface element and $dV$ an invariant, four-dimensional volume element.

There is no obstacle in evaluating the wall integral, for which $αβ T$ reduces to the field’s stress-energy tensor; for a wall of radius $r$ the integral scales as $1∕r2$ . The integrations over the caps, however, are problematic: While the particle’s contribution to the stress-energy tensor is integrable, the integration over the field’s contribution goes as $∫r(r′)−2dr′ 0$ and diverges. To properly regularize this integral requires great care, and the removal of all singular terms can be achieved by mass renormalization [24 ]. This issue arises also in flat spacetime [25 ], and while it is plausible that the rigourous distributional methods presented in [56 ] could be generalized to curved spacetimes, this remains to be done. More troublesome, however, is the interior integral, which does not appear in flat spacetime. Because $gμ α;β$ scales as $r$ , this integral goes as $∫ r ′−1 ′ 0 (r) dr$ and it also diverges, albeit less strongly than the caps integration. While simply discarding this integral produces the correct equations of motion, it would be desirable to go through a careful regularization of the interior integration, and provide a convincing reason to discard it altogether. To the best of my knowledge, this has not been done.

5.5.2 Averaging method

To identify the strengths and weaknesses of the averaging method it is convenient to adopt the Detweiler–Whiting decomposition of the retarded field into singular and radiative pieces. For concreteness I shall focus my attention on the electromagnetic case, and write

Recall that this decomposition is unambiguous, and that the retarded and singular fields share the same singularity structure near the world line. Recall also that the retarded and singular fields satisfy the same field equations (with a distributional current density on the right-hand side), but that the radiative field is sourcefree.

To formulate equations of motion for the point charge we temporarily model it as a spherical hollow shell, and we obtain the net force acting on this object by averaging $Fαβ$ over the shell’s surface. (The averaging is performed in the shell’s rest frame, and the shell is spherical in the sense that its proper distance from the world line is the same in all directions.) The averaged field is next evaluated on the world line, in the limit of a zero-radius shell. Because the radiative field is smooth on the world line, this yields

The equations of motion are then postulated to be $ν ma μ = e⟨Fμν⟩ u$ , where $m$ is the particle’s bare mass. With the preceding results we arrive at $mobsa μ = eF Ru ν μν$ , where $mobs ≡ m + δm$ is the particle’s observed (renormalized) inertial mass.

The averaging method is sound, but it is not immune to criticism. A first source of criticism concerns the specifics of the averaging procedure, in particular, the choice of a spherical surface over any other conceivable shape. Another source is a slight inconsistency of the method that gives rise to the famous “4/3 problem” [52 ]: The mass shift $δm$ is related to the shell’s electrostatic energy $E = e2∕(2s)$ by $4 δm = 3E$ instead of the expected $δm = E$ . This problem is likely due [45 ] to the fact that the field that is averaged over the surface of the shell is sourced by a point particle and not by the shell itself. It is plausible that a more careful treatment of the near-source field will eliminate both sources of criticism: We can expect that the field produced by an extended spherical object will give rise to a mass shift that equals the object’s electrostatic energy, and the object’s spherical shape would then fully justify a spherical averaging. (Considering other shapes might also be possible, but one would prefer to keep the object’s structure simple and avoid introducing additional multipole moments.) Further work is required to clean up these details.

The averaging method is at the core of the approach followed by Quinn and Wald [49 ], who also average the retarded field over a spherical surface surrounding the particle. Their approach, however, also incorporates a “comparison axiom” that allows them to avoid renormalizing the mass.

5.5.3 Detweiler–Whiting axiom

The Detweiler–Whiting decomposition of the retarded field becomes most powerful when it is combined with the Detweiler–Whiting axiom, which asserts that

the singular field exerts no force on the particle (it merely contributes to the particle’s inertia); the entire self-force arises from the action of the radiative field.

This axiom, which is motivated by the symmetric nature of the singular field, and also its causal structure, gives rise to the equations of motion $ma μ = eF Rμνuν$ , in agreement with the averaging method (but with an implicit, instead of explicit, mass shift). In this picture, the particle simply interacts with a free radiative field (whose origin can be traced to the particle’s past), and the procedure of mass renormalization is sidestepped. In the scalar and electromagnetic cases, the picture of a particle interacting with a radiative field removes any tension between the nongeodesic motion of the charge and the principle of equivalence. In the gravitational case the Detweiler–Whiting axiom produces the statement that the point mass $m$ moves on a geodesic in a spacetime whose metric $R gαβ + hαβ$ is nonsingular and a solution to the vacuum field equations. This is a conceptually powerful, and elegant, formulation of the MiSaTaQuWa equations of motion.

5.5.4 Matched asymptotic expansions

It is well known that in general relativity the motion of gravitating bodies is determined, along with the spacetime metric, by the Einstein field equations; the equations of motion are not separately imposed. This observation provides a means of deriving the MiSaTaQuWa equations without having to rely on the fiction of a point mass. In the method of matched asymptotic expansions, the small body is taken to be a nonrotating black hole, and its metric perturbed by the tidal gravitational field of the external universe is matched to the metric of the external universe perturbed by the black hole. The equations of motion are then recovered by demanding that the metric be a valid solution to the vacuum field equations. This method, which was the second starting point of Mino, Sasaki, and Tanaka [39 ], gives what is by far the most compelling derivation of the MiSaTaQuWa equations. Indeed, the method is entirely free of conceptual and technical pitfalls – there are no singularities (except deep inside the black hole) and only retarded fields are employed.

The introduction of a point mass in a nonlinear theory of gravitation would appear at first sight to be severely misguided. The lesson learned here is that one can in fact get away with it. The derivation of the MiSaTaQuWa equations of motion based on the method of matched asymptotic expansions does indeed show that results obtained on the basis of a point-particle description can be reliable, in spite of all their questionable aspects. This is a remarkable observation, and one that carries a lot of convenience: It is much easier to implement the point-mass description than to perform the matching of two metrics in two coordinate systems.

5.5.5 Evaluation of the gravitational self-force

The concrete evaluation of the scalar, electromagnetic, and gravitational self-forces is made challenging by the need to first obtain the relevant retarded Green’s function. Successes achieved in the past were reviewed in Section 1.10, and here I want to describe the challenges that lie ahead. I will focus on the specific task of computing the gravitational self-force acting on a point mass that moves in a background Kerr spacetime. This case is especially important because the motion of a small compact object around a massive (galactic) black hole is a promising source of low-frequency gravitational waves for the Laser Interferometer Space Antenna (LISA) [31 ]; to calculate these waves requires an accurate description of the motion, beyond the test-mass approximation which ignores the object’s radiation reaction.

The gravitational self-acceleration is given by the MiSaTaQuWa expression, which I write in the form

where $hR αβ$ is the radiative part of the metric perturbation. Recall that this equation is equivalent to the statement that the small body moves on a geodesic of a spacetime with metric $g + hR αβ αβ$ . Here $g αβ$ is the Kerr metric, and we wish to calculate $μ R a [h ]$ for a body moving in the Kerr spacetime. This calculation is challenging and it involves a large number of steps.

The first sequence of steps is concerned with the computation of the (retarded) metric perturbation $h αβ$ produced by a point particle moving on a specified geodesic of the Kerr spacetime. A method for doing this was elaborated by Lousto and Whiting [34 ] and Ori [44 ], building on the pioneering work of Teukolsky [57 ], Chrzanowski [18 ], and Wald [61 ]. The procedure consists of

It is well known that the Teukolsky equation separates when $ψ0$ or $ψ4$ is expressed as a multipole expansion, summing over modes with (spheroidal-harmonic) indices $l$ and $m$ . In fact, the procedure outlined above relies heavily on this mode decomposition, and the metric perturbation returned at the end of the procedure is also expressed as a sum over modes $hl αβ$ . (For each $l$ , $m$ ranges from $− l$ to $l$ , and summation of $m$ over this range is henceforth understood.) From these, mode contributions to the self-acceleration can be computed: $μ a [hl]$ is obtained from our preceding expression for the self-acceleration by substituting $hlαβ$ in place of $hRαβ$ . These mode contributions do not diverge on the world line, but $aμ [hl]$ is discontinuous at the radial position of the particle. The sum over modes, on the other hand, does not converge, because the “bare” acceleration (constructed from the retarded field $hαβ$ ) is formally infinite.

The next sequence of steps is concerned with the regularization of each $aμ[hl]$ by removing the contribution from $hS αβ$ [6 , 7 , 9 , 11 , 38 , 21 ]. The singular field can be constructed locally in a neighbourhood of the particle, and then decomposed into modes of multipole order $l$ . This gives rise to modes $aμ[hSl ]$ for the singular part of the self-acceleration; these are also finite and discontinuous, and their sum over $l$ also diverges. But the true modes $aμ[hRl ] = aμ[hl] − a μ[hSl ]$ of the self-acceleration are continuous at the radial position of the particle, and their sum does converge to the particle’s acceleration. (It might be noted that obtaining a mode decomposition of the singular field involves providing an extension of $S hαβ$ on a sphere of constant radial coordinate, and then integrating over the angular coordinates. The arbitrariness of the extension introduces ambiguities in each $aμ[hSl ]$ , but the ambiguity disappears after summing over $l$ .)

The self-acceleration is thus obtained by first computing $a μ[h ] l$ from the metric perturbation derived from $ψ0$ or $ψ4$ , then computing the counterterms $μ S a [hl ]$ by mode-decomposing the singular field, and finally summing over all $μ R μ μ S a [hl ] = a [hl] − a [hl ]$ . This procedure is lengthy and involved, and thus far it has not been brought to completion, except for the special case of a particle falling radially toward a nonrotating black hole [5 ]. In this regard it should be noted that the replacement of the central Kerr black hole by a Schwarzschild black hole simplifies the task considerably. In particular, because there exists a practical and well-developed formalism to describe the metric perturbations of a Schwarzschild spacetime [51 , 59 , 63 ], there is no necessity to rely on the Teukolsky formalism and the complicated reconstruction of the metric variables.

The procedure described above is lengthy and involved, but it is also incomplete. The reason is that the metric perturbations $l hαβ$ that can be recovered from $ψ0$ or $ψ4$ do not by themselves sum up to the complete gravitational perturbation produced by the moving particle. Missing are the perturbations derived from the other Newman–Penrose quantities: $ψ1$ , $ψ2$ , and $ψ3$ . While $ψ1$ and $ψ3$ can always be set to zero by an appropriate choice of null tetrad, $ψ 2$ contains such important physical information as the shifts in mass and angular-momentum parameters produced by the particle [60 ]. Because the mode decompositions of $ψ0$ and $ψ4$ start at $l = 2$ , we might colloquially say that what is missing from the above procedure are the “ $l = 0$ and $l = 1$ ” modes of the metric perturbations. It is not currently known how the procedure can be completed so as to incorporate all modes of the metric perturbations. Specializing to a Schwarzschild spacetime eliminates this difficulty, and in this context the low multipole modes have been studied for the special case of circular orbits [43 , 22 ].

In view of these many difficulties (and I choose to stay silent on others, for example, the issue of relating metric perturbations in different gauges when the gauge transformation is singular on the world line), it is perhaps not too surprising that such a small number of concrete calculations have been presented to date. But progress in dealing with these difficulties has been steady, and the situation should change dramatically in the next few years.

5.5.6 Beyond the self-force

The successful computation of the gravitational self-force is not the end of the road. After the difficulties reviewed in the preceding Section 5.5.5 have all been removed and the motion of the small body is finally calculated to order $m$ , it will still be necessary to obtain gauge-invariant information associated with the body’s corrected motion. Because the MiSaTaQuWa equations of motion are not by themselves gauge-invariant, this step will necessitate going beyond the self-force.

To see how this might be done, imagine that the small body is a pulsar, and that it emits light pulses at regular proper-time intervals. The motion of the pulsar around the central black hole modulates the pulse frequencies as measured at infinity, and information about the body’s corrected motion is encoded in the times-of-arrival of the pulses. Because these can be measured directly by a distant observer, they clearly constitute gauge-invariant information. But the times-of-arrival are determined not only by the pulsar’s motion, but also by the propagation of radiation in the perturbed spacetime. This example shows that to obtain gauge-invariant information, one must properly combine the MiSaTaQuWa equations of motion with the metric perturbations.

In the context of the Laser Interferometer Space Antenna, the relevant observable is the instrument’s response to a gravitational wave, which is determined by gauge-invariant waveforms, $h +$ and $h ×$ . To calculate these is the ultimate goal of this research programme, and the challenges that lie ahead go well beyond what I have described thus far. To obtain the waveforms it will be necessary to solve the Einstein field equations to second order in perturbation theory.

To understand this, consider first the formulation of the first-order problem. Schematically, one introduces a perturbation $h$ that satisfies a wave equation $□h = T[z]$ in the background spacetime, where $T[z]$ is the stress-energy tensor of the moving body, which is a functional of the world line $z (τ )$ . In first-order perturbation theory, the stress-energy tensor must be conserved in the background spacetime, and $z(τ)$ must describe a geodesic. It follows that in first-order perturbation theory, the waveforms constructed from the perturbation $h$ contain no information about the body’s corrected motion.

The first-order perturbation, however, can be used to correct the motion, which is now described by the world line $z(τ) + δz(τ)$ . In a naive implementation of the self-force, one would now re-solve the wave equation with a corrected stress-energy tensor, $□h = T [z + δz]$ , and the new waveforms constructed from $h$ would then incorporate information about the corrected motion. This implementation is naive because this information would not be gauge-invariant. In fact, to be consistent one would have to include all second-order terms in the wave equation, not just the ones that come from the corrected motion. Schematically, the new wave equation would have the form of $2 □h = (1 + h)T [z + δz ] + (∇h )$ , and this is much more difficult to solve than the naive problem (if only because the source term is now much more singular than the distributional singularity contained in the stress-energy tensor). But provided one can find a way to make this second-order problem well posed, and provided one can solve it (or at least the relevant part of it), the waveforms constructed from the second-order perturbation $h$ will be gauge invariant. In this way, information about the body’s corrected motion will have properly been incorporated into the gravitational waveforms.

5.5 Concluding remarks

5.5.1 Conservation of energy-momentum

5.5.2 Averaging method

5.5.3 Detweiler–Whiting axiom

5.5.4 Matched asymptotic expansions

5.5.5 Evaluation of the gravitational self-force

5.5.6 Beyond the self-force