24 Concluding remarks

We have presented a number of derivations of the equations that determine the motion of a point scalar charge q, a point electric charge e, and a point mass m in a specified background spacetime. In this concluding section we summarize these derivations and their foundations. We conclude by describing the next step in the gravitational case: obtaining an approximation scheme sufficiently accurate to extract the parameters of an extreme-mass-ratio inspiral from an observed gravitational waveform.

Our derivations are of two types. The first is based on the notion of an exact point particle. In this approach, we assume that the self-force on the particle arises from a particular piece of its field, either that which survives angle-averaging or the Detweiler–Whiting regular field. The second type is based on the notion of an asymptotically small body, and abandons the fiction of a point particle. In this approach, we don’t assume anything about the body’s equation of motion, but rather derive it directly from the field equations. Although we have presented such a derivation only in the gravitational case, analogous ones could be performed in the scalar and electromagnetic cases, using conservation of energy-momentum instead of the field equations alone. Such a calculation was performed by Gralla et al. [82] in the restricted case of an electric charge in a flat background.

Perhaps the essential result of our derivation based on an asymptotically small body is that it confirms all of the results derived using point particles: at linear order in the body’s mass, the field it creates is identical to that of a point particle, and its equation of motion is precisely that derived from physically motivated axioms for a point particle. In other words, at linear order, not only can we get away with the fiction of a point particle, but our assumptions about the physics governing its motion are also essentially correct.

24.1 The motion of a point particle

Spatial averaging

Our first means of deriving equations of motion for point particles is based on spatial averaging. In this approach, we assume the following axiom:

the force on the particle arises from the piece of the field that survives angle averaging.

For convenience in our review, we consider the case of a point electric charge and adopt the Detweiler–Whiting decomposition of the Faraday tensor into singular and regular pieces, R S F αβ = Fαβ + F αβ. We average F αβ over a sphere of constant proper distance from the particle. We then evaluate the averaged field at the particle’s position. Because the regular field is nonsingular on the world line, this yields

e⟨F ⟩uν = e⟨F S⟩u ν + eF R uν, μν μν μν

where

( 2 e2) e⟨F Sμν⟩u ν = − (δm )aμ, δm = lim ---- , s→0 3 s

and

( ) ∫ τ− R ν 2( ) 2-ν 1- ν λ 2 ν + ( ′) λ′ ′ eFμνu = e g μν + u μuν 3 ˙a + 3 R λu + 2e u −∞ ∇ [μG ν]λ′z (τ ),z (τ ) u dτ .

We now postulate that the equations of motion are ma = e⟨F ⟩uν μ μν, where m is the particle’s bare mass. With the preceding results we arrive at R ν mobsa μ = eFμνu, where mobs ≡ m + δm is the particle’s observed (renormalized) inertial mass.

In this approach, the fiction of a point particle manifests itself in the need for mass renormalization. Such a requirement can be removed, even within the point-particle picture, by adopting the “comparison axiom” proposed by Quinn and Wald [150Jump To The Next Citation Point]. If we consider extended (but small) bodies, no such renormalization is required, and the equations of motion follow directly from the conservation of energy-momentum. However, the essential assumption about the nature of the force is valid: only the piece of the field that survives angle-averaging exerts a force on the body.

The Detweiler–Whiting Axiom

Our second means of deriving equations of motion for point particles is based on the Detweiler–Whiting axiom, which asserts that

the singular field exerts no force on the particle; the entire self-force arises from the action of the regular field.

This axiom, which is motivated by the symmetric nature of the singular field, and also its causal structure, gives rise to the same equations of motion as the averaging method. In this picture, the particle simply interacts with a free 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 free radiation 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 a generalized equivalence principle (cf. Ref. [74Jump To The Next Citation Point]): up to order 𝜀2 errors, a point mass m moves on a geodesic of the spacetime with metric R gαβ + h αβ, which is nonsingular and a solution to the vacuum field equations. This is a conceptually powerful, and elegant, formulation of the MiSaTaQuWa equations of motion. And it remains valid for (non-spinning) small bodies.

Resolving historical ambiguities

Although they yield the correct physical description, the above axioms are by themselves insufficient, and historically, two problems have arisen in utilizing them: One, they led to ill-behaved equations of motion, requiring a process of order reduction; and two, in the gravitational case they led to equations of motion that are inconsistent with the field equations, requiring the procedure of gauge-relaxation. Both of these problems arose because the expansions were insufficiently systematic, in the sense that they did not yield exactly solvable perturbation equations. In the approach taken in our review, we have shown that these problems do not arise within the context of a systematic expansion. Although we have done so only in the case of an extended body, where we sought a higher degree of rigor, one could do the same in the case of point particles by expanding in the limit of small charge or mass (see, e.g., the treatment of a point mass in Ref. [144Jump To The Next Citation Point]).

Consider the Abraham–Lorentz–Dirac equation μ ma μ = fext + 23e2a˙μ. To be physically meaningful and mathematically well-justified, it must be thought of as an approximate equation of motion for a localized matter distribution with small charge e ≪ 1. But it contains terms of differing orders e0 and e2, and the acceleration itself is obviously a function of e. Hence, the equation has not been fully expanded. One might think that it is somehow an exact equation, despite its ill behaviour. Or one might replace it with the order-reduced equation ma μ = fμ + 2e2˙fμ ext 3m ext to eliminate that ill behaviour. But one can instead assume that μ a (e), like other functions of e, possesses an expansion in powers of e, leading to the two well-behaved equations μ ma (0)μ = fext and ma (1)μ = 23e2˙a(0)μ. However, the fact that such an equation can even arise indicates that one has not begun with a systematic expansion of the governing field equations (in this case, the conservation equation and the Maxwell equations). If one began with a systematic expansion, with equations exactly solvable at each order, no such ambiguity would arise.

The same can be said of the second problem. It is well known that in general relativity, the motion of gravitating bodies is determined by the Einstein field equations; the equations of motion cannot be separately imposed. And specifically, if we deal with the linearized Einstein equation (1) δG αβ[h ] = 8πT αβ[γ ], where T αβ is the energy-momentum tensor of a point particle in the background spacetime, then the linearized Bianchi identity requires the point particle to move on a geodesic of the background spacetime. This seems to contradict the MiSaTaQuWa equation and therefore the assumptions we made in deriving it. In order to remove this inconsistency, the earliest derivations [130Jump To The Next Citation Point, 150] invoked an a posteriori gauge-relaxation: rather than solving a linearized Einstein equation exactly, they solved the wave equation (1) Eαβ [h ] = − 16πT αβ in combination with the relaxed gauge condition (1) 2 Lα [𝜀h ] = O (𝜀 ). The allowed errors in the gauge condition carry over into the linearized Bianchi identity, such that it no longer restricts the motion to be geodesic. In this approach, one is almost solving the linearized problem one set out to solve. But again, such an a posteriori corrective measure is required only if one begins without a systematic expansion. The first-order metric perturbation is a functional of a world line; if we allow that world line to depend on 𝜀, then the metric has evidently not been fully expanded in powers of 𝜀. To resolve the problem, one needs to carefully deal with this fact.

In the approach we have adopted here, following Ref. [144Jump To The Next Citation Point], we have resolved these problems via a self-consistent expansion in which the world line is held fixed while expanding the metric. Rather than beginning with the linear field equation, we began by reformulating the exact equation in the form

eff Eαβ [h] = Tαβ , (24.1 ) Lα [h] = 0. (24.2 )
These equations we systematically expanded by (i) treating the metric perturbation as a functional of a fixed world line, keeping the dependence on the world line fixed while expanding, and (ii) expanding the acceleration of the world line. Since we never sought a solution to the equation δG [h (1)] = 8 πT [γ] αβ αβ, no tension arose between the equation of motion and the field equation. In addition, we dealt only with exactly solvable perturbative equations: rather then imposing the ad hoc a posteriori gauge condition L μ[𝜀h (1)] = O(𝜀2), our approach systematically led to the conditions [ ] L (0μ) h(1) = 0 and (1)[ (1)] (0)[ (2)] L μ h = − L μ h, which can be solved exactly. Like in the issue of order reduction, the essential step in arriving at exactly solvable equations is assuming an expansion of the particle’s acceleration.

Historically, these issues were first resolved by Gralla and Wald [83Jump To The Next Citation Point] using a different method. Rather than allowing an 𝜀-dependence in the first-order perturbation, they fully expanded every function in the problem in a power series, including the world line itself. In this approach, the world line of the body is found to be a geodesic, but higher-order effects arise in deviation vectors measuring the drift of the particle away from that geodesic. Such a method has the drawback of being limited to short timescales, since the deviation vectors will eventually grow large as the body moves away from the initial reference geodesic.

24.2 The motion of a small body

Although the above results for point particles require assumptions about the form of the force, their results have since been derived from first principles, and the physical pictures they are based on have proven to be justified: An asymptotically small body behaves as a point particle moving on a geodesic of the smooth part of the spacetime around it, or equivalently, it moves on a world line accelerated by the asymmetric part of its own field.

In addition to a derivation from first principles, we also seek a useful approximation scheme. Any such scheme must deal with the presence of multiple distinct scales. Most obviously, these are the mass and size of the body itself and the lengthscales of the external universe, but other scales also arise. For example, in order to accurately represent the physics of an extreme-mass-ratio system, we must consider changes over four scales: First, there is the large body’s mass, which is the characteristic lengthscale of the external universe. For convenience, since all lengths are measured relative to this one, we rescale them such that this global lengthscale is ∼ 1. Second, there is the small body’s mass ∼ 𝜀, which is the scale over which the gravitational field changes near the body; since the body is compact, this is also the scale of its linear size. Third, there is the radiation-reaction time ∼ 1∕𝜀; this is the time over which the small effects of the self-force accumulate to produce significant changes, specifically the time required for quantities such as the small body’s energy and angular momentum to accumulate order 1 changes. Fourth, there is the large distance to the wave zone. We will not discuss this last scale here, but dealing with it analytically would likely require matching a wave solution at null infinity to an expansion formally expected to be valid in a region of size ∼ 1∕𝜀.

Self-consistent and matched asymptotic expansions

In this review, we have focused on a self-consistent approximation scheme first presented in Refs. [144, 145Jump To The Next Citation Point, 143]. It deals with the small size of the body using two expansions. Near the body, to capture changes on the short distances ∼ 𝜀, we adopt an inner expansion in which the body remains of constant size while all other distances approach infinity. Outside of this small neibourhood around the body, we adopt an outer expansion in which the body shrinks to zero mass and size about an 𝜀-dependent world line that accurately reflects its long-term motion. The world line γ is defined in the background spacetime of the outer expansion. Its acceleration is found by solving the Einstein equation in a buffer region surrounding the body, where both expansions are valid; in this region, both expansions must agree, and we can use the multipole moments of the inner expansion to determine the outer one. In particular, we define γ to be the body’s world line if and only if the body’s mass dipole moment vanishes when calculated in coordinates centered on it. As in the point-particle calculation, the essential step in arriving at exactly solvable equations and a well-behaved equation of motion is an assumed expansion of the acceleration on the world line.

In order to construct a global solution in the outer expansion, we first recast the Einstein equation in a form that can be expanded and solved for an arbitrary world line. As in the point-particle case, we accomplish this by adopting the Lorenz gauge for the total metric perturbation. We can then write formal solutions to the perturbation equations at each order as integrals over a small tube surrounding the body (plus an initial data surface). By embedding the tube in the buffer region, we can use the data from the buffer-region expansion to determine the global solution. At first order, we find that the metric perturbation is precisely that of a point particle moving on the world line γ. While the choice of gauge is not essential in finding an expression for the force in terms of the field in the buffer-region expansion, it is essential in our method of determining the global field. Without making use of the relaxed Einstein equations, no clear method of globally solving the Einstein equation presents itself, and no simple split between the perturbation and the equation of motion arises.

Because this expansion self-consistently incorporates the corrections to the body’s motion, it promises to be accurate on long timescales. Specifically, when combined with the first-order metric perturbation, the first-order equation of motion defines a solution to the Einstein equation that we expect to be accurate up to order 2 𝜀 errors over times t ≲ 1∕𝜀. When combined with the second-order perturbation, it defines a solution we expect to be accurate up to order 𝜀3 errors on the shorter timescale ∼ 1.

This approach closely mirrors the extremely successful methods of post-Newtonian theory [74Jump To The Next Citation Point]. In particular, both schemes recast the Einstein equation in a relaxed form before expanding it. Also, our use of an inner limit near the body is analogous to the “strong-field point particle limit” exploited by Futamase and his collaborators [74Jump To The Next Citation Point]. And our calculation of the equation of motion is somewhat similar to the methods used by Futamase and others [74, 152], in that it utilizes a multipole expansion of the body’s metric in the buffer region.

Alternative methods

Various other approaches have been taken to deal with the multiple scales in the problem. In particular, even the earliest paper on the gravitational self-force [130Jump To The Next Citation Point] made use of inner and outer limits, which were also used in different forms in later derivations [142Jump To The Next Citation Point, 49Jump To The Next Citation Point, 72Jump To The Next Citation Point]. However, those early derivations are problematic. Specifically, they never adequately define the world line for which they derive equations of motion.

In the method of matched asymptotic expansions used in Refs. [130, 142, 49], the first-order perturbation in the outer expansion is assumed to be that of a point particle, with an attendant world line, and the inner expansion is assumed to be that of a perturbed black hole; by matching the two expansions in the buffer region, the acceleration of the world line is determined. Since the forms of the inner and outer expansions are already restricted, this approach’s conclusions have somewhat limited strength, but it has more fundamental problems as well. The matching procedure begins by expanding the outer expansion in powers of distance s (or r) from the world line, and the inner expansion for large spacial distances R ≫ 𝜀. But the two expansions begin in different coordinate systems with an unknown relationship between them. In particular, there is no given relationship between the world line and the “position” of the black hole. The two expansions are matched by finding a coordinate transformation that makes them agree in the buffer region. However, because there is no predetermined relationship between the expansions, this transformation is not in fact unique, and it does not yield a unique equation of motion. One can increase the strength of the matching condition in order to arrive at unique results, but that further weakens the strength of the conclusions. Refer to Ref. [145Jump To The Next Citation Point] for a discussion of these issues.

Rather than finding the equation of motion from the field equations, as in the above calculations, Fukumoto et al. [72Jump To The Next Citation Point] found the equation of motion by defining the body’s linear momentum as an integral over the body’s interior and then taking the derivative of that momentum. But they then required an assumed relationship between the momentum and the four-velocity of a representative world line in the body’s interior. Hence, the problem is again an inadequate definition of the body’s motion. Because it involves integrals over the body’s interior, and takes the world line to lie therein, this approach is also limited to material bodies; it does not apply to black holes.

The first reliable derivation of the first-order equation of motion for an asymptotically small body was performed by Gralla and Wald [83], who used a method of the same nature as the one presented here, deriving equations of motion by solving the field equations in the buffer region. As we have seen, however, their derivation is based on an expansion of the world line in powers of 𝜀 instead of a self-consistent treatment that keeps it fixed.

Using very different methods, Harte has also provided reliable derivations of equations of motion for extended bodies interacting with their own scalar and electromagnetic fields in fixed background spacetimes [89, 90, 91]. His approach is based on generalized definitions of momenta, the evolution of which is equivalent to energy-momentum conservation. The momenta are defined in terms of generalized Killing fields α ξ, the essential property of which is that they satisfy £ξgαβ|γ = ∇ γ£ ξgαβ|γ = 0 – that is, they satisfy Killing’s equation on the body’s world line and approximately satisfy it “nearby.” Here the world line can be defined in multiple ways using, for example, center-of-mass conditions. This approach is nonperturbative, with no expansion in the limit of small mass and size (though it does require an upper limit on the body’s size and a lower limit on its compactness). It has the advantage of very naturally deriving a generalization of the Detweiler–Whiting axiom: The field of an extended body can be split into (i) a solution to the vacuum field equations which exerts a direct self-force, and (ii) a solution to the equations sourced by the body, which shifts the body’s multipole moments. This approach has not yet been applied to gravity, but such an application should be relatively straightforward. However, as with the approach of Fukumoto et al. [72], this one does not apply to black holes.

Other methods have also been developed (or suggested) to accomplish the same goals as the self-consistent expansion. Most prominent among these is the two-timescale expansion suggested by Hinderer and Flanagan [94Jump To The Next Citation Point]. As discussed in Section 2, their method splits the orbital evolution into slow and fast dynamics by introducing slow and fast time variables. In the terminology of Section 22, this method constructs a general expansion by smoothly transitioning between regular expansions constructed at each value of the slow time variable, with the transition determined by the evolution with respect to the slow time. On the scale of the fast time, the world line is a geodesic; but when the slow time is allowed to vary, the world line transitions between geodesics to form the true, accelerated world line. This results in a global, uniform-in-time approximation. One should note that simply patching together a sequence of regular expansions, by shifting to a new geodesic every so often using the deviation vector, would not accomplish this: Such a procedure would accumulate a secular error in both the metric perturbation and the force, because the perturbation would be sourced by a world line secularly deviating from the position of the body, and the force would be calculated from this erroneous perturbation. The error would be proportional to the number of “shifts” multiplied by a nonlinear factor depending on the time between them. And this error would, formally at least, be of the same magnitude as the solution itself.

The fundamental difference between the self-consistent expansion and the two-timescale expansion is the following: In the two-timescale method, the Einstein equation, coupled to the equation of motion of the small body, is reduced to a dynamical system that can be evolved in time. The true world line of the body then emerges from the evolution of this system. In the method presented here, we have instead sought global, formal solutions to the Einstein equation, written in terms of global integrals; to accomplish this, we have treated the world line of the body as a fixed structure in the external spacetime. However, the two methods should agree. Note, though, that the two-timescale expansion is limited to orbits in Kerr, and it requires evolution equations for the slow evolution of the large black hole’s mass and spin parameters, which have not yet been derived. Since the changes in mass and spin remain small on a radiation-reaction timescale, in the self-consistent expansion presented here they are automatically incorporated into the perturbations h (n) αβ. It is possible that this incorporation leads to errors on long timescales, in which case a different approach, naturally allowing slow changes in the background, would be more advantageous.

24.3 Beyond first order

The primary experimental motivation for researching the self-force is to produce waveform templates for LISA. In order to extract the parameters of an extreme-mass-ratio binary from a waveform, we require a waveform that is accurate up to errors of order 𝜀 after a radiation-reaction time ∼ 1∕𝜀. If we use the first-order equation of motion, we will be neglecting an acceleration ∼ 𝜀2, which will lead to secular errors of order unity after a time ∼ 1∕𝜀. Thus, the second-order self-force is required in order to obtain a sufficiently accurate waveform template. In order to achieve the correct waveform, we must also obtain the second-order part of the metric perturbation; this can be easily done, at least formally, using the global integral representations outside a world tube. However, a practical numerical calculation may prove difficult, since one would not wish to excise the small tube from one’s numerical domain, and the second-order perturbation would diverge too rapidly on the world line to be treated straightforwardly.

A formal expression for the second-order force has already been derived by Rosenthal [155, 156]. However, he expresses the second-order force in a very particular gauge in which the first-order self-force vanishes. This is sensible on short timescales, but not on long timescales, since it forces secular changes into the first-order perturbation, presumably leading to the first-order perturbation becoming large with time. Furthermore, it is not a convenient gauge, since it does not provide what we wish it to: a correction to the nonzero leading-order force in the Lorenz gauge.

Thus, we wish to obtain an alternative to Rosenthal’s derivation. Based on the methods reviewed in this article, there is a clear route to deriving the second-order force. One would construct a buffer-region expansion accurate up to order 𝜀3. Since one would require the order 𝜀2s terms in this expansion, in order to determine the acceleration, one would need to increase the order of the expansion in r as well. Specifically, one would need terms up to orders 𝜀0s3, 𝜀s2, 𝜀2s, and 𝜀3s0. In such a calculation, one would expect the following terms to appear: the body’s quadrupole moment Qab, corrections δMi and δSi to its mass and spin dipoles, and a second-order correction δ2m to its mass. Although some ambiguity may arise in defining the world line of the body at this order, a reasonable definition appears to be to guarantee that δMi vanishes. However, at this order one may require some model of the body’s internal dynamics, since the equation of motion will involve the body’s quadrupole moment, for which the Einstein equation is not expected to yield an evolution equation. But if one seeks only the second-order self-force, one could simply neglect the quadrupole by assuming that the body is spherically symmetric in isolation. In any case, the force due to the body’s quadrupole moment is already known from various other methods: see, e.g., the work of Dixon [57, 58, 59]; more recent methods can be found in Ref. [165] and references therein.

Because such a calculation could be egregiously lengthy, one may consider simpler methods, perhaps requiring stronger assumptions. For example, one could straightforwardly implement the method of matched asymptotic expansions with the matching conditions discussed in Ref. [145], in which one makes strong assumptions about the relationship between the inner and outer expansions. The effective field theory method used by Galley and Hu [75] offers another possible route.

Alternatively, one could calculate only part of the second-order force. Specifically, as described in Section 2.5, Hinderer and Flanagan [94, 69] have shown that one requires in fact only the averaged dissipative part of the second-order force. This piece of the force can be calculated within an adiabatic approximation, in which the rates of change of orbital parameters are calculated from the radiative Green’s function, asymptotic wave amplitudes, and information about the orbit that sources them. Hence, we might be able to forgo a complete calculation of the second-order force, and use instead the complete first-order force in conjunction with an adiabatic approximation for the second-order force.

The self-force, however, is of interest beyond its relevance to LISA. The full second-order force would be useful for more general purposes, such as more accurate comparisons to post-Newtonian theory, and analysis of other systems, such as intermediate mass ratio binaries. Perhaps most importantly, it is of fundamental importance in our understanding of the motion of small bodies. For these reasons, proceeding to second order in a systematic expansion, and thereby obtaining second-order expressions for the force on a small body, remains an immediate goal.


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