Motion of small objects in curved spacetimes: An introduction to gravitational self-force

Motion of small objects in curved spacetimes: An introduction to gravitational self-force

Adam Pound111Email: a.pound@soton.ac.uk
Mathematical Sciences
University of Southampton
Southampton, SO17 1BJ, United Kingdom
July 14, 2019
Abstract

In recent years, asymptotic approximation schemes have been developed to describe the motion of a small compact object through a vacuum background to any order in perturbation theory. The schemes are based on rigorous methods of matched asymptotic expansions, which account for the object’s finite size, require no “regularization” of divergent quantities, and are valid for strong fields and relativistic speeds. Up to couplings of the object’s multipole moments to the external background curvature, these schemes have established that at least through second order in perturbation theory, the object’s motion satisfies a generalized equivalence principle: it moves on a geodesic of a certain smooth metric satisfying the vacuum Einstein equation. I describe the foundations of this result, particularly focusing on the fundamental notion of how a small object’s motion is represented in perturbation theory. The three common representations of perturbed motion are (i) the “Gralla-Wald” description in terms of small deviations from a reference geodesic, (ii) the “self-consistent” description in terms of a worldline that obeys a self-accelerated equation of motion, and (iii) the “osculating geodesics” description, which utilizes both (i) and (ii). Because of the coordinate freedom in general relativity, any coordinate description of motion in perturbation theory is intimately related to the theory’s gauge freedom. I describe asymptotic solutions of the Einstein equations adapted to each of the three representations of motion, and I discuss the gauge freedom associated with each. I conclude with a discussion of how gauge freedom must be refined in the context of long-term dynamics.

Contents

1 Preamble and survey

Consider a small object moving through a curved spacetime. What path does it follow? At the level of undergraduate physics, the answer is satisfyingly simple: if the object is sufficiently small and light, it can be idealized as a test particle, which does not affect the geometry around it and which moves on a geodesic of that geometry. But what if we do away with that idealization? The real object does perturb the geometry around it; how, then, does the object move in that geometry, which it itself affects?

In Newtonian gravity, we may ask the analogous question—how does a massive body move when it contributes to the gravitational field around it?—and we may happily provide an answer without yet leaving undergraduate physics: if the body is sufficiently spherical, it can be treated as a point particle located at its center of mass, and the motion of that center of mass is governed by the external gravitational fields produced by all other masses in the system; the object does not feel its own field, at least in so far as its center-of-mass motion is concerned.

However, in general relativity, the situation becomes radically more complicated. Because of the nonlinearity of the Einstein equations, an extended object cannot, in general, be modelled as a point particle without invoking post-hoc regularization procedures [1, 2]. Furthermore, in a curved background, an object, even an asymptotically small one, does feel its own field, for reasons discussed below (and elsewhere in these proceedings). Hence, the field is said to exert a gravitational self-force on the object.

One might wonder if this effect is actually relevant in any realistic scenarios. The answer, unequivocally, is “Yes”. Suppose we are interested in a bound binary of widely separated compact objects of comparable masses and moving slowly in each other’s weak mutual gravity. Each of the objects is small compared to the other scales in the system (for example, the typical orbital separation , or the radius of curvature of ’s field at ’s position). In a Newtonian approximation, is subject only to ’s Newtonian field, which at the position of exerts a force (per unit mass) . But if one requires anything more accurate than the Newtonian approximation, the moment one steps to the post-Newtonian level, self-forces can no longer be ignored: ’s field, which we can think of as scaling like , modifies the Newtonian fields, giving rise to a post-Newtonian force per unit mass that scales like  [3], similar in magnitude to any other post-Newtonian effect.222In a certain sense, an object’s mass affects its own motion even in a Newtonian binary. Each object follows a Keplerian orbit about the system’s center of mass, not about the center of the other object. Since the center of mass is shifted by the object’s own mass, the object affects its own motion; more plainly, influences its own motion by influencing that of . This is a more indirect effect than the type described above, but in practice, distinguishing it from any other post-test-body effect is nontrivial. See Ref. [4] for a discussion.

In these systems just described, however, each object, while small compared to the radius of curvature of the ambient external field it finds itself in, is not small compared to the other object. What if we are interested in a case where there are truly only two scales? Take a binary system of compact objects of mass and satisfying , and specifically focus on the regime in which the orbital separation is of order . In this case, it seems the gravity of must certainly have a very small effect on its own motion, and it must very nearly follow a geodesic of the metric of . And yet, even in this scenario, ’s gravitational self-force cannot be neglected. Although the effect is very small over a few orbits, the system continually radiates energy in the form of gravitational waves (or equivalently, the self-force does negative work), causing the orbit to shrink, and eventually causing to collide with (or plunge into , if is a black hole). In other words, the self-force has long-term, secular effects on the motion which make it impossible to ignore.

Both of these two types of systems—binaries of widely separated bodies whose mutual gravity is weak, and binaries of objects with very dissimilar masses, called extreme-mass-ratio inspirals (EMRIs)—are of increasing relevance to modern astrophysics. The prospect of directly detecting gravitational waves emitted from compact binaries, and extracting information about the binaries’ strong-field dynamics from those waves, has spurred an international effort to study them. In the case of widely separated bodies of comparable mass, the main method of analysis has been post-Newtonian theory,333Of course, once the two bodies are sufficiently close to each other, they interact in a highly nonlinear, highly relativistic way. In that regime, one must use numerical relativity to solve the fully nonlinear Einstein equations for the system. a historied subject modern overviews of which can be found in the review articles [5, 6] and the recent textbook [7]. In the case of EMRIs, the main method of analysis has been self-force theory; for summaries of efforts to model EMRIs, I refer the reader to the reviews [8, 9], the more up-to-date but brief survey [10], and the contributions of Babak et al. [11] and Wardell [12] elsewhere in this book.

In this paper, I seek to provide a single, unified description of gravitational self-force theory. Roughly speaking, this formalism consists of a perturbative expansion in powers of the small object’s mass . Although I will refer to EMRIs to motivate many of the methods and problems I discuss, my focus will instead be on foundational issues related to the problem of motion of a small object. My aim is to complement extant reviews [9, 8] by concentrating on three themes given limited attention in those reviews: (i) self-force theory at arbitrary perturbative order, (ii) differing ways to represent perturbed motion, and the asymptotic expansions of the metric corresponding to each, and (iii) the relationship between perturbed motion and gauge freedom. My presentation mostly follows the methods and viewpoint of Refs. [13, 14, 15, 16, 17, 18, 19], though it takes additional inspiration from the work of Gralla and Wald [20, 21, 22] and the classic papers of Mino, Sasaki, and Tanaka [23] and Detweiler and Whiting [24, 25]. To avoid excessive length, rather than striving for complete self-containment, I will often refer the reader to Refs. [9, 13, 17, 18] for mathematical tools and technical details.

Most of what I discuss could be applied to objects carrying scalar or electric charges, but for simplicity I restrict my attention to the purely gravitational case. I refer the reader to Refs. [26, 27, 28, 29, 30, 9] for discussions of self-forces due to scalar and electromagnetic fields on fixed background geometries, and to Refs. [31, 32] for recent work on the coupled system in which the metric reacts to both the mass and the scalar or electromagnetic field.

In the remainder of this introduction, I make an extensive survey of the main concepts and results of self-force theory, beginning with the standard picture of a point mass in linearized perturbation theory and proceeding to describe the more robust framework now available for studying the motion of extended (but small) objects at any order in perturbation theory.

Notation. Throughout this paper, Greek indices run from 0 to 4 and are raised and lowered with a background metric . Latin indices run from 1 to 3 and are freely raised and lowered with the Euclidean metric . Sans-serif symbols such as refer to tensors on the perturbed spacetime rather than on the background. A semicolon and refer to a covariant derivative compatible with , to the covariant derivative compatible with , and to the covariant derivative compatible with an “effective metric” . is used to count powers of , and labels such as the “” in refer to perturbative order ; I freely write these labels as either super- or subscripts. I work in geometric units with .

1.1 A point particle picture: the MiSaTaQuWa equation and its interpretations

1.1.1 Self-interaction with the tail of the perturbation

As mentioned above, a point particle stress-energy tensor is not a well-defined source in the nonlinear Einstein equations. However, it is a fine source in the linearized theory. For simplicity, assume our object of mass is isolated, such that in some large region, we can take it to be the sole source of stress-energy in the system. Now consider a metric , where the background metric is a vacuum solution to the nonlinear Einstein equations, and is the leading-order perturbation due to our small object. Linearizing the Einstein equations in , we obtain

(1)

where is the linearized Einstein tensor, and is the leading-order approximation to the object’s stress-energy tensor. Now suppose that the the object’s stress-energy can be approximated by

(2)

which is the stress-energy of a point mass moving on a worldline with coordinates in the background spacetime. Here is the particle’s four-velocity, is its proper time (as measured in ), and is a covariant delta distribution, with being the determinant of .

Unlike the nonlinear Einstein equations with a point particle source, Eq. (1) has a perfectly well-defined solution. If we introduce the trace-reversed perturbation and impose the Lorenz gauge condition , then the linearized Einstein equation takes the form of a wave equation,

(3)

where

(4)

, and is the Riemann tensor of the background. Assuming the existence of a global retarded Green’s function 444My conventions for the Green’s function are those of Ref. [9]; Ref. [9] also contains a pedagogical introduction to bitensors, objects which live in the tangent spaces of two different points and . for this wave equation, we can write the retarded solution as

(5)

where a primed index refers to the tangent space at the source point .

Now, if the background were flat, waves would propagate precisely on the light cone, and the retarded Green’s function would be supported only on points connected by a null curve: , where is a Cartesian coordinate system, is the retarded time, and is the spatial distance between the points and . However, in a curved spacetime, Huygen’s principle no longer holds. Waves scatter off the spacetime curvature, causing them to propagate from a source point both on the future null cone of and within that cone. Correspondingly, the Green’s function at a point has support both on the past null cone of and within it. This implies that if we look at the field at a point near the worldline, we can split it into two pieces: a direct piece, corresponding to the portion of the field that propagated to from a point along a null curve; and a so-called tail piece, , corresponding to the portion of the field that propagated from all earlier points on the worldline. The direct piece diverges like a Coulomb field, behaving as (where is a geodesic distance from the particle). The tail piece, on the other hand, is finite.

A detailed analysis (such as the one presented in the bulk of this paper) reveals that at leading order, the mass is constant, and the force on the particle vanishes; in other words, it behaves as a test particle in . The same analysis applied at subleading order reveals that the direct piece of the field exerts no force on the particle, but the tail piece does exert a force, and the equation of motion is found to be

(6)

where , projects orthogonally to the worldline, and the tail term is given by

(7)

the integral covers all of the worldline earlier than the point at which the force is evaluated. The bar atop again denotes a trace reversal.

Equation (6) is termed the MiSaTaQuWa equation, after Mino, Sasaki, and Tanaka [23], who first derived it, and Quinn and Wald [26], who re-derived it very shortly thereafter using a very different, independent method. The intuitive picture to glean from the MiSaTaQuWa result is that the direct piece of the field is analogous to a Coulomb field, moving with the particle and exerting no force on it, in the same way the self-field exerts no force on a body in Newtonian gravity. Very loosely speaking, from the perspective of the particle, the tail, consisting as it does of backscattered radiation, is indistinguishable from any other incoming radiation. In other loose words, it is effectively an external field, and like an external field, it exerts a force.

Much of this paper is devoted to showing how the MiSaTaQuWa result can be robustly justified, and higher-order corrections to it can be found, within a systematic expansion of the Einstein equations. As a byproduct of that analysis, in Sec. 4.1 I will show that the setup of the linearized system with a point particle source in this section is rigorously justified—even for a non-material object such as a black hole, and even though at nonlinear orders the field equations cannot be written in terms of such a source. However, before moving on to those matters, we may say significantly more on the basis of the point-particle result.

1.1.2 The Detweiler-Whiting description: a generalized equivalence principle

In the original MiSaTaQuWa papers [23, 26], the authors noted that Eq. (6) appears to have the form of the geodesic equation in a metric , when that geodesic equation is expanded to linear order in .555I refer the reader to Appendix A for the expansion of the geodesic equation in powers of a metric perturbation. However, is not in any way a nice field. It does not satisfy any particularly meaningful field equation, nor is it even differentiable at the particle [9] [despite superficial appearances in Eq. (6)].

Detweiler and Whiting provided a more compelling form of the MiSaTaQuWa result [24, 25]. Rather than splitting the retarded field into a direct piece and a tail, they split it as

(8)

The singular field can be interpreted as the bound self-field of the particle. Like the direct piece of the field, it exhibits a , Coulomb divergence at the particle; but unlike the direct piece, it satisfies the inhomogeneous linearized Einstein equation , bolstering its interpretation as a self-field. Similarly, the regular field improves on the interpretation of the tail: it includes all the backscattered radiation in the tail, but it is a smooth solution to the homogeneous wave equation . Hence, more than we could of the tail, we can think of as an effectively external field, propagating independently of the particle. From it we can define what I will variously call an effective metric or effectively external metric .

Fittingly, given this interpretation of , Detweiler and Whiting showed that the MiSaTaQuWa equation (6) can be equivalently written as

(9)

or (following Appendix A) explicitly as the geodesic equation in the metric ,

(10)

where is a covariant derivative compatible with , is the four-velocity normalized in , and is the proper time along as measured in . Equation (9) is equivalent to Eq. (6) because on the worldline, differs from only by (i) background Riemann terms that cancel in Eq. (9) and (ii) terms proportional to the worldline’s acceleration, which can be treated as effectively higher order because the acceleration is already .

Allow me to dwell longer on the interpretation of the regular field. Because is a smooth vacuum solution, at the particle’s position an observer cannot distinguish it from . Although a portion of comes from the retarded field sourced by the particle, to the observer on the worldline, it appears just as would any metric sourced by a distant object. However, this interpretation of the effective metric as an effectively external metric is delicate. To effect the desired split into and , both fields must be made acausal when evaluated off the worldline [25, 9]. More precisely, in addition to depending on the particle’s causal past, depends on the particle at spatially related points . So in that sense its interpretability as a physical external field is limited. Yet when evaluated on the worldline, and its derivatives are causal, and that is the sense in which appears as a physical metric on the worldline.

I impress upon the reader the significance of these properties of ; they are what makes Eq. (10) a meaningful result. Although it may not be an obvious fact at first glance, any equation of motion can be written as the geodesic equation in some smooth piece of the metric. This is most easily seen by writing the equation of motion in a frame that comoves with the particle. In locally Cartesian coordinates adapted to that frame, such as Fermi-Walker coordinates [9], the particle’s equation of motion reads , where is the particle’s covariant acceleration and is the force (per unit mass) acting on it. Now suppose we were to define some smooth field and a corresponding singular field . In the comoving coordinates, the linearized geodesic equation in the regular metric , in the form analogous to Eq. (9), reads . No matter what force acts on the particle, the equation of motion could be written as the geodesic equation in simply by choosing and , for example. Besides those two conditions, the regular field could be entirely freely specified, and regardless of the specification we made, we would have defined a split in which is singular and exerts no force, and is a regular metric in which the motion is geodesic.

Given this fact, it is of no special significance that the MiSaTaQuWa equation is equivalent to geodesic motion in some effective metric. But it is significant that the MiSaTaQuWa equation is equivalent to geodesic motion in the particular effective metric identified by Detweiler and Whiting, because of the particular, ‘physical’ properties of that metric: is a smooth vacuum solution that is causal on the particle’s worldline. Because of those properties, we may think of the MiSaTaQuWa equation as a generalized equivalence principle: any object, if it is sufficiently compact and slowly spinning, regardless of its internal composition, falls freely in a gravitational field that can be thought of (loosely if not precisely) as the physical ‘external’ gravitational field at its ‘position’. I stress that this is a derived result, not an assumption, as will be shown in Sec. 3. By that stage in the paper, the principle’s reference to extended, “sufficiently compact and slowly spinning” objects will have become clear.

Another aspect that will have become clearer is the non-uniqueness of the effective metric and the limitations of interpreting it in a strongly physical way. Nonetheless, the split of the metric into a self-field and an effective metric will be a recurring theme. In many ways, the generalized equivalence principle just described is both the central tool and the core result of self-force theory. It is conceptually more compelling than the expression for the force in terms of a tail,666Another compelling physical interpretation is provided by Quinn and Wald [26]. They showed that the MiSaTaQuWa equation follows from the assumption that the net force is equal to an average over a certain “bare” force over a sphere around the particle. (This assumption was later proved to be true in a large class of gauges [20, 21, 33].) In the language of Detweiler and Whiting fields, the force lines of the singular field are symmetric around the particle and vanish upon averaging, while the force lines of the regular field are asymmetric and add up to a net force on the particle. it is less tied to any particular choice of gauge [19], and it is often easier to use as a starting point from which to derive formal results. Perhaps most importantly, it is easily carried to nonlinear orders: as I describe in later sections, at least through second order in , the generalized equivalence principle stated above holds true [16, 18].

1.2 Extended bodies and the trouble with point particles

Although the point-particle picture seemingly works well within linearized theory, it is not obvious a priori how it fits within a systematic asymptotic expansion that goes to higher perturbative orders. For as I have reiterated above, one cannot use a point particle in the full nonlinear theory. Let me now expound on that point.

Suppose we attempt to model the extended object as a point particle in the exact spacetime, with a stress-energy tensor

(11)

where is proper time on as measured in . If we expand the metric as , the linearized Einstein equation is exactly as described in the previous section. But at second order, severe problems arise. The second-order term in the Einstein equation reads

(12)

where and is the second-order term in . There are two problems with Eq. (12). Most obviously, contains terms like

As described in the preceding section, diverges as near the particle; hence, the stress-energy diverges in the distributionally ill-defined manner . Even if we could somehow mollify the ill behavior of this piece of the source, the other source term in Eq. (12) would still present a problem. behaves schematically as . Therefore, it diverges as . Such a divergence is non-integrable, meaning it is well defined as a distribution only if it can be expressed as a linear operator acting on an integrable function. In the present case, there does not appear to be a unique way of so expressing it.

These arguments make clear that a point particle poses increasingly worsening difficulties at nonlinear orders in perturbation theory. And it is well known that in any well-behaved space of functions there exists no solution to the original, fully nonlinear equation with a point particle source [1, 2].

Despite these obstacles, one might suppose that the point-particle model could be adopted and the divergences resolved using post-hoc regularization methods. Evidence for this reasoning is given by the successful use of dimensional regularization in post-Newtonian theory [5]. In the fully relativistic context, the most promising route to such regularization appears to be effective field theory [34, 35].

However, at a fundamental level, there should be no need for such regularization in general relativity. Outside of curvature singularities inside black holes, everything in the problem should be perfectly finite. For that reason, in this paper I will be interested only in formalisms that deal with finite, well-defined quantities throughout.

So let us do away with the fiction of a point particle and think of an extended object. Perhaps the most obvious route, at least at first glance, to determining this object’s motion is to go to the opposite extreme from the linearized point particle model, by looking instead at a generic extended object in fully nonlinear general relativity; if one is interested in the case of a small object, one can always examine an appropriate limit of one’s generic results. This line of attack has been most famously pursued by Dixon [36] (inspired by early work by Mathisson [37]) and Harte [38]. Specializing to material bodies, Dixon showed that all information in the stress-energy can be encoded in a set of multipole moments, and all information in the conservation law can be encoded in laws of motion for the body. These laws take the form of evolution equations for some suitable representative worldline in the object’s interior and for the object’s spin about that worldline. A “good” choice of representative worldline may be made by, for example, defining the object’s mass dipole moment relative to any given worldline and then choosing the worldline for which the mass dipole moment vanishes, establishing the worldline as a center of mass [39]. However, in order to transform these general results into practical equations of motion, Dixon’s method requires an assumption that the metric and its derivatives do not vary much in the body’s interior. Given that assumption, the force and torque appearing on the right-hand side of the equations of motion can be written as a simple expansion composed of couplings of the metric’s curvature to the object’s higher multipole moments; the higher moments, beginning with the quadrupole, may be freely specified, and their specification is entirely equivalent to a specification of the object’s stress-energy tensor. Equations of motion of this form can be viewed in Eqs. (13)–(14) below.

Unfortunately, for a reasonably compact, strongly gravitating body, the physical metric does not vary slowly in its interior, due to the body’s own contribution to the metric. To make progress, one must treat the object as a test body, an extended object that is non-gravitating but is equipped with a multipole structure—or, as discussed in Sec. 13 of Ref. [36], one must find an efficacious means of separating the object’s self-field from the “external” field, analogous to the trivial split in Newtonian theory and to the Detweiler-Whiting split in the linearized point particle model. A well-chosen “external” field will vary little in the body’s interior, a well-chosen “self-field” will have minimal influence on the motion, and one can hope to arrive at equations of motion expressed in terms of couplings to the curvature of the external field alone.

In the fully nonlinear theory, finding such a split would seem to be highly nontrivial. Nevertheless, in a series of papers [28, 29, 40] culminating in Ref. [38] (see also Harte’s contribution to these proceedings [30]), Harte has succeeded in finding a suitable split by directly generalizing the Detweiler-Whiting decomposition. He has shown that the “self-field” he defines modifies the equation of motion only by shifting the values of the object’s multipole moments, and the object behaves as a test body moving in the effectively external metric he defines. This extends the Detweiler-Whiting result from the linearized model to the fully nonlinear problem. However, there is one caveat to this generalization: beyond linear order, Harte’s effective metric loses one of the compelling properties of the Detweiler-Whiting field: it is not a solution to the vacuum Einstein equation. Despite this feature, Harte’s work is a tour de force in the problem of motion.

The approach I take in this paper is complementary to Harte’s. Rather than beginning with the fully nonlinear problem, I will proceed directly to perturbation theory. There are several advantages to this. The perturbative approach naturally applies to black holes, while Harte’s formalism, because it is based on integrals over the object’s interior, is restricted to material bodies. The perturbative approach also naturally leads to a split into self-field and effective metric in which the effective metric satisfies the vacuum Einstein equation at all orders and is causal on the worldline. Most importantly, the perturbative approach provides a practical means of solving the Einstein equations. In the fully nonlinear approach, one arrives at equations of motion given the metric, but not a practical way to find that metric.

In the next section, I will begin to discuss the perturbative formalism. First, I note that with the work of Dixon and Harte, a new theme has been introduced: an object’s bulk motion can be expressed in terms of a set of multipole moments, and the moments are freely specifiable. Like the decomposition of the metric into a self-field and effectively external field, this second theme will appear prominently in the remainder of this paper.

1.3 When perturbation theory fails near a submanifold: the method of matched asymptotic expansions

As soon as we seek a perturbative description of the problem, we run into a new challenge. Say we assume an expansion of the exact spacetime about a background spacetime , as in , where all -dependent terms are created by the small object (or by nonlinear interactions of its field with itself). This expansion assumes the object has only a small effect on the metric. Clearly, if the object is compact, this cannot be true everywhere: sufficiently near the object, where , the object’s own gravitational field will contain a Coulomb term —just as large as the background , and because it varies on the spatial scale rather than , having much stronger curvature than .

This fact motivates the use of matched asymptotic expansions. At distances from the object (for example, in an EMRI), we expand the exact spacetime around , as above; I will call this the outer expansion. At distances from the object (or in dimensionful units), we introduce a second expansion, , where is the spacetime of the object were it isolated, and the perturbations are due to the fields of external objects (and to nonlinear interactions); I call this the inner expansion. In a buffer region around the object, defined by , we assume a matching condition is satisfied: if the outer expansion is re-expanded in the limit and the inner expansion is re-expanded in the limit , the two expansions must agree term by term in powers of and , since they both began as expansions of the same metric. The relationship between the various regions and expansions is shown schematically in Fig. 1, and it will be described precisely in Sec. 2.

Historically, matched asymptotic expansions have been a highly successful way of treating singular perturbation problems in which the behavior of the solution rapidly changes in a localized region. For general discussions of singular perturbation theory in applied mathematics, I refer readers to the textbook [41], and for more rigorous treatments, to Refs. [42, 43]. For general discussions in the context of general relativity, I refer them to [44, 14].

In the context of spacetimes containing small objects, matched expansions have been the standard method of tackling the problem; Refs. [45, 46, 47, 48, 23, 24, 20, 13, 16, 22] are but a small sample. When it comes to obtaining equations of motion, the method hearkens back to an early insight of Einstein and others [49, 50, 3, 51]: an object’s equations of motion can be determined from the Einstein equations in a region outside the object. Specifically, it can be determined from the field equations in the buffer region defined above. Relative to the object’s scale , the buffer region is at asymptotically large distances, allowing one to make an asympotic characterization of how well “centered” the buffer region is around the object. One such characterization is based on a definition of multipole moments in the buffer region. Again, because the buffer region is at asymptotic infinity in the object’s metric , we can define the object’s multipole moments by examining the form of the metric there, rather than having to refer to the object’s stress-energy. We can then use the metric’s mass dipole moment in the buffer region as a measure of centeredness: if we install a timelike curve in the background spacetime and define the mass dipole moment in coordinates centered on that worldline, then is a good representative worldline if the mass dipole moment vanishes. Centeredness conditions along these lines will be described in more detail in Secs. 2 and 7; for a schematic preview, see Figs. 1 and 2.

Prior to its application to self-force analyses [23, 24, 20, 13, 16, 22], this program was pursued furthest by Thorne and Hartle [48]. Where Dixon stood relative to the later non-perturbative work of Harte, Thorne and Hartle stand in the same position relative to perturbative self-force constructions. As did Dixon, they derived general laws of motion and precession for compact objects. They considered an object immersed in some external spacetime, say , and found forces and torques made up of couplings between the external curvature and the object’s multipole moments; specifically, in Cartesian coordinates that are at rest relative to a geodesic of the external spacetime (and in which the mass dipole moment vanishes), they found

(13)
(14)

where and are the object’s linear and angular momentum relative to , and are its “mass and current” quadrupole moments (see the ends of Secs. 3.2 and 3.4), is proper time (as measured in ) on , and is the flat-space, Cartesian Levi-Civita tensor. The quantities and are the electric-type and magnetic-type quadrupole tidal moments of the external universe, which describe the tidal environment the object is placed in; they are related to the Riemann tensor of according to and .

Physically, Eqs. (13)–(14) say that the object moves as a test body in the external metric; the equations have the same structure as the test-body equations of motion mentioned above in the context of Dixon’s work [36]. Up to the coupling of the object’s moments to the tidal moments of the external universe, the motion is geodesic, and the spin parallel-propagated, in the external metric. But just as in Dixon’s work, these results become useful only once one knows how to split the full metric into a self-field and an effectively external metric. What Thorne and Hartle call the “external metric” is not , but rather it is essentially defined to be whatever creates tidal fields across the object. As Thorne and Hartle themselves remarked, this “external” field includes a contribution from the object’s own field. (Likewise, the “object’s” multipole moments can be altered by the external field.)

In this sense, the self-force game is played by finding a useful split in which Thorne and Hartle’s general laws are valid. Doing so requires developing a systematic theory of matched expansions for spacetimes containing small objects, as described in the body of this paper. Using those matched expansions, we will find that at linear order, we can circle back to the point particle picture: as first shown by D’Eath [45, 46], the linearized perturbation in the outer expansion is identical to the linearized field sourced by a point particle (see also Refs.  [20, 13, 17] for more refined derivations). And at that linearized level, we will find that the effectively external metric of Thorne and Hartle can be taken to be the Detweiler-Whiting effective metric .

1.4 Gauge, motion, and long-term dynamics

Despite all the above preparation, we are still not ready to broach the problem of motion in perturbation theory. Two additional facts must first be understood: in perturbation theory, motion is intimately related to gauge freedom [52, 21, 19]; and in problems of astrophysical interest, the most important dynamical effects occur on the very long time scale  [53].

To the first point. At leading order, the object’s motion is geodesic in the background metric ; all deviation from that motion is driven by an order- force. Suppose the self-accelerated worldline is a smooth function of . Then we can write its coordinates as an expansion

(15)

where is a parameter on the worldline, and the zeroth-order term is a geodesic of . Now consider the effect of a gauge transformation. Under a transformation generated by a vector , a curve is shifted to a curve . Nothing prevents us from choosing , which leaves us with , entirely eliminating the first-order deviation from . This same idea can be carried to arbitrary order, meaning we can precisely set . In other words, the effect of the self-force appears to be pure gauge.

In one sense, this result is true. If we look at any finite region of spacetime and consider the limit in that region, the deviation from a background geodesic is, indeed, pure gauge. This does not mean it is irrelevant: in any given gauge, it must be accounted for to obtain the correct metric in that gauge. But it need not be accounted for in the linearized metric. We can always substitute the expansion (15) into to obtain

(16)

and we can then transfer the term into the second-order perturbation, . (An explicit expression for is given in Eq. 221.)

However, this analysis assumes we work in a fixed, finite domain—and as mentioned in the first paragraph of this section, we do not typically work in such a domain in problems of interest. Consider an EMRI. Gravitational waves carry away orbital energy from the EMRI at a rate . It follows that the inspiral occurs on the time scale , which is called the radiation-reaction time. So in practice, we are not looking at the limit on a finite interval of time , where is independent of ; instead, we are looking at the limit on a time interval that blows up.

This consideration forces us to adjust our thinking about motion and gauge. Loosely speaking, the deviation from geodesic motion, , is governed by an equation of the form 777More precisely, it is governed by Eq. (35).. On the radiation-reaction time scale, it therefore behaves as . In other words, it blows up in the limit . So on this domain, one cannot rightly write the worldline as a geodesic plus a self-forced correction, and one cannot use a small gauge transformation to shift the perturbed worldline onto a background geodesic; the gauge transformation would have to blow up in the limit .

Because we are solving partial differential equations and not ordinary ones, these arguments about time scales translate into arguments about spatial scales. For example, if we seek a solution in a Schwarzschild background spacetime, fields (including inaccuracies in them) propagates outward toward null infinity along curves of constant , or inward toward the future horizon along curves of constant , where is the tortoise coordinate. If one’s accuracy is limited to a time span , then it is also limited to a spatial region of similar size.

To organize our thinking, let us denote by a spacetime region roughly of size (both temporal and spatial). I call an asymptotic solution to the Einstein equations a “good” solution in if it is uniform in . That is, the asymptotic expansion must satisfy and uniformly (e.g., in a sup norm).

For the EMRI problem, we are interested in obtaining a good solution in a domain . Suppose we use an asymptotic expansion of the form (16) and incorporate into . In a gauge such as the Lorenz gauge, grows as , and so likewise grows as . Hence, on , its contribution to behaves at best as , comparable to . Clearly, this is not a good approximation. Suppose we instead eliminated using a gauge transformation generated by . This removes the offending growth in , but it commits a worse offense: it alters by an amount , which behaves at best as , or as on . Hence, if we are in a gauge where the self-force is nonvanishing, behaves poorly; if we are in a gauge where the self-force is vanishing, even behaves poorly.

Let us chase the consequences of this. To obtain a good approximation in , we need to work in a class of gauges compatible with uniformity in . This means, in particular, that if we obtain a good approximation in a particular gauge—call it a good gauge—we must confine ourselves to a class of gauges related to the good gauge by uniformly small gauge transformations. In turn, this means that the effects of the self-force are not pure gauge on . Due to dissipation, will deviate from any given geodesic by a very large amount in , but by using an allowed gauge transformation we may shift it only by a very small amount, of order , on that domain. In other words, although the self-forced deviation from is pure gauge on a domain like , it is no longer pure gauge in the domain .

1.5 Self-consistent, Gralla-Wald, and osculating-geodesics approximations

In the preceding sections, we encountered several core concepts pertaining to the motion of a small object: (i) the metric can be usefully split into a self-field and an effectively external metric in which the object behaves as a test body, (ii) the bulk motion of the object can be described in terms of forces and torques generated by freely specifiable multipole moments, (iii) by using matched expansions, all of this can be done in a vacuum region outside the object, where multipole moments can be defined from the metric and laws of motion can be derived from the vacuum Einstein equation, and (iv) the representation of the object’s bulk motion, the asymptotic expansion of the metric, and notions of gauge must all be tailored to suit the long timescales on which self-force effects accumulate.

Let me now combine this information into a cohesive framework, mostly following Ref. [13] (but see also Refs. [16, 17, 18, 19, 54]). The overarching method I describe consists of solving the Einstein equation with an “outer expansion” in some vacuum region outside the object, using only minimal information from the “inner expansion” to determine the behavior of the solution very near the object.

Since we work in the vacuum region , we seek an asymptotic solution to the vacuum Einstein equation

(17)

Following the lessons of post-Newtonian theory, I write this equation in a “relaxed” form [5, 6, 7]. I define , impose the Lorenz gauge condition888Reference [17] describes how the entirety of this section can be performed in any gauge in which the linearized Einstein tensor is hyperbolic. Section 6 below offers a more general discussion of gauge.

(18)

and write Eq. (17) with that condition imposed, making it read

(19)

where is the wave operator introduced in Eq. (4) (but here acting on the metric perturbation rather than on its trace reverse),999In the Lorenz gauge in a vacuum background, the linearized curvature tensors are related by . and the “source” is a nonlinear functional of . The background is chosen to be a smooth solution to (if matter exists outside the object, it is assumed to be sufficiently far away to lie outside ). This makes Eq. (19) a weakly nonlinear hyperbolic equation for the perturbation ; at this stage, that equation is still exact.

I wish to solve Eq. (19) in subject to two types of boundary conditions:

  • Global boundary conditions. Examples of these are retarded boundary conditions or specified Cauchy data.

  • The matching condition. In the buffer region, the solution must be compatible with an inner expansion.

Together these conditions ensure we are describing the correct physical situation. But they do not yet uniquely determine the solution. Equation (19) is called “relaxed” because, unlike Eq. (17), it can be solved no matter how the object moves; this relaxation arises because Eq. (19) is not constrained by the Bianchi identity, unlike Eq. (17). The object’s motion is determined only once the gauge condition is also imposed, thereby making the solution to the relaxed equation also a solution to the unrelaxed one. As discussed earlier, in the present problem, the motion of the object is defined by the mass dipole moment of the metric in the buffer region, and we will find that the evolution of that mass dipole moment is determined by the gauge condition.

Beyond these broad ideas, the specifics of the method, and the size of the region in which it applies, depend crucially on how one represents the object’s perturbed motion, which determines how one formulates the asymptotic solution to the Einstein equation. Here I describe three representations and corresponding asymptotic solutions: what I call “self-consistent”, “Gralla-Wald”, and “osculating geodesics” approximations.

1.5.1 Self-consistent approximation

In the self-consistent approximation, to avoid the secularly growing errors described in Sec. 1.4, I seek to directly determine an accelerated worldline that represents the object’s bulk motion; I do not wish to expand that worldline in powers of . To accommodate this, I write the perturbation as , where and the quantities after the semicolon denote a functional dependence. I expand this functional as

(20)

Despite the fact that and depend on , they are not expanded; in other words, I hold them fixed while taking the limit . Later, I will suppress the functional dependence on and simply write .

The self-consistent representation of motion is given its name because must be determined simultaneously with . It was the representation adopted in the original derivations of the MiSaTaQuWa equation [23, 26, 55], and it is the one I used in describing that equation in Sec. 1.1. It was first put on a sound and robust basis, as part of a systematic expansion of the Einstein equation, in Ref. [13]. In this section, I outline that expansion.

I first refine the region in which I seek a solution to the relaxed Einstein equation (19). Install the timelike curve (with coordinates in the background spacetime, and let be a region of size . I define to be a region of proper radius centered on , with . The region I seek a solution in is then . The inner boundary of this region lies in the buffer region, and there the solution must satisfy the matching condition (BC2).

1.5.2 Field equations

Since the coefficients in the expansion (20) depend on , it may seem they are not uniquely determined. However, here I define the functional to be the solution to the relaxed Einstein equation (19); the relaxed equation places no constraints on , and each function yields a different solution. (In the present context this means there are no constraints on the motion of the region .) The coefficients are then uniquely determined to be the solution to the th-order term in an ordinary power-series expansion of the relaxed Einstein equation. That th-order term has the form . Up to , it reads

(21a)
(21b)
(21c)

where I have defined the “th-order Ricci tensor” to be the th variation101010Cross terms like are as defined in Eq. (227).

(22)

In the concrete calculations in this paper, I will require only , which is given explicitly by

(23)

Solving Eqs. (21) subject to the boundary conditions (BC1)–(BC2) yields a functional-valued asymptotic solution to the relaxed Einstein equation (19). Now we must find a particular function for which is also an asymptotic solution to the unrelaxed equation (17). To do this, we must ensure also satisfies the gauge condition (18) to the same order.111111Equation (222) illustrates more explicitly, using a point-particle field, how the gauge condition implies an equation of motion.

I accomplish this in a systematic way by writing the accelerated equation of motion of as

(24)

and then assuming that like , the force (per unit mass) appearing on the right-hand side can be expanded as

(25)

I substitute this expansion, together with the one in Eq. (20), into the gauge condition (18) and solve order by order in while holding fixed. By holding fixed during this procedure, rather than expanding their dependence, I preserve the particular accelerated worldline that satisfies some appropriate mass-centeredness condition, such as the vanishing of a suitably defined mass dipole moment in the buffer region centered on . Solving the sequence of gauge conditions to higher and higher order yields a better and better approximation to the equation of motion of that particular worldline, without ever expanding the worldline itself. The first few of these gauge conditions are

(26a)
(26b)
(26c)

where I have defined and

(27)

This sequence determines the forces ,121212It also constrains other quantities in , particularly determining the evolution of the object’s mass and spin. thereby determining the equation of motion (24).

1.5.3 Solution method

We solve Eqs. (21) and (26) by working outward from the buffer region. Solving them in the buffer region using a local expansion, subject to (BC2), yields two things: the local form of the metric outside the object, and an equation of motion for the object.

The local form of the metric is described in Sec. 3. In line with the themes of the earlier sections, it allows a natural split into an effectively external metric and a self-field , where satisfies all the “nice” properties of the Detweiler-Whiting regular field.

Derivations of the equations of motion at first and second order are sketched in Secs. 3 and 7. If at leading order the object’s spin and quadrupole moment both vanish, then the equation of motion is [16, 18]

(28)

where . Following the steps of Appendix A.1, this equation of motion can also be written as , the geodesic equation in the effective (smooth, vacuum) metric . In other words, at least through second order in , the generalized equivalence principle described in Sec. 1.1.2 holds.

After obtaining the local results in the buffer region, one might think to solve Eqs. (21) and (26) globally in by imposing agreement with the local results on the inner boundary and then moving using the equation of motion. However, in practice, a global solution is instead obtained by analytically extending the buffer-region results into , replacing the physical metric there with the fictitious, analytically extended metric, while insisting that outside , the metric is unaltered. This procedure (described in detail in Sec. 4) allows us to work with field equations on the whole of . At order , the procedure reveals that in is precisely equal to the perturbation produced by a point mass moving on , as promised in Sec. 1.1.1. More generally, at all orders, it leads to a practical puncture scheme [56, 57, 58, 59, 17, 54], in which the puncture , a local approximation to , moves on , and the field equations in are recast as equations for a residual field that locally approximates .131313Note that although a puncture scheme utilizes approximations to and , it is designed to exactly obtain (and any finite number of its derivatives) on the worldline, meaning it does not introduce any approximation into the motion of . Nor does it introduce approximations into the physical field .

The setup of a puncture scheme is compactly summarized in Eqs. (98)–(100) below. Using this scheme, one can directly solve for the effective metric on the worldline and use it to evolve that worldline via the equation of motion, and at the same time one can obtain the physical metric outside . Although we begin with a potentially complicated extended object, this scheme illuminates the fact that in self-force theory, we do not need to know anything about the particularities of that object: at the end of the day, all necessary physical information about it is absorbed into the puncture and the motion of that puncture.

1.5.4 Accuracy estimates

How accurate will this self-consistent approximation be on a domain ? Let us make the reasonable assumption that the largest secularly growing error in the approximation arises from truncating the expansion (25) at some order , leading to an error in of order .141414Here I return to what will become my common practice of dropping the subscript on for simplicity, though I refer to the self-consistently determined center-of-mass worldline, not the freely specifiable worldline for which the relaxed Einstein equation can be solved. The largest error in is then

(29)

The order of accuracy depends on the size of the domain we work in. Suppose we work in , corresponding to the radiation-reaction time. On that domain, the error from neglecting is . Therefore, if we include only in the equation of motion, solving (via a puncture scheme) the coupled system comprising Eqs.  (21a) and (9), then the result contains errors —which is as large as our first-order perturbation. In other words, this approximation fails on . If in addition we include , solving the coupled system comprising Eq. (21a), (21b), and (28), then the error is ; hence, with this approximation, we can have faith in our field .

In the smaller domain , corresponding to the so-called dephasing time , the approximations are more accurate. Including only , by solving Eqs.  (21a) and (9), yields a first-order-accurate solution with errors