Motion of Small Objects in Curved Spacetimes: An Introduction to Gravitational Self-Force

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 desscription 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.

Appendices

Appendix 1: Expansions of the Geodesic Equation

1.1 Expansion in Powers of a Metric Perturbation

In this appendix, I examine the expansion of the geodesic equation in any sufficiently smooth metric ${\text{ g }}_{\mu \nu }$; the treatment is generic, not specialized to a spacetime containing a small object. I expand only in powers of a sufficiently smooth metric perturbation; I do not expand the worldline itself. Hence, the analysis is meant to apply to the self-consistent representation of motion, not to the Gralla-Wald representation.

The geodesic equation reads

$$\begin{aligned} \frac{d\dot{z}^\mu }{ds}+{}^{\text{ g }}\Gamma ^\mu _{\nu \rho }\dot{z}^\nu \dot{z}^\rho =\kappa \dot{z}^\mu , \end{aligned}$$

(183)

where $s$ is a potentially non-affine parameter on the curve $z^\mu (s)$, $\dot{z}^\mu \equiv \frac{dz^\mu }{ds}$ is its tangent vector field, ${}^{\text{ g }}\Gamma ^\mu _{\nu \rho }$ is the Christoffel symbol corresponding to ${\text{ g }}_{\mu \nu }$, and $\kappa =\frac{d}{ds}\ln \sqrt{-{\text{ g }}_{\mu \nu }\dot{z}^\mu \dot{z}^\nu }$.

If we now write the metric as the sum of two pieces, ${\text{ g }}_{\mu \nu }=g_{\mu \nu }+h_{\mu \nu }$, and if we take $s=\tau$, the proper time on $z^\mu$ in $g_{\mu \nu }$, and if we rewrite the geodesic equation in terms of covariant derivatives compatible with $g_{\mu \nu }$, we find

$$\begin{aligned} a^\mu = -\Delta \Gamma ^\mu _{\nu \rho }u^\nu u^\rho +\kappa u^\mu , \end{aligned}$$

(184)

where $a^\mu \equiv \frac{D^2z^\mu }{d\tau ^2}$, $u^\mu \equiv \frac{dz^\mu }{d\tau }$, and

$$\begin{aligned} \Delta \Gamma ^\alpha _{\beta \gamma }&\equiv {}^{\text{ g }}\Gamma ^\alpha _{\beta \gamma }-\Gamma ^\alpha _{\beta \gamma } = \frac{1}{2}{\text{ g }}^{\alpha \delta }\left( 2h_{\delta (\beta ;\gamma )}-h_{\beta \gamma ;\delta }\right) \end{aligned}$$

(185)

is the difference between the Christoffel symbol associated with the full metric ${\text{ g }}_{\mu \nu }$ and that associated with the background metric $g_{\mu \nu }$. With $\tau$ as a parameter, $\kappa$ becomes

$$\begin{aligned} \kappa = \frac{\frac{d}{d\tau }\sqrt{1-h_{\mu \nu }u^\mu u^\nu }}{\sqrt{1-h_{\mu \nu }u^\mu u^\nu }}. \end{aligned}$$

(186)

So far no approximation has been made; Eq. (184) is exact. If we now expand $\Delta \Gamma ^\mu _{\nu \rho }$ and $\kappa$ in powers of $h_{\mu \nu }$, we find

$$\begin{aligned} a^\alpha&= -\frac{1}{2}(g^{\alpha \delta }-h^{\alpha \delta })\!\left( 2h_{\delta (\beta ;\gamma )}-h_{\beta \gamma ;\delta }\right) \!u^\beta u^\gamma -\frac{1}{2}h_{\beta \gamma ;\delta }u^\alpha u^\beta u^\gamma u^\delta \nonumber \\&\quad -\frac{1}{2}h_{\mu \nu }h_{\beta \gamma ;\delta }u^\alpha u^\beta u^\gamma u^\delta u^\mu u^\nu -h_{\beta \gamma }u^\alpha a^\beta u^\gamma +\mathcal {O}(h^3). \end{aligned}$$

(187)

This equation is complicated by the fact that the acceleration appears on both sides in a nontrivial way. To disentangle the acceleration from the perturbation, I assume that $a^\mu$, too, has an expansion in powers of $h_{\mu \nu }$,

$$\begin{aligned} a^\mu = a^\mu _\mathrm{lin}+a^\mu _\mathrm{quad}+\mathcal {O}(h^3), \end{aligned}$$

(188)

where $a^\mu _\mathrm{lin}$ is linear in $h_{\mu \nu }$ and $a^\mu _\mathrm{quad}$ is quadratic in it. Substituting this expansion into Eq. (187), one finds

$$\begin{aligned} a^\alpha _\mathrm{lin}&= -\frac{1}{2}P^{\alpha \delta }\!\left( 2h_{\delta (\beta ;\gamma )}-h_{\beta \gamma ;\delta }\right) \!u^\beta u^\gamma ,\end{aligned}$$

(189)

$$\begin{aligned} a^\alpha _\mathrm{quad}&= -\frac{1}{2}P^{\alpha \mu }h^\delta {}_\mu \!\left( 2h_{\delta (\beta ;\gamma )}-h_{\beta \gamma ;\delta }\right) \!u^\beta u^\gamma , \end{aligned}$$

(190)

where $P^{\alpha \mu }\equiv g^{\alpha \mu }+u^\alpha u^\mu$. Summing these, we have

$$\begin{aligned} \frac{D^2z^\mu }{d\tau ^2} = -\frac{1}{2}P^{\alpha \mu } (g^\delta {}_\mu -h^\delta {}_\mu )\!\left( 2h_{\delta (\beta ;\gamma )}-h_{\beta \gamma ;\delta }\right) \!u^\beta u^\gamma +\mathcal {O}(h^3). \end{aligned}$$

(191)

As applied to the case of the effective metric $\tilde{g}_{\mu \nu }=g_{\mu \nu }+h^\mathrm{R}_{\mu \nu }$ (i.e., replacing $h_{\mu \nu }$ with $h^\mathrm{R}_{\mu \nu }$), Eq. (191) agrees with the second-order self-forced equation of motion derived in the body of the paper.

1.2 Expansion in Powers of a Metric Perturbation and a Worldline Deviation

In the last section, I expanded the geodesic equation while holding the solution $z^\mu _\epsilon (s)$ to that equation fixed. I now expand $z^\mu _\epsilon (s)$ as well. This procedure yields a sequence of equations for the terms in the expansion of $z^\mu _\epsilon (s)$, suitable for a Gralla-Wald approximation. My approach to the expansion closely follows the treatment of geodesic deviation in Sect. 1.10 in Ref. [86]

I first describe the geometry of the situation. Consider a family of worldlines $z^\mu (\tau ,\epsilon )$, with each member $z_\epsilon ^\mu (\tau )=z^\mu (\tau ,\epsilon )$ governed by the equation of motion (184). Each member satisfies

$$\begin{aligned} \frac{D^2 z_\epsilon ^\mu }{d\tau ^2}=F^\mu (\tau ,\epsilon ), \end{aligned}$$

(192)

where $\tau$ is proper time on $z_\epsilon ^\mu$, and $F^\mu$ is given by the right-hand side of Eq. (184). The family generates a two dimensional surface $\mathcal {S}$ with a tangent bundle spanned by $u^\mu \equiv \frac{\partial x^\mu }{\partial \tau }$ and $v^\mu \equiv \frac{\partial x^\mu }{\partial \epsilon }$. An important relation between these vector fields can be found from $\frac{\partial ^2 x^\mu }{\partial \tau \partial \epsilon }=\frac{\partial ^2 x^\mu }{\partial \epsilon \partial \tau }$, which implies $\mathcal {L}_u v^\mu =0=\mathcal {L}_v u^\mu$, and from there,

$$\begin{aligned} v^\mu {}_{;\nu }u^\nu = u^\mu {}_{;\nu }v^\nu . \end{aligned}$$

(193)

Now, we seek to describe the deviation of an accelerated worldline $z^\mu _\epsilon (\tau )$ from the zeroth order, geodesic worldline $z^\mu _0(\tau )\equiv z^\mu (\tau ,0)$. The first step is to expand the worldline in the power series

$$\begin{aligned} z^\mu (\tau ,\epsilon ) = z^\mu _0(\tau ) +\epsilon z^\mu _1(\tau ) + \epsilon ^2 z_2^\mu (\tau )+ O(\epsilon ^3), \end{aligned}$$

(194)

where

$$\begin{aligned} z^\mu _n(\tau ) = \frac{1}{n!}\epsilon ^n\frac{\partial ^n z^\mu }{\partial \epsilon ^n}(\tau ,0). \end{aligned}$$

(195)

We may also write the expansion as $z^\mu (\tau ,\epsilon )=\sum \frac{1}{n!}\epsilon ^n\mathcal {L}^n_v z^\mu |_{z_0(\tau )}$. Note that here $z^\mu$ is a scalar field equal to the $\mu$th coordinate field on the surface $\mathcal {S}$. The leading-order term is the family member $z_0^\mu (\tau )\equiv z^\mu (\tau ,0)$. The second term is $z_1^\mu (\tau )=\mathcal {L}_v z^\mu |_{z_0}=v^\mu (z_0(\tau ))$, a vector on $z^\mu _0$. But at second order and beyond, a subtlety arises: unlike the first derivative along a curve, which is a tangent vector, second and higher derivatives are not immediately vectorial quantities. The function $z^\mu (\tau ,\epsilon )$ describes a curve in a particular set of coordinates, and the corrections $z^\mu _n$ depend on the coordinate system in which one defines $z^\mu (\tau ,\epsilon )$. Since my notion of an object’s center is established with reference to a comoving normal coordinate system, I wish my covariant measure of the second-order deviation to agree, component by component, with $\frac{1}{2}\frac{\partial ^n z^\mu }{\partial \epsilon ^n}(\tau ,0)$ when evaluated in a normal coordinate system centered on $z^\mu _0$; Sect. 5 describes the utility of this choice when re-expanding a self-consistent approximation into Gralla-Wald or osculating-geodesics form. With that in mind, I define the vector

$$\begin{aligned} w^\alpha \equiv \frac{1}{2}\frac{Dv^\alpha }{d\epsilon } = \frac{1}{2}v^\beta \nabla _\beta v^\alpha \end{aligned}$$

(196)

and I seek an evolution equation for its restriction to $z_0^\mu$,

$$\begin{aligned} z^\alpha _{2\mathrm{F}}(\tau )\equiv w^\alpha |_{z_0(\tau )}. \end{aligned}$$

(197)

$z^\alpha _{2\mathrm{F}}$ is the second-order term in the expansion (194) when that expansion is performed in Fermi normal coordinates centered on $z_0^\mu$.

In addition to the choice of coordinates, the expansion (194) depends on the particular choice of parametrization $(\tau ,\epsilon )$ of the surface $\mathcal {S}$. A change of parametrization alters the direction of expansion away from $z_0^\mu$. Here, the parametrization is chosen such that $\tau$ is proper time along each curve $z^\mu _\epsilon (\tau )$, and a flow line generated by $v^\mu$ links points on different curves $z^\mu _\epsilon (\tau )$ at the same value of $\tau$. When restricted to $z_0^\mu$, the parameter $\tau$ is $\tau _0$, the proper time on $z_0^\mu$.

With all preliminaries established, I now proceed to find the evolution equations for $z_0^\mu$, $z_1^\mu$, and $z_2^\mu$. The leading term clearly satisfies

$$\begin{aligned} \frac{D^2z_0^\mu }{d\tau _0^2}=F^\mu (\tau ,0)=0. \end{aligned}$$

(198)

For the others, I first find evolution equations for $v^\mu$ and $w^\mu$ and then evaluate the results on $z^\mu _0$. At first order, using Eqs. (192) and (193), we have

$$\begin{aligned} \frac{D^2v^\alpha }{d\tau ^2}&= \left( v^\alpha {}_{;\beta }u^\beta \right) _{;\gamma }u^\gamma \end{aligned}$$

(199)

$$\begin{aligned}&= \left( u^\alpha {}_{;\beta }v^\beta \right) _{;\gamma }u^\gamma \end{aligned}$$

(200)

$$\begin{aligned}&= F^\alpha _{;\gamma }v^\gamma -R^\alpha {}_{\mu \beta \nu }u^\mu v^\beta u^\nu , \end{aligned}$$

(201)

where the second line follows from Eq. (193) and the third line follows from the Ricci identity and Eq. (192). Evaluating on $z_0^\mu$, I write this as

$$\begin{aligned} \frac{D^2z^\alpha _1}{d\tau _0^2} = \check{F}_1^\alpha (\tau _0)-R^\alpha {}_{\mu \beta \nu }u_0^\mu z^\beta _1 u_0^\nu , \end{aligned}$$

(202)

where $\check{F}^\alpha _1\equiv \frac{DF^\alpha }{d\epsilon }|_{\gamma _0}$. This is a generalization from the usual geodesic deviation equation to the deviation between neighbouring accelerating worldlines; it is valid even if $F^\mu (\tau ,0)\ne 0$.

At second order, repeated use of Eq. (193) and Ricci’s identity leads to

$$\begin{aligned} \frac{D^2w^\alpha }{d\tau ^2}&= \frac{1}{2}\left[ \left( v^\alpha {}_{;\beta }v^\beta \right) _{;\mu }u^\mu \right] _{;\nu }u^\nu \end{aligned}$$

(203)

$$\begin{aligned}&= \frac{1}{2}\left[ \left( u^\alpha {}_{;\beta }u^\beta \right) _{;\gamma }v^\gamma \right] _{;\delta }v^\delta +\frac{1}{2}R^\alpha {}_{\mu \beta \nu ;\gamma }\left( v^\mu u^\beta v^\nu u^\gamma -u^\mu v^\beta u^\nu v^\gamma \right) \nonumber \\&\quad -R^\alpha {}_{\mu \beta \nu }\left( u^\mu w^\beta u^\nu +2u^\mu {}_{;\gamma }v^\gamma v^\beta u^\nu +\tfrac{1}{2}v^\mu v^\beta u^\nu _{;\gamma }u^\gamma \right) . \end{aligned}$$

(204)

Evaluating on $z^\mu _0$ and using Eq. (192), we can write this as

$$\begin{aligned} \frac{D^2z_{2\mathrm{F}}^\alpha }{d\tau _0^2}&= \check{F}^\alpha _2(\tau _0) -R^\alpha {}_{\mu \beta \nu }\left( u_0^\mu z_{2\mathrm{F}}^\beta u_0^\nu +2u_1^\mu z_1^\beta u_0^\nu \right) +2R^\alpha {}_{\mu \beta \nu ;\gamma }z_1^{(\mu } u_0^{\beta )} z_1^{[\nu } u_0^{\gamma ]} \end{aligned}$$

(205)

where $\check{F}^\alpha _2\equiv \frac{1}{2}\frac{D^2F^\alpha }{d\epsilon ^2}|_{\gamma _0}$ and $u_1^\mu \equiv \frac{Dz_1^\mu }{d\tau }$. Equation (205) describes the second deviation between neighbouring accelerating worldlines. In the case of neighbouring geodesics, it agrees with “Bazanski’s equation” in the form given in Eq. (5.9) of Ref. [87].

The quantities $\check{F}^\mu _n$ appearing in Eqs. (202) and (205) can be straightforwardly evaluated by performing the expansion $h_{\mu \nu }(x,\epsilon )=\epsilon \check{h}^1_{\mu \nu }(x)+\epsilon ^2\check{h}^2_{\mu \nu }(x)+\mathcal {O}(\epsilon ^3)$ in Eq. (191) and then taking covariant derivatives with respect to $v^\mu$. The results are

$$\begin{aligned} \check{F}_1^\mu = \frac{1}{2}P^{\mu \nu }_0\left( \check{h}^1_{\sigma \lambda ;\rho }-2\check{h}^1_{\rho \sigma ;\lambda }\right) u^\sigma _0 u^\lambda _0 \end{aligned}$$

(206)

and

$$\begin{aligned} \check{F}_2^\mu&= -\frac{1}{2}P_0^{\mu \nu }\left( 2\check{h}^2_{\nu \sigma ;\lambda }-\check{h}^2_{\sigma \lambda ;\nu }\right) u^\sigma _0 u^\lambda _0 -\frac{1}{2}P_0^{\mu \nu }\left( 2\check{h}^1_{\nu \sigma ;\lambda \delta }-\check{h}^1_{\sigma \lambda ;\nu \delta }\right) u^\sigma _0 u^\lambda _0z_1^\delta \nonumber \\&\quad -\left( 2\check{h}^1_{\nu \sigma ;\lambda }-\check{h}^1_{\sigma \lambda ;\nu }\right) \left( u^{(\mu }_1 u^{\nu )}_0u^\sigma _0u_0^\lambda +P_0^{\mu \nu }u^{(\sigma }_1 u^{\lambda )}_0\right) \nonumber \\&\quad +P_0^{\mu \nu }\check{h}^1_\nu {}^\rho \left( 2\check{h}^1_{\rho \sigma ;\lambda }-\check{h}^1_{\sigma \lambda ;\rho }\right) u^\sigma _0 u^\lambda _0. \end{aligned}$$

(207)

As applied to the case of the effective metric $\tilde{g}_{\mu \nu }=g_{\mu \nu }+h^\mathrm{R}_{\mu \nu }$ (i.e., replacing $\check{h}^n_{\mu \nu }$ with $\check{h}^{\mathrm{R}n}_{\mu \nu }$), Eqs. (202) and (205), together with Eqs. (206) and (207), are the second-order expansion of the motion that apply in a Gralla-Wald approximation.

Appendix 2: Expansion of Point-Particle Fields in Powers of a Worldline Deviation

In this appendix, I derive the linear terms in expansions of the point particle stress-energy $T_1^{\mu \nu }(x;z)$ and the Lorenz-gauge retarded field $h^1_{\mu \nu }(x;z)$ when the worldline is expanded as $z^\mu (s,\epsilon )=z_0^\mu (s)+\epsilon z_1^\mu (s)+\mathcal {O}(\epsilon ^2)$, where $s$ is an arbitrary parameter. I also establish the identification of these linear terms with the mass dipole moment terms found from the local analysis in Sect. 3.

1.1 Stress-Energy

I write the stress-energy in the parametrization-invariant form [9]

$$\begin{aligned} T^{\alpha \beta }_1(x;z) = m\int _\gamma g^\alpha _{\alpha '}(x,z)g^\beta _{\beta '}(x,z) \dot{z}^{\alpha '}\dot{z}^{\beta '} \delta (x,z) \frac{ds}{\sqrt{-g_{\mu '\nu '}(z)\dot{z}^{\mu '}\dot{z}^{\nu '}}}, \end{aligned}$$

(208)

where $g^\alpha _{\alpha '}(x;z)$ is a parallel propagator from the source point $x'=z(s,\epsilon )$ to the field point $x$, and $\dot{z}^\mu \equiv \frac{dz^\mu }{ds}$.

Substituting the expansion (30) into this stress-energy tensor, we obtain

$$\begin{aligned} T_1^{\alpha \beta }(x;z)&= m\int _{\gamma _0} \left[ g^\alpha _{\alpha '}(x,z_0)g^\beta _{\beta '}(x,z_0) \dot{z}_0^{\alpha '}\dot{z}_0^{\beta '} +\epsilon z_1^{\mu '}\nabla _{\mu '}(g^\alpha _{\alpha '}g^\beta _{\beta '} \dot{z}^{\alpha '}\dot{z}^{\beta '})|_{\epsilon =0}\right] \nonumber \\&\quad \times \left[ \delta (x,z_0)+\epsilon z^{\nu '}_1\nabla _{\nu '}\delta (x,z)|_{\epsilon =0}\right] \left[ 1-\epsilon \frac{\dot{z}_{0\delta '}z_1^{\gamma '}\nabla _{\gamma '}\dot{z}_0^{\delta '}}{\dot{z}^{\kappa '}_0\dot{z}_{0\kappa '}}\right] \frac{ds}{\sqrt{-\dot{z}^{\rho '}_0\dot{z}_{0\rho '}}}\nonumber \\&\quad +O(\epsilon ^2). \end{aligned}$$

(209)

In each instance, the evaluation at $\epsilon =0$ occurs after taking the derivative.

I simplify this expression using the distributional identities $\nabla _{\mu '}\delta (x,z)=-g^\mu _{\mu '}\nabla _\mu \delta (x,z)$ and $g^\alpha _{\alpha ';\beta '}\delta (x,z)=0$ [9]. I also use the identity

$$\begin{aligned} z_1^{\mu '}(\nabla _{\mu '}\dot{z}^{\alpha '})\big |_{\epsilon =0}=\frac{Dz_1^{\alpha '}}{ds}\equiv \dot{z}_1^{\alpha '}, \end{aligned}$$

(210)

which follows in the same manner as Eq. (193).

The result of those simplifications is

$$\begin{aligned} \epsilon T_1^{\alpha \beta }(x;z) = \epsilon T^{\alpha \beta }_1(x;z_0)+\epsilon ^2 \delta T_1^{\alpha \beta }(x;z_0,z_1)+O(\epsilon ^3), \end{aligned}$$

(211)

with

$$\begin{aligned} T^{\alpha \beta }_1(x;z_0)&= m\int _{\gamma _0} g^\alpha _{\alpha '}g^\beta _{\beta '} u_0^{\alpha '}u_0^{\beta '}\delta (x,z_0)d\tau _0',\end{aligned}$$

(212)

$$\begin{aligned} \delta T^{\alpha \beta }_1(x;z_0,z_1)&= m\int _{\gamma _0} g^\alpha _{\alpha '}g^\beta _{\beta '} \Bigl [\left( 2u_0^{(\alpha '}u_{1}^{\beta ')}-u_0^{\alpha '}u_0^{\beta '}u_{0\gamma '}u_{1}^{\gamma '}\right) \delta (x,z_0)\nonumber \\&\quad -u_0^{\alpha '}u_0^{\beta '}z_{1}^{\gamma '}g^\gamma _{\gamma '}\nabla _{\gamma }\delta (x,z_0)\Bigr ]d\tau _0', \end{aligned}$$

(213)

where $u_1^\mu (\tau _0)\equiv \frac{Dz_{1}^{\mu }}{d\tau _0}$, and the parallel propagators are evaluated at $(x,z_0(\tau _0'))$. I have simplified these expressions by reparametrizing $z_0^\mu$ in terms of $\tau _0$, the proper time on $\gamma _0$, but note that this does not correspond to choosing the original parameter $s=\tau _0$. Equations (211)–(213) are valid for any choice of parameter $s$, and $z^\mu _1(\tau _0)$ actually depends on the original choice of $s$: a change of parametrization $s\rightarrow s'( s,\epsilon )$ will change the direction of $z^\mu _1$, in particular changing whether or not $z^\mu _1$ is orthogonal to $u_0^\mu$.

To eliminate this dependence on the initial choice of parametrization, I rewrite $\delta T^{\alpha \beta }_1$ in terms of the orthogonal part of $z^\mu _1$, $z^\mu _{1\perp }\equiv (\delta ^\mu _\nu +u^\mu _0u_{0\nu })z^\nu _1$. The result is

$$\begin{aligned} \delta T^{\alpha \beta }_1 = m\!\int _{\gamma _0}\! g^\alpha _{\alpha '}g^\beta _{\beta '} \left[ 2u_0^{(\alpha '}u_{1\perp }^{\beta ')}\delta (x,z_0) -u_0^{\alpha '}u_0^{\beta '}z_{1\perp }^{\gamma '}g^\gamma _{\gamma '}\nabla _{\gamma }\delta (x,z_0)\right] d\tau _0', \end{aligned}$$

(214)

where $u_{1\perp }^\mu \equiv \frac{Dz_{1\perp }^{\mu }}{d\tau _0}$. Note that the part of $z^\mu _1$ parallel to $u_0^\mu$ does not appear in this expression. Again, this result does not depend on the initial choice of parametrization. One need not choose a parametrization by hand that enforces $z_1^\mu u_{0\mu }=0$; no matter the choice, only the perpendicular part plays a role in the field equations.

The quantity $\delta T^{\mu \nu }_1(x;z_0)$ is equal to , where $v^\mu =\frac{\partial z^\mu (s,\epsilon )}{\partial \epsilon }$ is introduced in Appendix 1, and is the Lie derivative introduced in Sect. 5. In fact, for any vector $\xi ^\mu$, is given by Eq. (214) with the replacement $z_1^\mu \rightarrow \xi ^\mu$ and $u^\mu _0\rightarrow u^\mu$. This quantity is useful when considering gauge transformations in the self-consistent approximation. Also useful is the ordinary Lie derivative of $T^{\mu \nu }_1$; taking similar steps as above, one finds

$$\begin{aligned} \mathcal {L}_\xi T^{\alpha \beta }_1(x;z)&= -m\!\int _\gamma \! g^\alpha _{\alpha '}g^\beta _{\beta '} \bigg \{\left[ 2u^{(\alpha '}\frac{D\xi _{\perp }^{\beta ')}}{d\tau '} +u^{\alpha '}u^{\beta '}\left( \frac{d\xi _{\parallel }}{d\tau }+\xi ^{\rho '}{}_{;\rho '}\right) \right] \delta (x,z)\nonumber \\&\quad -u^{\alpha '}u^{\beta '}\xi _{\perp }^{\gamma '}g^\gamma _{\gamma '}\nabla _{\gamma }\delta (x,z)\bigg \}d\tau ', \end{aligned}$$

(215)

where $\xi _{\perp }^{\beta '}\equiv P^{\beta '}{}_{\alpha '}\xi ^{\alpha '}$ and $\xi _{\parallel }\equiv u_{\mu '}\xi ^{\mu '}$. The sum of the two Lie derivatives yields the simple result

(216)

A $\xi ^\mu \nabla _\mu \delta (x,z)$ term signals that the mass $m$ is displaced from $z^\mu$ by an amount $\xi ^\mu$; the lack of any $\nabla _\mu \delta (x,z)$ term in Eq. (216) signals that the displacements due to the two derivatives cancel one another, leaving the mass $m$ moving on $z^\mu$.

1.2 Metric Perturbation

In the Lorenz gauge, the first-order self-consistent field, given some global boundary conditions, is given by

$$\begin{aligned} h^1_{\mu \nu }(x;z) = 4m\int \bar{G}_{\mu \nu \mu '\nu '}u^{\mu '}u^{\nu '}d\tau ', \end{aligned}$$

(217)

where $G_{\mu \nu \mu '\nu '}$ is the Green’s function that comports with the global boundary conditions. I wish to expand this about $z^\mu =z_0^\mu$ to obtain something of the form

$$\begin{aligned} \epsilon h^1_{\mu \nu }(x;z) = \epsilon h^1_{\mu \nu }(x;z_0)+\epsilon ^2 \delta h^1_{\mu \nu }(x;z_0,z_1)+\mathcal {O}(\epsilon ^3). \end{aligned}$$

(218)

There are two methods available for achieving this: directly, following steps analogous to those in the previous section; or by making use of the result of the previous section.

Here I adopt the second method. Noting that , that , and that commutes with derivatives acting at $x$, we have

(219)

Hence,

(220)

From Eq. (214), this evaluates to

$$\begin{aligned} \delta h^1_{\mu \nu }(x;z_0,z_1) = 4m\!\int _{\gamma _0}\!\! \left( 2\bar{G}_{\mu \nu \mu '\nu '}u_0^{(\mu '}u_{1\perp }^{\nu ')} + \bar{G}_{\mu \nu \mu '\nu ';\gamma '}u^{\mu '}_0u^{\nu '}_0z_{1\perp }^{\gamma '}\right) d\tau _0'. \end{aligned}$$

(221)

1.2.1 Gauge Condition

It is worth examining how $h^1_{\alpha \beta }(x;z)$ and $\delta h^1_{\alpha \beta }(x;z_0,z_1)$ contribute to the Lorenz gauge condition. Those contributions are easily found by invoking the identity $\nabla ^\nu G_{\mu \nu \mu '\nu '}=-G_{\mu (\mu ';\nu ')}$ [9], where $G_{\mu \mu '}$ is a Green’s function for the vector wave equation $\Box V_\mu =S_\mu$, and both Green’s functions must satisfy the same boundary conditions. Performing a trace-reversal on Eq. (217), taking the divergence of the result, using the Green’s-function identity, and integrating by parts yields

$$\begin{aligned} \nabla ^\beta \bar{h}^1_{\alpha \beta }(x;z) = 4\int G_{\alpha \alpha '}\frac{D}{d\tau '}(mu^{\alpha '})d\tau '. \end{aligned}$$

(222)

(Earlier results in this section assumed constant $m$, but here I momentarily leave it arbitrary for generality.) The contribution to the gauge condition is determined entirely by $\frac{dm}{d\tau }$ and the acceleration of $z^\mu$. In a Gralla-Wald expansion, one has $\nabla ^\beta \check{\bar{h}}^1_{\alpha \beta }(x;z_0)=0$, from which one can read off $\frac{dm}{d\tau }=0$ and $a_0^\mu =0$. In a self-consistent expansion, one instead has $\nabla ^\beta [\epsilon \bar{h}^1_{\alpha \beta }(x;z)+\epsilon ^2\bar{h}^2_{\alpha \beta }(x;z)]=\mathcal {O}(\epsilon ^3)$ [or more precisely, Eq. (26)], which determines Eq. (9).

Doing the same with Eq. (221) yields

$$\begin{aligned} \nabla ^\beta \delta \bar{h}^1_{\alpha \beta }(x;z_0,z_1) = 4m\int G_{\alpha \alpha '}\left( \frac{D^2z^{\alpha '}_{1\perp }}{d\tau '^2_0}+R^{\alpha '}{}_{\beta '\gamma '\delta '}u_0^{\beta '}z_{1\perp }^{\gamma '}u_0^{\delta '}\right) d\tau '_0. \end{aligned}$$

(223)

The contribution to the gauge condition is determined entirely by the acceleration of the deviation from $z_0^\mu$ (together with the geodesic-deviation term). In a Gralla-Wald expansion, $\delta h^1_{\alpha \beta }(x;z_0,z_1)$ is included in $\check{h}^2_{\alpha \beta }(x;z_0)$, and $\nabla ^\beta \check{\bar{h}}^2_{\alpha \beta }(x;z_0)=0$ determines Eq. (35).

1.3 Local Expansion and Identification of Mass Dipole Moment

In Sect. 4.1, I showed that the mass dipole moment of the object creates a term (96) [and contributes to the term (94)] in the object’s skeletal stress-energy. By comparing that result to Eq. (214), we can make the identification $M^i=mz^i_1$, exactly as concluded elsewhere in the paper. Since the two stress-energy tensors are the same, it follows that $\delta h^1_{\mu \nu }(x;z_0,z_1)$ includes the entire contribution to $\check{h}^2_{\mu \nu }$ coming from the mass dipole moment.

Here, I complete the circle by performing a local expansion of Eq. (221) near $z_0^\mu$ and showing $\delta h^1_{\mu \nu }(x;z_0,z_1)$ reproduces the order-$1/r^2$ and $1/r$ terms for the mass dipole seed solution in Fermi normal coordinates. The method of local expansion is elaborated in Ref. [9]. In particular, I follow steps analogous to those in Sect. 23.2 of that reference. To avoid belabouring the details, here I provide only the barest sketch; I leave it to the interested reader to fill in the gaps using the tools of Ref. [9].

The starting point is the Hadamard decomposition of the retarded Green’s function, $G_{\mu \nu \mu '\nu '}=U_{\mu \nu \mu '\nu '}\delta _+(\sigma ) + V_{\mu \nu \mu '\nu '}\theta _+(-\sigma )$. Here $\delta _+(\sigma )$ is a Dirac delta function supported on the past light cone of $x$, and $\theta _+(-\sigma )$ is a Heaviside step function supported in the interior of that light cone. $\sigma (x,x')$ is one-half the squared geodesic distance from $x'$ to $x$, such that it vanishes when a null geodesic connects the two points.

From this starting point, the final result is obtained by a two-step process: (i) evaluating the integrals over $\tau _0'$ by changing the integration variable to $\sigma$, and (ii) writing the retarded distance from $x'$ to $x$ in terms of the Fermi radial coordinate $r$. The results for the two terms in Eq. (221) are

$$\begin{aligned} 8m\int \bar{G}_{\mu \nu \mu '\nu '}u_0^{(\mu '}\frac{Dz_{1\perp }^{\nu ')}}{d\tau '_0} d\tau '_0 = \frac{8m}{r}u_{0(\mu }e^i_{\nu )}\dot{z}_{1i} +\mathcal {O}(r^0), \end{aligned}$$

(224)

where $\dot{z}^i_1 = \frac{dz^i_1}{dt}$, and

$$\begin{aligned} 4m\int \bar{G}_{\mu \nu \mu '\nu ';\gamma '}u^{\mu '}_0u^{\nu '}_0z_{1\perp }^{\gamma '}d\tau _0' = \frac{2m}{r^2}z_1^an_a\delta _{\mu \nu }+\mathcal {O}(r^0). \end{aligned}$$

(225)

The components in Fermi coordinates are

$$\begin{aligned} \delta h^1_{tt}&= \frac{2mz^i_1n_i}{r^2}+\mathcal {O}(r^0),\end{aligned}$$

(226a)

$$\begin{aligned} \delta h^1_{ta}&= \frac{4m \dot{z}_{1a}}{r}+\mathcal {O}(r^0),\end{aligned}$$

(226b)

$$\begin{aligned} \delta h^1_{ab}&= \frac{2mz^i_1n_i}{r^2}\delta _{ab}+\mathcal {O}(r^0). \end{aligned}$$

(226c)

Here we see that with the identification $M^i=mz_1^i$, the $tt$ and $ab$ components are precisely those in Eq. (105), and the $ta$ component is precisely the term proportional to $\dot{M}^i$ in Eq. (106). In other words, the integral (224) corresponds to the mass dipole moment’s contribution to the monopole seed $h^\mathrm{seed}_{\mu \nu }(x;z_0,\delta m)$, while the integral (225) corresponds to the mass-dipole seed $h^\mathrm{seed}_{\mu \nu }(x;z_0,M)$.

Appendix 3: Identities for Gauge Transformations of Curvature Tensors

Let $A[g]$ be a tensor of any rank constructed from a metric $g$. (To streamline the presentation, I adopt index-free notation throughout most of this appendix.) Now define

$$\begin{aligned} \delta ^n A[f_1,\ldots ,f_n] \equiv \frac{1}{n!}\frac{d^n}{d\lambda _1\cdots d\lambda _n}A[g+\lambda _1 f_1 +\cdots + \lambda _n f_n]\big |_{\lambda _1=\cdots =\lambda _n=0}. \end{aligned}$$

(227)

This tensor is linear in each of its arguments $f_1,\ldots ,f_n$; it is also symmetric in them. In the case that all the arguments are the same, we have $\delta ^n A[h,\ldots ,h] = \frac{1}{n!}\frac{d^n}{d\lambda ^n}A[g+\lambda h]\big |_{\lambda =0}$, the piece of $A[g+h]$ containing precisely $n$ factors of $h$ and its derivatives.

The following identities are easily proved by writing Lie derivatives as ordinary derivatives:

$$\begin{aligned} \mathcal {L}_\xi A[g]&=\delta A[\mathcal {L}_\xi g], \end{aligned}$$

(228)

$$\begin{aligned} \tfrac{1}{2}\mathcal {L}^2_\xi A[g]&=\tfrac{1}{2}\delta A[\mathcal {L}^2_\xi g] + \delta ^2A[\mathcal {L}_\xi g,\mathcal {L}_\xi g],\end{aligned}$$

(229)

$$\begin{aligned} \mathcal {L}_\xi \delta A[h]&= \delta A[\mathcal {L}_\xi h] + 2\delta ^2 A[\mathcal {L}_\xi g, h]. \end{aligned}$$

(230)

Note that $\delta ^2 A[\mathcal {L}_\xi g,h]= \delta ^2 A[h,\mathcal {L}_\xi g]=\frac{1}{2}\left( \delta ^2 A[h,\mathcal {L}_\xi g]+\delta ^2 A[\mathcal {L}_\xi g,h]\right)$. As an example, if $A$ is the Ricci tensor, then

$$\begin{aligned} \mathcal {L}_\xi R_{\mu \nu }[g]&=\delta R_{\mu \nu }[\mathcal {L}_\xi g], \end{aligned}$$

(231)

$$\begin{aligned} \tfrac{1}{2}\mathcal {L}^2_\xi R_{\mu \nu }[g]&=\tfrac{1}{2}\delta R_{\mu \nu }[\mathcal {L}^2_\xi g] + \delta ^2R_{\mu \nu }[\mathcal {L}_\xi g,\mathcal {L}_\xi g],\end{aligned}$$

(232)

$$\begin{aligned} \mathcal {L}_\xi \delta R_{\mu \nu }[h]&= \delta R_{\mu \nu }[\mathcal {L}_\xi h] + 2\delta ^2 R_{\mu \nu }[h,\mathcal {L}_\xi g], \end{aligned}$$

(233)

where I have restored indices to avoid confusion with the Ricci scalar.

To establish Eq. (228), one can write the metric as a function of a parameter $\lambda$ along the flow generated by $\xi$ and then perform a Taylor expansion:

$$\begin{aligned} \mathcal {L}_\xi A[g] = \frac{d}{d\lambda }A\!\left[ g(0)+\lambda \frac{dg}{d\lambda }\Big |_{\lambda =0}\right] \!\!\bigg |_{\lambda =0} = \delta A\left[ \frac{dg}{d\lambda }\big |_{\lambda =0}\right] = \delta A[\mathcal {L}_\xi g]. \end{aligned}$$

(234)

Similarly, to establish Eq. (229), one can write

$$\begin{aligned} \tfrac{1}{2}\mathcal {L}^2_\xi A[g]&= \tfrac{1}{2}\frac{d^2}{d\lambda ^2} A\!\left[ g(0)+\lambda \frac{dg}{d\lambda }\Big |_{\lambda =0}+\tfrac{1}{2}\lambda ^2\frac{dg}{d\lambda ^2}\Big |_{\lambda =0}\right] \!\!\bigg |_{\lambda =0}\end{aligned}$$

(235a)

$$\begin{aligned}&= \tfrac{1}{2}\delta A[\mathcal {L}^2_\xi g] + \delta ^2A[\mathcal {L}_\xi g,\mathcal {L}_\xi g], \end{aligned}$$

(235b)

and to establish Eq. (230), one can write $g$ as a function of parameters $(\lambda ,\epsilon )$, where $h\equiv \frac{dg}{d\epsilon }\big |_{\epsilon =0}$, and then write

$$\begin{aligned} \mathcal {L}_\xi \delta A[h]&= \frac{d^2}{d\lambda d\epsilon }A\!\left[ g(\lambda ,0)+\epsilon \frac{d g}{d\epsilon }(\lambda ,0)\right] \!\!\bigg |_{\lambda =\epsilon =0}\end{aligned}$$

(236a)

$$\begin{aligned}&= \frac{d^2}{d\lambda d\epsilon }A\!\left[ g(0,0)+\lambda \frac{d g}{d\lambda }(0,0) + \epsilon \frac{d g}{d\epsilon }(0,0)+\lambda \epsilon \frac{d^2 g}{d\lambda d\epsilon }(0,0)\right] \!\!\bigg |_{\lambda =\epsilon =0}\nonumber \\&= \delta A[\mathcal {L}_\xi h] + 2\delta ^2 A[h,\mathcal {L}_\xi g]. \end{aligned}$$

(236b)