Matched Expansions and the Small-Body Limit

Gravitational self-force theory does not begin by declaring a compact object to be a point particle. It begins with a one-parameter family of smooth vacuum spacetimes containing a body whose mass and size shrink together relative to an external curvature scale. Matched asymptotic expansions then identify the body’s exterior multipoles, select a representative worldline, and derive its motion. The leading worldline is a geodesic of the background; the first correction is geodesic motion in a regular effective metric.

Required background. PDEs and Green Functions for Gravitation supplies distributional sources and retarded boundary-value problems. Helpful background. The Einstein Equation supplies the nonlinear vacuum equations; Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge transformations; Presymplectic Potentials and Currents supplies the variational meaning of reduced motion; Black-Hole Perturbation Theory supplies the fixed-background field equations used after matching.

Green-function bridge. A point-supported source is a distribution: an equation such as $L\Phi=-4\pi q\int\delta_4(x,z(\tau))\,d\tau$ means equality after integration against every smooth compactly supported test function. A retarded Green function $G_{\mathrm{ret}}(x,x')$ is a bidistribution satisfying the corresponding delta-normalized equation in its first argument and vanishing unless $x$ lies in the causal future of $x'$. Convolution with it selects the retarded solution; adding a smooth homogeneous solution changes boundary data without changing the local source.

Two expansions and one buffer region

Let the external curvature radius be $\mathcal R$. Describe the family with a fixed reference mass $m_0$ and reference radius $\rho_0$, while the physical body parameters shrink as

$$m(\varepsilon)=\varepsilon m_0, \qquad R_{\mathrm b}(\varepsilon)=\varepsilon\rho_0, \qquad \varepsilon\ll1,$$

using $G=c=1$ throughout this local matching calculation, so masses have length units. With physical masses restored, the monopole below is $2Gm_{\mathrm{phys}}/r$ and compactness is $Gm_{\mathrm{phys}}/R_{\mathrm b}$. The fixed ratio $m_0/\mathcal R$ is absorbed into the normalization of $\varepsilon$; equivalently, choose $m_0=\mathcal R$. The small-body limit holds compactness $m(\varepsilon)/R_{\mathrm b}(\varepsilon)=m_0/\rho_0$ fixed while shrinking both mass and size relative to $\mathcal R$. It therefore includes black holes and strongly self-gravitating stars; it is not a weak-internal-gravity approximation.

Far from the body on the scale of its radius, use outer coordinates $x^\mu$ and expand

$$g_{\mu\nu}^{\mathrm{full}}(x,\varepsilon) =g_{\mu\nu}(x) +\varepsilon h_{\mu\nu}^{(1)}(x) +\varepsilon^2 h_{\mu\nu}^{(2)}(x) +O(\varepsilon^3).$$

Here $g_{\mu\nu}$ is a smooth vacuum background with curvature scale $\mathcal R$. The coefficients $h_{\mu\nu}^{(n)}$ are singular on a worldline $\gamma$, because the outer expansion has replaced the finite body by its multipoles. Near the body, introduce scaled coordinates

$$\bar x^i=\frac{x^i-z^i}{\varepsilon}$$

and $\bar r=r/\varepsilon$. The scaled variables hold the body’s size and monopole finite as $\varepsilon\to0$.

The inner expansion is

$$g_{\mu\nu}^{\mathrm{full}} =g_{\mu\nu}^{\mathrm{body}}(\bar x) +\varepsilon H_{\mu\nu}^{(1)}(\bar x) +\varepsilon^2 H_{\mu\nu}^{(2)}(\bar x) +\cdots .$$

The leading inner metric is that of the isolated compact body, while the $H_{\mu\nu}^{(n)}$ describe slowly varying external tidal fields. The two approximations overlap in the buffer region

$$R_{\mathrm b}\ll r\ll\mathcal R, \qquad \varepsilon\ll\frac{r}{\mathcal R}\ll1.$$

The outer solution is expanded for small $r/\mathcal R$ and the inner solution for large $R_{\mathrm b}/r$. Equality of the common powers and angular multipoles is the matching condition. This double expansion and its role in deriving, rather than assuming, a point-particle description are developed in Poisson, Pound, and Vega 2011, §§ 20–23 and summarized in Pound and Wardell 2021, § 5.1, pp. 40–42.

What matching determines

Fermi coordinates around an accelerated candidate worldline include an inertial term $g_{tt}=-1-2a_i x^i+O(r^2)$. Once leading geodesic motion has been established, this linear term vanishes and the background begins as

$$g_{\mu\nu} =\eta_{\mu\nu} +O\!\left(\frac{r^2}{\mathcal R^2}\right).$$

The large-$\bar r$ inner metric contains the body monopole,

$$g_{tt}^{\mathrm{body}} =-1+\frac{2m_0}{\bar r} +O\!\left(\frac{m_0^2}{\bar r^2}\right).$$

together with spin and higher multipoles if they are retained. In common outer coordinates the same term is $2m(\varepsilon)/r=2\varepsilon m_0/r$. The small-distance outer perturbation must contain precisely that monopole with the same normalization. Matching therefore identifies the fixed inner coefficient $m_0$ with the coefficient of the first-order outer singular field, while $m(\varepsilon)=\varepsilon m_0$ is the physical mass in the family. At higher orders matching identifies spin and induced multipoles and separates them from external tidal moments.

The mass dipole has a special role. A translation of the inner coordinates changes the dipole; choosing it to vanish defines a center-of-mass worldline. Matching the dipolar field equation then constrains that worldline’s acceleration. At leading order an acceleration term would create an unmatched dipole, so

$$a_{(0)}^\mu \equiv u^\nu\nabla_\nu u^\mu =0.$$

Thus the limiting worldline is a geodesic of $g_{\mu\nu}$. This conclusion requires a nonzero leading mass monopole, the body’s mass to shrink with its size, a smooth vacuum exterior, and controlled multipoles. It is not obtained by evaluating the body’s divergent field “at its center.” Gralla and Wald formulate the one-parameter limit in § II and derive geodesic leading motion in Gralla and Wald 2008, § IV, eqs. (47)–(50), especially eq. (49).

At the next order, the outer field separates locally into singular and regular pieces,

$$h_{\mu\nu}^{(1)} =h_{\mu\nu}^{\mathrm S(1)} +h_{\mu\nu}^{\mathrm R(1)}.$$

The singular piece contains the multipoles required to match the inner body. The regular piece is smooth on $\gamma$ and alters the external tidal field seen in the inner expansion. For a nonspinning monopolar body, the acceleration through first order is

$$a^\mu =-\frac{\varepsilon}{2} \left(g^{\mu\nu}+u^\mu u^\nu\right) \left( 2h_{\nu\rho;\sigma}^{\mathrm R(1)} -h_{\rho\sigma;\nu}^{\mathrm R(1)} \right) u^\rho u^\sigma +O(\varepsilon^2).$$

All fields are evaluated on the worldline. Equivalently, through the same order, the body follows a geodesic of

$$\tilde g_{\mu\nu} =g_{\mu\nu} +\varepsilon h_{\mu\nu}^{\mathrm R(1)}.$$

This compact effective-metric statement is meaningful only after the gauge, regular field, and representative worldline have been specified. Pound derives the self-consistent first-order motion while retaining the accelerated worldline in Pound 2010, §§ III–V, especially eqs. (88)–(90).

Regular expansion versus self-consistent motion

There are two related perturbative descriptions. In a Gralla–Wald expansion, one expands the worldline itself,

$$z^\mu(\tau,\varepsilon) =z_0^\mu(\tau) +\varepsilon z_1^\mu(\tau) +\cdots,$$

about a background geodesic $z_0^\mu$. The deviation $z_1^\mu$ obeys a forced geodesic-deviation equation. This is well suited to local comparisons and gauge-invariant conservative quantities, but a fixed expansion around $z_0^\mu$ becomes nonuniform over the radiation-reaction time because the true orbit drifts by order unity.

In a self-consistent expansion, $h_{\mu\nu}^{(1)}$ is sourced by an accelerated worldline $z^\mu$, and the regular field updates that same worldline. The resulting equation captures secular drift without Taylor-expanding the orbit about one geodesic. Its perturbative accuracy remains order by order in $\varepsilon$; “self-consistent” does not mean exact. Long inspirals still require a multiscale organization, which is developed in the EMRI and radiation-reaction chapters.

The two descriptions agree over their common domain after the worldline, gauge vector, and initial data are related. A gauge transformation generated by $\xi^\mu$ moves both the metric perturbation and the coordinate representation of the worldline. Comparing $z_1^\mu$ at fixed coordinate labels across gauges is therefore not a physical test. Frequency relations, redshift variables, or scattering data provide cleaner comparisons, as developed on Gauge-Invariant Self-Force Observables.

Independent checks and limitations

The first check is scale matching: the $m/r$ monopole must appear with identical normalization in the large-radius inner and small-radius outer expansions. The second is the geodesic limit: sending $\varepsilon\to0$ while holding the background fixed must eliminate the self-acceleration. The third is orthogonality, $u_\mu a^\mu=0$, which follows from the projector $g^{\mu\nu}+u^\mu u^\nu$. The fourth is gauge consistency: an admissible smooth gauge transformation must change the regular field and worldline together so that an invariant relation is unchanged.

The construction has boundaries. The buffer region disappears if the body size is not small compared with $\mathcal R$. A rapidly varying internal mode invalidates a quasistationary inner expansion. Spin and higher multipoles enter at orders set by their scaling and cannot be silently omitted when comparable to the desired correction. A material body moving through nonvacuum external matter needs additional matching data. Finally, the expansion does not supply global retarded boundary conditions by local matching alone; those conditions enter the outer field problem.

The next page, Detweiler–Whiting Singular and Regular Fields, makes the local field split precise. Mode-Sum Regularization and Effective-Source and Puncture Methods then turn that split into practical calculations.

Exercise

Assume the outer first-order monopole scales as $\varepsilon h_{tt}^{(1)}\sim2\varepsilon m_0/r$, with $m(\varepsilon)=\varepsilon m_0$, while the leading inner metric has $g_{tt}^{\mathrm{body}}\sim-1+2m_0/\bar r$ at large scaled radius. Set $r=\varepsilon\bar r$ in the buffer region. Show that the two expansions have the same monopole scaling, and explain why an $O(1)$ acceleration cannot survive as $\varepsilon\to0$.

Solution

In the outer expansion,

$$\varepsilon h_{tt}^{(1)} \sim\frac{2\varepsilon m_0}{r} =\frac{2m_0}{\bar r}.$$

In the inner expansion the leading coefficient is already written in fixed scaled variables:

$$g_{tt}^{\mathrm{body}}+1 =\frac{2m_0}{\bar r}.$$

Equivalently, returning to common coordinates gives $2m_0/\bar r=2m(\varepsilon)/r$. The shared $1/\bar r$ coefficient identifies the same mass monopole in both descriptions. An order-unity acceleration would enter the local outer/background expansion as a dipolar inertial term that has no counterpart in the centered, stationary leading inner metric. Matching with vanishing mass dipole therefore forces $a^\mu_{(0)}=0$. Self-acceleration can begin only at the next order, where the regular external perturbation supplies the matching tidal field.

References

• Gralla, Samuel E., and Robert M. Wald. “A Rigorous Derivation of Gravitational Self-Force.” Classical and Quantum Gravity 25, 205009 (2008). doi:10.1088/0264-9381/25/20/205009.
• Poisson, Eric, Adam Pound, and Ian Vega. “The Motion of Point Particles in Curved Spacetime.” Living Reviews in Relativity 14, 7 (2011). doi:10.12942/lrr-2011-7.
• Pound, Adam, and Barry Wardell. “Black Hole Perturbation Theory and Gravitational Self-Force.” In Handbook of Gravitational Wave Astronomy, edited by Cosimo Bambi, Stavros Katsanevas, and Konstantinos D. Kokkotas, 1–119. Singapore: Springer (2021). doi:10.1007/978-981-15-4702-7_38-1.
• Pound, Adam. “Self-Consistent Gravitational Self-Force.” Physical Review D 81, 024023 (2010). doi:10.1103/PhysRevD.81.024023.