Motion of a point mass, or a black hole, in a background spacetime

1.9 Motion of a point mass, or a black hole, in a background spacetime

The case of a point mass moving in a specified background spacetime presents itself with a serious conceptual challenge, as the fundamental equations of the theory are nonlinear and the very notion of a “point mass” is somewhat misguided. Nevertheless, to the extent that the perturbation $hαβ(x )$ created by the point mass can be considered to be “small”, the problem can be formulated in close analogy with what was presented before.

We take the metric $gαβ$ of the background spacetime to be a solution of the Einstein field equations in vacuum. (We impose this condition globally.) We describe the gravitational perturbation produced by a point particle of mass $m$ in terms of trace-reversed potentials $γαβ$ defined by

where $hαβ$ is the difference between $g αβ$ , the actual metric of the perturbed spacetime, and $gαβ$ . The potentials satisfy the wave equation

together with the Lorenz gauge condition $αβ γ ;β = 0$ . Here and below, covariant differentiation refers to a connection that is compatible with the background metric, $□ = gαβ∇ α∇ β$ is the wave operator for the background spacetime, and $Tαβ$ is the stress-energy tensor of the point mass; this is given by a Dirac distribution supported on the particle’s world line $γ$ . The retarded solution is

where $G αβ (x, z) + μν$ is the retarded Green’s function associated with Equation (43

). The perturbation $h αβ(x)$ can be recovered by inverting Equation (42

Equations of motion for the point mass can be obtained by formally demanding that the motion be geodesic in the perturbed spacetime with metric $gαβ = g αβ + h αβ$ . After a mapping to the background spacetime, the equations of motion take the form of

The acceleration is thus proportional to $m$ ; in the test-mass limit the world line of the particle is a geodesic of the background spacetime.

We now remove $hSαβ(x)$ from the retarded perturbation and postulate that it is the radiative field $hSαβ(x)$ that should act on the particle. (Note that $γSαβ$ satisfies the same wave equation as the retarded potentials, but that $γR αβ$ is a free gravitational field that satisfies the homogeneous wave equation.) On the world line we have

where the tail term is given by

When Equation (46

) is substituted into Equation (45

) we find that the terms that involve the Riemann tensor cancel out, and we are left with

Only the tail integral appears in the final form of the equations of motion. It involves the current position $z(τ)$ of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions $z(τ′)$ , at which tensors with primed indices are evaluated. As before the integral is cut short at $τ′ = τ− ≡ τ − 0+$ to avoid the singular behaviour of the retarded Green’s function at coincidence.

The equations of motion of Equation (48 ) were first derived by Mino, Sasaki, and Tanaka [39 ], and then reproduced with a different analysis by Quinn and Wald [49 ]. They are now known as the MiSaTaQuWa equations of motion. Detweiler and Whiting [23 ] have contributed the compelling interpretation that the motion is actually geodesic in a spacetime with metric $gαβ + hRαβ$ . This metric satisfies the Einstein field equations in vacuum and is perfectly smooth on the world line. This spacetime can thus be viewed as the background spacetime perturbed by a free gravitational wave produced by the particle at an earlier stage of its history.

While Equation (48 ) does indeed give the correct equations of motion for a small mass $m$ moving in a background spacetime with metric $gαβ$ , the derivation outlined here leaves much to be desired – to what extent should we trust an analysis based on the existence of a point mass? Fortunately, Mino, Sasaki, and Tanaka [39 ] gave two different derivations of their result, and the second derivation was concerned not with the motion of a point mass, but with the motion of a small nonrotating black hole. In this alternative derivation of the MiSaTaQuWa equations, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: It must move according to Equation (48 ). This alternative derivation is entirely free of conceptual and technical pitfalls, and we conclude that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored).

It is important to understand that unlike Equations (33 ) and (40 ), which are true tensorial equations, Equation (48 ) reflects a specific choice of coordinate system and its form would not be preserved under a coordinate transformation. In other words, the MiSaTaQuWa equations are not gauge invariant, and they depend upon the Lorenz gauge condition $γαβ;β = 0$ . Barack and Ori [8 ] have shown that under a coordinate transformation of the form $xα → xα + ξα$ , where $x α$ are the coordinates of the background spacetime and $α ξ$ is a smooth vector field of order $m$ , the particle’s acceleration changes according to $μ μ μ a → a + a[ξ]$ , where

is the “gauge acceleration”; $D2 ξν∕dτ2 = (ξν u μ);ρuρ ;μ$ is the second covariant derivative of $ξν$ in the direction of the world line. This implies that the particle’s acceleration can be altered at will by a gauge transformation; $ξα$ could even be chosen so as to produce $aμ = 0$ , making the motion geodesic after all. This observation provides a dramatic illustration of the following point: The MiSaTaQuWa equations of motion are not gauge invariant and they cannot by themselves produce a meaningful answer to a well-posed physical question; to obtain such answers it shall always be necessary to combine the equations of motion with the metric perturbation $hαβ$ so as to form gauge-invariant quantities that will correspond to direct observables. This point is very important and cannot be over-emphasized.