19 Motion of a point mass

19.1 Dynamics of a point mass


In this section we consider the motion of a point particle of mass m subjected to its own gravitational field in addition to an external field. The particle moves on a world line γ in a curved spacetime whose background metric gαβ is assumed to be a vacuum solution to the Einstein field equations. We shall suppose that m is small, so that the perturbation h αβ created by the particle can also be considered to be small. In the final analysis we shall find that hαβ obeys a linear wave equation in the background spacetime, and this linearization of the field equations will allow us to fit the problem of determining the motion of a point mass within the general framework developed in Sections 17 and 18. We shall find that γ is not a geodesic of the background spacetime because hαβ acts on the particle and produces an acceleration proportional to m; the motion is geodesic in the test-mass limit only.

While we can make the problem fit within the general framework, it is important to understand that the problem of motion in gravitation is conceptually very different from the versions encountered previously in the case of a scalar or electromagnetic field. In these cases, the field equations satisfied by the scalar potential Φ or the vector potential Aα are fundamentally linear; in general relativity the field equations satisfied by h αβ are fundamentally nonlinear, and this makes a major impact on the formulation of the problem. (In all cases the coupled problem of determining the field and the motion of the particle is nonlinear.) Another difference resides with the fact that in the previous cases, the field equations and the law of motion could be formulated independently of each other (because the action functional could be varied independently with respect to the field and the world line); in general relativity the law of motion follows from energy-momentum conservation, which is itself a consequence of the field equations.

The dynamics of a point mass in general relativity must therefore be formulated with care. We shall describe a formal approach to this problem, based on the fiction that the spacetime of a point particle can be constructed exactly in general relativity. (This is indeed a fiction, because it is known [80Jump To The Next Citation Point] that the metric of a point particle, as described by a Dirac distribution on a world line, is much too singular to be defined as a distribution in spacetime. The construction, however, makes distributional sense at the level of the linearized theory.) The outcome of this approach will be an approximate formulation of the equations of motion that relies on a linearization of the field equations, and which turns out to be closely analogous to the scalar and electromagnetic cases encountered previously. We shall put the motion of a small mass on a much sounder foundation in Part V, where we take m to be a (small) extended body instead of a point particle.

Exact formulation

Let a point particle of mass m move on a world line γ in a curved spacetime with metric gαβ. This is the exact metric of the perturbed spacetime, and it depends on m as well as all other relevant parameters. At a later stage of the discussion gαβ will be expressed as sum of a “background” part gαβ that is independent of m, and a “perturbation” part hα β that contains the dependence on m. The world line is described by relations μ z (λ) in which λ is an arbitrary parameter – this will later be identified with proper time τ in the background spacetime. We use sans-serif symbols to denote tensors that refer to the perturbed spacetime; tensors in the background spacetime will be denoted, as usual, by italic symbols.

The particle’s action functional is

∫ ∘ ---------- Sparticle = − m − gμνz˙μ˙zνdλ (19.1 ) γ
where ˙zμ = dzμ∕d λ is tangent to the world line and the metric is evaluated at z. We assume that the particle provides the only source of matter in the spacetime – an explanation will be provided below – so that the Einstein field equations take the form of
αβ αβ G = 8πT , (19.2 )
where Gα β is the Einstein tensor constructed from gαβ and
αβ ∫ g αμ(x, z)gβν(x,z )z˙μ˙zν T (x) = m -----∘-------μ-ν-----δ4(x,z)dλ (19.3 ) γ − gμν ˙z z˙
is the particle’s energy-momentum tensor, obtained by functional differentiation of Sparticle with respect to gαβ(x ); the parallel propagators appear naturally by expressing gμν as α β g μg νgαβ.

On a formal level the metric gαβ is obtained by solving the Einstein field equations, and the world line is determined by the equations of energy-momentum conservation, which follow from the field equations. From Eqs. (5.14View Equation), (13.3View Equation), and (19.3View Equation) we obtain

∫ ( ) αβ -d- ---gαμ˙zμ---- ∇ βT = m dλ ∘ −-g--˙zμ˙zν- δ4(x,z)dλ, γ μν

and additional manipulations reduce this to

∫ ( ) αβ gαμ D ˙zμ μ ∇ βT = m ∘--------μ-ν- -dλ--− kz˙ δ4(x,z)dλ, (19.4 ) γ − gμνz˙ ˙z
where D ˙zμ∕dλ is the covariant acceleration and k is a scalar field on the world line. Energy-momentum conservation therefore produces the geodesic equation
μ D-z˙-= k˙zμ, (19.5 ) dλ
1 d ∘ ---------- k := ∘-------μ-ν--- − gμνz˙μz˙ν (19.6 ) − gμν ˙z ˙z d λ
measures the failure of λ to be an affine parameter on the geodesic γ.

Decomposition into background and perturbation

At this stage we begin treating m as a small quantity, and we write

gαβ = gαβ + hαβ, (19.7 )
with gαβ denoting the m → 0 limit of the metric gαβ, and hαβ containing the dependence on m. We shall refer to gαβ as the “metric of the background spacetime” and to hαβ as the “perturbation” produced by the particle. We insist, however, that no approximation is introduced at this stage; the perturbation h αβ is the exact difference between the exact metric g αβ and the background metric gαβ. Below we shall use the background metric to lower and raise indices.

We introduce the tensor field

Cαβγ := Γ αβγ − Γ αβγ (19.8 )
as the exact difference between Γ αβγ, the connection compatible with the exact metric gα β, and Γ αβγ, the connection compatible with the background metric g αβ. A covariant differentiation indicated by ;α will refer to Γ α βγ, while a covariant differentiation indicated by ∇ α will continue to refer to α Γ βγ.

We express the exact Einstein tensor as

G αβ = G αβ[g] + δG αβ[g,h] + ΔG αβ[g,h], (19.9 )
where Gαβ is the Einstein tensor of the background spacetime, which is assumed to vanish. The second term δGαβ is the linearized Einstein operator defined by
( ) ( ) δG αβ := − 1 □ γαβ + 2R αγ δβγ γδ + 1- γαγ;γβ+ γ βγ;γα − gαβγ γδ;γδ , (19.10 ) 2 2
where □γ αβ := gγδγ αβ;γδ is the wave operator in the background spacetime, and
1 ( ) γαβ := hαβ − --gαβ gγδhγδ (19.11 ) 2
is the “trace-reversed” metric perturbation (with all indices raised with the background metric). The third term ΔG αβ contains the remaining nonlinear pieces that are excluded from δG αβ.

Field equations and conservation statement

The exact Einstein field equations can be expressed as

δG αβ = 8 πTαeffβ, (19.12 )
where the effective energy-momentum tensor is defined by
αβ αβ 1-- αβ T eff := T − 8πΔG . (19.13 )
Because αβ δG satisfies the Bianchi-like identities αβ δG ;β = 0, the effective energy-momentum tensor is conserved in the background spacetime:
αβ T eff;β = 0. (19.14 )
This statement is equivalent to αβ ∇ βT = 0, as can be inferred from the equations αβ β ∇ βG αβ = G ;β + CαγβG γβ + CγβG αγ, αβ β ∇ βTαβ = T ;β + CαγβT γβ + CγβT αγ, and the definition of αβ Teff. Equation (19.14View Equation), in turn, is equivalent to Eq. (19.5View Equation), which states that the motion of the point particle is geodesic in the perturbed spacetime.

Integration of the field equations

Eq. (19.12View Equation) expresses the full and exact content of Einstein’s field equations. It is written in such a way that the left-hand side is linear in the perturbation hαβ, while the right-hand side contains all nonlinear terms. It may be viewed formally as a set of linear differential equations for h αβ with a specified source term αβ Teff. This equation is of mixed hyperbolic-elliptic type, and as such it is a poor starting point for the selection of retarded solutions that enforce a strict causal link between the source and the field. This inadequacy, however, can be remedied by imposing the Lorenz gauge condition

γαβ = 0, (19.15 ) ;β
which converts δG αβ into a strictly hyperbolic differential operator. In this gauge the field equations become
□ γαβ + 2R αγ δβγ γδ = − 16πT αeβff . (19.16 )
This is a tensorial wave equation formulated in the background spacetime, and while the left-hand side is manifestly linear in hαβ, the right-hand side continues to incorporate all nonlinear terms. Equations (19.15View Equation) and (19.16View Equation) still express the full content of the exact field equations.

A formal solution to Eq. (19.16View Equation) is

αβ ∫ αβ ′ γ′δ′ ′ ∘ ---- 4 ′ γ (x) = 4 G + γ′δ′(x,x )Teff (x ) − g′d x, (19.17 )
where G+αβγ′δ′(x,x ′) is the retarded Green’s function introduced in Section 16. With the help of Eq. (16.21View Equation), it is easy to show that
∫ αβ α γ′δ′∘ ---′4 ′ γ ;β = 4 G +γ′Teff ;δ′ − g d x (19.18 )
follows directly from Eq. (19.17View Equation); α ′ G+ γ′(x,x ) is the electromagnetic Green’s function introduced in Section 15. This equation indicates that the Lorenz gauge condition is automatically enforced when the conservation equation Teαffβ;β = 0 is imposed. Conversely, Eq. (19.18View Equation) implies that □(γ αβ;β) = − 16πT αeffβ;β, which indicates that imposition of αβ γ ;β = 0 automatically enforces the conservation equation. There is a one-to-one correspondence between the conservation equation and the Lorenz gauge condition.

The split of the Einstein field equations into a wave equation and a gauge condition directly tied to the conservation of the effective energy-momentum tensor is a most powerful tool, because it allows us to disentangle the problems of obtaining hαβ and determining the motion of the particle. This comes about because the wave equation can be solved first, independently of the gauge condition, for a particle moving on an arbitrary world line γ; the world line is determined next, by imposing the Lorenz gauge condition on the solution to the wave equation. More precisely stated, the source term Tαβ eff for the wave equation can be evaluated for any world line γ, without demanding that the effective energy-momentum tensor be conserved, and without demanding that γ be a geodesic of the perturbed spacetime. Solving the wave equation then returns h αβ[γ ] as a functional of the arbitrary world line, and the metric is not yet fully specified. Because imposing the Lorenz gauge condition is equivalent to imposing conservation of the effective energy-momentum tensor, inserting h αβ[γ] within Eq. (19.15View Equation) finally determines γ, and forces it to be a geodesic of the perturbed spacetime. At this stage the full set of Einstein field equations is accounted for, and the metric is fully specified as a tensor field in spacetime. The split of the field equations into a wave equation and a gauge condition is key to the formulation of the gravitational self-force; in this specific context the Lorenz gauge is conferred a preferred status among all choices of gauge.

An important question to be addressed is how the wave equation is to be integrated. A method of principle, based on the assumed smallness of m and hαβ, is suggested by post-Minkowskian theory [180, 26]. One proceeds by iterations. In the first iterative stage, one fixes γ and substitutes αβ h0 = 0 within T αeffβ; evaluation of the integral in Eq. (19.17View Equation) returns the first-order approximation h αβ[γ ] = O (m ) 1 for the perturbation. In the second stage hαβ 1 is inserted within T αβ eff and Eq. (19.17View Equation) returns the second-order approximation αβ 2 h2 [γ] = O(m, m ) for the perturbation. Assuming that this procedure can be repeated at will and produces an adequate asymptotic series for the exact perturbation, the iterations are stopped when the nth-order approximation h αnβ[γ ] = O (m, m2, ⋅⋅⋅,mn ) is deemed to be sufficiently accurate. The world line is then determined, to order mn, by subjecting the approximated field to the Lorenz gauge condition. It is to be noted that the procedure necessarily produces an approximation of the field, and an approximation of the motion, because the number of iterations is necessarily finite. This is the only source of approximation in our formulation of the dynamics of a point mass.

Equations of motion

Conservation of energy-momentum implies Eq. (19.5View Equation), which states that the motion of the point mass is geodesic in the perturbed spacetime. The equation is expressed in terms of the exact connection α Γβγ, and with the help of Eq. (19.8View Equation) it can be re-written in terms of the background connection Γ αβγ. We get D z˙μ∕d λ = − C μz˙νz˙λ + k ˙zμ νλ, where the left-hand side is the covariant acceleration in the background spacetime, and k is given by Eq. (19.6View Equation). At this stage the arbitrary parameter λ can be identified with proper time τ in the background spacetime. With this choice the equations of motion become

aμ = − Cνμλu νuλ + kuμ, (19.19 )
where uμ := dzμ∕dτ is the velocity vector in the background spacetime, aμ := Du μ∕dτ the covariant acceleration, and
k = ∘------1-------d-∘1 -−--h--uμu-ν. (19.20 ) 1 − hμνuμuν dτ μν
Eq. (19.19View Equation) is an exact statement of the equations of motion. It expresses the fact that while the motion is geodesic in the perturbed spacetime, it may be viewed as accelerated motion in the background spacetime. Because hαβ is calculated as an expansion in powers of m, the acceleration also is eventually obtained as an expansion in powers of m. Here, in keeping with the preceding sections, we will use order-reduction to make that expansion well-behaved; in Part V of the review, we will formulate the expansion more clearly as part of more systematic approach.

Implementation to first order in m

While our formulation of the dynamics of a point mass is in principle exact, any practical implementation will rely on an approximation method. As we saw previously, the most immediate source of approximation concerns the number of iterations involved in the integration of the wave equation. Here we perform a single iteration and obtain the perturbation hαβ and the equations of motion to first order in the mass m.

In a first iteration of the wave equation we fix γ and set ΔG αβ = 0, T αβ = T αβ, where

∫ αβ α β μ ν T = m g μ(x,z)g ν(x,z)u u δ4(x, z)dτ (19.21 ) γ
is the particle’s energy-momentum tensor in the background spacetime. This implies that Teαffβ= Tα β, and Eq. (19.16View Equation) becomes
αβ α β γδ αβ 2 □ γ + 2R γ δ γ = − 16πT + O(m ). (19.22 )
Its solution is
∫ αβ αβ μ ν 2 γ (x) = 4m G + μν(x, z)u u dτ + O (m ), (19.23 ) γ
and from this we obtain h αβ. Equation (19.8View Equation) gives rise to C αβγ = 12(hαβ;γ + hαγ;β − h βγ;α) + O (m2 ), and from Eq. (19.20View Equation) we obtain k = − 1h uνuλuρ − h u νaλ + O(m2 ) 2 νλ;ρ νλ; we can discard the second term because it is clear that the acceleration will be of order m. Inserting these results within Eq. (19.19View Equation), we obtain
μ 1 ( μ μ ;μ μ ρ) ν λ 2 a = − -- h ν;λ + h λ;ν − h νλ + u hνλ;ρu u u + O (m ). (19.24 ) 2
We express this in the equivalent form
aμ = − 1-(gμν + uμuν)(2h − h )u λuρ + O(m2 ) (19.25 ) 2 νλ;ρ λρ;ν
to emphasize the fact that the acceleration is orthogonal to the velocity vector.

It should be clear that Eq. (19.25View Equation) is valid only in a formal sense, because the potentials obtained from Eqs. (19.23View Equation) diverge on the world line. To make sense of these equations we will proceed as in Sections 17 and 18 with a careful analysis of the field’s singularity structure; regularization will produce a well-defined version of Eq. (19.25View Equation). Our formulation of the dynamics of a point mass makes it clear that a proper implementation requires that the wave equation of Eq. (19.22View Equation) and the equations of motion of Eq. (19.25View Equation) be integrated simultaneously, in a self-consistent manner.

Failure of a strictly linearized formulation

In the preceding discussion we started off with an exact formulation of the problem of motion for a small mass m in a background spacetime with metric gαβ, but eventually boiled it down to an implementation accurate to first order in m. Would it not be simpler and more expedient to formulate the problem directly to first order? The answer is a resounding no: By doing so we would be driven toward a grave inconsistency; the nonlinear formulation is absolutely necessary if one wishes to contemplate a self-consistent integration of Eqs. (19.22View Equation) and (19.25View Equation).

A strictly linearized formulation of the problem would be based on the field equations δG αβ = 8πT αβ, where Tαβ is the energy-momentum tensor of Eq. (19.21View Equation). The Bianchi-like identities δG αβ = 0 ;β dictate that αβ T must be conserved in the background spacetime, and a calculation identical to the one leading to Eq. (19.5View Equation) would reveal that the particle’s motion must be geodesic in the background spacetime. In the strictly linearized formulation, therefore, the gravitational potentials of Eq. (19.23View Equation) must be sourced by a particle moving on a geodesic, and there is no opportunity for these potentials to exert a self-force. To get the self-force, one must provide a formulation that extends beyond linear order. To be sure, one could persist in adopting the linearized formulation and “save the phenomenon” by relaxing the conservation equation. In practice this could be done by adopting the solutions of Eq. (19.23View Equation), demanding that the motion be geodesic in the perturbed spacetime, and relaxing the linearized gauge condition to αβ 2 γ ;β = O (m ). While this prescription would produce the correct answer, it is largely ad hoc and does not come with a clear justification. Our exact formulation provides much more control, at least in a formal sense. We shall do even better in Part V.

An alternative formulation of the problem provided by Gralla and Wald [83Jump To The Next Citation Point] avoids the inconsistency by refraining from performing a self-consistent integration of Eqs. (19.22View Equation) and (19.25View Equation). Instead of an expansion of the acceleration in powers of m, their approach is based on an expansion of the world line itself, and it returns the equations of motion for a deviation vector which describes the offset of the true world line relative to a reference geodesic. While this approach is mathematically sound, it eventually breaks down as the deviation vector becomes large, and it does not provide a justification of the self-consistent treatment of the equations.

The difference between the Gralla–Wald approach and a self-consistent one is the difference between a regular expansion and a general one. In a regular expansion, all dependence on a small quantity m is expanded in powers:

∑∞ hαβ(x,m ) = mnh (αnβ)(x ). (19.26 ) n=0
In a general expansion, on the other hand, the functions (n) hαβ are allowed to retain some dependence on the small quantity:
∑∞ h αβ(x,m ) = mnh (nα)β(x,m ). (19.27 ) n=0
Put simply, the goal of a general expansion is to expand only part of a function’s dependence on a small quantity, while holding fixed some specific dependence that captures one or more of the function’s essential features. In the self-consistent expansion that we advocate here, our iterative solution returns
∑N hNαβ(x,m ) = mnh (αnβ)(x; γ(m )), (19.28 ) n=0
in which the functional dependence on the world line γ incorporates a dependence on the expansion parameter m. We deliberately introduce this functional dependence on a mass-dependent world line in order to maintain a meaningful and accurate description of the particle’s motion. Although the regular expansion can be retrieved by further expanding the dependence within γ(m ), the reverse statement does not hold: the general expansion cannot be justified on the basis of the regular one. The notion of a general expansion is at the core of singular perturbation theory [63Jump To The Next Citation Point, 96Jump To The Next Citation Point, 109Jump To The Next Citation Point, 111Jump To The Next Citation Point, 178Jump To The Next Citation Point, 145Jump To The Next Citation Point]. We shall return to these issues in our treatment of asymptotically small bodies, and in particular, in Section 22.5 below.

Vacuum background spacetime

To conclude this subsection we should explain why it is desirable to restrict our discussion to spacetimes that contain no matter except for the point particle. Suppose, in contradiction with this assumption, that the background spacetime contains a distribution of matter around which the particle is moving. (The corresponding vacuum situation has the particle moving around a black hole. Notice that we are still assuming that the particle moves in a region of spacetime in which there is no matter; the issue is whether we can allow for a distribution of matter somewhere else.) Suppose also that the matter distribution is described by a collection of matter fields Ψ. Then the field equations satisfied by the matter have the schematic form E[Ψ; g] = 0, and the background metric is determined by the Einstein field equations G [g ] = 8πM [Ψ; g], in which M [Ψ; g] stands for the matter’s energy-momentum tensor. We now insert the point particle in the spacetime, and recognize that this displaces the background solution (Ψ,g) to a new solution (Ψ + δΨ, g + δg). The perturbations are determined by the coupled set of equations E [Ψ + δΨ; g + δg] = 0 and G [g + δg ] = 8πM [Ψ + δΨ; g + δg] + 8πT [g ]. After linearization these take the form of

( ) E Ψ ⋅ δΨ + Eg ⋅ δg = 0, Gg ⋅ δg = 8π M Ψ ⋅ δΨ + Mg ⋅ δg + T ,

where E Ψ, Eg, M Φ, and Mg are suitable differential operators acting on the perturbations. This is a coupled set of partial differential equations for the perturbations δΨ and δg. These equations are linear, but they are much more difficult to deal with than the single equation for δg that was obtained in the vacuum case. And although it is still possible to solve the coupled set of equations via a Green’s function technique, the degree of difficulty is such that we will not attempt this here. We shall, therefore, continue to restrict our attention to the case of a point particle moving in a vacuum (globally Ricci-flat) background spacetime.

19.2 Retarded potentials near the world line

Going back to Eq. (19.23View Equation), we have that the gravitational potentials associated with a point particle of mass m moving on world line γ are given by

∫ γαβ(x) = 4m G +αβμν(x,z)u μuνdτ, (19.29 ) γ
up to corrections of order 2 m; here μ z (τ) gives the description of the world line, μ μ u = dz ∕dτ is the velocity vector, and G αβ′′(x,x′) + γ δ is the retarded Green’s function introduced in Section 16. Because the retarded Green’s function is defined globally in the entire background spacetime, Eq. (19.29View Equation) describes the gravitational perturbation created by the particle at any point x in that spacetime.

For a more concrete expression we take x to be in a neighbourhood of the world line. The manipulations that follow are very close to those performed in Section 17.2 for the case of a scalar charge, and in Section 18.2 for the case of an electric charge. Because these manipulations are by now familiar, it will be sufficient here to present only the main steps. There are two important simplifications that occur in the case of a massive particle. First, it is clear that

μ μ a = O (m ) = a˙ , (19.30 )
and we will take the liberty of performing a pre-emptive order-reduction by dropping all terms involving the acceleration vector when computing γαβ and γαβ;γ to first order in m; otherwise we would arrive at an equation for the acceleration that would include an antidamping term 11 μ − 3 m ˙a [92Jump To The Next Citation Point, 93Jump To The Next Citation Point, 150Jump To The Next Citation Point]. Second, because we take gαβ to be a solution to the vacuum field equations, we are also allowed to set
Rμν(z) = 0 (19.31 )
in our computations.

With the understanding that x is close to the world line (refer back to Figure 9View Image), we substitute the Hadamard construction of Eq. (16.7View Equation) into Eq. (19.29View Equation) and integrate over the portion of γ that is contained in 𝒩 (x). The result is

∫ u ∫ τ< γαβ(x) = 4m--Uα β′′(x, x′)u γ′uδ′ + 4m V αβ(x, z)uμuνdτ + 4m G αβ (x,z )u μuνdτ,(19.32 ) r γδ τ< μν −∞ + μν
in which primed indices refer to the retarded point ′ x := z(u) associated with x, α′ r := σ α′u is the retarded distance from ′ x to x, and τ< is the proper time at which γ enters 𝒩 (x ) from the past.

In the following subsections we shall refer to γ (x) αβ as the gravitational potentials at x produced by a particle of mass m moving on the world line γ, and to γαβ;γ(x ) as the gravitational field at x. To compute this is our next task.

19.3 Gravitational field in retarded coordinates

Keeping in mind that ′ x and x are related by ′ σ(x,x ) = 0, a straightforward computation reveals that the covariant derivatives of the gravitational potentials are given by

4m ′ ′ 4m ′ ′ 4m ′ ′ ′ γαβ;γ(x) = − --2 U αβα′β′uα uβ ∂γr +---U αβα′β′;γu αu β + ---U αβα′β′;γ′uα uβ uγ ∂γu r α ′β′ rtail r + 4mV αβα′β′u u ∂γu + γαβγ(x ), (19.33 )
where the “tail integral” is defined by
∫ u ∫ τ< γtαaβilγ(x ) = 4m ∇ γVαβμν(x,z)uμu νdτ + 4m ∇ γG+ αβμν(x,z)uμu νdτ τ< −∞ ∫ u− = 4m ∇ γG+ αβμν(x,z)uμu νdτ. (19.34 ) −∞
The second form of the definition, in which the integration is cut short at τ = u− := u − 0+ to avoid the singular behaviour of the retarded Green’s function at σ = 0, is equivalent to the first form.

We wish to express γαβ;γ(x) in the retarded coordinates of Section 10, as an expansion in powers of r. For this purpose we decompose the field in the tetrad (eα0,eαa) that is obtained by parallel transport of (uα′,eα′) a on the null geodesic that links x to x′; this construction is detailed in Section 10. We recall from Eq. (10.4View Equation) that the parallel propagator can be expressed as α′ α′ 0 α′ a gα = u e α + ea eα. The expansion relies on Eq. (10.29View Equation) for ∂γu and Eq. (10.31View Equation) for ∂γr, both simplified by setting aa = 0. We shall also need

[ ] α′ β′ α′ β′ 3 U αβα′β′u u = g(αg β) uα′uβ′ + O (r ) , (19.35 )
which follows from Eq. (16.13View Equation),
[ ] U ′′ u α′uβ′ = gα′ gβ′gγ′ − r(R ′ ′ + R ′ ′ Ωd)u ′ + O (r2) , (19.36 ) αβαβ ;γ (α β) γ α0γ 0 α0γ d β α′ β′ γ′ α′ β′[ d 2 ] Uαβα′β′;γ′u u u = g (αgβ) rR α′0d0Ω uβ′ + O (r ) , (19.37 )
which follow from Eqs. (16.14View Equation) and (16.15View Equation), respectively, as well as the relation α′ α′ a α′ σ = − r(u + Ω ea ) first encountered in Eq. (10.7View Equation). And finally, we shall need
′ ′ ′ ′ [ ] V αβα′β′uα uβ = gα(αgββ) R α′0β′0 + O (r) , (19.38 )
which follows from Eq. (16.17View Equation).

Making these substitutions in Eq. (19.3View Equation) and projecting against various members of the tetrad gives

γ (u,r,Ωa ) := γ (x)eα(x)eβ(x )e γ(x ) = 2mR ΩaΩb + γtail + O (r ), (19.39 ) 000 αβ;γ 0 0 0 a0b0 000 γ0b0(u,r,Ωa ) := γ αβ;γ(x)eα0(x)eβb(x )e γ0(x ) = − 4mRb0c0 Ωc + γta0bi0l+ O (r), (19.40 ) a α β γ tail γab0(u,r,Ω ) := γ αβ;γ(x)ea(x)eb(x )e 0(x ) = 4mRa0b0 + γ ab0 + O (r ), (19.41 ) γ00c(u,r,Ωa ) := γ αβ;γ(x)eα0(x)eβ0(x )e γc(x ) [ ( ) ] = − 4m -1 + 1-Ra0b0ΩaΩb Ωc + 1-Rc0b0Ωb − 1Rca0bΩa Ωb + γt0a0ilc + O(r), (19.42 ) r2 3 6 6 γ (u,r,Ωa ) := γ (x)eα(x)eβ(x )e γ(x ) 0bc αβ;γ( 0 b dc d ) tail = 2m Rb0c0 + Rb0cdΩ + Rb0d0Ω Ωc + γ0bc + O (r), (19.43 ) γ (u,r,Ωa ) := γ (x)eα(x)eβ(x )e γ(x ) = − 4mR Ω + γtail+ O (r), (19.44 ) abc αβ;γ a b c a0b0 c abc
where, for example, ′ ′ ′ ′ Ra0b0(u) := Rα ′γ′β′δ′eαa uγ eβb u δ are frame components of the Riemann tensor evaluated at x′ := z (u ). We have also introduced the frame components of the tail part of the gravitational field, which are obtained from Eq. (19.34View Equation) evaluated at x ′ instead of x; for example, tail α′ β′ γ′ tail ′ γ000 = u u u γα′β′γ′(x ). We may note here that while γ00c is the only component of the gravitational field that diverges when r → 0, the other components are nevertheless singular because of their dependence on the unit vector Ωa; the only exception is γab0, which is regular.

19.4 Gravitational field in Fermi normal coordinates

The translation of the results contained in Eqs. (19.39View Equation) – (19.44View Equation) into the Fermi normal coordinates of Section 9 proceeds as in Sections 17.4 and 18.4, but is simplified by setting aa = ˙a0 = ˙aa = 0 in Eqs. (11.7View Equation), (11.8View Equation), (11.4View Equation), (11.5View Equation), and (11.6View Equation) that relate the Fermi normal coordinates (t,s,ωa) to the retarded coordinates. We recall that the Fermi normal coordinates refer to a point ¯x := z(t) on the world line that is linked to x by a spacelike geodesic that intersects γ orthogonally.

The translated results are

¯γ (t,s,ωa) := γ (x )¯eα(x)¯eβ(x)¯eγ(x) = ¯γtail+ O (s), (19.45 ) 000 αβ;γ 0 0 0 000 ¯γ0b0(t,s,ωa) := γαβ;γ(x )¯eα0(x)¯eβb(x)¯eγ0(x) = − 4mRb0c0 ωc + ¯γt0aib0l+ O (s), (19.46 ) a α β γ tail ¯γab0(t,s,ω ) := γαβ;γ(x )¯ea(x)¯eb(x)¯e0(x) = 4mRa0b0 + ¯γab0 + O (s), (19.47 ) ¯γ00c(t,s,ωa) := γαβ;γ(x )¯eα0(x)¯eβ0(x)¯eγc(x) [( ) ] = − 4m 1-− 1Ra0b0ωa ωb ωc + 1-Rc0b0ωb + ¯γt0ai0cl+ O (s), (19.48 ) s2 6 3 ¯γ (t,s,ωa) := γ (x )¯eα(x)¯eβ(x)¯eγ(x) = 2m (R + R ωd ) + ¯γtail+ O (s), (19.49 ) 0bc αβ;γ 0 b c b0c0 b0cd 0bc ¯γabc(t,s,ωa) := γαβ;γ(x )¯eαa(x)¯eβb(x)¯eγc(x) = − 4mRa0b0 ωc + ¯γtaaibcl+ O (s), (19.50 )
where all frame components are now evaluated at ¯x instead of ′ x.

It is then a simple matter to average these results over a two-surface of constant t and s. Using the area element of Eq. (17.24View Equation) and definitions analogous to those of Eq. (17.25View Equation), we obtain

tail ⟨¯γ000⟩ = γ¯000 + O (s), (19.51 ) ⟨¯γ0b0⟩ = γ¯ta0bi0l+ O (s), (19.52 ) tail ⟨¯γab0⟩ = 4mRa0b0 + ¯γab0 + O (s), (19.53 ) ⟨¯γ00c⟩ = γ¯ta00icl+ O (s), (19.54 ) tail ⟨¯γ0bc⟩ = 2mRb0c0 + ¯γ0bc + O(s), (19.55 ) ⟨¯γabc⟩ = γ¯taabicl+ O (s). (19.56 )
The averaged gravitational field is regular in the limit s → 0, in which the tetrad α α (¯e0,¯ea) coincides with α¯ ¯α (u ,ea). Reconstructing the field at ¯x from its frame components gives
( ) ¯δ ¯𝜖 tail ⟨γ¯α¯β;¯γ⟩ = − 4m u(α¯R β¯)¯δ¯γ¯𝜖 + R ¯α¯δ¯β¯𝜖u ¯γ u u + γ¯α¯β¯γ, (19.57 )
where the tail term can be copied from Eq. (19.34View Equation),
∫ t− γtail(¯x) = 4m ∇ G ¯ (¯x,z)uμu νdτ. (19.58 ) ¯α¯β¯γ −∞ ¯γ +¯αβμν
The tensors that appear in Eq. (19.57View Equation) all refer to the simultaneous point ¯x := z(t), which can now be treated as an arbitrary point on the world line γ.

19.5 Singular and regular fields

The singular gravitational potentials

∫ γ αSβ(x) = 4m G Sα μβν(x, z)uμuνdτ (19.59 ) γ
are solutions to the wave equation of Eq. (19.22View Equation); the singular Green’s function was introduced in Section 16.5. We will see that the singular field S γ αβ;γ reproduces the singular behaviour of the retarded solution near the world line, and that the difference, γR = γαβ;γ − γS αβ;γ α β;γ, is smooth on the world line.

To evaluate the integral of Eq. (19.59View Equation) we take x to be close to the world line (see Figure 9View Image), and we invoke Eq. (16.31View Equation) as well as the Hadamard construction of Eq. (16.37View Equation). This gives

∫ v γαβ (x ) = 2m-U αβ uγ′uδ′ +-2m-U αβ uγ′′uδ′′ − 2m V αβ (x,z)uμu νdτ, (19.60 ) S r γ′δ′ radv γ′′δ′′ u μν
where primed indices refer to the retarded point ′ x := z (u ), double-primed indices refer to the advanced point ′′ x := z(v), and where α′′ radv := − σ α′′u is the advanced distance between x and the world line.

Differentiation of Eq. (19.60View Equation) yields

γS (x) = − 2m--U ′ ′u α′u β′∂ r − -2m--U ′′ ′′uα′′uβ′′∂ r + 2m--U ′ ′ uα′uβ′ αβ;γ r2 αβα β γ radv2 αβα β γ adv r αβα β;γ 2m ′ ′ ′ 2m ′′ ′′ 2m ′′ ′′ ′′ + ----Uαβα′β′;γ′uα uβ uγ ∂γu + ----U αβα′′β′′;γuα u β + ----Uαβα′′β′′;γ′′u α uβ uγ ∂γv r radv ∫ radvv ′′ α′ β′ ′′′′ α′′ β′′ μ ν + 2mV αβα β u u ∂ γu − 2mV αβα β u u ∂γv − 2m u ∇ γVαβμν(x,z)u u dτ, (19.61 )
and we would like to express this as an expansion in powers of r. For this we will rely on results already established in Section 19.3, as well as additional expansions that will involve the advanced point x ′′. We recall that a relation between retarded and advanced times was worked out in Eq. (11.12View Equation), that an expression for the advanced distance was displayed in Eq. (11.13View Equation), and that Eqs. (11.14View Equation) and (11.15View Equation) give expansions for ∂γv and ∂ γradv, respectively; these results can be simplified by setting aa = a˙0 = a˙a = 0, which is appropriate in this computation.

To derive an expansion for U ′′ ′′uα′′uβ′′ αβα β we follow the general method of Section 11.4 and introduce the functions μ ν Uαβ(τ) := U αβμν(x, z)u u. We have that

′′ ′′ 1 ( ) U αβα′′β′′uα u β := U αβ(v) = Uαβ(u ) + U˙αβ(u)Δ ′ +-¨Uαβ(u)Δ ′2 + O Δ ′3 , 2

where overdots indicate differentiation with respect to τ and Δ ′ := v − u. The leading term α′β′ U αβ(u) := Uαβα′β′u u was worked out in Eq. (19.35View Equation), and the derivatives of U αβ(τ) are given by

′ ′ ′ ′ β′ [ ] ˙Uαβ(u ) = U αβα′β′;γ′uα uβ uγ = gα(αg β) rR α′0d0Ωdu β′ + O (r2)


¨ ′ ′ ′′ α′ β′ γ′ δ′ U αβ(u) = Uαβα β;γδ u u u u = O (r),

according to Eqs. (19.37View Equation) and (16.15View Equation). Combining these results together with Eq. (11.12View Equation) for ′ Δ gives

′[ ] Uαβα′′β′′u α′′uβ′′ = gα′(αg ββ) uα′uβ′ + 2r2Rα′0d0Ωduβ′ + O(r3) , (19.62 )
which should be compared with Eq. (19.35View Equation). It should be emphasized that in Eq. (19.62View Equation) and all equations below, all frame components are evaluated at the retarded point x′, and not at the advanced point. The preceding computation gives us also an expansion for
α′ β′′ γ′′ ′ ′2 U αβα′′β′′;γ′′u u u = ˙Uαβ(u) + ¨Uαβ (u )Δ + O (Δ ),

which becomes

′[ ] U αβα′′β′′;γ′′uα′′u β′′uγ′′ = gα′(αg ββ) rR α′0d0Ωdu β′ + O (r2) , (19.63 )
and which is identical to Eq. (19.37View Equation).

We proceed similarly to obtain an expansion for U αβα′′β′′;γu α′′uβ′′. Here we introduce the functions μ ν U αβγ(τ) := U αβμν;γu u and express ′′′′ α′′β′′ Uα βαβ ;γu u as ˙ ′ ′2 U αβγ(v) = Uαβγ(u) + U αβγ(u )Δ + O (Δ ). The leading term α ′β′ Uαβγ(u ) := U αβα′β′;γu u was computed in Eq. (19.36View Equation), and

α′ β′ γ′ α′ β′ γ′[ ] U˙αβγ(u) = U αβα′β′;γγ′u u u = g (αg β)g γ R α′0γ′0uβ′ + O (r)

follows from Eq. (16.14View Equation). Combining these results together with Eq. (11.12View Equation) for Δ′ gives

[ ] α′′β′′ α′ β′ γ′ ( d) 2 Uα βα′′β′′;γu u = g(αg β)gγ r Rα′0γ′0 − R α′0γ′dΩ uβ′ + O(r ) , (19.64 )
and this should be compared with Eq. (19.36View Equation). The last expansion we shall need is
[ ] V ′′ ′′uα′′uβ′′ = g α′gβ′ R ′ ′ + O (r) , (19.65 ) αβα β (α β) α 0β0
which is identical to Eq. (19.38View Equation).

We obtain the frame components of the singular gravitational field by substituting these expansions into Eq. (19.61View Equation) and projecting against the tetrad (eα0,e αa). After some algebra we arrive at

S a S α β γ a b γ000(u,r,Ω ) := γαβ;γ(x )e0(x)e0(x)e0(x) = 2mRa0b0 Ω Ω + O (r ), (19.66 ) γS (u,r,Ωa) := γS (x )eα(x)eβ(x)eγ(x) = − 4mR Ωc + O (r), (19.67 ) 0b0 αβ;γ 0 b 0 b0c0 γSab0(u,r,Ωa) := γSαβ;γ(x )eαa(x)eβb(x)eγ0(x) = O (r), (19.68 ) S a S α β γ γ00c(u,r,Ω ) := γαβ;γ(x[ )e0(x)e0(x)ec(x) ] ( 1 1 a b) 1 b 1 a b = − 4m -2 + -Ra0b0Ω Ω Ωc + -Rc0b0Ω − --Rca0bΩ Ω + O(r), (19.69 ) r 3 ( 6 6 ) γS0bc(u,r,Ωa) := γSαβ;γ(x )eα0(x)eβb(x)eγc(x) = 2m Rb0cdΩd + Rb0d0Ωd Ωc + O (r), (19.70 ) S a S α β γ γabc(u,r,Ω ) := γαβ;γ(x )ea(x)eb(x)ec(x) = − 4mRa0b0 Ωc + O (r), (19.71 )
in which all frame components are evaluated at the retarded point x′. Comparison of these expressions with Eqs. (19.39View Equation) – (19.44View Equation) reveals identical singularity structures for the retarded and singular gravitational fields.

The difference between the retarded field of Eqs. (19.39View Equation) – (19.44View Equation) and the singular field of Eqs. (19.66View Equation) – (19.71View Equation) defines the regular gravitational field γR αβ;γ. Its frame components are

R tail γ000 = γ000 + O (r), (19.72 ) γR0b0 = γta0ib0l+ O (r), (19.73 ) R tail γab0 = 4mRa0b0 + γ ab0 + O (r ), (19.74 ) γR00c = γta0i0cl+ O (r), (19.75 ) γR = 2mR + γtail+ O(r), (19.76 ) 0bc b0c0 0bc γRabc = γtaaibcl+ O (r), (19.77 )
and we see that γR αβ;γ is regular in the limit r → 0. We may therefore evaluate the regular field directly at ′ x = x, where the tetrad α α (e 0,ea) coincides with α′ α′ (u ,ea ). After reconstructing the field at ′ x from its frame components, we obtain
( ) ′ ′ γRα′β′;γ′(x ′) = − 4m u (α′R β′)δ′γ′𝜖′ + R α′δ′β′𝜖′uγ′ uδ u𝜖 + γtαa′ilβ′γ′, (19.78 )
where the tail term can be copied from Eq. (19.34View Equation),
∫ u− γtaαi′βl′γ′(x′) = 4m ∇ γ′G+ α ′β′μν(x ′,z)uμu νdτ. (19.79 ) − ∞
The tensors that appear in Eq. (19.79View Equation) all refer to the retarded point x′ := z (u ), which can now be treated as an arbitrary point on the world line γ.

19.6 Equations of motion

The retarded gravitational field γ αβ;γ of a point particle is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and influence its motion. The field’s singularity structure was analyzed in Sections 19.3 and 19.4, and in Section 19.5 it was shown to originate from the singular field γSαβ;γ; the regular field γRαβ;γ was then shown to be regular on the world line.

To make sense of the retarded field’s action on the particle we can follow the discussions of Section 17.6 and 18.6 and postulate that the self gravitational field of the point particle is either ⟨γμν;λ⟩, as worked out in Eq. (19.57View Equation), or γRμν;λ, as worked out in Eq. (19.78View Equation). These regularized fields are both given by

( ) reg ρ ξ tail γμν;λ = − 4m u(μR ν)ρλξ + Rμρνξuλ u u + γμνλ (19.80 )
∫ τ− ( ) ′ ′ γtμaνilλ = 4m ∇ λG+ μνμ′ν′z (τ ),z(τ′) u μuν dτ′, (19.81 ) −∞
in which all tensors are now evaluated at an arbitrary point z(τ) on the world line γ.

The actual gravitational perturbation h α β is obtained by inverting Eq. (19.10View Equation), which leads to 1 ρ h μν;λ = γμν;γ − 2g μνγρ;λ. Substituting Eq. (19.80View Equation) yields

reg ( ) ρ ξ tail hμν;λ = − 4m u(μR ν)ρλξ + R μρνξu λ u u + hμνλ, (19.82 )
where the tail term is given by the trace-reversed counterpart to Eq. (19.81View Equation):
∫ τ− ( ) htail = 4m ∇ G ′ ′ − 1-g G ρ ′ ′ (z(τ),z(τ′))uμ′uν′d τ′. (19.83 ) μνλ −∞ λ +μνμ ν 2 μν + ρμ ν
When this regularized field is substituted into Eq. (19.13View Equation), we find that the terms that depend on the Riemann tensor cancel out, and we are left with
Du-μ- 1-( μν μ ν)( tail tail) λ ρ dτ = − 2 g + u u 2h νλρ − h λρν u u . (19.84 )
We see that only the tail term is involved in the final form of the equations of motion. The tail integral of Eq. (19.83View Equation) 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 all tensors with primed indices are evaluated. The tail integral is cut short at τ′ = τ− := τ − 0+ to avoid the singular behaviour of the retarded Green’s function at coincidence; this limiting procedure was justified at the beginning of Section 19.3.

Eq. (19.84View Equation) was first derived by Yasushi Mino, Misao Sasaki, and Takahiro Tanaka in 1997 [130Jump To The Next Citation Point]. (An incomplete treatment had been given previously by Morette-DeWitt and Ging [133].) An alternative derivation was then produced, also in 1997, by Theodore C. Quinn and Robert M. Wald [150Jump To The Next Citation Point]. These equations are now known as the MiSaTaQuWa equations of motion, and other derivations [83Jump To The Next Citation Point, 144Jump To The Next Citation Point], based on an extended-body approach, will be reviewed below in Part V. It should be noted that Eq. (19.84View Equation) is formally equivalent to the statement that the point particle moves on a geodesic in a spacetime with metric gαβ + hRαβ, where hRαβ is the regular metric perturbation obtained by trace-reversal of the potentials γR := γ − γS αβ αβ αβ; this perturbed metric is regular on the world line, and it is a solution to the vacuum field equations. This elegant interpretation of the MiSaTaQuWa equations was proposed in 2003 by Steven Detweiler and Bernard F. Whiting [53]. Quinn and Wald [151] have shown that under some conditions, the total work done by the gravitational self-force is equal to the energy radiated (in gravitational waves) by the particle.

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