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C.2 Equations of motion

Let us start explaining again our situation and discuss the strong field point particle limit [81]. A small spherical compact object with mass m moves at an arbitrary speed around a massive body with mass M. We would like to find the equations of motion for the object including radiation reaction. We assume that the object is stationary except for higher order tidal effects, so that we can safely neglect the emission of gravitational waves from the object itself, but of course we cannot neglect the gravitational waves emitted by the orbital motion of the object. We denote the world line of the center of mass of the object as zμ(τ) and define the body zone of the object as follows. We imagine a spherical region around μ z (τ) and a radius that scales as ε. At the same time we scale the linear dimension of the object as 2 ε so that the boundary of the body zone is located at the far zone of the object. Namely we are able to have a multipole expansion of the field generated by the object at the surface of the body zone. We also implicitly assume that the mass of the object scales as ε so that the compactness of the object remains constant in the point particle limit as ε → 0. This is why we call this limit the strong field point particle limit. Then we calculate the metric perturbation induced by the small object in this limit. The smallness parameter ε has a dimension of length which characterizes the smallness of the object. One may regard it as the ratio between the physical scale of the object and the characteristic scale of the background curvature. We assume that the background metric g μν satisfies the Einstein equations in vacuum. So the Ricci tensor of the background vanishes. Since we have assumed that the mass scale of the small object is much smaller than the scale of the gravitational field of the background geometry, we approximate the metric perturbation by the linear perturbation of the small particle h μν.

We will work in the harmonic gauge,

μν ¯h ν = 0, (240 )
where the semicolon means that the covariant derivative with respect to the background metric and the trace-reversed variable is defined as usual,
¯hμν = h μν − 1gμνgρλhρλ. (241 ) 2
Then the linearized Einstein equations take the following form:
− 1-¯hμν;ξξ(x ) − R μξνρ(x )¯h ξρ(x ) = 8 πT μν (x). (242 ) 2
This can be solved formally as follows,
∫ ¯hμν(x) = 8π d4y√ −-g Gμν (x,y )T αβ(y), (243 ) αβ
where we have used the retarded tensor Green’s function defined by
μναβ -1- [ μναβ μναβ ] G (x,y) = 4π θ(Σ (x ),y) u (x,y )δ(σ (x,y)) + v (x,y)θ(− σ(x,y)) . (244 )
For the general tensor Green’s function and the definition of σ (x,y), μα g¯ (x,y), μναβ u (x,y), μναβ v (x, y), and Σ(x), please refer to Mino et al. [121] and DeWitt and Brehme [68].

Now we take the point particle limit. In this limit the above metric perturbation contains terms of different ε dependence which makes the calculation of the equations of motion simple. For example, terms with negative powers of ε appear from the Dirac delta distribution part of the Green’s function:

∫ ¯μν 4 √ --- μν αβ sh (x) = 2 d y − gu αβ(x,y )δ(σ (x,y))T (y ). (245 )

As explained below we only need the field on the boundary of the body zone which is the far zone of the body itself, and thus we may make use of the multipole expansion for the field. For this purpose we choose our coordinate system as follows,

yα = z α(τ) + δy α, (246 ) z0(τ ) = τ, (247 ) δy0 = 0, (248 )
where zα(τ) is the world line of the center of the object. The center is assumed to be always inside the body in the point particle limit, and thus there is no ambiguity for the choice of the center. In this coordinate system the volume element satisfies d4y = dτ d3δy. Thus we have
∫ √ --- s¯h μν(x ) = 2 d3δy − g uμναβ(x,z(τy) + δy )-1---T αβ(y), (249 ) ˙σ|τ=τy
where τy is the retarded time of each point y. Then the multipole expansion is obtained by expanding the above expression at the retarded time of the center of the object τz defined by σ (x, z(τz)) = 0. This can be easily done by noticing the condition σ(x, z(τy) + δy) = 0. Then the difference between τz and τy is given by
σ;α(x,z (τz)) δyα ( 2) δτ = − -----------------+ 𝒪 δy . (250 ) σ˙(x,z (τz))
Using this δτ we can expand uμναβ(x,z (τy) + δy) around τz. In principle we can calculate arbitrarily high multipole moments in this way. Here we only calculate the leading term. Then we only need μν μν u αβ(x,z (τy) + δy) = u αβ(x,z (τz)) + 𝒪 (δy), and ˙σ(x,z(τy) + δy) = ˙σ(x,z(τz)) + 𝒪 (δy ). By defining the mass as follows,
∫ α β 3 √ --- α β mz˙ (τz)z˙ (τz) = d δy − g T (y ), (251 )
we finally obtain the following expression for μν s¯h:
2m s¯h μν(x) = ----------u μναβ(x,z (τz))z˙α(τz)˙zβ(τz). (252 ) ˙σ(x,z (τz))

Now we derive the equations of motion using this expression. First we define the ε dependent four-momentum of the object as the volume integral of the effective stress-energy tensor μν Θ over the body zone B (τ),

∫ Pμ (τ ) = − Θ μν dΣ . (253 ) B(τ) ν
Since the effective stress-energy tensor satisfies the conservation law Θμν,ν = 0, the change of the four-momentum defined above may be expressed as the surface integral over the boundary of the body zone ∂B,
μ ∮ dP-- = − dΩ ε2n (1 + εa ⋅ n )(− g)tμν, (254 ) dτ ∂B ν LL
where μ n is the unit normal to the surface and μ μ a = du ∕d τ is the four-acceleration. The equations of motion are obtained by taking the point particle limit of ε → 0. Thus we need to calculate μν (− g)tLL on the body zone boundary (field points) on which the multipole expansion of the self-field and the Taylor expansion of the nonsingular part of the field are available, and to pick up only terms of order ε−2 in the expression. All the other terms but one term, which is proportional to 2 μ m a ∕ ε and can be renormalized to the mass of the small object, vanish in the limit or the angular integration. Remembering that the LL tensor is bilinear in the Christoffel symbols and the Christoffel symbols are the derivatives of the metric tensor, one may realize that the only remaining terms come from the combination of the 0th order of the smooth part of the metric and 1∕ε part of the self-field which is given by Equation (252View Equation). The remaining part of the self-field is the so-called tail part that is regular at the object which is the only relevant point in our calculation. The field point (x,τx) is now on the body zone boundary, which is defined by ∘ ------------ 2σ (x,z(τx)) = ε and
α [σ;α(x, z(τ))˙z (τ)]τ= τx = 0. (255 )

Using this expression we have the following result in the point particle limit,

μ dP-- = − m Γ μ (gs)uαu β − m 1Γ ααβ(gs)u βuμ, (256 ) dτ αβ 2
where the Christoffel symbols here are calculated in terms of the smooth part of the metric gs. Then the ADM mass is related to the four-momentum as follows, which is supported by the higher order post-Newtonian approximation [91Jump To The Next Citation Point9591]:
Pμ = √−-g-mu μ. (257 ) s
Finally we have
du μ μ α β ---- = − Γαβ (gs)u u , (258 ) dτ
which is the geodesic equation on the geometry determined by the smooth part of the metric around the compact object.

In fact, the spin effect on the equations of motion can be derived in a similar way and the standard result [152] can be obtained.

We have proved that a small compact object moves on the geodesic determined by only the smooth part of the geometry around the object. Thus the equations of motion are automatically obtained by determining the geometry around the object which is of course an implicit functional of the world line of the object. The smooth part contains the gravitational waves emitted by the orbital motion so that this equation includes the damping force due to radiation reaction. Our method avoids using a singular source in the first place by making use of the strong field point particle limit. All the quantities should be evaluated at the surface of the body zone boundary and thus we only need the dependence of the distance from the center of the object, namely the ε dependence of the field. In this way we are able to avoid using any divergent quantities in any part of our calculation. This strongly suggests that our method may be used to get unique equations of fast motion with radiation reaction. This will be investigated in future publications.

In this section, we have assumed spherical symmetry of the compact object except for the tidal effect. It is straightforward to generalize the case to multipole moments in our formalism. This will also be studied in future publications.

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