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2.3 Explicit calculation in harmonic coordinates

Here we shall use the above formalism to make an explicit calculation in harmonic coordinates. The reduced Einstein equations in the harmonic condition are written as
˜gαβ˜gμν = 16πΘ μν − ˜gμα ˜gνβ , (14 ) μν ,ααβ ,β ,α ∂μ[˜g ∂νx ] = 0, (15 )
˜gμν = (− g)1∕2gμν, (16 ) αβ αβ αβ Θ = (− g)(T + tLL), (17 )
where tμν LL is the Landau–Lifshitz pseudotensor [115Jump To The Next Citation Point]. In this section we shall choose an isentropic perfect fluid for αβ T which is enough for most applications,
T αβ = (ρ + ρΠ + P )uαu β + Pg αβ, (18 )
where ρ is the rest mass density, Π the internal energy, P the pressure, and μ u the four-velocity of the fluid with normalization
g uαu β = − 1. (19 ) αβ
The conservation of energy and momentum is expressed as
αβ ∇ βT = 0. (20 )

Defining the gravitational field variable as

μν μν 1∕2 μν h = η − (− g) g , (21 )
where ημν is the Minkowski metric, the reduced Einstein equations (14View Equation) and the gauge condition (15View Equation) take the following form:
αβ αβ μν μν μα νβ (η − h )h ,αβ = − 16 πΘ + h ,βh ,α, (22 ) hμν,ν = 0. (23 )
Thus the characteristics are determined by the operator αβ αβ (η − h )∂α∂β, and thus the light cone deviates from that in the flat spacetime. We may use this form of the reduced Einstein equations in the calculation of the waveform far away from the source because the deviation plays a fundamental role there [12]. However, in the study of the gravitational field near the source it is not necessary to consider the deviation of the light cone from the flat one and thus it is convenient to use the following form of the reduced Einstein equations [5]:
ημνh αβ,μν = − 16πΛ αβ, (24 )
αβ αβ αβμν Λ = Θ + χ ,μν, (25 ) χ αβμν = (16π)−1(h ανhβμ − hαβhμν). (26 )
Equations (23View Equation) and (24View Equation) together imply the conservation law
αβ Λ ,β = 0. (27 )
We shall take as our variables the set {ρ,P,vi,hαβ }, with the definition
i i 0 v = u∕u . (28 )
The time component of four-velocity u0 is determined from Equation (19View Equation). To make a well-defined system of equations we must add the conservation law for the number density n, which is some function of the density ρ and pressure P:
α ∇ α(nu ) = 0. (29 )
Equations (27View Equation) and (29View Equation) imply that the flow is adiabatic. The role of the equation of state is played by the arbitrary function n(ρ,p).

Initial data for the above set of equations are αβ αβ h ,h ,0,ρ,P, and i v, but not all these data are independent because of the existence of the constraint equations. Equations (23View Equation) and (24View Equation) imply the four constraint equations among the initial data for the field,

Δh α0 + 16π Λα0 − δijh α,0 = 0, (30 ) i,j
where Δ is the Laplacian in the flat space. We shall choose hij and hij,0 as free data and solve Equation (30View Equation) for h α0(α = 0,...,3) and Equation (23View Equation) for hα0,0. Of course these constraints cannot be solved explicitly, since Λα0 contains hα0, but they can be solved iteratively as explained below. As discussed above, we shall assume that the free data ij h and ij h ,0 for the field vanish. One can show that such initial data satisfy the O’Murchadha and York criterion for the absence of radiation far away from the source [124].

In the actual calculation, it is convenient to use an expression with explicit dependence of ε. The harmonic condition allows us to have such an expression in terms of the retarded integral,

∫ μν i 3 μν i μν i h (ε,τ,x ) = 4 i d yΛ (τ − εr,y,ε)∕r + hH (ε,τ,x ), (31 ) C(ε,τ,x )
where r = |yi − xi| and C (ε,τ,xi) is the past flat light cone of the event (τ,xi) in the spacetime given by ε, truncated where it intersects with the initial hypersurface τ = 0. hμν H is the unique solution of the homogeneous wave equation in the flat spacetime,
□h μHν = ηαβh μHν,αβ = 0. (32 )
μν h H evolves from a given initial data on the τ = 0 initial hypersurface which are subject to the constraint equations (30View Equation). The explicit form of μν hH is available via the Poisson formula (see e.g. [144Jump To The Next Citation Point]),
[ ] μν i τ ∮ μν i 1 ∂ ∮ μν i h H (τ,x ) = --- h ,τ(τ = 0,y ) dΩy + ------ τ h (τ = 0, y) dΩy . (33 ) 4π ∂C(τ,xi) 4π ∂τ ∂C (τ,xi)

We shall henceforth ignore the homogeneous solutions because they play no important role. Because of the ε dependence of the integral region, the domain of integral is finite as long as ε ⁄= 0 and their diameter increases like ε−1 as ε → 0.

Given the formal expression (31View Equation) in terms of initial data (9View Equation), we can take the Lie derivative and evaluate these derivatives at ε = 0. The Lie derivative is nothing but a partial derivative with respect to ε in the coordinate system for the fiber bundle given by i (ε,τ,x ). Accordingly one should convert all the time indices to τ indices. For example, Tττ = ε2T tt which is of order ε4, since Ttt ∼ ρ is of order ε2. Similarly the other components of stress-energy tensor Tτi = εTti and Tij are of order ε4 as well. Thus we expect that the first nonvanishing derivative in Equation (31View Equation) will be the forth derivative. In fact we find

∫ i hττ(τ,xi) = 4 2ρ(τ,y-) d3y, (34 ) 4 R3 r ∫ k i k 4hτi(τ,xi) = 4 2ρ(τ,y-)1v-(τ,x-) d3y, (35 ) R3 r ∫ 2ρ(τ,yk)1vi(τ,yk)1vj(τ,yk) + 4tij (τ,yk ) 4hij(τ,xk) = 4 ------------------------------LL-------d3y, (36 ) R3 r (37 )
where we have adopted the notation
1 ∂n nf (τ,xi) = --lim --n-f(ε,τ,xi), (38 ) n!ε→0 ∂ε
1 ( 1 ) 4tiLjL = ---- 4h ττ,i4h ττ,j − -δij4h ττ,k4hττ,k . (39 ) 64 π 2

In the above calculation we have taken the point of view that h μν is a tensor field, defined by giving its components in the assumed harmonic coordinates as the difference between the tensor density √ --- μν − gg and ημν.

The conservation law (27View Equation) also has its first nonvanishing derivatives at this order, which are

2ρ,τ + (2ρ1vi),i = 0, (40 ) (2ρ1vi),τ + (2ρ1vi1vj),j + 4P,i − 12ρ4h ττ,i = 0. (41 ) 4
Equations (34View Equation), (40View Equation), and (41View Equation) constitute Newtonian theory of gravity. Thus the lowest nonvanishing derivative with respect to ε is indeed Newtonian theory, and the 1 PN and 2 PN equations emerge from the sixth and eighth derivatives, respectively, in the conservation law (27View Equation). At the next derivative, the quadrupole radiation reaction term emerges.

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