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4.3 Einstein’s equations in “relaxed” form

The Einstein equations G μν = 8πTμν are elegant and deceptively simple, showing geometry (in the form of the Einstein tensor G μν, which is a function of spacetime curvature) being generated by matter (in the form of the material stress-energy tensor T μν). However, this is not the most useful form for actual calculations. For post-Newtonian calculations, a far more useful form is the so-called “relaxed” Einstein equations:
□h αβ = − 16π ταβ, (62 )
where 2 2 2 □ ≡ − ∂ ∕∂t + ∇ is the flat-spacetime wave operator, αβ h is a “gravitational tensor potential” related to the deviation of the spacetime metric from its Minkowski form by the formula h αβ ≡ ηαβ − (− g )1∕2gαβ, g is the determinant of gαβ, and a particular coordinate system has been specified by the de Donder or harmonic gauge condition ∂h αβ∕∂xβ = 0 (summation on repeated indices is assumed). This form of Einstein’s equations bears a striking similarity to Maxwell’s equations for the vector potential α A in Lorentz gauge: α α □A = − 4 πJ, α α ∂A ∕∂x = 0. There is a key difference, however: The source on the right hand side of Equation (62View Equation) is given by the “effective” stress-energy pseudotensor
ταβ = (− g)T αβ + (16π)−1Λ αβ, (63 )
where Λαβ is the non-linear “field” contribution given by terms quadratic (and higher) in h αβ and its derivatives (see [189], Eqs. (20.20, 20.21) for formulae). In GR, the gravitational field itself generates gravity, a reflection of the nonlinearity of Einstein’s equations, and in contrast to the linearity of Maxwell’s equations.

Equation (62View Equation) is exact, and depends only on the assumption that spacetime can be covered by harmonic coordinates. It is called “relaxed” because it can be solved formally as a functional of source variables without specifying the motion of the source, in the form

∫ αβ ταβ(t − |x − x ′|,x′) 3 ′ h (t,x) = 4 ------------′------d x , (64 ) 𝒞 |x − x |
where the integration is over the past flat-spacetime null cone 𝒞 of the field point (t,x). The motion of the source is then determined either by the equation ∂ ταβ∕∂x β = 0 (which follows from the harmonic gauge condition), or from the usual covariant equation of motion αβ T ;β = 0, where the subscript ;β denotes a covariant divergence. This formal solution can then be iterated in a slow motion (v < 1) weak-field (||hαβ|| < 1) approximation. One begins by substituting αβ h 0 = 0 into the source ταβ in Equation (64View Equation), and solving for the first iterate hα1β, and then repeating the procedure sufficiently many times to achieve a solution of the desired accuracy. For example, to obtain the 1PN equations of motion, two iterations are needed (i.e. αβ h 2 must be calculated); likewise, to obtain the leading gravitational waveform for a binary system, two iterations are needed.

At the same time, just as in electromagnetism, the formal integral (64View Equation) must be handled differently, depending on whether the field point is in the far zone or the near zone. For field points in the far zone or radiation zone, − ′ |x | > λ > |x | (− λ is the gravitational wavelength divided by 2π), the field can be expanded in inverse powers of R = |x | in a multipole expansion, evaluated at the “retarded time” t − R. The leading term in 1∕R is the gravitational waveform. For field points in the near zone or induction zone, |x| ∼ |x′| < λ−, the field is expanded in powers of |x − x′| about the local time t, yielding instantaneous potentials that go into the equations of motion.

However, because the source ταβ contains h αβ itself, it is not confined to a compact region, but extends over all spacetime. As a result, there is a danger that the integrals involved in the various expansions will diverge or be ill-defined. This consequence of the non-linearity of Einstein’s equations has bedeviled the subject of gravitational radiation for decades. Numerous approaches have been developed to try to handle this difficulty. The “post-Minkowskian” method of Blanchet, Damour, and Iyer [353637763833] solves Einstein’s equations by two different techniques, one in the near zone and one in the far zone, and uses the method of singular asymptotic matching to join the solutions in an overlap region. The method provides a natural “regularization” technique to control potentially divergent integrals (see [34Jump To The Next Citation Point] for a thorough review). The “Direct Integration of the Relaxed Einstein Equations” (DIRE) approach of Will, Wiseman, and Pati [291Jump To The Next Citation Point208] retains Equation (64View Equation) as the global solution, but splits the integration into one over the near zone and another over the far zone, and uses different integration variables to carry out the explicit integrals over the two zones. In the DIRE method, all integrals are finite and convergent. Itoh and Futamase have used an extension of the Einstein–Infeld–Hoffman matching approach combined with a specific method for taking a point-particle limit [134], while Damour, Jaranowski, and Schäfer have pioneered an ADM Hamiltonian approach that focuses on the equations of motion [139140777978].

These methods assume from the outset that gravity is sufficiently weak that ||hαβ|| < 1 and harmonic coordinates exists everywhere, including inside the bodies. Thus, in order to apply the results to cases where the bodies may be neutron stars or black holes, one relies upon the SEP to argue that, if tidal forces are ignored, and equations are expressed in terms of masses and spins, one can simply extrapolate the results unchanged to the situation where the bodies are ultrarelativistic. While no general proof of this exists, it has been shown to be valid in specific circumstances, such as at 2PN order in the equations of motion, and for black holes moving in a Newtonian background field [70].

Methods such as these have resolved most of the issues that led to criticism of the foundations of gravitational radiation theory during the 1970s.

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