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2 Einstein’s Field Equations

The field equations of general relativity form a system of ten second-order partial differential equations obeyed by the space-time metric gab,
Gab[g,@g, @2g] = 8pG-T ab[g], (9) c4
where the Einstein curvature tensor Gab =_ Rab - 1R gab 2 is generated, through the gravitational coupling k = 8pG/c4, by the matter stress-energy tensor T ab. Among these ten equations, four govern, via the contracted Bianchi identity, the evolution of the matter system,
\~/ Gam =_ 0 ===> \~/ T am = 0. (10) m m
The space-time geometry is constrained by the six remaining equations, which place six independent constraints on the ten components of the metric gab, leaving four of them to be fixed by a choice of a coordinate system.

In most of this paper we adopt the conditions of harmonic, or de Donder, coordinates. We define, as a basic variable, the gravitational-field amplitude

hab = V~ --g gab- jab, (11)
where ab g denotes the contravariant metric (satisfying am a g gmb = db), where g is the determinant of the covariant metric, g = det(gab), and where jab represents an auxiliary Minkowskian metric. The harmonic-coordinate condition, which accounts exactly for the four equations (10View Equation) corresponding to the conservation of the matter tensor, reads
@mham = 0. (12)
Equations (11View Equation, 12View Equation) introduce into the definition of our coordinate system a preferred Minkowskian structure, with Minkowski metric jab. Of course, this is not contrary to the spirit of general relativity, where there is only one physical metric gab without any flat prior geometry, because the coordinates are not governed by geometry (so to speak), but rather are chosen by researchers when studying physical phenomena and doing experiments. Actually, the coordinate condition (12View Equation) is especially useful when we view the gravitational waves as perturbations of space-time propagating on the fixed Minkowskian manifold with the background metric jab. This view is perfectly legitimate and represents a fruitful and rigorous way to think of the problem when using approximation methods. Indeed, the metric jab, originally introduced in the coordinate condition (12View Equation), does exist at any finite order of approximation (neglecting higher-order terms), and plays in a sense the role of some “prior” flat geometry.

The Einstein field equations in harmonic coordinates can be written in the form of inhomogeneous flat d’Alembertian equations,

16pG []hab = ------tab, (13) c4
where [] =_ []j = jmn@m@n. The source term t ab can rightly be interpreted as the stress-energy pseudo-tensor (actually, tab is a Lorentz tensor) of the matter fields, described by Tab, and the gravitational field, given by the gravitational source term ab /\, i.e.
c4 t ab = |g|T ab + -----/\ab. (14) 16pG
The exact expression of /\ab, including all non-linearities, reads5
ab mn 2 ab an bm 1-ab mt nc /\ = -h @mnh + @mh @nh + 2g gmn@ch @th -gamg @ hbt@ hnc - gbmg @ hat@ hnc + g gct@ ham@ hbn nt c m nt c m mn c t + 1(2gamgbn - gabgmn)(2gctgep - gtegcp)@mhcp@nhte. (15) 8
As is clear from this expression, /\ab is made of terms at least quadratic in the gravitational-field strength h and its first and second space-time derivatives. In the following, for the highest post-Newtonian order that we consider (3PN), we need the quadratic, cubic and quartic pieces of ab /\. With obvious notation, we can write them as
/\ab = N ab[h,h] + M ab[h,h, h] + Lab[h, h,h,h] + O(h5). (16)
These various terms can be straightforwardly computed from Equation (15View Equation); see Equations (3.8) in Ref. [38Jump To The Next Citation Point] for explicit expressions.

As said above, the condition (12View Equation) is equivalent to the matter equations of motion, in the sense of the conservation of the total pseudo-tensor ab t,

@ tam = 0 <====> \~/ T am = 0. (17) m m
In this article, we look for the solutions of the field equations (13View Equation, 14View Equation, 15View Equation, 17View Equation) under the following four hypotheses:
  1. The matter stress-energy tensor ab T is of spatially compact support, i.e. can be enclosed into some time-like world tube, say r < a, where r = |x| is the harmonic-coordinate radial distance. Outside the domain of the source, when r > a, the gravitational source term, according to Equation (17View Equation), is divergence-free,
    am @m/\ = 0 (when r > a). (18)
  2. The matter distribution inside the source is smooth6: ab o o 3 T (- C (R ). We have in mind a smooth hydrodynamical “fluid” system, without any singularities nor shocks (a priori), that is described by some Eulerian equations including high relativistic corrections. In particular, we exclude from the start any black holes (however we shall return to this question when we find a model for describing compact objects).
  3. The source is post-Newtonian in the sense of the existence of the small parameter defined by Equation (1View Equation). For such a source we assume the legitimacy of the method of matched asymptotic expansions for identifying the inner post-Newtonian field and the outer multipolar decomposition in the source’s exterior near zone.
  4. The gravitational field has been independent of time (stationary) in some remote past, i.e. before some finite instant -T in the past, in the sense that
    @ [ ab ] --- h (x,t) = 0 when t < - T . (19) @t

The latter condition is a means to impose, by brute force, the famous no-incoming radiation condition, ensuring that the matter source is isolated from the rest of the Universe and does not receive any radiation from infinity. Ideally, the no-incoming radiation condition should be imposed at past null infinity. We shall later argue (see Section 6) that our condition of stationarity in the past, Equation (19View Equation), although much weaker than the real no-incoming radiation condition, does not entail any physical restriction on the general validity of the formulas we derive.

Subject to the condition (19View Equation), the Einstein differential field equations (13View Equation) can be written equivalently into the form of the integro-differential equations

16pG hab = ---4--[] -re1ttab, (20) c
containing the usual retarded inverse d’Alembertian operator, given by
integral integral integral -1 1 d3x' ' ' ([] retf )(x, t) =_ ---- ------'-f(x ,t- |x - x |/c), (21) 4p |x - x |
extending over the whole three-dimensional space R3.

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