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
where
where is the Landau–Lifshitz pseudotensor [115]. In this section we shall choose an isentropic perfect fluid for which is enough for most applications,
where is the rest mass density, the internal energy, the pressure, and the four-velocity of the fluid with normalization
The conservation of energy and momentum is expressed as

Defining the gravitational field variable as

where is the Minkowski metric, the reduced Einstein equations (14) and the gauge condition (15) take the following form:
Thus the characteristics are determined by the operator , 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]:
where
Equations (23) and (24) together imply the conservation law
We shall take as our variables the set , with the definition
The time component of four-velocity is determined from Equation (19). To make a well-defined system of equations we must add the conservation law for the number density , which is some function of the density and pressure :
Equations (27) and (29) imply that the flow is adiabatic. The role of the equation of state is played by the arbitrary function .

Initial data for the above set of equations are , and , but not all these data are independent because of the existence of the constraint equations. Equations (23) and (24) imply the four constraint equations among the initial data for the field,

where is the Laplacian in the flat space. We shall choose and as free data and solve Equation (30) for and Equation (23) for . Of course these constraints cannot be solved explicitly, since contains , but they can be solved iteratively as explained below. As discussed above, we shall assume that the free data and 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 . The harmonic condition allows us to have such an expression in terms of the retarded integral,

where and is the past flat light cone of the event in the spacetime given by , truncated where it intersects with the initial hypersurface . is the unique solution of the homogeneous wave equation in the flat spacetime,
evolves from a given initial data on the initial hypersurface which are subject to the constraint equations (30). The explicit form of is available via the Poisson formula (see e.g. [144]),

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

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

where we have adopted the notation
and

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

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

Equations (34), (40), and (41) constitute Newtonian theory of gravity. Thus the lowest nonvanishing derivative with respect to is indeed Newtonian theory, and the 1 PN and 2 PN equations emerge from the sixth and eighth derivatives, respectively, in the conservation law (27). At the next derivative, the quadrupole radiation reaction term emerges.