 |
 |
 |
|
|
|
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,
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,
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
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
), (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.
|
|
|
| http://www.livingreviews.org/lrr-2007-2 | 
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |
![˜gαβ˜gμν = 16πΘ μν − ˜gμα ˜gνβ , (14 ) μν ,ααβ ,β ,α ∂μ[˜g ∂νx ] = 0, (15 )](article100x.gif){kind=small}
![[ ] 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)](article150x.gif){kind=small}
