First, the metric is decomposed into a conformal factor
multiplying an auxiliary 3-metric [68,
107,
108]:
The auxiliary 3-metric
is often called the conformal or background 3-metric, and it
carries five degrees of freedom. Its natural definition is given
by
leaving
, but we are free to choose any normalization for
. Using (16
), we can rewrite the Hamiltonian constraint (14
) as
where
is the scalar Laplace operator, and
and
are the covariant derivative and Ricci scalar associated with
. Equation (18
) is a quasilinear elliptic equation for the conformal factor
, and we see that the Hamiltonian constraint naturally
constrains the 3-metric.
The conformal decomposition of the Hamiltonian constraint was proposed by Lichnerowicz. But, the key to the full decomposition is the treatment of the extrinsic curvature introduced by York [109, 110]. This begins by splitting the extrinsic curvature into its trace and tracefree parts,
The decomposition proceeds by using the fact that we can covariantly split any symmetric tracefree tensor as follows:
Here,
is a symmetric, transverse-traceless tensor (i. e.,
and
) and
After separating out the transverse-traceless portion of
, what remains,
, is referred to as its ``longitudinal'' part. We now want to
apply this transverse-traceless decomposition to the tracefree
part of the extrinsic curvature
. However, the conformal decomposition of the metric leaves us
with at least two ways to proceed.
The goal of the decomposition is to produce a coupled set of
elliptic equations to be solved with some prescribed boundary
conditions. We have already reduced the Hamiltonian constraint to
an elliptic equation being solved on a
background
space in terms of differential operators that are compatible
with the conformal 3-metric. In the end, we want to reduce the
momentum constraints to a set of elliptic equations based on
differential operators that are compatible with the same
conformal 3-metric. But, the longitudinal operator (21) can be defined with respect to
any
metric space. In particular, it is natural to consider
decompositions with respect to both the physical and conformal
3-metrics.
Next, the transverse-traceless decomposition is applied to the conformal extrinsic curvature,
Note that the longitudinal operator
and the symmetric, transverse-tracefree tensor
are both defined with respect to covariant derivatives
compatible with
.
Applying equations (16), (19
), (21
), (22
), and (23
) to the momentum constraints (15
), we find that they simplify to
[A1
]
and we have used the fact that
for any symmetric tracefree tensor
.
In deriving equation (24), we have also used the fact that
is transverse (i. e.
). However, in general, we will not know if a given symmetric
tracefree tensor, say
, is transverse. By using (20
) we can obtain its transverse-traceless part
via
and using the fact that if
is transverse, we find
Thus, Eqs. (27) and (28
) give us a general way of
constructing
the required symmetric transverse-traceless tensor from a
general symmetric traceless tensor.
Using the linearity of
, we can rewrite (23
) as
where
Similarly, using the linearity of
, we can rewrite (24
) as
By solving directly for
, we can combine the steps of decomposing
with that of solving the momentum constraints.
After applying (19) and (22
) to the Hamiltonian constraint (18
), we obtain the following full decomposition, which I will list
together here for convenience:
In the decomposition given by (32), we are free to specify a symmetric tensor
as the conformal 3-metric, a symmetric tracefree tensor
, and a scalar function
K
. Then, with given matter energy and momentum densities,
and
, and appropriate boundary conditions, the coupled set of
constraint equations for
and
are solved. Finally, given the solutions, we can construct the
physical initial data,
and
.
The decomposition outlined above has the interesting property
that if we choose
K
to be constant and if the momentum density vanishes
, then the momentum constraint equations fully decouple from the
Hamiltonian constraint. As we will see later, this simplification
has proven to be useful.
In this case, the longitudinal operator
and the symmetric transverse-tracefree tensor
are both defined with respect to covariant derivatives
compatible with
.
Applying equations (16), (19
), (21
), (33
), and (26
) to the momentum constraint (15
), we find that it simplifies to
where we have used the fact that
As in the previous section, we will obtain the symmetric
transverse-traceless tensor
from a general symmetric tracefree tensor
by using (20
). In this case, we take
and use the fact that
is transverse, to obtain
Again, we can define
and use the linearity of
and
to combine the process of obtaining the transverse-traceless
part of
and solving the momentum constraints. We obtain the following
full decomposition, which I will list together here for
convenience
:
In the decomposition given by (39), we are again free to specify a symmetric tensor
as the conformal 3-metric, a symmetric tracefree tensor
, and a scalar function
K
. Then, with given matter energy and momentum densities,
and
, and appropriate boundary conditions, the coupled set of
constraint equations for
and
are solved. Finally, given the solutions, we can construct the
physical initial data,
and
.
Notice that, while very similar to the decomposition from
Section
2.2.1, the sets of equations are distinctly different. In general, if
we make the same choices for the freely specifiable data in both
decompositions (i. e., we choose
,
, and
K
the same), we will produce two different sets of initial data.
Both will be equally valid solutions of the constraint equations,
but they will have distinct physical properties.
There is at least one exception to this. Assume we have a
valid set of initial data
and
, which satisfies the constraint equations (14
) and (15
). For any everywhere-positive function
, we define our freely specifiable data as follows:
Then the solution to both sets of equations, assuming we use
correct boundary conditions, will be
and
, which yields the original data as the solution for each
decomposition.
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Initial Data for Numerical Relativity
Gregory B. Cook http://www.livingreviews.org/lrr-2000-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |