Given the foliation, and the triad
we can reconstruct the metric as
. Given another foliation,
, but the same triad, we get another metric tensor, the relation
between both is a diffeomorphism which leaves invariant
and is generated by the integral curves of
. Alternatively we can choose another pair
and so construct another metric tensor, the relation between
both metric thus obtained is also a diffeomorphism. Since one is
interested in geometries, that is in metric tensors up to
diffeomorphisms, both metric obtained are equivalent, but one
needs to pick up an specific one in order to write down, (and
solve), Einstein's equations.
Using the above splitting vacuum Einstein's equations become two evolution equations:
where the dot means a Lie derivative with respect to
,
and
is the
extrinsic curvature
of
with respect to the four-geometry, that is,
.
is the covariant derivative associated with
on each
, and
its Ricci tensor.
And two constraint equations:
Note that there are two ``dynamical'' variables,
, and
, while the lapse-shift pair
, although necessary to determine the evolution, is undetermined
by the equations. Note also that they do not enter into the
constraint equations, for as said above a change on the
lapse-shift pair leave the fields at initial surface
unchanged.
It is important step back now and see that these equations can
be thought as ``living'' in a structure completely detached from
space-time. To see this, identify all points laying in the same
integral curve of
, thus the equivalence class is a three dimensional manifold
S, homeomorphic to any
. On it, for each
t
we can induce space-contravariant tensors, such as
,
,
, and scalars, as
N
(t). As long as the surfaces are space-like, the induced metric is
negative definite and we can invert it, thus we can perform all
kinds of contractions and write the above equations as dynamical
equations on the parameter
t
on fields on the same manifold,
S
. This is of course the setting in which one sets most of the
schemes to solve the equations, and it is hard to keep control,
even awareness, that the surfaces defining the foliation can
become null or nearly so. Einstein's equations ``feel'' that
effect since they are causal, and this is abruptly fed back via
the development of singularities on the solutions . They do not
have any thing to do with real singularities of space-time, but
rather with foliations becoming null.
The first application of this idea to Einstein's equations
appears to have been [39]. There a hyperbolic system consisting of wave equations for the
time derivative of the variable
is obtained when the shift is taken to vanish and the lapse is
chosen so as to impose the time component of the harmonic gauge.
The shift is set to zero, but, as stated in the paper, this is an
unnecessary condition. Thus, it is clear that one gains in
flexibility compared with the standard method above. This seems
to correspond with the fact that in one of the equations forming
the system, the time derivative of the evolution equation for the
momentum, the momentum constraint has been suitably added, thus
modifying the evolution flow outside the constraint sub-manifold.
As stated in the paper, they could not use the other constraint,
the Hamiltonian one, to modify that equation.
The condition for that system to be (symmetric) hyperbolic is
that the term below should not have second derivatives of
.
where
is the induced three metric on a hypersurface,
is the covariant derivative at that hypersurface compatible with
,
,
N
is the lapse function, and
, with
the momentum field conjugated to
.
The simplest condition to guarantee this is:
which in view of the definition of
, which implies
, has as a solution,
where
is the determinant of the metric
with respect to a constant in time background metric
. This is precisely the harmonic condition for the time
component of
in notation introduced in the paper's introduction. That is,
, where
it the normal to the foliation. If the determinant of
is not taken to be constant in time, then one gets,
and so the system remains hyperbolic. Thus, we see that, up to the determinant of the metric, the lapse can be prescribed freely. This freedom is very important because it gives ductility to the approach, since this function can be specified according to the needs of applications. We shall call this a generalized harmonic time gauge .
Although in the introduction of [39] there is a remark dedicated to numerical relativists about the
possible importance of having a stable system, the paper did not
spark interest until recently, when applications required these
results to proceed. In recent years, a number of papers have
appeared which further elaborate on this system, [40,
42
,
43]. In particular I would like to mention [42], where the authors look at the system in detail, writing it as
a first order system, and introduce all variables which are
needed for that. In these recent papers, the generalized harmonic
time gauge has been included, as well as arbitrarily prescribed
shift vectors. If one attempts attempts to write down the system
as a first order one, that is, to give new names to the
derivatives of the basic fields until bringing the system to the
form of equation
1
, the resulting system is rather big, it has fifty four
variables, without counting the lapse-shift pair. We shall see
that there are first order hyperbolic systems with half that
number of variables.
Two similar results are of interest: In [44] a system is introduced with basically the same properties, but
of lower order, that is, only first derivatives of the basic
variables are taken as new independent variables in making the
system first order. In this paper, it is realized that the same
trick of modifying the evolution equations using the constraints
can be done by modifying, instead of the second time derivative
of the momentum, the extra equation which appears when making the
ADM equations a first order system, that is the equation which
fixes the time evolution of the space derivatives of the metric,
or alternatively the time evolution of the Christoffel symbols.
When this equation is suitably modified by adding a term
proportional to the momentum constraint, and when the harmonic
gauge in the generalized sense used above is imposed, a symmetric
hyperbolic system results. In [45] the generalized harmonic time gauge is included, as well as
arbitrarily prescribed shift vectors. For the latest on this
approach see, [47
]. I shall comment more on this in next section, §
4
.
In [48] a similar system is presented. In this case, the focus is on
establishing some rigorous results in the Newtonian limit. So a
conformal rescaling of the metric is employed using the lapse
function as conformal factor. The immediate consequence of this
transformation is to eliminate from the evolution equation for
the term with second space derivatives of the lapse function,
precisely the term giving rise to one of the terms in equation
4.1
. The end consequence is that the conformal metric is flatter to
higher order. With this re-scaling, and using the same type of
modification of the evolution equation for the space derivatives
of the metric that the above two approaches use, a symmetric
hyperbolic system is found, for arbitrary shift and lapse.
This freedom of the lapse and shift was used to cancel several
divergent terms of the energy integrals in the Newtonian limit by
imposing an elliptic gauge condition on the shift, which also
determined uniquely the lapse. This resulted in a mixed symmetric
hyperbolic-elliptic system of equations. In [49
] an attempt is made to explore what other possibilities there
are of making symmetric hyperbolic systems for general relativity
with arbitrarily prescribed lapse-shift pairs. A set of
parametrized changes of field variables and of linear
combinations of equations are made, and it is shown that there
exists at least a one parameter family of symmetric hyperbolic
systems. In these systems generalized harmonic time gauge is
replaced by:
So, the dependence of the lapse on the determinant of the metric can be modified, but never suppressed. Since the changes in the parameter imply changes in the dynamical variables, while the factor proportional to the momentum added to the evolution equation for the connection is unique, and so fixed, it is not clear whether this can be of help for improving numerical algorithms. We shall see this type of dependence arise in one of the developments of one of the above mentioned approaches.
In [35] a similar system, in the sense of using variables from the 3+1
decomposition, is obtained by imposing also the same generalized
harmonic time gauge. This system, as is the original system of [39
], is of higher order because it includes the electric and
magnetic parts of the Weyl tensor in the 3+1 decomposition. As
such, it contains more variables (fifty) than the two discussed
above (thirty).
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |