Here,
is called the
lapse function
(frequently denoted
N
in the literature). The scalar lapse function sets the proper
interval measured by observers as they move between slices on a
path that is normal to the hypersurface (so-called normal
observers):
Of course, there is no reason that observers must move along a path normal to the hypersurface. In general, we can define the time vector as
where
Here,
is called the
shift vector
(frequently denoted
in the literature).
Because of (9),
has only three independent components and is a
spatial
vector, tangent to the hypersurface on which it resides. At this
point, it is convenient to introduce a coordinate system adapted
to the foliation
. Let
be the spatial coordinates in the slice. The fourth coordinate,
t, is the parameter labeling each slice. With this adapted
coordinate system, we find that 3-dimensional coordinate values
remain constant as we move between slices along the
direction (8
). The four parameters,
and
, are a manifestation of the 4-dimensional coordinate
invariance, or gauge freedom, in Einstein's theory. If we let
represent the metric of the spacelike hypersurfaces, then we can
rewrite the interval (5
) as
In the Cauchy formulation of Einstein's equations,
is regarded as the fundamental variable and values for its
components must be given as part of a well-posed initial-value
problem. Since Einstein's equations are second order, we must
also specify something like a time derivative of the metric. For
this, we use the second fundamental form, or extrinsic curvature,
of the slice,
, defined
by
where
denotes the Lie derivative along the
direction.
Together,
and
are the minimal set of initial data that must be specified for a
Cauchy evolution of Einstein's equations. The metric
on a hypersurface is induced on that surface by the 4-metric
. This means that the values
receives depend on how
is embedded in the full spacetime. In order for the foliation of
slices
to fit into the higher-dimensional space, they must satisfy the
Gauss-Codazzi-Ricci conditions. Combining these conditions with
Einstein's equations, and using (10
), the six evolution equations become
Here,
is the spatial covariant derivative compatible with
,
is the Ricci tensor associated with
,
,
is the matter energy density,
is the matter stress tensor, and
.
We have also used the fact that in our adapted coordinate
system,
. The set of second-order evolution equations is completed by
rewriting the definition of the extrinsic curvature (11
) as
Equations (12) and (13
) are a first-order representation of a complete set of evolution
equations for given initial data
and
. However, the data cannot be freely specified in their
entirety. The four constraint equations, following the same
procedure outlined above for the evolution equations, become
and
Here,
and
is the matter momentum density. Equation (14
) is referred to as the Hamiltonian or scalar constraint, while (15
) are referred to as the momentum or vector constraints. Valid
initial data for the evolution equations (12
) and (13
) must satisfy this set of constraints. And, as mentioned
earlier, the Bianchi identities (4
) guarantee that the evolution equations will preserve the
constraints on future slices of the evolution.
As we will see below, the Hamiltonian constraint (14) most naturally constrains the 3-metric
, while the momentum constraints (15
) naturally constrain the extrinsic curvature
. Taking the constraints into consideration, it seems that the
3-metric has five degrees of freedom remaining, while the
extrinsic curvature has three. But we know that the gravitational
field in Einstein's theory has two dynamical degrees of freedom,
so we expect that both
and
should each have only two free components. The answer to this
problem is, once again, the coordinate invariance of Einstein's
theory. This seems strange at first, because we have already used
the coordinate invariance of the theory to narrow our scope from
the ten components of the 4-metric to the six components of the
3-metric. However, the lapse and shift do not completely specify
the coordinate gauge. Rather, they specify how an initial choice
of gauge will evolve with the foliation. The metric on a given
hypersurface retains full 3-dimensional coordinate invariance,
reducing the number of freely specifiable components to
two [107
,
108
]. There also remains one degree of gauge freedom associated with
the time coordinate which must be fixed. Each hypersurface
represents a
slice of the spacetime, so how the initial hypersurface is
embedded in the full 4-dimensional manifold represents our
temporal gauge choice. There is no unique way to specify this
choice, but it is often convenient to let the trace of the
extrinsic curvature
K
represent this temporal gauge choice [108
]. Thus, we find that we
are
allowed to choose freely five components of the 3-metric and
three components of the extrinsic curvature. However, only
two
of the components for each field represent dynamical degrees of
freedom, the remainder are gauge degrees of freedom.
The four constraint equations, (14) and (15
), represent conditions which the 3-metric and extrinsic
curvature must satisfy. But, they do not specify which components
(or combination of components) are constrained and which are
freely specifiable. In the weak field limit where Einstein's
equations can be linearized, there are clear ways to determine
which components are dynamic, which are constrained, and which
are gauge. However, in the full nonlinear theory, there is no
unique decomposition. In this case, one must choose a method for
decomposing the constraint equations. The goal is to transform
the equations into standard elliptic forms which can be solved
given appropriate boundary conditions [80,
82
,
111
]. Each different decomposition yields a unique set of elliptic
equations to be solved
and
a unique set of freely specifiable parameters which must be
fixed somehow. Seemingly similar sets of assumptions applied to
different decompositions can lead to physically different initial
conditions.
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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 |