, represent ten independent equations. Since there are ten
equations and ten independent components of the 4-metric
, it seems that we have the same number of equations as
unknowns. From the definition of the Einstein tensor (3
), we see that these ten equations are linear in the second
derivatives and quadratic in the first derivatives of the metric.
We might expect that these ten second-order equations represent
evolution equations for the ten components of the metric.
However, a close inspection of the equations reveals that only
six of the ten involve second time-derivatives of the metric. The
remaining four equations are not evolution equations. Instead,
they are constraint equations. The full system of equations is
still well posed, however, because of the Bianchi identities
The four constraint equations appear as a result of the general covariance of Einstein's theory, which gives us the freedom to apply general coordinate transformations to each of the four coordinates and leave the interval
unchanged.
If we consider Einstein's equations as a Cauchy problem
, we find that the ten equations separate into a set of four
constraint or initial-value equations, and six evolution or
dynamical equations. If the four initial-value equations are
satisfied on some spacelike hypersurface, which we can label with
t
=0, then the Bianchi identities (4) guarantee that the evolution equations preserve the constraints
on neighboring spacelike hypersurfaces.
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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 |
