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 |