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About 50 years ago it was shown that one can formulate Einstein's equations
of General Relativity as an initial value problem. One specifies data
at one moment in time and then one reconstructs the spacetime from this
information. Needless to say, it is very difficult to do this in general.
Consequently, it is natural to make additional assumptions before trying
to deal with the problem. In cosmology for instance, one standard assumption
is that the universe is spatially homogeneous and isotropic. What this means
is that at a fixed time, the universe looks the same at different points
in space (spatial homogeneity) and it also looks the same in different
directions (isotropy). When one has made these stringent assumptions the
only freedom left is a scale factor, and Einstein's equations reduce to
an ordinary differential equation for the scale factor. It is of some
interest to try to relax these conditions. If one first drops the isotropy
condition, the mathematical problem one ends up with is a system of
ordinary differential equations. If one also relaxes the condition of
spatial homogeneity, one is confronted with non-linear wave equations.
One can still demand that there be no spatial variation in certain spatial
directions, so that one gets non-linear wave equations in 1+1 dimensions,
2+1 dimensions or 3+1 dimensions depending on the symmetry
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kth.se).