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In my studies I faced "Einstein evolution equations". I know about "Einstein equation", but I don't know what are "Einstein evolution equations"? Unfortunately, my searches did not result in much. Can anyone help me?

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As is said in Carroll's Lecture Notes on General Relativity :

There are ten independent components in Einstein’s equations. The four represented by $$G^{0\nu}=8\pi GT^{0\nu}\tag{4.82}$$ cannot be used to evolve the initial data $(g_{\mu \nu},\partial_t g_{\mu \nu})_\sum$. Rather, they serve as constraints on this initial data; we are not free to specify any combination of the metric and its time derivatives on the hypersurface , since they must obey the relations $(4.82)$. The remaining equations, $$G^{ij}=8\pi GT^{ij}$$ are the dynamical evolution equations for the metric.

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