Abstract
When Lagrangians are restricted to subspaces of the phase space satisfying certain symmetry conditions, they do not always yield the correct Einstein equations. An example was found by Hawking in the Lagrangian for spatially homogeneous universes. First we show that this problem is not limited to general relativity. Then we “visualize” certain terms which do not appear in Lagrangians on symmetry-restricted phase spaces but which nevertheless contribute to the Euler-Lagrange derivatives. We apply our general formula to the Schwarzschild metric, the Friedmann universes, and spatially homogeneous universes.
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Fushiki, I. Einstein equations and Lagrangians on symmetry-restricted phase spaces. Gen Relat Gravit 17, 1017–1024 (1985). https://doi.org/10.1007/BF00774204
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DOI: https://doi.org/10.1007/BF00774204