A New Proof of Birkhoo's Theorem
نویسنده
چکیده
Assuming SO(3)-spherical symmetry, the 4{dimensional Einstein equation reduces to an equation conformally related to the eld equation for 2{dimensional gravity following from the Lagrangian L = jRj 1=3. Solutions for 2{dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoo's theorem; contrary to other proofs, no coordinates must be introduced. The SO(m)-spherically symmetric solutions of the (m+1){dimen-sional Einstein equation can be found by considering L = jRj 1=m in two dimensions. This yields several generalizations of Birkhoo's theorem in an arbitrary number of dimensions, and to an arbitrary signature of the metric.
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