  Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation
reduces to an equation conformally related to the field equation for
2-dimensional gravity following from the Lagrangian L = R^(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 Birkhoff's theorem; contrary to other proofs, no
coordinates must be introduced.
  The SO(m)-spherically symmetric solutions of the (m+1)-dimensional Einstein
equation can be found by considering L = R^(1/m) in two dimensions. This yields
several generalizations of Birkhoff's theorem in an arbitrary number of
dimensions, and to an arbitrary signature of the metric.
