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In this worksheet I demonstrate three versions of Birkhoff's theorem, which is a characterization of spherically symmetric solutions of the Einstein equations. The three versions considered here correspond to taking the "Einstein equations" to be: (1) the vacuum Einstein equations; (2) the Einstein equations with a cosmological constant (3) the Einstein-Maxwell equations. I will restrict my attention to 4-dimensional spacetimes.
Torre, Charles G., "Examples of the Birkhoff Theorem and its Generalizations" (2018). Tutorials on... in 1 hour or less. Paper 9.
Additional FilesBirkhoff.pdf (1245 kB)