MG14 - Talk detail

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A uniqueness theorem for static and asymptotically flat Einstein-Maxwell spacetimes with a photon sphere $P^3$ is considered. To do this we first modify the definition of the photon sphere for electrically charged spacetimes by adding the property that the one form $\iota_{\xi}F$ is normal to the photon sphere. We then set the magnetic charge to zero and assume that the lapse function regularly foliates the spacetime outside the photon sphere. As an auxiliary (and important in itself) result we prove that $P^3$ has constant mean and scalar curvatures. Then, after deriving a few more equations, we give a proof of the main uniqueness theorem, i. e. the static asymptotically flat Einstein-Maxwell spacetimes with a non-extremal photon sphere are isometric to the Reissner-Nordstr\"om one with mass $M$ and electric charge $Q$ subject to $\frac{Q^2}{M^2}\le \frac{9}{8}$.

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