Analytic continuation of Spacetime Metrics

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Kerr analytic extension In Kerr-Schild coordinates ,the metric takes the form r determined implicitly, up to a sign, in terms of x, y, z The function r can in fact be analytically continued from positive to negative values through the interior of the disc x 2 + y 2 < a 2 , z = 0, to to obtain a maximal analytic extension of the solution. attach another plane (x‘, y', z') where a point on the top side of the disc x 2 + y 2 < a 2 , z = 0 in the (x, y, z) plane is identified with a point with the same x and у coordinates on the bottom side of the corresponding disc in the (x‘, y', z') plane. Solution geodesically incomplete at the ring singularity The metric on the (x’ y', z') region has the same form but with negative values of r. At large negative values of r, the space is again asymptotically flat but with negative mass. The circles {t = constant, r = constant, theta = constant) are closed timelike curves. The metric extends to a larger manifold.
Maximal extension diagrams Penrose diagram of maximally extended Reissner-Nordstrom BH at the border the null energy condition will start to be violated
Page 1 and 2: Analytic continuation of Spacetime
Page 3 and 4: Formulation of the problem -BH: reg
Page 5 and 6: Exact solutions any space-time metr
Page 7 and 8: Beyond the Event Horizon (M′, g
Page 9 and 10: Closed trapped surfaces Cauchy hori
Page 11 and 12: Compacted conformal infinity by cho
Page 13 and 14: Revisiting the problem Time slices,
Page 15 and 16: The Schwarzschild black hole The me
Page 17: Cross-section of a Kerr black hole
Page 21 and 22: Where are we? -if singularities can

Analytic continuation of Spacetime Metrics

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