Read e-book online A relativist's toolkit: the mathematics of black-hole PDF

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By Eric Poisson

ISBN-10: 0521830915

ISBN-13: 9780521830911

This textbook fills a spot within the latest literature on common relativity via supplying the complicated scholar with useful instruments for the computation of many bodily attention-grabbing amounts. The context is equipped by means of the mathematical concept of black holes, essentially the most winning and suitable functions of normal relativity. subject matters lined contain congruences of timelike and null geodesics, the embedding of spacelike, timelike and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of basic relativity.

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The fact that the entropy of black holes is connected with a topological invariant, the Euler number, is a strong argument that it will remain even if we have to go to a more fundamental theory. This idea is anathema to most particle physicists, who are a very conservative lot and want to make everything like the Yang-Mills theory. They agree that the radiation from black holes seems to be thermal and independent of how the hole was formed if the hole is large compared to the Planck length. But they would claim that when the black hole loses mass and gets down to the Planck size, quantum general relativity will break down and all bets will be off.

Thus, it is the future event horizon, the epithet future being added to distinguish it from the past event horizon which goes from bottom right to top left. Let us now return to the Schwarzschild metric in the original rand t coordinates. If one puts t = iT one gets a positive definite metric. I shall refer to such positive definite metrics as Euclidean, even though they may be curved. In the Euclidean-Schwarzschild metric there is again an apparent singularity at r = 2M. However, one can define a new radial coordinate x to be 4M(1 - 2Mr- 1 ) ~ .

The fact that the path integral is on a periodic background implies that all physical quantities such as expectation values will be thermal. This would have been very difficult to establish in the frequency mixing approach. One can extend these interactions to include interactions with the gravitational field itself. One starts with a background metric go such as the Euclidean-Schwarzschild metric that is a solution of the classical field equations. One can then expand the action I in a power series in the perturbations ~g about go: The linear term vanishes because the background is a solution of the field equations.

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A relativist's toolkit: the mathematics of black-hole mechanics by Eric Poisson

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