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In general relativity, Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere ...
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(In this example, only four of the six are nonvanishing.) We can collect these one-forms into a matrix of one-forms, or even better an SO(1,3)-valued one-form. Note that the resulting matrix of one-forms will not quite be antisymmetric as for an SO(4)-valued one-form; we need to use instead a notion of transpose arising from the Lorentzian adjoint.
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Working in a coordinate chart with coordinates (,,,) labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum.
"The Kruskal-Szekeres coordinates do not describe a coordinate patch that covers a part of the gravitational manifold that is not otherwise covered - they describe a completely different pseudo-Riemannian manifold that has nothing to do with Einstein’s gravitational field (Abrams 1980; Loinger 2002; Crothers 2006).
Kruskal count principle: Image title: Explanation of the Kruskal Count mathematical magic trick, by CMG Lee. A volunteer picks a number on a clock face. Starting from 12, we move clockwise the same number of spaces as letters in the number spelled out, with wraparound. We move clockwise again the same number of spaces as letters in the new number.