Kruskal–Szekeres coordinates

Kruskal- Szekeres coordinates, introduced by Martin Kruskal and George Szekeres, are coordinates for the Schwarzschild metric, the metric that describes the exterior of a spherically symmetric, non-rotating and electrically neutral mass distribution.

In contrast to the Schwarzschild coordinates for this often used the Kruskal- Szekeres coordinates at the event horizon () be non-singular, and are therefore often used for the description of black holes (more precisely: to be described by co-moving, internal observers as opposed to external observers, which are "fixed", for example, in the outer region of a star. ).

The line element of the Schwarzschild metric in Kruskal- Szekeres coordinates is ( is the mass of the central body and the gravitational constant and the speed of light are placed here ):

.

Is equal to the Schwarzschild coordinates and implicitly given by:

And arise from the Schwarzschild coordinates and by:

And wherein for ( that is, in outer space)

And wherein for ( that is, in the interior).

Exterior and interior are visible " seamlessly " connected to each other via the diagonals of time. The expression ( -ds2/c2 ) corresponds to the measured with an on-board clock proper time; c is the speed of light, which has been replaced here by one.