Kruskal-Szekeres coordinates

Kruskal diagram. In the animation below, each blue hyperbola represents a position at a constant radius. The event horizon corresponds to the drawn diagonals. There the metric is not singular . Rather, the hyperbolas only disappear at the borders of the hatched area. The shaded area above corresponds in the Schwarzschild coordinates to the entire singularity of a so-called black hole at r = 0.

Kruskal diagram - animation.

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

In contrast to the Schwarzschild coordinates that are often used for this , the Kruskal-Szekeres coordinates at the event horizon ( ) are not singular and are therefore often used to describe black holes (more precisely: for description by internal observers who are moving with them, as opposed to external observers, For example, they are "fixed" to a star outside.). ${\ displaystyle r = 2M}$

presentation

In the following formulas, the value for the gravitational constant and the speed of light is used for simplification ; be the mass of the central body. The line element of the Schwarzschild metric in Kruskal-Szekeres coordinates is then: ${\ displaystyle G}$ ${\ displaystyle c}$ ${\ displaystyle 1}$ ${\ displaystyle M}$

{\ displaystyle ds ^ {2} = (32M ^ {3} / r) \, e ^ {- r / 2M} \, (- dv ^ {2} + du ^ {2}) + r ^ {2} (d \ theta ^ {2} + \ sin ^ {2} \ theta \, d \ phi ^ {2})}

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${\ displaystyle r}$ is equal to that of the Schwarzschild coordinates and is implicitly given by: ${\ displaystyle r}$

{\ displaystyle \ left ({\ frac {r} {2M}} - 1 \ right) \, e ^ {r / 2M} = u ^ {2} -v ^ {2}}

${\ displaystyle u}$ and result from the Schwarzschild coordinates and from: ${\ displaystyle v}$ ${\ displaystyle t}$ ${\ displaystyle r}$

{\ displaystyle u = x \, \ cosh (t / 4M)}

and , where for (i.e. outside),

{\ displaystyle v = x \, \ sinh (t / 4M)}

{\ displaystyle x = {\ sqrt {{\ frac {r} {2M}} - 1}} \, e ^ {r / 4M}}

{\ displaystyle r \ geq 2M}

{\ displaystyle u = x \, \ sinh (t / 4M)}

and , where for (i.e. in the interior).

{\ displaystyle v = x \, \ cosh (t / 4M)}

{\ displaystyle x = {\ sqrt {1 - {\ frac {r} {2M}}}} \, e ^ {r / 4M}}

{\ displaystyle r <2M}

The exterior and interior are visibly "seamlessly" connected to one another across the diagonals. The expression corresponds to the proper time measured with a watch that is carried along. ${\ displaystyle -ds ^ {2} / c ^ {2}}$

Research history

The Kruskal-Szekeres coordinates were found by Martin Kruskal in the mid-1950s, but only made known by John Archibald Wheeler around 1959 . George Szekeres found them in 1961. They were also found independently by Christian Fronsdal in 1959 and by David Finkelstein .

literature

Charles Misner , Kip S. Thorne , John A. Wheeler : Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0

Web links

Andreas Muller: Black holes - The darkest secret of gravitation Page 34 (PDF file; 64 kB)
Andreas Müller: Black holes - Schwarzschild solution

Individual evidence

↑ Werner Israel: Dark stars: the evolution of an idea . In: Stephen Hawking, Werner Israel (Ed.): 300 years of Gravitation . Cambridge University Press, 1987, ISBN 978-0-521-37976-2 .

[1] Werner Israel: Dark stars: the evolution of an idea . In: Stephen Hawking, Werner Israel (Ed.): 300 years of Gravitation . Cambridge University Press, 1987, ISBN 978-0-521-37976-2 .