In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...
, Kruskal–Szekeres coordinates, named after
Martin Kruskal and
George Szekeres, are a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
for the
Schwarzschild geometry for a
black hole
A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can defo ...
. These coordinates have the advantage that they cover the entire spacetime
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity. There is no misleading coordinate singularity at the horizon.
The Kruskal–Szekeres coordinates also apply to space-time around a spherical object, but in that case do not give a description of space-time inside the radius of the object. Space-time in a region where a star is collapsing into a black hole is approximated by the Kruskal–Szekeres coordinates (or by the
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordi ...
). The surface of the star remains outside the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
in the Schwarzschild coordinates, but crosses it in the Kruskal–Szekeres coordinates. (In any
"black hole" which we observe, we see it at a time when its matter has not yet finished collapsing, so it is not really a black hole yet.) Similarly, objects falling into a black hole remain outside the event horizon in Schwarzschild coordinates, but cross it in Kruskal–Szekeres coordinates.
Definition
Kruskal–Szekeres coordinates on a
black hole
A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can defo ...
geometry are defined, from the
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordi ...
, by replacing ''t'' and ''r'' by a new timelike coordinate ''T'' and a new spacelike coordinate
:
:
:
for the exterior region
outside the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
and:
:
:
for the interior region