The Duality of Time Theory

For ordinary stars and planets, the radiusof the gravitating mass, is much larger than its Schwarzschild radius. For example, the radius of the Sun is approximately 700,000 km, while its Schwarzschild radius is only 3 km. The radius of the Earth is 6,371 km, while its Schwarzschild radius is only 8.9 mm. This means that if the mass of the Earth is compressed to a sphere with 8.9 mm radius, it becomes a black hole with a singularity at.

The sphere atdivides the Schwarzschild coordinates in two disconnected patches with two different solutions. The exterior solution withis the one that is related to the gravitational fields of stars and planets, while the interior solution with, which contains the singularity at, is completely separated from the outer patch.

The singularity atarises from a bad choice of coordinates or coordinate conditions. Using a different coordinate transformation, such as: Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates, Novikov coordinates, or Gullstrand-Painleve coordinates, one can then relate the extended external patch to the inner patch.

However, the case whenis a true physical singularity at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant, which is given by:

Atthe curvature becomes infinite, indicating the presence of a singularity. At this point, the metric, and space-time itself, is no longer well-defined. For a long time it was thought that such a solution was non-physical. However, a greater understanding of General Relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case.

The Schwarzschild solution, taken to be valid for all, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although it has some rather bizarre properties. Forthe Schwarzschild radial coordinatebecomes time-like and the time coordinatebecomes space-like. A curve at constantis no longer a possible world-line of a particle or observer, not even if a force is exerted to try to keep it there; this occurs because space-time has been curved so much that the direction of cause and effect points into the singularity.

The surfacedemarcates the event horizon of the black hole, which represents the point past which light can no longer escape the gravitational field. Any physical object whose radius becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole.