The Duality of Time Theory

After less than one year of their publication, Karl Schwarzschild (1873-1916) found the first non-trivial exact solution to the Einstein Field Equations. This solution is commonly called as the Schwarzschild metric, which laid the groundwork for the description of the final stages of gravitational collapse, and the objects such as singularities and black holes.

The Schwarzschild metric, or vacuum, is the most general spherically symmetric vacuum that describes the gravitational field of a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including the Earth and the Sun.

A Schwarzschild black hole, which is a static black hole, is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. This boundary, however, is only a mathematical surface which is significant in determining the black hole’s properties. In general, any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole, as we shall describe below.

In space-time coordinates, withsignature, the line element for the Schwarzschild metric takes the form:

Whenis positive,is the proper time, whereis the time coordinate,is Lorentz factor,is the radial coordinate,is the colatitude,is the longitude, andis the Schwarzschild radius of the massive body, whileis the speed of light.

This solution is analogous to the classical Newtonian theory of gravity corresponding to the gravitational field around a point particle. However, the ratiois extremely small in most classical cases, when the mass density is not very large, so the corrections to Newtonian gravity are negligible. The ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars.