Gravitation

Pure mathematics

General relativity

Mathematics

Mathematical physics

Asymmetry

Theoretical physics

Quantum mechanics

Singularity

Classical mechanics

Field (mathematics)

Gravitational field

Point particle

Schwarzschild radius

Mathematical analysis

Schwarzschild metric

Deriving the Schwarzschild solution

Kerr metric

Physics

Physical Review

The analytic extension of the Schwarzschild exterior solution is given in a closed form valid throughout empty space-time and possessing no irregularities except that at the origin. The gravitational field of a spherical point particle is then seen not to be invariant under time reversal for any admissible choice of time coordinate. The Schwarzschild surface $r=2m$ is not a singularity but acts as a perfect unidirectional membrane: causal influences can cross it but only in one direction. The apparent violation of the principle of sufficient reason seems similar to that which is associated with instabilities in other nonlinear phenomena.

Past-Future Asymmetry of the Gravitational Field of a Point Particle