On the Schwarzschild field

Abstract

General relativity is a non-linear theory with the distinguishing feature that gravitational field energy also acts as gravitational charge density. In the well-known Schwarzschild solution describing field of an isolated massive body at rest, the scalar function $ϕ$ characterising the field acts as a gravitational potential as well as it curves space part of spacetime. We demonstrate explicitly that it is the latter property that accounts for the non-linear (gravity as its own source) aspect which is not explicit in usual derivations. It is worth noting that the Einstein vacuum equations ultimately reduce to the Laplace equation and its first integral which fixes zero of $ϕ$ at infinity. Thus the Schwarzschild field alongwith its asymptotic flat character is completely determined without application of any boundary condition by the field equations themselves. That means non-zero constant value of $ϕ$ will have non-vacuous effect. It in fact produces stresses exactly of the form required to represent a global monopole. By retaining freedom of choosing zero of $ϕ$, which will break asymptotic flatness, we can obtain the Schwarzschild black hole with global monopole charge. It is the non-linear aspect responsible for ``curving'' space, which has no Newtonian analogue, survives even when $ϕ$ is constant but not zero.