The Einstein equation solution inside a ball with uniform density

Abstract

A great number of solutions of the Einstein field equation are known. They describe the gravitational field in the empty spacetime, in the spacetime with electromagnetic field and for a ball filled with a liquid under pressure. The present work is devoted to gravitational field generated by some mass distribution. One of the simplest cases is considered, when mass is uniformely distributed inside a ball and is not moving. The boundary problem for the Einstein equation is formulated. Solution outside the ball is the Schwartzschild solution in vacuum. The coordinates at which the Schwartzschild solution is written are different from the coordinates used in equations for components of the metric tensor inside the ball. Relations between internal and external coordinates are found on the ball surface. They allow to use the Schwartzschild solution for formulation of boundary conditions for internal solution. The solution of the boundary problem is found for the case of weak field. This solution can be used as an example in the analysis of laws of conservation for the gravitational field, in which interaction of mass with field generated by the mass gives a contribution to momentum and energy of the gravitational field.