Revisiting Weyl's calculation of the gravitational pull in Bach's two-body solution

Abstract

When the mass of one of the two bodies tends to zero, Weyl's definition of the gravitational force in an axially symmetric, static two-body solution can be given an invariant formulation in terms of a force four-vector. The norm of this force is calculated for Bach's two-body solution, that is known to be in one-to-one correspondence with Schwarzschild's original solution when one of the two masses l, l' is made to vanish. In the limit when, say, l' goes to zero, the norm of the force divided by l' and calculated at the position of the vanishing mass is found to coincide with the norm of the acceleration of a test body kept at rest in Schwarzschild's field. Both norms happen thus to grow without limit when the test body (respectively the vanishing mass l') is kept at rest in a position closer and closer to Schwarzschild's two-surface.