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

TL;DR

This paper revisits Weyl's formulation of gravitational force in two-body solutions, analyzing its invariant properties and behavior in the limit where one mass vanishes, linking it to Schwarzschild's acceleration.

Contribution

It provides an invariant formulation of Weyl's gravitational force in two-body solutions and explores its behavior as one mass tends to zero, connecting it to Schwarzschild's acceleration.

Findings

The force norm coincides with Schwarzschild's test body acceleration in the zero-mass limit.

Both norms grow unbounded as the test body approaches the Schwarzschild two-surface.

The invariant force formulation remains consistent in the limit of vanishing mass.

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.