The Schwarzschild Metric Violates the Weak Principle of Equivalence

Raymond W. Jensen
Department of Mathematics, University of Notre Dame Notre Dame

Abstract: It is shown that for subluminal observers rotating with constant angular velocity under the Schwarzschild metric at the radius in which light orbits about the gravitating mass located at the origin, the locally-measured tangential speed of (orbital) light depends on direction and angular velocity of the observer. Further, it can be shown that all such observers experience the same acceleration regardless of the angular velocity. Thus by transitivity of the weak principle of equivalence (WEP), all of the observers should measure the local speed of light as the same, which is a contradiction. Hence the WEP or the Schwarzschild metric is at fault.

Simply, it is shown that:

(a) The weak principle of equivalence (WEP) states that two observers under equivalent local gravitational fields
need observe the same behavior of photons within small enough regions of spacetime.
(b) Under the Schwarzschild metric, there exist two observers under equivalent local gravitational fields which
observe photons to behave differently from one another within their respective small regions of spacetime.
(c) Thus there is a contradiction between statements (a) and (b).
(d) Hence the WEP is false or the Schwarzschild metric is false.

Further, it has been suggested here that it is the Schwarzschild metric which is at fault; i.e. the metric is not entirely physically valid, although a good approximation in the weak-field limit. This is in order to save the WEP. Which is actually at fault however will need to be determined by experiment.