Inertial forces and photon surfaces in arbitrary spacetimes.

*(English)*Zbl 1050.83009Over the years, the possibility of a repelling centrifugal force in general relativity near the horizon of a black hole has been subject to several debates. The question is not a fully trivial one, because it is not a priori clear, how to give the notions of acceleration and other 3-dimensional objects a true 4-dimensional meaning. The present authors made considerable efforts to clarify the situation by showing that the notions under debate strongly depend on the chosen systems of reference.

Here is their own abstract: Given, in an arbitrary spacetime \((M, g)\), a two-dimensional timelike submanifold \(\Sigma\) and an observer field \(n\) on \(\Sigma\), we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline \(\lambda\) in \(\Sigma\) with respect to \(n\). We prove that centrifugal and Coriolis forces vanish, for all \(\lambda\) in \(\Sigma\) with respect to any \(n\), if and only if \(\Sigma\) is a photon 2-surface, i.e., generated by two families of lightlike geodesics. We further demonstrate that a photon 2-surface can be characterized in terms of gyroscope transport and we give several mathematical criteria for the existence of photon 2-surfaces. Finally, examples of photon 2-surfaces in conformally flat spacetimes, in Schwarzschild and Reissner-Nordström spacetimes, and in Gödel spacetime are worked out.

Here is their own abstract: Given, in an arbitrary spacetime \((M, g)\), a two-dimensional timelike submanifold \(\Sigma\) and an observer field \(n\) on \(\Sigma\), we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline \(\lambda\) in \(\Sigma\) with respect to \(n\). We prove that centrifugal and Coriolis forces vanish, for all \(\lambda\) in \(\Sigma\) with respect to any \(n\), if and only if \(\Sigma\) is a photon 2-surface, i.e., generated by two families of lightlike geodesics. We further demonstrate that a photon 2-surface can be characterized in terms of gyroscope transport and we give several mathematical criteria for the existence of photon 2-surfaces. Finally, examples of photon 2-surfaces in conformally flat spacetimes, in Schwarzschild and Reissner-Nordström spacetimes, and in Gödel spacetime are worked out.

Reviewer: Hans-Jürgen Schmidt (Potsdam)