Inertial forces and photon surfaces in arbitrary spacetimes

Abstract

Given, in an arbitrary spacetime (M, g), a 2-dimensional timelike submanifold Σ and an observer field n on Σ, we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline λ in Σ with respect to n. We prove that centrifugal and Coriolis forces vanish, for all λ in Σ with respect to any n, if and only if Σ 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.

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

“…We have chosen the standard observers; a different choice would lead to different formulas for the inertial accelerations (19), (20) and (21), but to the same Ψ ± . For the sake of comparison, the reader may consult Nayak and Vishveshwara [20] where the inertial accelerations are calculated with respect to the zero angular momentum observers.…”

Inertial forces and the foundations of optical geometry

“…First we transport it along the curved projected trajectory, then we Lie-transport it up along the congruence as depicted in figure 1 3 . Alternatively, we may in the style of [4], consider the worldsheet spanned by the congruence lines that are crossed by the test particle worldline. On this sheet we can uniquely extend the forward vector t µ , defined along the test particle worldline, into a vector field that is tangent to the sheet, normalized and orthogonal to η µ .…”

On the Radar Method in General-Relativistic Spacetimes

“…These equations hold for all t and allt in R. By considering the special case t =t − R(t) we find R(t) = R t − 2R(t) (11) for allt in R. To ease notation, we drop the tilde in the following. By induction, (11) yields…”

“…Note that any timelike 2-surface is ruled by two families of lightlike curves; in general, however, these will not be geodesics. Foertsch, Hasse and Perlick [11] have shown that a timelike 2-surface is ruled by two families of lightlike geodesics if and only if its second fundamental form is a multiple of its first fundamental form. In the mathematical literature, such surfaces are called totally umbilic.…”

“…In the mathematical literature, such surfaces are called totally umbilic. Some constrution methods and examples of timelike totally umbilic 2-surfaces are discussed in Foertsch, Hasse and Perlick [11]. Note that in an arbitrary spacetime totally umbilic 2-surfaces need not exist.…”