On timelike surfaces in Lorentzian manifolds

TL;DR

This paper classifies timelike surfaces in Lorentzian manifolds, linking their geometric properties to physical interpretations in general relativity, including visual appearance, gyroscopic transport, and inertial forces, with applications to Kerr-Newman spacetime.

Contribution

It provides a novel classification of timelike surfaces based on algebraic properties of the second fundamental form, connecting geometry with physical phenomena in relativity.

Findings

Classification of timelike surfaces into four algebraic cases

Relation of surface properties to inertial forces and gyroscopic transport

Application to timelike surfaces in Kerr-Newman spacetime

Abstract

We discuss the geometry of timelike surfaces (two-dimensional submanifolds) in a Lorentzian manifold and its interpretation in terms of general relativity. A classification of such surfaces is presented which distinguishes four cases of special algebraic properties of the second fundamental form from the generic case. In the physical interpretation a timelike surface can be viewed as the worldsheet of a ``track'', and timelike curves in this surface can be viewed as the worldlines of observers who are bound to the track, like someone sitting in a roller-coaster car. With this interpretation, our classification turns out to be closely related to (i) the visual appearance of the track, (ii) gyroscopic transport along the track, and (iii) inertial forces perpendicular to the track. We illustrate our general results with timelike surfaces in the Kerr-Newman spacetime.