An Observer's View on Relativity: Space-Time Splitting and Newtonian Limit

Abstract

We motivate and construct a mathematical theory for the separation of space and time in general relativity. The formalism only requires a single observer and an optional choice of reference frame at each instant. As the splitting is done via the observer's past light cone, it is both closer to the experimental situation and mathematically less restrictive than the splitting via observer vector fields or spacelike hypersurfaces. Indeed, the theory can in principle be applied to all spacetimes and adapted to other `metric' theories of gravity. Instructive examples are developed along with the general theory. In particular, we obtain an alternative description for accelerated frames of reference in Minkowski spacetime. Further, we use the splitting formalism to motivate a new mathematical approach to the Newtonian limit of the motion of mass points. This employs a general formula for their observed motion, distinguishing between `actual' forces (i.e. those detectable via an accelerometer) and pseudo-forces. Via this formula we show that for inertial frames of reference in Minkowski spacetime the essential laws of non-gravitational Newtonian mechanics can be derived. Physically relevant, related, open problems are indicated throughout the text. These include the proof, that the Newtonian limit gives rise to the central pseudo-forces known from Newtonian mechanics (`constant gravity', Euler, Coriolis and centrifugal force) for non-inertial frames of reference in Minkowski spacetime, as well as the derivation of Newton's law of gravitation in the Schwarzschild spacetime under said limit. This is a slightly corrected version of a master's thesis in mathematical relativity, written at TU Berlin in 2016/2017. Comments by the reviewers have been taken into account. If there are any remaining errors, they are solely due to the author.