Relativity and Hyperbolic Space

the relativity of hyperbolic space

The longitudinal Doppler shift is a measure of hyperbolic distance. Transformations of uniform motion are determined by the Doppler shift, while its square root transforms to a uniformly accelerated frame. A time-velocity space metric is derived, by magnifying the Beltrami coordinates with the geometric time, which is similar to the one obtained by Friedmann using Einstein’s equations in which ...

Hyperbolic Formulation of General Relativity

Einstein’s theory, viewed mathematically as a system of second-order partial differential equations for the metric, is not a hyperbolic system without modification and is not manifestly well-posed, though physical information propagates at the speed of light. A well-posed hyperbolic system admits unique solutions which depend continuously on the initial data and seems to be required for robust,...

Strongly Hyperbolic Systems in General Relativity *

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that is a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any near by one. ...

3 + 1 General Relativity In Hyperbolic Form

First order hyperbolic formalism for Numerical Relativity

The causal structure of Einstein’s evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution equations, that can lead to numerical inaccuracies, can be eliminated by using the Hamiltonian constraint. Furthermore, we show that the entire system is hyper...