Gravity's Rainbow

TL;DR

This paper extends non-linear special relativity to include curvature, proposing an energy-dependent metric and modified Einstein's equations, with solutions addressing cosmological and black hole scenarios.

Contribution

It introduces a framework for doubly general relativity with energy-dependent geometry, expanding non-linear relativity to curved space-time.

Findings

Energy-dependent metric modifies Einstein's equations.

Cosmological solutions may resolve the horizon problem.

Energy-dependent horizons in black hole solutions.

Abstract

Non-linear special relativity (or doubly special relativity) is a simple framework for encoding properties of flat quantum space-time. In this paper we show how this formalism may be generalized to incorporate curvature (leading to what might be called ``doubly general relativity''). We first propose a dual to non-linear realizations of relativity in momentum space, and show that for such a dual the space-time invariant is an energy-dependent metric. This leads to an energy-dependent connection and curvature, and a simple modification to Einstein's equations. We then examine solutions to these equations. We find the counterpart to the cosmological metric, and show how cosmologies based upon our theory of gravity may solve the ``horizon problem''. We discuss the Schwarzchild solution, examining the conditions for which the horizon is energy dependent. We finally find the weak field limit.