Gravity's rainbow

Nonlinear special relativity (or doubly special relativity) is a simple framework for encoding properties of flat quantum spacetime. 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 nonlinear realization of relativity in momentum space, and show that for such a dual the spacetime 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 Schwarzschild solution, examining the conditions for which the horizon is energy dependent. Finally, we find the weak field limit.