Space-time thermodynamics in momentum dependent geometries

TL;DR

This paper explores how a momentum-dependent spacetime metric, inspired by quantum gravity ideas, affects the thermodynamics and causal structure experienced by accelerated observers, linking it to Einstein's equations.

Contribution

It provides a geometrical interpretation of deformed relativistic kinematics in doubly special relativity with a momentum-dependent metric affecting Rindler observers.

Findings

Local Rindler wedge description is affected by the momentum-dependent metric.

The causal structure remains unchanged despite the metric modification.

Einstein's equations are derived as thermodynamic equations of state in this framework.

Abstract

A possible way to capture the effects of quantum gravity in spacetime at a mesoscopic scale, for relatively low energies, is through an energy dependent metric, such that particles with different energies probe different spacetimes. In this context, a clear connection between a geometrical approach and modifications of the special relativistic kinematics has been shown in the last few years. In this work, we focus on the geometrical interpretation of the relativistic deformed kinematics present in the framework of doubly special relativity, where a relativity principle is present. In this setting, we study the effects of a momentum dependence of the metric for a uniformly accelerated observer. We show how the local Rindler wedge description gets affected by the proposed observer dependent metric, while the local Rindler causal structure is not, leading to a standard local causal horizon thermodynamic description. For the proposed modified metric, we can reproduce the derivation of Einstein's equations as the equations of state for the thermal Rindler wedge. The conservation of the Einstein tensor leads to the same privileged momentum basis obtained in other works of some of the present authors, so supporting its relevance.