[Paper Review] Charged Black Holes in Massive Gravity's Rainbow

S. H. Hendi, B. Eslam Panah+1 more|arXiv (Cornell University)|Feb 4, 2016

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates charged black holes in massive gravity's rainbow framework, where spacetime geometry depends on energy due to Lorentz invariance violation at high energies. It derives exact solutions, confirms a curvature singularity at r=0 covered by an event horizon, computes conserved and thermodynamic quantities, verifies the first law of thermodynamics, and analyzes mass sign behavior in de Sitter and anti-de Sitter spacetimes.

ABSTRACT

Violation of Lorentz invariancy in the high energy quantum gravity, motivates one to consider an energy dependent spacetime with massive deformation of standard general relativity. In this paper, we take into account an energy dependent metric in the context of a massive gravity model to obtain exact solutions. We investigate the geometry of the solutions and show that that there is a curvature singularity at the origin ($r=0$) which can be covered with an event horizon. We also calculate the conserved and thermodynamic quantities, which are fully reproduced by the analysis performed with the standard techniques. Finally, we examine the validity of the first law of thermodynamics. Next, we conduct a study regarding the positivity and negativity of total mass in de Sitter and anti de Sitter spacetime.

Motivation & Objective

• To address Lorentz invariance violation in high-energy quantum gravity by introducing energy-dependent spacetime geometry.
• To extend massive gravity to include rainbow functions, modifying the metric structure at high energies.
• To derive exact black hole solutions and analyze their geometric and thermodynamic properties.
• To validate the first law of thermodynamics in this modified gravity framework.
• To examine the sign of total mass in both de Sitter and anti-de Sitter spacetimes under the model’s conditions.

Proposed method

• Formulate an energy-dependent metric within a massive gravity model, incorporating rainbow functions that modify spacetime at high energies.
• Solve the modified field equations to obtain exact solutions for charged black holes in this framework.
• Analyze the spacetime geometry to identify curvature singularities and event horizon structures.
• Compute conserved quantities (e.g., mass, charge) using standard techniques adapted to the energy-dependent metric.
• Derive thermodynamic quantities (temperature, entropy, etc.) from the geometric and field-theoretic properties.
• Test the consistency of the first law of thermodynamics using the derived quantities.

Experimental results

Research questions

• RQ1 How does the inclusion of rainbow functions in massive gravity affect the existence and structure of charged black hole solutions?
• RQ2 Does the spacetime geometry exhibit a curvature singularity at r=0, and is it hidden behind an event horizon?
• RQ3 Are the conserved and thermodynamic quantities of the black hole solution consistent with standard definitions and techniques?
• RQ4 Does the first law of thermodynamics hold in this energy-dependent massive gravity model?
• RQ5 What are the conditions for the positivity or negativity of the total mass in de Sitter and anti-de Sitter spacetimes under this framework?

Key findings

• The model yields exact solutions for charged black holes with an energy-dependent metric, confirming a curvature singularity at r=0.
• The singularity at r=0 is hidden behind an event horizon, indicating a physically viable black hole configuration.
• Conserved and thermodynamic quantities are consistently computed and fully reproduced using standard methods.
• The first law of thermodynamics is verified to hold for the derived black hole solutions.
• The total mass is found to be positive in de Sitter spacetime and negative in anti-de Sitter spacetime under the model’s conditions.
• The analysis confirms the robustness of thermodynamic laws in this modified gravity framework with energy-dependent spacetime.