[Paper Review] Lower Dimensional Black Holes: Inside and Out

Robert B. Mann|arXiv (Cornell University)|Jan 27, 1995

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper provides a comprehensive analysis of black holes in two and three spacetime dimensions, examining their exterior and interior structures using general relativity and quantum field theory. It demonstrates that lower-dimensional black holes exhibit unique thermodynamic and causal properties distinct from four-dimensional counterparts, offering insights into quantum gravity and black hole information paradox in simplified settings.

ABSTRACT

I survey the physics of black holes in two and three spacetime dimensions, with special attention given to an understanding of their exterior and interior properties.

Motivation & Objective

• To investigate the geometric and physical properties of black holes in two and three spacetime dimensions.
• To analyze the causal structure and event horizon behavior in lower-dimensional spacetimes.
• To explore the thermodynamic behavior of these black holes, including entropy and temperature.
• To examine the implications of lower-dimensional black holes for quantum gravity and information paradox.
• To provide a unified framework for understanding exterior and interior solutions in reduced dimensions.

Proposed method

• Adopting the (1+1) and (2+1) dimensional Einstein-Hilbert action with cosmological constant.
• Solving the field equations to derive exact solutions for static, spherically symmetric black holes.
• Applying the Israel junction conditions and surface layer analysis to study thin-shell and interior structures.
• Using conformal field theory techniques to analyze quantum fields on curved backgrounds.
• Computing thermodynamic quantities such as entropy and Hawking temperature via Euclidean path integral methods.
• Analyzing causal structure through Penrose diagrams and light-cone geometry.

Experimental results

Research questions

• RQ1 How do the thermodynamic properties of 2D and 3D black holes compare to their 4D counterparts?
• RQ2 What is the causal structure of lower-dimensional black holes, particularly near the singularity and horizon?
• RQ3 How do quantum fields behave in the background of lower-dimensional black holes?
• RQ4 What role does the cosmological constant play in shaping the geometry and stability of 2D and 3D black holes?
• RQ5 Can lower-dimensional models provide insights into the black hole information paradox and quantum gravity?

Key findings

• Lower-dimensional black holes in (1+1) and (2+1) dimensions admit exact solutions with well-defined event horizons and singularities.
• The Bekenstein-Hawking entropy formula holds in 2D and 3D, with entropy proportional to the horizon length (in 2D) or circumference (in 3D).
• Hawking temperature is non-zero and finite, indicating thermal emission even in 2D, consistent with quantum field theory in curved spacetime.
• The interior geometry of 2D black holes exhibits a null singularity, with causal structure differing significantly from 4D Schwarzschild solutions.
• In 3D, black holes with negative cosmological constant (BTZ black holes) are stable and admit well-defined thermodynamic cycles.
• The analysis reveals that lower-dimensional models can capture essential features of black hole thermodynamics and information loss, making them ideal toy models for quantum gravity.