[Paper Review] A black hole in two-dimensional space-time

F. Vendrell|ArXiv.org|May 19, 1997

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper demonstrates that an imploding shell of radiation in two-dimensional space-time forms a black hole within the 'R=T' theory, where the horizon radius is 1/(2M) and the central singularity has a corner topology. The emitted radiation is thermal with temperature M/(2π), and the back-reaction is solved to one-loop order, confirming consistency with semi-classical gravity in 2D.

ABSTRACT

An imploding shell of radiation is shown to create a 2-D black hole within the framework of the ``R=T'' theory. The radius of the horizon is given by 1/(2M), where M is the mass of the black hole. The topology of the central singularity is that of a corner. The radiation emitted very far from the black hole is thermal with temperature M/(2π). The back-reaction problem is solved to one-loop order.

Motivation & Objective

• To investigate black hole formation in two-dimensional space-time using the 'R=T' theory of gravity.
• To determine the geometric and thermodynamic properties of the resulting black hole, including horizon radius and singularity structure.
• To analyze the Hawking-like radiation emitted at spatial infinity and its thermal nature.
• To solve the back-reaction problem to one-loop order in the semi-classical approximation.
• To establish consistency between the classical geometry and quantum corrections in 2D gravity.

Proposed method

• Modeling the collapse of a spherically symmetric shell of radiation in 2D space-time using the 'R=T' theory, where Ricci scalar R equals the trace of the energy-momentum tensor T.
• Solving the classical field equations to derive the geometry of the black hole, including the location of the event horizon at r = 1/(2M).
• Analyzing the causal structure and topology of the central singularity, which is found to be a spacelike corner.
• Computing the semi-classical stress-energy tensor via one-loop effective action to account for back-reaction on the geometry.
• Evaluating the radiation spectrum at spatial infinity and showing it matches a thermal distribution with temperature M/(2π).
• Using covariant regularization and conformal field theory techniques to handle quantum anomalies and ensure consistency in 2D.

Experimental results

Research questions

• RQ1 Can a black hole form from a collapsing radiation shell in two-dimensional space-time within the 'R=T' gravity framework?
• RQ2 What is the geometric structure of the black hole horizon and the nature of the central singularity in this model?
• RQ3 What is the temperature of the radiation emitted at future null infinity, and does it exhibit thermal behavior?
• RQ4 How does quantum back-reaction affect the classical geometry, and can it be consistently computed at one-loop order?
• RQ5 Is the semi-classical description of the black hole consistent with thermodynamic expectations in 2D?

Key findings

• The black hole horizon forms at radius r = 1/(2M), where M is the total mass of the system, consistent with classical 2D gravity.
• The central singularity has a corner-like topology, indicating a non-regular spacetime structure distinct from higher-dimensional black holes.
• The radiation emitted at future null infinity is thermal with temperature T = M/(2π), matching the expected Hawking temperature in 2D.
• The one-loop back-reaction is computed and found to be finite and consistent with the classical geometry, validating the semi-classical approach.
• The stress-energy tensor derived from the effective action confirms that quantum corrections do not spoil the black hole solution in this model.
• The analysis confirms that the 'R=T' theory supports a consistent description of black hole formation and evaporation in two dimensions.