[Paper Review] Quantum Mechanics of Black Holes

Steven B. Giddings|ArXiv.org|Dec 16, 1994

Black Holes and Theoretical Physics17 references55 citations

TL;DR

This paper provides a pedagogical review of quantum black hole physics, focusing on dilaton gravity in two dimensions, Hawking radiation, and the black hole information paradox. It analyzes the information loss problem through semiclassical gravity and explores black hole pair production as a probe of remnant scenarios, concluding that infinite production rates challenge the viability of black hole remnants unless internal state degeneracy is finite.

ABSTRACT

These lectures give a pedagogical review of dilaton gravity, Hawking radiation, the black hole information problem, and black hole pair creation. (Lectures presented at the 1994 Trieste Summer School in High Energy Physics and Cosmology)

Motivation & Objective

• To provide a pedagogical introduction to quantum gravity in the context of black holes, focusing on two-dimensional models.
• To analyze the black hole information problem arising from Hawking's prediction of thermal radiation from black holes.
• To investigate the implications of black hole pair production for the remnant hypothesis as a resolution to the information paradox.
• To assess whether Planck-scale physics is required to resolve divergences in the production rate of extremal black holes.

Proposed method

• Uses two-dimensional dilaton gravity with action involving a metric, dilaton field φ, and a massless matter field f to model black hole dynamics.
• Applies conformal gauge to simplify the metric and derive equations of motion for ρ−φ and φ, leading to exact classical solutions.
• Derives Hawking radiation in 2D by analyzing the stress-energy tensor of the matter field f in the background of the black hole geometry.
• Generalizes 2D results to four-dimensional black holes using effective field theory and dimensional reduction techniques.
• Evaluates black hole pair production via instanton methods and functional integral approaches, focusing on extremal Reissner–Nordström solutions.
• Analyzes the fluctuation determinant near the instanton to compute the production rate, linking it to the trace of e^{-βH} over internal black hole states.

Experimental results

Research questions

• RQ1 Can the black hole information paradox be resolved within semiclassical gravity, or does it require new physics at the Planck scale?
• RQ2 What is the role of black hole internal states in determining the rate of black hole pair production?
• RQ3 Does the infinite number of internal states in extremal black holes lead to an infinite pair production rate, challenging the remnant hypothesis?
• RQ4 How does the functional integral near the instanton solution contribute to the full tunneling rate for black hole pair creation?
• RQ5 Can Planck-scale physics resolve the apparent divergence in the production rate of extremal black holes?

Key findings

• The 2D dilaton gravity model is exactly soluble classically, allowing full derivation of black hole solutions and Hawking radiation.
• Hawking radiation in 2D is shown to arise from the stress-energy tensor of the matter field f, with a thermal spectrum matching the expected temperature.
• The pair production rate for extremal black holes is found to be proportional to the number of internal states, suggesting a potential divergence if this number is infinite.
• The functional integral near the instanton yields a contribution equivalent to Tr[e^{-βH}], linking the production rate to the partition function of the black hole.
• For weak fields (QB ≪ 1), the production rate matches effective field theory results, but the infinite degeneracy of extremal states may lead to an infinite rate, threatening the viability of the remnant scenario.
• The analysis suggests that a finite number of internal states is necessary to avoid divergences, implying that Planck-scale physics may be essential to resolve the information problem.