[Paper Review] On the Gravitational Back Reaction to Hawking Radiation

Serge Massar, Renaud Parentani|ArXiv.org|Jan 13, 1998

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper proposes adding a surface term to the Einstein-Hilbert action to correctly describe gravitational back reaction in Hawking radiation. By incorporating this boundary term, the authors derive that the probability of particle emission is $e^{-\Delta A/4}$, directly linking quantum emission amplitudes to the first law of black hole mechanics through horizon area change.

ABSTRACT

We show that a surface term should be added to the Einstein-Hilbert action in order to properly describe quantum transitions occurring around a black hole. The introduction of this boundary term has been advocated by Teitelboim and collaborators and it has been used in the computation of the black hole entropy. Here, we use it to compute the gravitational corrections to the transition amplitudes giving rise to Hawking radiation. This surface term implies that the probability to emit a particle is given by $e^{- ΔA/4}$ where $ΔA$ is the change in the area of the black hole horizon induced by the emission. Its inclusion at the level of the amplitudes therefore relates quantum black hole radiation to the first law of black hole dynamics. In both cases indeed, the term expressing the change in area directly results from the same boundary term introduced for the same reason: to obtain a well defined action principle.

Motivation & Objective

• To address inconsistencies in describing quantum transitions near black holes using standard general relativity.
• To resolve the issue of ill-defined transition amplitudes in black hole radiation due to missing boundary terms.
• To unify the description of Hawking radiation with the first law of black hole thermodynamics.
• To establish a consistent action principle for quantum processes involving black hole horizons.
• To demonstrate that the probability of particle emission is governed by the change in horizon area, $\Delta A$, via $e^{-\Delta A/4}$.

Proposed method

• Incorporates a surface term in the Einstein-Hilbert action, previously advocated by Teitelboim and collaborators for entropy calculations.
• Applies the modified action to compute transition amplitudes for particle emission from black holes.
• Uses the boundary term to ensure a well-defined variational principle in the presence of horizons.
• Derives the emission probability as $e^{-\Delta A/4}$, where $\Delta A$ is the change in black hole horizon area.
• Establishes a direct link between the quantum amplitude and the first law of black hole dynamics.
• Ensures consistency between thermodynamic laws and quantum field theory in curved spacetime.

Experimental results

Research questions

• RQ1 How can the gravitational back reaction in Hawking radiation be consistently described using a modified action principle?
• RQ2 What role does the surface term play in ensuring a well-defined action for quantum transitions near black hole horizons?
• RQ3 How is the emission probability of a particle related to the change in black hole horizon area?
• RQ4 Can the first law of black hole thermodynamics be derived from quantum amplitudes using this formalism?
• RQ5 What is the physical significance of the $e^{-\Delta A/4}$ factor in the context of black hole radiation?

Key findings

• The inclusion of a surface term in the Einstein-Hilbert action leads to a well-defined action principle for quantum processes near black hole horizons.
• The probability of emitting a particle is given by $e^{-\Delta A/4}$, where $\Delta A$ is the change in horizon area due to emission.
• This result directly links quantum emission amplitudes to the first law of black hole thermodynamics.
• The surface term ensures consistency between the dynamics of horizon area change and quantum transition amplitudes.
• The formalism provides a unified framework where both the first law and Hawking radiation emerge from the same boundary term.
• The derivation confirms that the $\Delta A/4$ factor in the emission probability is not an ad hoc assumption but a consequence of a properly defined action principle.