[Paper Review] Corrections to Unruh effect in tunneling formalism and mapping with Hawking effect

Rabin Banerjee, Bibhas Ranjan Majhi+1 more|ArXiv.org|Jan 5, 2009

Quantum Electrodynamics and Casimir Effect58 references17 citations

TL;DR

This paper computes quantum corrections to the Unruh effect using the tunneling formalism in a generalized Rindler metric, demonstrating that the corrected Unruh temperature maps precisely to the corrected Hawking temperature via the Global Embedding Minkowski Spacetime (GEMS) method. The analysis extends semiclassical tunneling beyond the WKB approximation, revealing a direct correspondence between corrections in accelerated frames and black hole horizons.

ABSTRACT

We consider coordinate systems adapted to accelerated observers, construct a generalised Rindler metric and then adopt a specific form of it to compute the semiclassical Unruh temperature, using a WKB approximation within a Hamilton Jacobi analysis in the tunneling picture. Corrections to this temperature are next computed by going beyond the usual WKB approximation. A connection of the corrected Unruh temperature with the Hawking temperature is established. This connection is also explained through the method of Global Embedding Minkowski Spacetime (GEMS).

Motivation & Objective

• To compute quantum corrections to the Unruh effect beyond the semiclassical WKB approximation.
• To establish a direct mapping between corrected Unruh and Hawking temperatures using the tunneling formalism.
• To generalize the Rindler metric to unify various forms found in the literature and enable consistent tunneling analysis.
• To demonstrate that the tunneling process in Rindler spacetime mirrors particle emission from black holes, validating the physical picture.
• To use the Global Embedding Minkowski Spacetime (GEMS) method to connect flat-space accelerated observer effects with curved-space black hole radiation.

Proposed method

• Construct a generalized Rindler metric $ ds^2 = -a^2 F(x)^2 dt^2 + F'(x)^2 dx^2 + dy^2 + dz^2 $, where $ F(x) $ is an arbitrary analytic, monotonic, positive function.
• Apply the Hamilton-Jacobi method to the Klein-Gordon equation for a massless scalar particle in the generalized Rindler metric.
• Use the WKB ansatz $ \phi = e^{-i S / \hbar} $ and expand the action $ S = S_0 + \hbar S_1 + \cdots $, retaining higher-order $ \hbar $ corrections beyond the semiclassical limit.
• Compute the imaginary part of the action for outgoing and ingoing modes, enforcing unitary ingoing probability to extract the tunneling amplitude.
• Apply the principle of detailed balance to relate the outgoing probability to a thermal distribution, yielding the Unruh temperature.
• Map the corrected Unruh temperature to the Hawking temperature using the GEMS method, showing equivalence under redshift and horizon structure.

Experimental results

Research questions

• RQ1 How do quantum corrections beyond the WKB approximation modify the semiclassical Unruh temperature in accelerated frames?
• RQ2 Can the corrected Unruh temperature be mapped to the corrected Hawking temperature via a geometric embedding method?
• RQ3 Does the tunneling formalism in a generalized Rindler metric reproduce the standard Unruh effect and its corrections consistently?
• RQ4 What is the role of the function $ F(x) $ in defining the horizon and acceleration in the generalized Rindler metric?
• RQ5 How does the GEMS method facilitate a direct correspondence between the Unruh and Hawking effects in the presence of quantum corrections?

Key findings

• The corrected Unruh temperature derived via the Hamilton-Jacobi tunneling method with higher-order $ \hbar $ corrections matches the form expected from quantum back-reaction effects.
• The imaginary part of the time component of the action yields $ \text{Im}~{}t = -\frac{\pi}{a} $, leading to the standard Unruh temperature $ T_U = \frac{\hbar \alpha}{2\pi} $, where $ \alpha = \frac{1}{F(x_0)} $ is the local acceleration at the horizon.
• The tunneling process in the Rindler metric corresponds to emission from a black hole-like region across the Rindler horizon, validating the physical analogy with Hawking radiation.
• The corrected Unruh temperature maps exactly to the corrected Hawking temperature when the near-horizon Rindler metric is embedded into GEMS, confirming a deep connection between the two effects.
• The generalized Rindler metric unifies multiple standard forms (e.g., $ ds^2 = -x^2 dt^2 + dx^2 $, $ ds^2 = e^{2\alpha x}(-dt^2 + dx^2) $) through appropriate choices of $ F(x) $, enabling a consistent framework.
• The analysis shows that the horizon is located at the zero of $ F(x) $, and the acceleration $ \alpha $ is determined by $ \alpha = 1/F(x_0) $, ensuring coordinate-dependent temperature.