[Paper Review] Casimir-Lifshitz force between moving plates at different temperatures

T. G. Philbin, Ulf Leonhardt|ArXiv.org|Apr 14, 2009

Quantum Electrodynamics and Casimir Effect3 references3 citations

TL;DR

This paper derives the exact Casimir-Lifshitz force between two parallel plates with arbitrary electric permittivity and magnetic permeability, where one plate moves at constant speed and both are at different temperatures. Using a stress tensor formalism based on Antezza et al., it shows that thermal radiation induces a lateral force opposing motion, while quantum vacuum contributions remain unaffected by motion—confirming the absence of quantum friction due to Lorentz invariance of vacuum energy.

ABSTRACT

The zero-temperature Casimir-Lifshitz force between two plates moving parallel to each other at arbitrary constant speed was found in [New J. Phys. 11, 033035 (2009)]. The solution is here generalized to the case where the plates are at different temperatures. The Casimir-Lifshitz force is obtained by calculating the electromagnetic stress tensor, using the method employed by Antezza et al. [Phys. Rev. A 77, 022901 (2008)] for non-moving plates at different temperatures. The perpendicular force on the plates has contributions from the quantum vacuum and from the thermal radiation; both of these contributions are influenced by the motion. In addition to the perpendicular force, thermal radiation from the moving plates gives rise to a lateral component of the Casimir-Lifshitz force, an effect with no quantum-vacuum contribution. The zero-temperature results are reproduced, in particular the non-existence of a quantum-vacuum friction between the plates.

Motivation & Objective

• To generalize the zero-temperature Casimir-Lifshitz force between moving plates to the case of different temperatures.
• To resolve contradictions in the literature regarding lateral (frictional) forces between moving plates due to quantum vacuum and thermal radiation.
• To compute the electromagnetic stress tensor for moving plates at different temperatures using a relativistically consistent framework.
• To demonstrate that thermal radiation, not vacuum fluctuations, is responsible for lateral forces in non-equilibrium conditions.
• To confirm that the absence of quantum vacuum friction is a consequence of Lorentz invariance of zero-point energy.

Proposed method

• Adapts the stress tensor formalism of Antezza et al. for non-moving plates at different temperatures to include relativistic motion.
• Computes the electromagnetic stress tensor using the Green's function of the vector potential, incorporating both dielectric and magnetic responses.
• Derives the contribution of the non-moving plate using a Green tensor formalism with reflection coefficients and polarization vectors.
• Transforms the Green tensor for the moving plate by accounting for its velocity and differing material properties via Lorentz transformation of the radiation spectrum.
• Uses the inverse matrix method to expand the Green tensor in terms of basis vectors derived from the plate's polarization states.
• Evaluates the stress tensor and Poynting vector from the full Green tensor expression, including off-diagonal components that yield lateral forces.

Experimental results

Research questions

• RQ1 Does lateral Casimir-Lifshitz force arise between moving plates at different temperatures due to thermal radiation?
• RQ2 How does motion affect the quantum vacuum contribution to the Casimir-Lifshitz force?
• RQ3 Is the absence of quantum vacuum friction between moving plates preserved when thermal effects are included?
• RQ4 How does the Planck spectrum of thermal radiation transform under Lorentz boosts, and what is its impact on the force?
• RQ5 Can the exact force be derived in full generality for arbitrary permittivity, permeability, and relative motion?

Key findings

• The perpendicular Casimir-Lifshitz force has contributions from both quantum vacuum fluctuations and thermal radiation, both of which are modified by plate motion.
• A lateral force arises exclusively from thermal radiation and acts to oppose the relative motion, even when both plates are at the same non-zero temperature.
• The lateral force vanishes at zero temperature, confirming that quantum vacuum fluctuations do not induce friction, consistent with Lorentz invariance.
• The thermal radiation from each plate has a modified Planck spectrum in the frame of the other plate due to Doppler and relativistic effects, leading to asymmetric radiation pressure.
• The zero-temperature result of no quantum friction is recovered as a limiting case, validating the general solution.
• The stress tensor contains non-zero off-diagonal components due to thermal radiation in moving frames, which directly yield the lateral force component.