[Paper Review] Thermodynamics of massless particles in curved spacetime

A. A. Araújo Filho|arXiv (Cornell University)|Dec 31, 2021

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates the thermodynamics of massless particles in curved spacetime using the Einstein-aether theory within Friedmann-Robertson-Walker cosmology. It derives analytical and numerical expressions for key thermodynamic quantities—spectral radiance, Helmholtz free energy, pressure, entropy, mean energy, and heat capacity—showing explicit dependence on the scale factor $ C( heta) $ and the Riemann zeta function $ \zeta(s) $, with corrections to the Stefan-Boltzmann law and equations of state derived across three distinct models and three cosmological temperature regimes.

ABSTRACT

This work is devoted to study the behavior of massless particles within the context of curved spacetime. In essence, we investigate the consequences of the scale factor $C(η)$ of the Friedmann-Robertson-Walker metric in the Einstein-aether formalism to study photon-like particles. To do so, we consider the system within the canonical ensemble formalism in order to derive the following thermodynamic state quantities: spectral radiance, Helmholtz free energy, pressure, entropy, mean energy and the heat capacity. Moreover, the correction to the Stefan-Boltzmann law and the equation of states are also provide. Particularly, we separate our study within three distinct cases, i.e., $s=0,p=0$; $s=1,p=1$; $s=2,p=1$. In the first one, the results are derived numerically. Nevertheless, for the rest of the cases, all the calculations are accomplished analytically showing explicitly the dependence of the scale factor $C(η)$ and the Riemann zeta function $ξ(s)$. Furthermore, our analyses are accomplished in general taking into account three different regimes of temperature of the universe, i.e., the inflationary era ($T=10^{13}$ GeV), the electroweak epoch ($T=10^{3}$ GeV) and the cosmic microwave background ($T=10^{-13}$ GeV).

Motivation & Objective

• To investigate the thermodynamic behavior of massless particles in curved spacetime under Lorentz symmetry breaking via the Einstein-aether formalism.
• To derive exact thermodynamic state functions—spectral radiance, Helmholtz free energy, pressure, entropy, mean energy, and heat capacity—within the canonical ensemble framework.
• To analyze the impact of the scale factor $ C( heta) $ in the FRW metric on thermodynamic quantities across three distinct model configurations: $ s=0,p=0 $, $ s=1,p=1 $, and $ s=2,p=1 $.
• To examine the system under three cosmological temperature regimes: inflationary era ($ T=10^{13} $ GeV), electroweak epoch ($ T=10^3 $ GeV), and CMB ($ T=10^{-13} $ GeV).
• To provide corrections to the Stefan-Boltzmann law and derive equations of state, with analytical solutions under specific asymptotic limits for complex models.

Proposed method

• Formalism is based on the canonical ensemble in the Einstein-aether theory, where the gravitational action includes a timelike unit vector field (aether), breaking Lorentz symmetry.
• The Friedmann-Robertson-Walker metric is used with a scale factor $ C( heta) $, which governs the expansion of spacetime and influences all thermodynamic quantities.
• Thermodynamic functions are derived via statistical mechanics, integrating over energy states with a modified density of states dependent on $ C( heta) $ and the Riemann zeta function $ \zeta(s) $.
• For the $ s=0,p=0 $ case, results are obtained numerically due to complexity; for $ s=1,p=1 $ and $ s=2,p=1 $, analytical solutions are derived under specific asymptotic limits.
• The spectral radiance is computed via the Bose-Einstein distribution modified by the scale factor and curvature effects, with corrections to the Stefan-Boltzmann law emerging from the integration.
• Heat capacity is derived from the second derivative of the free energy, with analytical forms obtained under the condition $ \left(\sqrt{3}\sqrt{27E^4C(\theta)^4 - 4C(\theta)^6} - 9E^2C(\theta)^2\right)^{1/3} \ll 1 $ for the $ s=2,p=1 $ model.

Experimental results

Research questions

• RQ1 How does the scale factor $ C(\theta) $ in the FRW metric influence the thermodynamic properties of massless particles in the Einstein-aether theory?
• RQ2 What are the analytical expressions for spectral radiance, free energy, entropy, and heat capacity in the $ s=1,p=1 $ and $ s=2,p=1 $ models under cosmological temperature regimes?
• RQ3 How do the thermodynamic functions behave in the high-temperature (inflationary) and low-temperature (CMB) limits, and do they exhibit instabilities or phase transitions?
• RQ4 To what extent do the derived thermodynamic quantities depend on the Riemann zeta function $ \zeta(s) $, and how does this affect the Stefan-Boltzmann law correction?
• RQ5 Can analytical solutions for heat capacity be obtained, and under what physical limits do they emerge?

Key findings

• For the $ s=0,p=0 $ model, numerical results show instability in spectral radiance at the CMB temperature ($ T=10^{-13} $ GeV), suggesting non-physical behavior in that regime.
• In the $ s=1,p=1 $ model, analytical solutions are derived under the limit $ \sqrt{4E^2 + C(\theta)} \ll 1 $, and the system respects the second law of thermodynamics, with no explicit spectral radiance in the CMB regime.
• The $ s=2,p=1 $ model exhibits spectral radiance behavior resembling the Rayleigh-Jeans law at high temperatures ($ T=10^{13} $ GeV and $ T=10^3 $ GeV), but shows a well-behaved peak at low temperature ($ T=10^{-13} $ GeV).
• Analytical expressions for heat capacity in the $ s=2,p=1 $ model are derived under the condition $ \left(\sqrt{3}\sqrt{27E^4C(\theta)^4 - 4C(\theta)^6} - 9E^2C(\theta)^2\right)^{1/3} \ll 1 $, yielding $ C_{V3}(\beta,C(\theta)) = \frac{3C(\theta)^2(\sqrt[3]{2} - 2\sqrt[3]{3}C(\theta)^2)\sqrt{\sqrt[3]{2} + 2\sqrt[3]{3}C(\theta)^2}(-\beta^2\zeta(3) + 60\sqrt{3}\zeta(5)C(\theta)^2)}{\pi^2\beta^4} $.
• All thermodynamic functions depend explicitly on the Riemann zeta function $ \zeta(s) $, particularly $ \zeta(3) $ and $ \zeta(5) $, indicating deep number-theoretic structure in the thermodynamics of massless modes.
• No dark energy-like behavior or phase transitions were observed in the derived models, and all thermodynamic functions remain well-behaved across temperature regimes, with no suppression of logarithmic or exponential terms in high or low temperature limits.