Effects of Spatial Curvature on Blackbody Radiation: Modifications to Energy Distribution and Fundamental Laws

TL;DR

This work addresses how spatial curvature alters blackbody radiation by adopting an analog general-relativity model with harmonic oscillators on a circle, yielding curvature-dependent energy spectra $E_n(\Lambda)=\hbar\omega\left[\Gamma\left(n+\tfrac{1}{2}\right)+\tfrac{\Lambda}{2}n^{2}\right]$ where $\Gamma=\dfrac{\Lambda+\sqrt{\Lambda^{2}+4}}{2}$. The authors derive a curvature-dependent Planck distribution $u(\omega,T,\Lambda)$ and show that increasing curvature reduces the Planck peak height and width, and redshifts the peak; they further obtain a curvature-modified Stefan–Boltzmann constant $\sigma_{\Lambda}$ and asymptotic forms for the Rayleigh-Jeans and Wien regimes, including a generalized Wien displacement law whose numeric solutions indicate $\lambda_{\max}T$ grows with $\Lambda$. These results imply that spatial curvature can mimic a temperature-like redshift in blackbody spectra and quantify curvature’s impact on fundamental radiative laws within a tractable analog framework. The study provides a controlled pathway to explore curvature effects on thermal radiation and connects to broader GR-thermodynamics concepts in a simplified, analyzable setting.

Abstract

In this paper, we investigate the effects of spatial curvature on blackbody radiation. By employing an analog model of general relativity, we replace the conventional straight-line harmonic oscillators used to model blackbody radiation with oscillators on a circle. This innovative approach provides an effective framework for describing blackbody radiation influenced by spatial curvature. We derive the curvature-dependent Planck energy distribution and find that moving from flat to curved space results in a reduction in both the height and width of the Planck function. Moreover, increasing the curvature leads to a pronounced redshift in the peak frequency. We also analyze the influence of spatial curvature on the Stefan-Boltzmann law, Rayleigh-Jeans law, and Wien law.

Figures (6)

• Figure 1: Energy density $u(\omega,T,\Lambda)$ versus $\omega$ for $T=6000K$; the thin-solid blue curve corresponds $\Lambda=0$, the dashed red curve to $\Lambda=0.1$, the dotted green curve to $\Lambda=0.2$, the dotted-dashed magenta curve to $\Lambda=0.3$.
• Figure 2: Energy density $u(\omega,T,\Lambda)$ versus $\omega$ and $\Lambda$ for $T=6000K$.
• Figure 3: Energy density $u(\omega,T,\Lambda)$ versus $\omega$, for $\Lambda=0.3$; the thin-solid blue curve corresponds $T=6000K=0$, the dashed red curve to $T=5500K$, the dotted green curve to $T=5000K$, the dotted-dashed magenta curve to $T=4500K$ .
• Figure 4: The spectral distribution of energy in the blackbody radiation; The thin-solid red curve corresponds the quantum theoretical formula of generalized Planck; the Dashed blue curve to the short-wavelength approximation of generalized Wien; the dotted green curve to the long-wavelength approximation of generalized Rayleigh-Jeans law are also shown.
• Figure 5: Energy density versus wavelength, $\lambda$, $T=6000K$; the thin-solid blue curve corresponds $\Lambda=0$, the dashed red curve to $\Lambda=0.1$, the dotted green curve to $\Lambda=0.2$, the dotted-dashed magenta curve to $\Lambda=0.3$.