Heat propagation in rotating relativistic bodies

TL;DR

This work provides a covariant, first-order, gradient-expansion formulation for heat propagation in rigidly rotating relativistic media, delivering a unique hyperbolic evolution equation for the temperature $T$ that respects the Tolman-Ehrenfest effect and remains causal and stable. By constructing the redshifted heat current and enforcing equilibrium constraints, the authors derive the principal equation $nc_v u^ u abla_ u({ m K}T)= abla_ u[\kappa\nabla^ u({ m K}T)]+O(\nabla^3)$, and show that any other first-order theory reduces to this form up to higher-gradient corrections. The theory is illustrated across several settings, including non-rotating and rotating relativistic stars, spinning cylinders in Minkowski spacetime, and black-body emission cooling, highlighting how relativistic effects like time dilation and length contraction modify heat transport and cooling laws. The results offer a universal, well-posed framework that supersedes parabolic approximations in relativistic contexts and provides practical tools for simulating heat flow in neutron stars and other rapidly rotating relativistic objects, while clarifying the relation to Eckart's temperature and the relativistic Fourier law.

Abstract

We investigate heat propagation in rigidly rotating bodies within the theory of general relativity. Using a first-order gradient expansion, we derive a universal partial differential equation governing the temperature evolution. This equation is hyperbolic, causal, and stable, and it naturally accounts for both rotational and gravitational Tolman-Ehrenfest effects. Any other first-order theory consistent with established physics (including the parabolic theories used in neutron star cooling models) must be equivalent to our formulation within an error that is of higher order in gradients. As a case study, we analyze heat transfer in solid cylinders rotating around their symmetry axis, so that the tangential speed approaches the speed of light on the surface. We also compute the relativistic rotational corrections to the cooling law of black bodies.

Figures (3)

• Figure 1: Temperature (left panel) and heat flux (right panel) across a cylindrical shell rotating with outer tangential speed $\Omega \varrho_{\text{out}}=0$ (dashed), 0.7 (blue), 0.9 (magenta), 0.99 (red), and 0.99999 (orange). The corresponding analytical expressions are provided in equations \ref{['ToutTin']} and \ref{['OhmacheHeat!']}. For the sake of illustration, the boundary conditions have been fixed as follows: $T_{\text{out}}/T_{\text{in}}=0.5$, and $\varrho_{\text{in}}/\varrho_{\text{out}}=0.1$. We observe that, at high rotation rates, $T(\varrho)$ tends to form a local minimum near the outer boundary.
• Figure 2: Left panel: Stationary temperature profile \ref{['ToutTin']} of a cylindrical shell with $\varrho_{\text{in}}/\varrho_{\text{out}}=0.1$ and $\Omega=0.999$, in thermal contact with an inner and an outer thermostat having the same $T$. Right panel: Thermal evolution \ref{['TrrrEqaziamo']} of the same shell if the thermostats are quickly removed, and replaced with an adiabatic insulator. The various curves are snapshots at different times following the change in boundary conditions, namely $Dt/\varrho_{\text{out}}^2=0$ (blue), 0.05 (magenta), 0.1 (red), 0.2 (orange), $\infty$ (dashed). The temperature profile spontaneously evolves to the Tolman-Ehrenfest prediction \ref{['Tolmaniamo']}, as expected.
• Figure 3: Thermal evolution \ref{['Cauchyiamostrani']} of a hot infinite cylinder whose surface is in contact with a thermostat with temperature $T_{\text{ther}}$. The dashed line is the non-rotating case, while the blue line refers to a cylinder spinning with surface speed $\Omega\varrho_{\text{cyl}}=0.999$. At high rotation rates, the temperature is initially driven outward by the centrifugal term ($q^\mu \sim -u^\nu \nabla_\nu u^\mu$). Subsequently, the temperature profile develops a local minimum (as in figure \ref{['fig:stationary']}), before relaxing to the Tolman–Ehrenfest solution \ref{['Tolmaniamo']}.

Theorems & Definitions (1)

• Theorem 1