The thermal view of singularity-free scalar-tensor spacetimes

TL;DR

This work analyzes the Pimentel two-parameter Brans-Dicke solution through a thermal perspective in which GR is the zero-temperature equilibrium. It uncovers novel thermal phenomena, including an anomalous $\omega\to\infty$ limit where the metric tends to Mars with a surviving Einstein-frame scalar rather than a true vacuum GR limit, and it extends the solution space with a 3-parameter family generated by a symmetry. By tracking the expansion scalar $\Theta$ and the gravity temperature product $${\cal K}{\cal T} = {\sqrt{-\nabla^{\mu}\phi \nabla_{\mu}\phi}}/(8\pi \phi)$$, the paper reveals complex heating and cooling dynamics and fixed points on the line $8\pi{\cal K}{\cal T}=\Theta$, illustrating how scalar degrees of freedom can drive departures from GR in non-singular, inhomogeneous spacetimes. The results broaden the thermal framework for scalar-tensor gravity and highlight that the approach yields qualitatively new behaviours beyond those previously known for exact solutions.

Abstract

The two-parameter inhomogeneous and time-dependent Pimentel solution of Brans-Dicke theory is analyzed to probe the new thermal view in which General Relativity is the zero-temperature (equilibrium) state of scalar-tensor gravity. As the parameters vary, we uncover phenomenology not found before with other exact solutions, nor contemplated thus far in the general theory. In the process, we also discuss the anomalous limit to General Relativity of the Pimentel geometry and show how the Mars solution of the Einstein equations is its Einstein frame version.

Figures (1)

• Figure 1: In the $\left( \Theta, {\cal K}{\cal T}\right)$ plane, the trajectory of the Pimentel spacetime (solid) begins in the $\Theta<0$ region, crosses the vertical axis and continues in the region $\Theta>0$, $8\pi {\cal K}{\cal T}>\Theta$, where it approaches asymptotically a fixed point on the critical half-line $8\pi {\cal K}{\cal T}=\Theta$, with horizontal tangent and vanishing speed. A disconnected trajectory (dashed) describes the evolution for $t>t_3$, in the region below the critical half-line. (For illustration, $s=0$, $p=-2h$, the time parameter $t$ is in units of $h^{-1}$, while $\Theta$ and ${\cal K}{\cal T}$ are in units of $h$ and different scales are used on the axes for clarity.)