Thermal nature of the causal diamond horizon: A hidden property of the inertial propagator

TL;DR

This work shows that thermality of horizon physics can emerge directly from the causal structure of spacetime by analyzing the inertial Feynman propagator in Minkowski space through a causally consistent time-Fourier transform. By comparing frequency-space propagators for horizon-crossing paths within a causal diamond and in the Rindler wedge, the authors demonstrate that the emission-absorption ratio across past and future horizons yields Boltzmann factors with temperatures $T = 1/(π α)$ for a finite-lifetime observer and $T = a/(2π)$ for uniformly accelerated observers. The key result is that horizon thermality can be attributed to causal geometry itself rather than to dynamics like acceleration or gravity, with causal diamonds acting as minimal units encoding quantum thermodynamic behavior. The framework unites propagator-based horizon thermodynamics with conformal mappings between Minkowski, Rindler, and diamond geometries, and it extends Padmanabhan’s thermality idea to a causally consistent, horizon-thermodynamics setting that has potential implications for quantum gravity and entanglement studies.

Abstract

Inspired by the novel idea proposed by T.~Padmanabhan in \textit{Phys.\ Rev.\ D 100, 045024 (2019)}, we develop a method to uncover the hidden thermal properties of the inertial Feynman propagator in Minkowski spacetime in a causally consistent manner. This, in turn, enables a coherent interpretation based on future-directed propagation. In our approach, the Fourier transform is implemented following the convention used in the analysis of vacuum fluctuations. As a result, future-directed propagation across causal horizons can be consistently interpreted, from the perspective of an observer confined to a causally disconnected region, as the emission of scalar quanta at the past horizon and their absorption at the future horizon. Moreover, we find that the ratio between emission and absorption processes reproduces the characteristic Boltzmann factor of a thermal ensemble. We first apply this analysis to a causal diamond of length $2α$, performing a detailed study of the near-horizon geometry and thereby obtaining the temperature associated with the thermal behavior of the Minkowski vacuum as perceived by an observer with finite lifetime $2α$. For completeness, we also apply the method to the right Rindler wedge, recovering the well-known Unruh temperature, $T = a/(2π)$. Our results demonstrate that thermality can emerge directly from causal structure, independently of acceleration or gravity, with causal diamonds encoding intrinsic thermodynamic behavior in quantum field theory.

Figures (10)

• Figure 1: (a) Causal diamond $D_R$ defined as the intersection of a future and a past light cone. (b) Uniformly accelerated trajectories in the right Rindler wedge $R$.
• Figure 2: (a) Rindler wedges, $R$, $L$, $P$, and $F$, in Minkowski spacetime. (b) Diamond regions $D_R$, $D_L$, $D_P$, and $D_F$ in Minkowski spacetime.
• Figure 3: Constant-$\rho$ trajectories for (a) $\rho_{r,p,f} \ll 1$, and (b) $\rho_{r,p,f} \gg 1$.
• Figure 4: Causal characterization of the boundary of the diamond $D_R$.
• Figure 5: Geometric flow associated with the Killing vector field $\partial_\eta$, which is timelike within the regions $D_R$ and $D_L$, and spacelike in the regions $D_F$ and $D_P$.