Light Cone Thermodynamics

TL;DR

This paper shows that radial Minkowski conformal Killing fields define bifurcate conformal Killing horizons formed by light-cone segments with an $S^2\times\mathbb{R}$ topology, mirroring stationary black-hole horizons. It proves a conformal zeroth law with constant $\kappa_{SG}$, a first-law balance $\delta M = (\kappa_{SG}/8\pi)\delta A + \delta M_{\infty}$, a second law $\delta A\ge0$, and a third-law-like extremal limit $r_H\to0$ giving $\kappa_{SG}\to0$ and $A\to0$, while quantum fields render the Minkowski vacuum thermal to CKF observers with temperature $T=\kappa_{SG}/(2\pi)$; the temperature is derived via Bogoliubov transformations yielding a Planck spectrum with $T=\sqrt{\Delta}/(2\pi)$. A conformal-energy interpretation ties the temperature to a conserved, conformally invariant energy, and Euclidean continuation yields a Hartle–Hawking–like state for Region II. The analysis relies on mapping to a static FRW region and on near-horizon expansions, providing a clean flat-spacetime setting in which thermality, geometry, and conformal symmetry reproduce black-hole thermodynamics. These results illuminate how horizon thermodynamics and entropy concepts may emerge from conformal structures in flat spacetime and offer a simplified arena to probe the gravity–thermality–geometry nexus.

Abstract

We show that null surfaces defined by the outgoing and infalling wave fronts emanating from and arriving at a sphere in Minkowski spacetime have thermodynamical properties that are in strict formal correspondence with those of black hole horizons in curved spacetimes. Such null surfaces, made of pieces of light cones, are bifurcate conformal Killing horizons for suitable conformally stationary observers. They can be extremal and non-extremal depending on the radius of the shining sphere. Such conformal Killing horizons have a constant light cone (conformal) temperature, given by the standard expression in terms of the generalisation of surface gravity for conformal Killing horizons. Exchanges of conformally invariant energy across the horizon are described by a first law where entropy changes are given by $1/(4\ell_p^2)$ of the changes of a geometric quantity with the meaning of horizon area in a suitable conformal frame. These conformal horizons satisfy the zeroth to the third laws of thermodynamics in an appropriate way. In the extremal case they become light cones associated with a single event; these have vanishing temperature as well as vanishing entropy.

Figures (5)

• Figure 1: The Penrose diagram of the Reissner-Nordstrom black hole on the left compared with the causal structure of the radial CKF in Minkowski spacetime on the right, in both the non-extremal $\Delta>0$ and extremal $\Delta =0$ case. The letters $S$ and $T$ designate the regions where the Killing or conformal Killing fields are spacelike or timelike respectively. The light cone emanating from the points $O^\pm$ (and $O$ in the extremal case) are the hypersurface where the MCKF is null.
• Figure 2: The flow of the radial MCKF, depending on the value of the parameter $\Delta$, Eq. \ref{['Delta']}. When $\Delta < 0$: the MCKF is timelike everywhere. a) $\Delta > 0$: the Minkowski spacetime is divided into 6 different regions, where the norm of the MCKF changes from being timelike to being spacelike, through being null along the four light rays $u_\pm$, $v_\pm$. b) $\Delta = 0$: the MCKF is everywhere timelike except for the two null rays $u_0=v_0$ where it is null.
• Figure 3: A $(2+1)$ dimensional diagram depicting the regions of interest in Minkowski spacetime. The two cones truncated at the sphere of radius $r=r_{ H}$ represent the bifurcate conformal Killing horizon. They meet future and past null infinity on spherical cross-sections, represented by the two bigger rings. The central ring is the bifurcate sphere $r = r_H$, where $\xi=0$. The horizon is therefore a sphere with radius growing at the speed of light. In the center, one can also see the Cauchy development of the bifurcate sphere, the diamond, Region I. Shaded in yellow is Region II, the region representing the outside of the horizon. Geometrically, it is the Cauchy development of the complement of Region I. The remaining part of Minkowski spacetime is occupied by Regions III to VI, which, to simplify the picture, are not clearly depicted here.
• Figure 4: Three dimensional representation of the flow of the conformal Killing field in the Euclidean spacetime ${\mathbb R}^4$. The orbits in this one-dimension-less representation are non-concentric tori around the bifurcate sphere $r=r_H$---here represented as a circle. They degenerate into the $t_E$ axis for $R=2r_H$.
• Figure 5: The Penrose diagram for the static FRW spacetime of Eq. \ref{['eq:FRWmetric']}. The shaded region is the one conformally related to Region II in Minkowski spacetime. Its boundaries are in correspondence with pieces of Minkowskian future and past null infinities ${\mathfs {J}}^\pm_M$, as well as with the bifurcate conformal Killing horizon $H^+ \cup H^-$. The light grey hyperbolas are radial flow lines of the field $\partial_\tau$, or in another words $\rho = const$ lines.