On the stable configuration of ultra-relativistic material spheres. The solution for the extremely hot gas

TL;DR

This paper analyzes the internal state of an ultra-relativistic ideal gas in extremely strong gravitational fields near a compact object, revealing a balance between gravity and pressure that resembles general relativity conditions.

Contribution

It derives the equation of state for an ultra-relativistic gas in strong gravity and links it to the equilibrium condition in general relativity.

Findings

High internal energy and pressure near the object's surface.

Equation of state matches the general relativity equilibrium condition.

Gravity can be balanced by pressure gradient in ultra-relativistic regime.

Abstract

During the last stage of collapse of a compact object into the horizon of events, the potential energy of its surface layer decreases to a negative value below all limits. The energy-conservation law requires an appearance of a positive-valued energy to balance the decrease. We derive the internal-state properties of the ideal gas situated in an extremely strong, ultra-relativistic gravitational field and suggest to apply our result to a compact object with the radius which is slightly larger than or equal to the Schwarzschild's gravitational radius. On the surface of the object, we find that the extreme attractivity of the gravity is accompanied with an extremely high internal, heat energy. This internal energy implies a correspondingly high pressure, the gradient of which has such a behavior that it can compete with the gravity. In a more detail, we find the equation of state in the case when the magnitude of the potential-type energy of constituting gas particles is much larger than their rest energy. This equation appears to be identical with the general-relativity condition of the equilibrium between the gravity and pressure gradient. The consequences of the identity are discussed.