Appendix9

## Black Shell Temperature

### Introduction

I assume a quantized spacetime, containing spacetime quanta spheres of Planck-length diameter and Planck-time time interval, tightly packed into a universal matrix. Particle transfer from one quantum to the next occurs at the speed of light. While rigid to the standard forces. the quanta compress in the direction of a gravitational field. A corresponding decrease in transfer time keeps light speed constant. In gravitational collapse of a star, the field ultimately compresses quanta into a terminal shell of circular disks with zero thickness. These form a featureless spherical event horizon and prevent further inward radial movement of particles. This paper shows the elementary particles filling the shell quanta are photons with the temperature of a black hole of the same mass. Externally, a black shell is indistinguishable from a black hole.

### The Calculation

Once formed, a black shell has a surface temperature determined by the energy of particles in its quanta. To calculate this, the shell is assumed to have a Schwarzschild radius, rs, determined by its mass, M. If G is the gravitational constant and c is the speed of light,

From this, the area of black shell’s event horizon, As, is given by

The mass collapsing into a black shell becomes distributed uniformly throughout the quanta in the shell, maintaining the inherent symmetry and simplicity of gravitationally collapsed objects. The following equation provides the number of Planck areas, Np, by dividing the shell area by the Planck area, Ap.

Ap is not the area of the circular disk of a collapsed spacetime quantum. It is the square of the Planck length, being the sum of the disk area plus the average area of the dimensionless space between quanta. This square tessellation of the shell surface determines the number of quanta present.

Dividing the black shell mass by the number of Planck areas provides the mass in each spacetime quantum, Mq.

Multiplying the mass, Mq, by the square of the velocity of light provides the energy per quantum, Eq.

This mass per particle is difficult to attribute to any particle other than a photon. A black shell five times the mass of the sun’s has a particle mass of 3.79x10-46 kg. In comparison, a 1.1 eV neutrino has a mass of 1.78x10-36 kg.

This indicates that star material is crushed into protons as it approaches the event horizon. These are shot out sideways when forward progress is blocked by the zero width of the quantum disks forming the terminal spherical shell.

In terms of universal constants Ap is given by

The energy per Planck area becomes

To obtain the temperature, I assume the photons have only one degree of freedom. Although each quantum has four contacts with other quanta, movement of any photon in a two-dimensional symmetrically spherical matrix of photons does not change the nature of the array. All photons are identical. The one degree of freedom left is that individual photons can exchange spins in the spin-neutral array required to produce a uniform shell surface. So, the conversion factor for energy is

where T is the absolute temperature and k is the Boltzmann constant.

The expression for temperature of the Planck area and therefore that of the
black shell, T_{p}, then becomes

The expression for the temperature of a black hole, T'_{p}, is

The two equations are identical.

Residual randomness in photon velocity allows those above escape velocity to leave tangentially, causing the shell to shrink.

7/30/2020 5:26