# Randomness = Platonic Aeon

1. Apr 25, 2004

### Russell E. Rierson

An isometry on spacetime is a diffeomorphism which leaves the metric g_ab invariant. The action on a connected spacetime is determined by its action on a single point, along with the induced action on the tangent space at the point, since isometries map geodesics onto geodesics.

For a two dimensional manifold, the antisymmetric tensor determines the value of the field at the point of two intersecting null planes. If the metric has a Riemannian signature, the equations of action are equal to the action of an infinitesimal rotation in flat Euclidean space. It follows that the map is simply an ordinary rotation on a tangent space.

In a Schwarzchild spacetime via an analogue of the Schwarzchild spacetime of the Rindler vacuum state, for which static observers detect no particles, the expected stress-energy tensor becomes singular on two distinct portions of the intersecting null planes. This is known as the Hartle-Hawking vacuum, and the vacuum state will become a thermal state with respect to the notion of time translations with temperature T = hbar*c^3 / 8pi*k*G*M .

The vacuum state gives rise to a generalized entropy law, where the entropy S never decreases:

S' = S_m + A/4

The area of a spacetime surface and the maximum amount of information contained in a finite region of space, cannot be greater than one quarter of the area in Planck units. Spin networks can describe the quantum geometry of space at the intersection of horizon boundaries, where the spin networks intersect with the boundary at a finite number of points.
There is a finite amount of energy contained by a given region of spacetime. A finite amount of information. A finite number of quantum phase entanglements and random fluctuations.

A phenomenon is random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions.

The Hawking-Unruh effect is therefore the consequence of the noise spectrum for a massless scalar field along an accelerated trajectory in Minkowski space. It would appear to be a Fermi-Dirac form. The Unruh effect then becomes an integral part of a quantum field theory.