From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Quantum Gravity 1
Organization: Whole Earth 'Lectronic Link
Date: Tue, 6 Oct 1992 06:19:38 GMT


Zeh continues p.123 "The existence of horizons is not bound to situations
with singularities of the spacetime curvature. For example, there exists a
future as well as a past horizon for a uniformly accelerated observer in
flat Minkowski spacetime. His orbit is a hyperbola, and the asymptotes
(caused by the relativistic limit to his velocity) represent horizons to
him. ...There exists a whole ... ensemble ,,, of 'parallelly accelerated'
local observers who share the same horizons. They possess different
accelerations in their [common] co-moving rest frames ... such that they
stay at fixed distances [from each other] in these [simultaneous] frames...
a new [Rindler] space coordinate is defined as being constant along their
orbits... and a new [Rindler] time coordinate [ the locally measurable
constant acceleration a of the observer self-referentially measured by the
observer in his co-moving frame is related to the Rindler space coordinate
R by a = (2R^1/2)^-1. The common asymptotes (at +&- 45 degrees on spacetime
diagram in the spacetime diagram of all of these non-inertial hyperbolic
observers that measure fixed distances from each other, like the US Navy
Blue Angels in a tight fixed formation,divide the flat Minkowski spacetime
into 4 regions I, III on other side of I where observers are. The
transformation between flat Minkowski coordinates and curved Rindler
coordinates is via cosh and sinh so it is highly nonlinear - details don't
matter here JS] ..the classical electromagnetic radiation of a uniformly
accelerated point charge. The radiation propagating along the forward light
cone of an event on the accelerated orbit will asymptotically completely
intersect the future horizon, and therefore completely disappear from the
point of view of each of the parallelly accelerated non-inertial observers
[for which R = constant but a different value of R for each non-inertial
observer who sees the charge at "rest" at fixed distance d from her- JS]...
This illustrates why the accelerated charge may radiate for an inertial
observer, but not for a parallelly accelerated one.

... Unruh (1976) has demonstrated, an accelerated counter in the vacuum of
a quantum field must register isotropic thermal radiation corresponding to
[T = ah/4pi^2ck , a = acceleration of counter, c = speed of light, h =
Planck's constant, k = Boltzmann's constant - JS] ...According to the
principle of equivalence [ accelerating counter equivalent to counter fixed
in gravity field - locally JS] this is physically equivalent to the Hawking
radiation. It can also be understood from the expansion of the Minkowski
vacuum |Om> in the basis of 'particle' states defined by means of the
Rindler coordinates: If the [Minkowski flat spacetime]plane waves.... are
expressed by these nopn-Lorentzian [Rindler] coordinates, their
corresponding creation and annihilation operators [second quantization in
which the field modes of frequency f are formal quantum harmonic
oscillators and 'particle number' n is level of excited state of energy
E(n)  = (n + 1/2)hf with 1/2hf the zero-point energy of the virtual
particle of the vacuum mode n = 0 - JS] are given by the .. Bogoljubov
transformation [well known to solid state theorists JS]...the non-linear
transformation does not preserve the sign of the frequency [f Minkowski or
w Rindler - Hawking shows that this permits faster-than-light or 'acausal
propagation' in quantum gravity- JS] which distinguish particle and anti-
particle modes. p.125

In terms of the [curved] Rindler particles, the [flat] Minkowski vacuum
assumes the [well known to solid state theorists JS]. the [pair] correlated
form of a BCS [Bardeen, Cooper, Schrieffer] state known from the theory of
superconductivity [the pair correlated Rindler particles are far apart one
in region I and the other in region III defined by the effective double
light cone family of parallel future and past horizons of the parallelly
accelerated Rindler observers at fixed distances from each other-JS]. The
Rindler vacuum... is therfore (physically) different from the Minkowski
vacuum. Since the Rindler modes are relevant for the accelerated counter,
the concept of the vacuum (together with that of particles a global
concept) is thus not invariant under non-Lorentzian transformations. It may
be expected that this observer-relatedness of the operational meaning of a
quantum state (in contrast to the invariance of the 'real' state itself!)
is of more general importance , and not limited to a dependence on the
observer's acceleration. p.125 Zeh The Direction of Time, Springer-Verlag
1992.