Re: EP and Unruh - effect (Layman)

markwh04@xxxxxxxxx wrote:
> A good example is the one I've been looking at, lately, in Minkowski
> space which takes two spacetime points (0,0,0,0); (1,0,0,0) (which any
> 2 points can be arranged to become, up to a scale factor), and writes
> down the coordinates
> r = R (1 - T^2)/(1 - (RT)^2)
> t = T (1 - R^2)/(1 - (RT)^2)
> for the region defined by |r| + |t| <= 1.

That should be (-1,0,0,0) and (1,0,0,0).

The horizon for the Unruh effect is the boundary of the Rindler wedge,
which would have the form
{ (t,x,y,z): x = c |t| }.
The thermal states are associated with the coarse graining done on this
surface. Since the Rindler Wedge
{ (t,x,y,z): x >= c |t| }
is open, then the modes for the wave equation densely pack frequency
space, and one has to extract out a delta(0) somewhere down the line to
get the final result.

The integral curves of the time-like field d/dT are just
(R,theta,phi) = constant.
The one for R = 0 is geodesic, as are those (approximately) in its
vicinity. So, this represents INERTIAL motion of the observer on the
worldline connecting (-1,0,0,0) to (1,0,0,0).

This region and coordinate system has a Cauchy problem whose initial
value surface is compact -- the 3-surface T = 0, which is the unit ball
|R| <= 1.

Since it's compact, the associated modes have a discrete spectrum, so
(unlike the case of the Unruh) everything in the integrals involved
remains finite. The area of the horizon is finite, the entropy
associated with the coarse-graining is finite. There WILL be a thermal
state or some other kind of mixed state associated with the Minkowski
vacuum, but the observer at the center is inertial.

The coarse-graining, in specific, comes from the fact that the densely
packed modes of the infinite space are only partially approximated by
the discretely spaced modes of the 3-ball R <= 1. What this does is
project the function space C^{infinity}(R^3) to the space of the
functions on the unit ball C^{infinity}(B^3) and you lose information
due to the spacing between the modes in the latter case.

So, the result is that you'll STILL get something like a Rindler state,
even with the observer following the worldline in the center being
inertial.

.

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