From: columbus@strident.think.com (Michael Weiss)
Newsgroups: sci.physics
Subject: The accelerating charge meets the EPR paradox.
Date: 18 Dec 92 17:04:21
Organization: Thinking Machines Corporation, Cambridge MA, USA
NNTP-Posting-Host: strident.think.com


No, the subject field is not a joke, just a jazzier title for the essay
"Some Simple Consequences of the Loss of Information in a Spacetime with a
Horizon", by Shin Takagi, which is in the collection "Complexity, Entropy,
and the Physics of Information", ed. by Wojciech H. Zurek, volume 8 in the
Sante Fe Institute Studies in the Sciences of Complexity (pub.
Addison-Wesley).  This is my second post on this collection.

First, some old business.  My last post on this collection inadvertently
omitted three of Wheeler's "clues":

        The super-Copernican principle.
        "Consciousness."
        More is different.

That second one is a biggy.  It says something for the rest of the
contributions that they rarely mention "consciousness", even when
discussing the interpretation of QM.  (I think Wheeler is also the only
contributor who cites Thomas Mann.)

On to the accelerating charge.  Takagi actually considers an accelerating
atom, not a charge, but there seem to be connections to the "accelerating
charge" threads that come and go on sci.physics like tracks in a bubble
chamber.  I'm a little out of my depth on this one, so I appeal to the net
gods to fill in the gaps.

OK.  Say we take an atom and accelerate it uniformly through "empty space".
That is, at any instant we can pick an inertial frame of reference in which
the atom is momentarily at rest; measured in this frame, the acceleration
is a constant g.  (Geometrically, we draw a tangent to the atom's
worldline.)

The atom will find itself bathed in black-body radiation at a temperature T
proportional to g.  I've seen this called Rindler radiation, though Takagi
calls it the Unruh effect.  The energy spectrum is (writing w for omega)

                 2
            1   w
    F(w) =  _ __________
            w exp(w/T)-1


Takagi says "Here 1/w is a kinematical factor, and w^2 comes from the
density of states."  I am not quite sure what he means by this.  For
"density of states" I have a clue.  (In the usual derivation of the
black-body formula, the w^2 factor comes from looking at standing waves---
eigenstates--- in a box, and asking how many standing waves there are with
frequency in the range w to w+dw--- i.e., counting density of eigenstates.)
But kinematical? 

Now the fun begins.  Where is the energy coming from?  And perhaps just as
important: where is the entropy coming from?  Answer: beyond the horizon!
The worldline of the atom is a hyperbola z^2 - t^2 = (1/g)^2, with
asymptotes z = +-t.  If we ask, "who can send signals to the atom?" the
answer is: everyone with z>t.  Conversely, the atom can send signals only
to those events with z<t.  Space is drawn and quartered into 4 regions by
the hyperplanes z = +-t:

                     \ III/
                      \  /
                       \/
                   II  /\   I
                      /  \
                     / IV \


(Why hyperplanes and not light-cones?  This one is not hard to figure out,
but the answer is nice--- if you like conic sections.)  The atom's
worldline is in, say, region I.

Here's where the EPR paradox rears its head---by way of analogy.  In the
Bohm-Bell version of the EPR paradox, we start with a spin-zero state (say
a positronium atom) and pry it apart into two spin-one states (say by
letting the positronium decay into two photons).  The photons separate, we
measure the spin-state of the photons along some axis, and voila!
correlated randomness, and Einstein spins in his grave.  (Couldn't resist.)

Here, we are prying the quantum vacuum state apart between regions I and
II.  Takagi puts it this way (Unruh, Wald and others are credited with the
insight): 

    The definition of the vacuum requires information on the field in both
    regions I and II.  However, the uniformly accelerated observer "sees"
    only those degrees of freedom in region I.  Therefore, results of his
    measurements can be described by an ensemble of states.  Since regions
    I and II are space-like apart from each other, this is a typical case
    of the underlying EPR correlation.

Some of this should look familiar if you've seen Kurt Sonnenmoser's summary
of Boulware's (non-quantum-field-theory) calculations for the uniformly
accelerating charge.  Sonnenmoser writes (modifying his notation slightly):

    In order to satisfy Maxwells equations everywhere (and thereby ensure
    energy conservation), one has to introduce a (Dirac-)delta-function
    field on the boundary z=-t, i.e. a function delta(z+t) times an
    ordinary function depending on other space-time coordinates. [...]
    Thus the energy flows as follows: the source of the energy is the
    delta-function field supported by the boundary z=-t, then it is
    absorbed and reemitted by the charge, and finally it ends up in region
    III. In particular, there is no energy source in region I.

At this point I have a warm fuzzy feeling that everything fits
together--- just don't ask me to explain it!  Takegi now goes into high
gear, reporting on his own work.  I will hit the high spots, giving
Takagi's section headings, and my impressions as to what's going on.

                     Apparent inversion of statistics

"Let us ask, just out of curiosity, what the case would be in a fictition
three-dimensional spacetime.  The answer to this question is another
surprise:

            1   w
    F(w) =  _ __________
            w exp(w/T)+1


[...] the distribution function here is that of Fermi-Dirac, although we
are dealing with photons (i.e., bosons) in both cases."

                            Huygens' principle

"It is well known that Huygens' principle is valid only in even-dimensional
spacetimes."  Huygens' principle (in case anyone who doesn't know has
tagged along this far) tells us that if we make a sharp pulse disturbance
to a field (a flash of light, a hand-clap) the pulse remains sharp as it
spreads out--- so we see a flash and not a gradual increase/decrease in
illumination.  Contrast this with dropping a stone in a pond.  (There is,
incidentally, a nice intuitive argument that *because* Huygens' principle
holds in 3+1 dimensions, it will fail in 2+1.  I don't know a simple
intuitive reason though why it should hold in 3+1 dimensions.)

Takagi remarks that the inversion of statistics is closely related to the
failure of Huygens' principle.  He gives a reference, but no details.
Any net-gods have intuition about this?


		  Density of States in the Rindler Wedge
			  Open Einstein Universe
		Quantum Statistical Mechanics of a Dumbbell

In these last three sections, Takagi plays around with the coordinate
system and then the metric of the Rindler wedge (i.e., region I) for the
2+1 dimensional case.  The basic idea is to define a local density of
states D(w) so that the formula for F(w) becomes

            1   D(w)
    F(w) =  _ __________
            w exp(w/T)-1

in 2+1 dimensions.  I.e., back to bosons again--- blame the inversion of
statistics on a strange density of states!  I won't go through his algebra,
but I will mention what are his concerns, and how he deals with it:

         problem                        solution

    a temperature-dependent         Re-coordinatize spacetime
    density of states does  	    by marshalling an army of accelerating
    not make sense--- but   	    observers.  The observers have different
    if you work out D(w),   	    accelerations, hence different T's, and
    you get a formula involving	    so their coordinates can be used in place
    T	    	    	    	    of T.

    After this 
    re-coordinatization,space-like  Replace the old metric with a new, conformally
    sections of region I (with	    equivalent metric.  This turns the Rindler
    respect to the new coords)	    wedge into the open Einstein universe.
    are spatially inhomogeneous

Finally, he uses analytic continuation to turn the problem of computing the
density of states for the open Einstein universe into the quantum
statistical mechanics of a particle on an ordinary sphere (the Dumbbell).
Transforming back, he ends up with the right formula for D(w).

I don't know if I follow this or not.  Maybe if I reread Wheeler's clues...

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