What is the 'Unruh Effect'?
the softrat
enlighten me? Thank you.
the softrat
Sometimes I get so tired of the taste of my own toes.
mailto:sof...@pobox.com
--
So tell me, are those cookies made with real Girl Scouts?
Jonathan Thornburg -- remove -animal to reply
> I have been out of physics research for a while. Will someone please
> enlighten me? Thank you.
The Wikipedia article
http://en.wikipedia.org/wiki/Unruh_effect
is a good starting point.
ciao,
--
-- "Jonathan Thornburg -- remove -animal to reply" <jth...@aei.mpg-zebra.de>
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam
Spud
> I have been out of physics research for a while. Will someone please
> enlighten me? Thank you.
30 years is a lomg time, just google "Unruh Effect"
Spud
the softrat
30 years? Tell me about it!
Thanks for the pointer. However the information one can pick up from
the net in general and from the 'Wikipedia' is laced with kooks and
other mistakes.
Hmmmmmm. Looks warmer, huh ....
PS: You don't suppose that the Pioneer Anomaly is caused by dark
matter, do you? Or is the order of magnitude *way* off?
the softrat
Sometimes I get so tired of the taste of my own toes.
mailto:sof...@pobox.com
--
A little rudeness and disrespect can elevate a meaningless
interaction to a battle of wills and add drama to an otherwise
dull day.
mark...@yahoo.com
> I have been out of physics research for a while. Will someone please
> enlighten me? Thank you.
The "Minkowski" or "inertial" vacuum state seen in an "accelerating
frame" is a thermal state at a temperature proportional to the
"acceleration"; i.e., an heat bath containing an infinite number of
particles (finite density) distributed in a fashion consistent with a
gas at a particular temperature.
The words in quotes are very misleading however, and require a large
amount of clarification, because the effect has little or nothing per
se to do with acceleration or being inertial; but rather with the
occurrrence of a causal horizon.
A quantum field theory requires you first to define a "frame", the word
which -- unlike in Relativity -- does NOT a coordinate system; but a
"flow of time". Quantum theory, is you recall, treats time as a
process, not a dimension.
The flow of time is represented by a vector field which is timelike.
The Minkowski or inertial frame has associated with it a constant
time-like field which (by suitable Lorentz transformation) can be
represented as T = d/dt -- i.e. the 4-vector T = T^{mu} d/dx^{mu} whose
only non-zero component is T^0 (with x^0 = t).
The Unruh frame uses a time-like field which does NOT cover all of
space. The flow lines are all hyperbolic, each naturally associated
with an observer at a given acceleration. The hyperbolas all have, as
asymptotes, the 2+1 boundary given by an equation of the form x = c|t|;
the region associated with the field being x > c|t|.
The "acceleration" a is normally defined as that associated with one of
the worldlines in the Unruh frame. Different worldlines have different
accelerations associated with them.
At this boundary, the timelike field T becomes null. A second, mirror
region, x < -c|t| has the boundary x = -c|t|. Both boundaries meet at
t = 0. In this region, the field T "flows" in the opposite direction.
The boundary x = c|t| is the causal horizon mentioned before.
A field is uniquely determined by its values at t = 0, and the space of
all states of a system is generally always associated with the initial
values of whatever system is in question. Here, that means, there is a
natural split of the underlying state space H into H1 + H2, with H1
being the state space associated with the region x > c|t|, and H2 being
that associated with the region x < -c|t|.
(Solving the field equation by taking its initial values (and the
initial values of its time derivative) comprises what's called a Cauchy
problem. For the Klein-Gordon field, the initial values play the
analogous role of coordinates, the initial time derivative the
congugate momenta. The state space is then a Hilbert space in which
these quantities act as operators satisfying the usual Heisenberg
relations).
H1, here, is the only one of physical relevance. But a full
description of the Minkowski frame requires both H1 and H2. In
particular, the vacuum state |0> of a Klein Gordon field -- as seen in
the Minkowski frame -- when expressed in terms of the H1 & H2 states
--becomes:
|0> = sum |n>_1 |n>_2 exp(-pi a).
This is readily identifiable in the language of finite-temperature
quantum field theory. The states |n>_1 can be thought of as particle
states, those |n>_2 can be thought of as states associated with vacuum
fluctuations of the corresponding heat bath (i.e. "holes"). So, the
superposition |n>_1 & |n>_2 has total energy 0, since |n>_2 reflects
|n>_1. All the states |n>_2 are negative energy since the time flow in
region 2, x < -c|t|, goes the other way.
Since only region 1, x > c|t|, is physically relevant (you can't see
past the boundary, the causal horizon), then the actual quantum state
associated with it is arrived at by phase-averaging over the states of
region 2. This turns the Minkowski state into the region 1 state:
|0><0| --> Trace_2(|0><0|) = V_1
with
|0><0| = sum |n>_1 <n|_1 |n>_2 <n|_2 exp(-2 pi a)
which, after being traced over give you
Trace_2(|0><0|) = sum |n>_1 <n|_1 exp(-2 pi a)
which is a MIXED (and thermal) state, no longer a pure state,
associated with a temperature proportional to a.
Having a mixed state means you've lost information -- this loss being
represented by the coefficients of the mixture
exp(-2 pi a)
which represent (up to proportion) probabilities ... and probabilities
always mean you lost information somewhere.
In fact, this general process of tracing over a causal horizon of some
sort is GENERALLY how you get probabilities out of quantum theory.
Everything is a pure state, until you do a partial trace
phase-averaging cut-off on a horizon somewhere, and the horizon,
itself, can be thought of as nothing less than a way of quantifying the
word "observer".
The loss of information is readily identified with the loss of
information of what's going on in the other parts of spacetime outside
the region x > c|t|.
The general lesson is that what appears as quantum noise in one frame
becomes thermal noise seen from another frame; and there is no longer
any covariant distinction between quantum and thermal noise.
There's an unlimited number of ways to define time-like fields in a
region of spacetime, and you can always have time-like fields defined
in such a way that they have causal horizons somewhere. A more
dramatic example of this is where the region in question is actually
finite in size: |r| < 1 - |t| (using units where c = 1). Then defining
coordinates (R, T) by
r = R (1 - T^2)/(1 - (RT)^2)
t = T (1 - R^2)/(1 - (RT)^2)
this produces a metric
ds^2 = dT^2 ((1 - R^2)/(1 - (RT)^2))^2
- ds_3^2 ((1 - T^2)/(1 - (RT)^2)^2
where
ds_3 = dR^2 + R^2 ((d theta)^2 + (sin theta d psi)^2)
The worldlines associated with (R, theta, psi) = constant are
hyperbolic worldlines that meet at (t,r) = (-1,0), (t,r) = (+1,0), and
cross the t = 0 hyperplane at r = R; and the timelike field is just
d/dT, that associated with the coordinate "T", itself.
Here, the causal horizon is |r| = 1 - |t| and the region enclosed is
finite, which means that the modes associated with the Klein-Gordon
field will form a discretely spaced set, rather than continuous (just
as if you were to quantize "in a box"). So, there is a HUGE cut-off of
field modes here, since the globally defined quantum field has a
continuously distributed set of modes.
As of yet, I still don't know what the Minkowski vacuum looks like in
this frame. The wave equation has a particularly bizarre feature that
half of the initial values become redundant, when the initial values
are taken on the 1/2 causal horizon |r| = 1 - t, 0 < t < 1; or the
other 1/2; |r| = 1 + t, -1 < t < 0; and the Cauchy problem for the
field becomes a Dirichlet problem (in part because there are no
time-derivatives to define, since "time is frozen" on the causal
horizon).
Daryl McCullough
>The "Minkowski" or "inertial" vacuum state seen in an "accelerating
>frame" is a thermal state at a temperature proportional to the
>"acceleration"; i.e., an heat bath containing an infinite number of
>particles (finite density) distributed in a fashion consistent with a
>gas at a particular temperature.
>
>The words in quotes are very misleading however, and require a large
>amount of clarification, because the effect has little or nothing per
>se to do with acceleration or being inertial; but rather with the
>occurrence of a causal horizon.
I have a hard time understanding, physically, why this should be.
I understand that for the classical (non-quantum) accelerated
observer, there is a "horizon" in the sense that there is a
region of spacetime such that it is impossible to send a signal
from that region to the accelerated observer. However, that is
only strictly speaking true if the accelerated observer *never*
stops accelerating. If the observer accelerates for only a finite
length of time, and then shuts off his rocket motors, then that
region of spacetime becomes accessible again. The fact that it is
a horizon depends on assumptions about the *future* trajectory of
the accelerated observer.
Okay, now suppose that the accelerated observer is carrying a
very, very, sensitive thermometer. In his accelerated frame, he
sees Unruh radiation, and so his thermometer shows a nonzero
value. But what if he stops accelerating immediately *after*
registering a nonzero temperature?
It doesn't make sense to me that a physically measurable effect
(recording a nonzero temperature on your thermometer) could depend
on the existence of a horizon, which in turn depends on your
trajectory *after* the measurable effect.
--
Daryl McCullough
Ithaca, NY
mark...@yahoo.com
> >amount of clarification, because the effect has little or nothing per
> >se to do with acceleration or being inertial; but rather with the
> >occurrence of a causal horizon.
>
> I have a hard time understanding, physically, why this should be.
Actually, that's the EASY part. Cutting off at horizons constitutes a
form of "phase-averaging" or "coarse-graining" which is the universal
recipe for obtaining thermal (or more generally, mixed) states --
classical or quantum (by the way) from pure states.
> I understand that for the classical (non-quantum) accelerated
> observer
I've already clarified that the very use of the term "accelerating" is
misleading. The term "frame" -- in the context of quantum theory --
does not mean coordinate grid, but locally defined flow of time (a
time-like field). Different parts of the field can be geodesic or
non-geodesic, none of which has to do with the occurrence of a horizon.
The example I provided here, involving a region comprising an
Alexandroff Interval, in a related article shows a case in point. The
flow in the central part of the region is geodesic (i.e. "inertial").
But there's a horizon and, consequently, a mixed state associated with
the frame.
Daryl McCullough
>
>Daryl McCullough wrote:
>> mark...@yahoo.com (Mark Hopkins) says...
>> >The words in quotes are very misleading however, and require a large
>> >amount of clarification, because the effect has little or nothing per
>> >se to do with acceleration or being inertial; but rather with the
>> >occurrence of a causal horizon.
>>
>> I have a hard time understanding, physically, why this should be.
>
>Actually, that's the EASY part. Cutting off at horizons constitutes a
>form of "phase-averaging" or "coarse-graining" which is the universal
>recipe for obtaining thermal (or more generally, mixed) states --
>classical or quantum (by the way) from pure states.
You say it's the easy part, but I can't make any sense of your explanation.
Presumably, "cutting off at horizons" and "phase-averaging" and
"coarse-graining" are activities of the *theorist*. I'm trying to
understand it at the level of an electron getting excited by a thermal
photon.
Physically, what does it mean to say that there is a horizon?
>I've already clarified that the very use of the term "accelerating" is
>misleading.
You've already *said* it, but I wouldn't say that you've already *clarified*
it. Does an (idealized, extremely sensitive) accelerated thermometer measure
an Unruh temperature, or not?
we_p...@yahoo.com
> I have been out of physics research for a while. Will someone please
> enlighten me? Thank you.
>
I found a Google-article that may give hand-wavy understanding:
http://groups-beta.google.com/group/sci.physics/msg/6cfadf6d7f02a47f
I Googled "unruh cosmological horizon" to find out
if the cosmic background radiation is the unruh radiation
because of the cosmological horizon but did not find the answer.
We Pretty
mi...@despammed.com
> [...] Will someone please enlighten me [about Unruh effect]?
The following two papers may be helpful/interesting:
Alsing & Milonni:
"Simplified derivation of the Hawking-Unruh temperature
for an accelerated observer in vacuum."
- Available as quant-ph/0401170.
This paper gives a simplified derivation for a scalar
field in 1 spatial dimension and is useful for initially
getting one's head around the basic concepts.
Clifton & Halvorson:
"Are Rindler Quanta Real? Inequivalent Particle
Concepts in QFT."
- Available as quant-ph/0008030
This paper delves into the philosophical questions
of what "particle" means to different observers
who are using disjoint Hilbert spaces, i.e: an inertial
observer and an accelerating observer whose state
spaces correspond to unitarily inequivalent
representations, and who therefore have inequivalent
notions of "particle".
tes...@um.bot
> I have been out of physics research for a while. Will someone please
> enlighten me?
I was going to say
http://en.wikipedia.org/wiki/Unruh_radiation
when I noticed that Jonathan Thornburg already said the same thing! Then
I saw that "softrat" commented
> Thanks for the pointer. However the information one can pick up from
> the net in general and from the 'Wikipedia' is laced with kooks and
> other mistakes.
Was this a general comment, or did you have doubts about this particular
article?
No cure for the net in general, but you can help slow the (probably
inevitable) cranky infestation of the Wikipedia by writing quality
articles on topics you know well.
A few tips may be helpful:
* Using their common sense "bs detector", almost any intelligent reader
can usually quickly recognize the kookier pages on Wikipedia.
* Like any other open resource, the more you use it, the more you will
become aware of who the knowledgeable/trustworthy contributors are. Note
that you can click on "history" at the top of any article and from there
click on user pages to possibly find out something useful about
contributors to that article. Sadly, the ever increasing hostility of the
web is reflected in the fact that most of these are rather short on
qualifications, but even so you may recongnize some names, e.g. at least
one contributor to the "Unruh radiation" article is a former(?) occasional
poster here, and a former student of our former comoderator, John Baez.
* Wikipedia encourages anyone to edit, which would seem to invite chaos,
but so far, to a suprising extent, knowledgeable people have managed to
keep the physics pages of at least some value. In fact, overall, I think
that overall the signal-to-noise ratio in the Wikipidea physics pages is
currently as good as this newsgroup, maybe better.
"T. Essel" (lurqing somewhere in cyberspace)
the softrat
Thank you. I will request the papers from the Library.
the softrat
Sometimes I get so tired of the taste of my own toes.
mailto:sof...@pobox.com
--
"Hear the pulse, and vibration, and the rumblin' force;
someone is out there, beating on a dead horse."
--bob
mi...@despammed.com
>> The following two papers may be helpful/interesting:
>> [...] quant-ph/0401170 quant-ph/0008030
The Softrat said:
> [...] I will request the papers from the Library.
Why wait for the library? You can download them easily
from the LANL archive.
For example: http://xxx.lanl.gov/abs/quant-ph/0401170
- MikeM
the softrat
Thank you!
the softrat
Sometimes I get so tired of the taste of my own toes.
mailto:sof...@pobox.com
--
No matter where you go, there you are