gravity from spacetime thermodynamics
Jay Olson
"Gravity from Spacetime Thermodynamics" by T. Padmanabhan where the
Einstein-Hilbert action for gravity is obtained using only the
assumptions that 1) space-time is a manifold (principle of
equivalence) and 2) that the entropy of a horizon (in a Rindler frame)
is proportional to its area.
This seems interesting in the context of quantum gravity, since it
seems to imply that ANY quantum theory of space-time must imply
general relativity, as long as it predicts something like area
quantization and reproduces the principle of equivalence in the
classical limit. This seems strange, since the usual rout to quantum
gravity is to start with the classical phase space to find appropriate
operators in the quantum theory. But doesn't this paper imply that
the fundamental theory that we should quantize may not actually be
general relativity on the microscopic level? Am I reading too much
into this paper?
BTW, the paper is at
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0209/0209088.pdf
A.J. Tolland
On Mon, 30 Sep 2002, Jay Olson wrote:
> An interesting paper appeared on the preprint server the other day,
> "Gravity from Spacetime Thermodynamics" by T. Padmanabhan where the
> Einstein-Hilbert action for gravity is obtained using only the
> assumptions that 1) space-time is a manifold (principle of
> equivalence) and 2) that the entropy of a horizon (in a Rindler frame)
> is proportional to its area.
So far as I know, this idea first appeared in Ted Jacobson's
gr-qc/9504004. Padmanabhan didn't do his homework. He obviously should
read s.p.r. more; we've discussed it here before.
In any case, I'm not sure you can safely argue from the existence
of a thermodynamic derivation of the Einstein equations to the
inappropriateness of quantizing gravity. (Regular readers of my posts
will know what's coming next. Come on, everyone, sing along!) The
Einstein-Hilbert action + Cosmological Constant is the unique leading
order term in _any_ Lagrangian you construct with diff invariance and a
dynamical metric. Once you start with these ingredients, you already have
the Einstein equations. The thermodynamic argument is an unnecessary
step.
If you want to do something interesting, try to derive the
next-to-leading order corrections from these thermodynamic considerations.
Comparing these with string theory predictions ought to be amusing.
--A.J.
Arkadiusz Jadczyk
Olson) wrote:
>An interesting paper appeared on the preprint server the other day,
>"Gravity from Spacetime Thermodynamics" by T. Padmanabhan where the
>Einstein-Hilbert action for gravity is obtained using only the
>assumptions that 1) space-time is a manifold (principle of
>equivalence) and 2) that the entropy of a horizon (in a Rindler frame)
>is proportional to its area.
After looking through the paper (though not very deep, because the
paper is not instantly clear to me) it seems to that there is an
additional, rather strong, I would say, assumption, that is not listed
above. Namely that gravity is described by (pseudo-)Riemannian metric
tensor and by nothing but that?
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
Robert C. Helling
>
> If you want to do something interesting, try to derive the
>next-to-leading order corrections from these thermodynamic considerations.
>Comparing these with string theory predictions ought to be amusing.
If this works out this would surprise me: Stringtheorists know some of
the higher order terms in the Lagrangian. And Cardoso, de Wit, and
Mohaupt have calculated the microscopic entropy including those
corrections in
CORRECTIONS TO MACROSCOPIC SUPERSYMMETRIC BLACK HOLE ENTROPY.
By Gabriel Lopes Cardoso, Bernard de Wit (Utrecht U.),
Thomas Mohaupt (Martin Luther U., Halle-Wittenberg). SPIN-1998-11,
THU-98-44,
Dec 1998. 10pp.
Published in Phys.Lett.B451:309-316,1999
e-Print Archive: hep-th/9812082
They find their result _not_ agreeing with the usual area law but with
a modified area law that was suggested by Wald for higher derivative
theories. This is somewhat the opposite line of argument but it shows
that string theory is not compatible with something that give the pure
area law.
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/764263
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Uncle Al
Jay Olson wrote:
>
> An interesting paper appeared on the preprint server the other day,
> "Gravity from Spacetime Thermodynamics" by T. Padmanabhan where the
> Einstein-Hilbert action for gravity is obtained using only the
> assumptions that 1) space-time is a manifold (principle of
> equivalence) and 2) that the entropy of a horizon (in a Rindler frame)
> is proportional to its area.
> http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0209/0209088.pdf
Newton said gravitational force upon a test mass is
F = GmM/r^2
where G is the gravitational constant, m is te test mass, M is the
other body providing the gravitational field, and r is the radius of
separation of their respective centers of mass. One might then
naively expect that force is directly proportional to (test
mass)/area. One would like to know the fundamental scale of area.
http://arXiv.org/abs/gr-qc/0209088
Does a silicon single crystal javelin gravitate identically to the
same number of atoms as a solid single crystal ball? Does the
apparent mass of a collection of atoms in small part depend upon their
configuration (compactness - ratio of area to perimeter)?
One imagines two identically massed balls of soft dense metal (e.g.,
lead d=11.34 g/cm^3 or lead loaded with micronized tungsten d=19.3
g/cm^3), one being reworked into a long thin cylinder. (Mercury has
vapor pressure and evaporation problems.) Would they still mass and
weigh the same? Kg masses may be determined to 10 ng, one in 10^11,
with existing equipment
Phys. Rev. Lett. 80(6) 1142 (1998)
Or go small with an atomic force microscope and a needle crystal that
is melted into a spontaneous sphere.
Stretched further, zinc has its atoms hexagonally close packed,
d=7.134 g/cm^3 with atomic weight = 65.39. Tellurium has its atoms
concatenated into macromolecular 3-fold helices that are then loosely
packed in parallel, d=6.243 g/cm^3 with atomic weight = 127.60.
http://www.mazepath.com/uncleal/tecyl.gif
Does lattice configuration, given a conveniently sized scale of
gravitational area, anomalously affect the apparent mass of a
collection of atoms?
Of course not! Maybe.
--
Uncle Al
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)
Urs Schreiber
"A.J. Tolland" wrote:
> So far as I know, this idea first appeared in Ted Jacobson's
> gr-qc/9504004. Padmanabhan didn't do his homework. He obviously should
> read s.p.r. more; we've discussed it here before.
Under his reference number 3 Padmanabhan writes:
"For an earlier attempt, similar in spirit to the current
paper, see Jacobson, T. ..."
> In any case, I'm not sure you can safely argue from the existence
> of a thermodynamic derivation of the Einstein equations to the
> inappropriateness of quantizing gravity.
Since both the (Euclidean) classical action and the canonical
free energy are the saddle points of
Tr exp(- something H)
it is really a tautology, that they are related, I'd say.
> (Regular readers of my posts
> will know what's coming next. Come on, everyone, sing along!) The
> Einstein-Hilbert action + Cosmological Constant is the unique leading
> order term in _any_ Lagrangian you construct with diff invariance and a
> dynamical metric. Once you start with these ingredients, you already have
> the Einstein equations. The thermodynamic argument is an unnecessary
> step.
As far as I understand, Padmanabhan shows that one can replace
diff invariance + dynamical metric
with
diff invariance + holography
and still uniquely arrive at the EH action plus cosmological
constant.
> If you want to do something interesting, try to derive the
> next-to-leading order corrections from these thermodynamic considerations.
> Comparing these with string theory predictions ought to be amusing.
For that one would need the next-to-leading order corrections
of the horizon entropy, no?
A.J. Tolland
On Thu, 3 Oct 2002, Urs Schreiber wrote:
> "A.J. Tolland" wrote:
> > So far as I know, this idea first appeared in Ted Jacobson's
> > gr-qc/9504004. Padmanabhan didn't do his homework. He obviously should
> > read s.p.r. more; we've discussed it here before.
>
> Under his reference number 3 Padmanabhan writes:
>
> "For an earlier attempt, similar in spirit to the current
> paper, see Jacobson, T. ..."
Ah. I take it back. I didn't scan the references.
> As far as I understand, Padmanabhan shows that one can replace
>
> diff invariance + dynamical metric
>
> with
>
> diff invariance + holography
What? He states on page 3 that "the principe of equivalence leads
to a geometrical description of gravity in which g_ab are the fundamental
variables" and then writes the action for gravity as some unknown
reasonably local functional of g_ab. It looks to me like he has replaced
"diff invariance + dynamical metric" with "diff invariance, dynamical
metric, & holography".
> > If you want to do something interesting, try to derive the
> > next-to-leading order corrections from these thermodynamic considerations.
> > Comparing these with string theory predictions ought to be amusing.
>
> For that one would need the next-to-leading order corrections
> of the horizon entropy, no?
I think so. In any case, I wouldn't bet that thermodynamic
considerations would be sufficient to determine the higher order terms.
(Caveat: I have not looked into this proposal of Wald's that Robert
Helling mentioned.) I meant my remark to be sarcastic; I guess I should
have used an emoticon. :)
--A.J.
Urs Schreiber
erroneously wrote:
> As far as I understand, Padmanabhan shows that one can replace
>
> diff invariance + dynamical metric
>
> with
>
> diff invariance + holography
>
> and still uniquely arrive at the EH action plus cosmological
> constant.
but what I meant is that Padmanabhan uses
dynamical metric + holography
to arrive at the EH action without assuming diff invariance
beforehand. Sorry.
Urs Schreiber
"A.J. Tolland" <a...@math.berkeley.edu> schrieb im Newsbeitrag
news:Pine.SOL.4.44.0210021955210.28708-100000@pub-708c-6...
> On Thu, 3 Oct 2002, Urs Schreiber wrote:
> > As far as I understand, Padmanabhan shows that one can replace
> >
> > diff invariance + dynamical metric
> >
> > with
> >
> > diff invariance + holography
>
> What?
Yes, that's nonsense. What I meant to write is that Padmanabhan uses
dynamical metric + holography
to derive the EH action without assuming diff invariance beforehand.
But you write:
> He states on page 3 that "the principe of equivalence leads
> to a geometrical description of gravity in which g_ab are the fundamental
> variables" and then writes the action for gravity as some unknown
> reasonably local functional of g_ab. It looks to me like he has replaced
> "diff invariance + dynamical metric" with "diff invariance, dynamical
> metric, & holography".
Does he really assume diff invariance? His starting point is equation (4),
which gives the action A' as a functional of the metric and its first and
second derivatives (hence dynamical metric is assumed, of course) taking some
reasonable general form. Diffeomorphism invariance does not seem to be imposed
at this point, instead Padmanabhan writes "... we shall later see that the
action A' is indeed generally covariant." He then goes on to determine the
boundary part (A' - A) by invoking holography (equation (13)), which then also
fixes the remaining part of the action (equation (15)). It turns out that the
sum of both parts is indeed the EH action (16).
So I'd say he does not use diff invariance for his derivation. Unfortunately
this is not the case for his short remark on the cosmological constant (pp
7-8).
Urs Schreiber
Jay Olson wrote:
>
> An interesting paper appeared on the preprint server the other day,
> "Gravity from Spacetime Thermodynamics" by T. Padmanabhan where the
> Einstein-Hilbert action for gravity is obtained using only the
> assumptions that 1) space-time is a manifold (principle of
> equivalence) and 2) that the entropy of a horizon (in a Rindler frame)
> is proportional to its area.
I am wondering if the author furthermore has to assume that
the boundary part of the action is the one that gives the
entropy. Below equation (7) it says
"I will demand that the surface term in equation (4) should
be related
to the entropy S by S = - A_surface ..."
Since the bulk term of the Euclidean action gives the ADM
energy E/T one might argue that any further terms must
necessarily contribute to S, without requiring further
assumptions.
This leads me to the following question:
The author emphasizes that the (Euclidean) Einstein-Hilbert
action for a field configuration periodic in imaginary time is
the canonical free energy of the field configuration (i.e. of
the spacetime geometry in the case of gravity). But isn't this
true for every field theory, pretty much by definition of the
action, the free energy, the path integral and the partition
sum? The nontrivial part seems to me instead to be the
explicit splitting of the Euclidean EH action into the form S
- E/T.
Also, I am confused about the following, more general aspect:
In the case of black holes, how do we know that we have to
choose the event horizons as our boundary for integrations
over spacetime when computing the (Euclidean) action? Since
there is no true singularity at the horizon one could
presumeably just as well choose any surface inside the event
horizon, closer to the singularity. Does choosing any surface
surrounding the singularity lead to the same result for the
Euclidean EH action?
> This seems interesting in the context of quantum gravity, since it
> seems to imply that ANY quantum theory of space-time must imply
> general relativity, as long as it predicts something like area
> quantization and reproduces the principle of equivalence in the
> classical limit.
I believe area quantization is not the essential point. The
essential ingredient for Padmanabhan's argument to apply is
(besides the equivalence principle) the holographic principle,
i.e. that the maximal entropy of a spacetime region is
proportional to its surface area. This is of course the
content of equation (7):
S = A / A_P.
I think for this formula to make sense and for Padmanabhan's
reasoning to apply it is not necessary that area differences
smaller than a multiple of A_P are strictly unobservable (even
though this may well be the case, of course).
> This seems strange, since the usual rout to quantum
> gravity is to start with the classical phase space to find appropriate
> operators in the quantum theory. But doesn't this paper imply that
> the fundamental theory that we should quantize may not actually be
> general relativity on the microscopic level? Am I reading too much
> into this paper?
> BTW, the paper is at
> http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0209/0209088.pdf
Jay Olson
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote >
> I believe area quantization is not the essential point. The
> essential ingredient for Padmanabhan's argument to apply is
> (besides the equivalence principle) the holographic principle,
> i.e. that the maximal entropy of a spacetime region is
> proportional to its surface area. This is of course the
> content of equation (7):
>
> S = A / A_P.
>
> I think for this formula to make sense and for Padmanabhan's
> reasoning to apply it is not necessary that area differences
> smaller than a multiple of A_P are strictly unobservable (even
> though this may well be the case, of course).
Yes, I definately see your point. When I mentioned area quantization,
I had spin networks in mind, and the nice result that S = A follows
pretty naturally from the fact that spin networks quantize area. The
question I was moving towards was the following: "What, in principle,
could we change or remove in a hypothetical quantum theory of GR, and
still end up with GR in the classical limit, if we take this 'metric +
holography' principle as fundamental?"
For example, as you've been discussing with Dr. Tolland, perhaps
diffeomorphism invariance need not hold at the fundamental level. My
intuition for such matters still isn't up to the point where I feel
comfortable asking, but would something terrible happen if we didn't
mod out by all space-time diffeomorphisms to get the space of states
in the quantum theory? What if these constraints appear as a
manifestation of thermodynamics on the classical level?
Urs Schreiber
> When I mentioned area quantization,
> I had spin networks in mind, and the nice result that S = A follows
> pretty naturally from the fact that spin networks quantize area. The
> question I was moving towards was the following: "What, in principle,
> could we change or remove in a hypothetical quantum theory of GR, and
> still end up with GR in the classical limit, if we take this 'metric +
> holography' principle as fundamental?"
>
> For example, as you've been discussing with Dr. Tolland, perhaps
> diffeomorphism invariance need not hold at the fundamental level.
I am not entitled to make a conclusive statement about this,
but I gather that the general consensus is that "at the
fundamental level" indeed diffeomorphism invariance is not
expected to hold, if only because spacetime is not expected to
be a true manifold anymore at that level.
On the other hand, we might take the calculations that we have
been discussing in this thread as one more indication that
some sort of effective diff invariance on non-fundamental
levels may be pretty generic.
> My
> intuition for such matters still isn't up to the point where I feel
> comfortable asking, but would something terrible happen if we didn't
> mod out by all space-time diffeomorphisms to get the space of states
> in the quantum theory? What if these constraints appear as a
> manifestation of thermodynamics on the classical level?
I don't know. But this reminds me of something that may be
relevant: I seem to recall that for the BH entropy
calculations in LQG people have used merely the kinematical
properties of this theory, i.e. the fact that spin networks
form a complete basis of the (kinematical) space of states, as
well as some arguments about how these spin networks look on
the boundary given by the BH horizon. If I recall correctly,
at no point in the derivation does one solve the Hamiltonian
constraint, simply because it's too difficult (I forget if one
solves the spatial diff constraints). If memory serves one
expects that the result of the entropy calculation carries
over to the set of states that do satisfy all the constraints.
Do I remember that correctly?
Here is another related question:
In the paper
H. Kastrup, Canonical Quantum Statistics of an Isolated
Schwarzschild Black Hole with a Spectrum E_n = sigma sqrt{n}
E_p, gr-qc/9707009
the author presents a long list of literature which supports
the idea that the spectrum of a quantum BH should be indexed
by a quantum number n such that the energy is
E_n ~ sqrt{n}
and the level degeneracy is
d_n = g^n
for some g > 1, so that the entropy of a given leven n is
S ~ n .
If I use this to get the spectrum of the horizon area A ~ M^2
~ E^2 (in D=3+1) I obtain
A ~ E^2 ~ n ~ S .
The author writes that all other thermodynamic properties of
BHs can be obtained from this quantum spectrum.
I have not looked at any of the given references yet, so this
may be an unnecessary question, but: How is this approach
related to the LQG framework?
For one, in LQG the area spectrum is usually said to be
A_n = sqrt{n^2 + n}.
For large n this is approximately
A'_n = n + 1/2
which coincides with the spectrum mentioned above. Is the
spectrum discussed by Kastrup supposed to be the large n limit
of the true LQG spectrum?
On the other hand, I have come across the paper
A. Alekseev, A. Polychronakos & M. Smedbaeck, On Area and
entropy of a black hole, hep-th/0004036,
which claims to present "evidence" that the area operator of
LQG, when *properly* quantized (please see there for the
details) actually does have the spectrum
A'_n = n + 1/2
*exactly*, i.e. also for small n.
Now, I am not sure what to think of this paper. On the one
hand side, the second part of it, which I think I understand
better, seems dubious to me, on the other hand the authors
acknowledge discussion with John Baez... :-)
Ilja Schmelzer
Urs Schreiber <Urs.Sc...@uni-essen.de> writes:
> Also, I am confused about the following, more general aspect:
>
> In the case of black holes, how do we know that we have to
> choose the event horizons as our boundary for integrations
> over spacetime when computing the (Euclidean) action? Since
> there is no true singularity at the horizon one could
> presumeably just as well choose any surface inside the event
> horizon, closer to the singularity. Does choosing any surface
> surrounding the singularity lead to the same result for the
> Euclidean EH action?
Why only inside? Why not take a delta-neighbourhood around some
solipsistic observer? Whatever is observable for this observer comes
through this surface.
The question is IMHO related to quantum version of the hole argument.
We have two metrics g_ij g'_ij which are identical on the side of the
observer and equivalent but different on the other side (the "hole" or
"behind the horizon" or "outside of the delta-neighbourhood of the
only observer" does not matter IMHO).
Now, from classical point of view, we cannot distinguish these states,
and nothing changes if we mingle or identify them. But from quantum
point of view? Here we have the Pauli principle, it matters if states
are /really/ different or not, and not only what we think about them.
If we identify these metrics, we obtain a completely different path
integral.
Ilja
--
I. Schmelzer, <il...@ilja-schmelzer.net>, http://ilja-schmelzer.net
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>Jay Olson wrote:
>> The question I was moving towards was the following: "What, in
>> principle, could we change or remove in a hypothetical quantum
>> theory of GR, and still end up with GR in the classical limit, if
>> we take this 'metric + holography' principle as fundamental?"
An interesting question, but it's too hard for me.
>> For example, as you've been discussing with Dr. Tolland, perhaps
>> diffeomorphism invariance need not hold at the fundamental level.
>I am not entitled to make a conclusive statement about this,
>but I gather that the general consensus is that "at the
>fundamental level" indeed diffeomorphism invariance is not
>expected to hold, if only because spacetime is not expected to
>be a true manifold anymore at that level.
That's right: we have no idea what spacetime is at the
fundamental level, so we can only guess that the probability
it's a manifold equals 1 divided by the number of possible
options. :-)
>On the other hand, we might take the calculations that we have
>been discussing in this thread as one more indication that
>some sort of effective diff invariance on non-fundamental
>levels may be pretty generic.
The success of general relativity strongly suggests that
in some sort of classical limit, diffeomorphisms act as gauge
symmetries. Since diffeomorphisms naturally act as gauge
symmetries in any background-free theory formulated on a manifold,
I'd guess the real lesson is this: the laws of physics are
background-free! If so, in the limit where spacetime can be
approximated by a manifold, diffeomorphisms would presumably
act as gauge symmetries.
>> My
>> intuition for such matters still isn't up to the point where I feel
>> comfortable asking, but would something terrible happen if we didn't
>> mod out by all space-time diffeomorphisms to get the space of states
>> in the quantum theory? What if these constraints appear as a
>> manifestation of thermodynamics on the classical level?
I'm not sure what you mean by this, or why you think it might
happen.
>I don't know. But this reminds me of something that may be
>relevant: I seem to recall that for the BH entropy
>calculations in LQG people have used merely the kinematical
>properties of this theory, i.e. the fact that spin networks
>form a complete basis of the (kinematical) space of states, as
>well as some arguments about how these spin networks look on
>the boundary given by the BH horizon. If I recall correctly,
>at no point in the derivation does one solve the Hamiltonian
>constraint, simply because it's too difficult (I forget if one
>solves the spatial diff constraints). If memory serves one
>expects that the result of the entropy calculation carries
>over to the set of states that do satisfy all the constraints.
>Do I remember that correctly?
Not exactly. We - Ashtekar, Krasnov and I - don't just work
with kinematical states and "expect" that the calculation still
works for states that satisfy all the constraints. We carefully
show that under a very weak hypothesis on the Hamiltonian constraint,
we can count the *physical* states of the black hole horizon -
that is, states of the horizon geometry that extend to states
satisfying the Hamiltonian constraint.
We are able do this because while the Hamiltonian constraint
is poorly understood in the "bulk" - i.e., in the region outside
the horizon - we have a good understanding of the geometry of
the horizon itself. Our very weak hypothesis is that for any
state of the horizon geometry, there is at least one state of
the bulk which satisfies the Hamiltonian constraint.
>In the paper
>
>H. Kastrup, Canonical Quantum Statistics of an Isolated
>Schwarzschild Black Hole with a Spectrum E_n = sigma sqrt{n}
>E_p, gr-qc/9707009
>
>the author presents a long list of literature which supports
>the idea that the spectrum of a quantum BH should be indexed
>by a quantum number n such that the energy is
>
> E_n ~ sqrt{n}
>
>and the level degeneracy is
>
> d_n = g^n
>
>for some g > 1, so that the entropy of a given leven n is
>
> S ~ n .
>
>If I use this to get the spectrum of the horizon area A ~ M^2
>~ E^2 (in D=3+1) I obtain
>
> A ~ E^2 ~ n ~ S .
>
>The author writes that all other thermodynamic properties of
>BHs can be obtained from this quantum spectrum.
>
>I have not looked at any of the given references yet, so this
>may be an unnecessary question, but: How is this approach
>related to the LQG framework?
If I could answer that, I would be a happy man. At least
for a while, anyway! This is one of the big puzzles about loop
quantum gravity.
>For one, in LQG the area spectrum is usually said to be
>
> A_n = sqrt{n^2 + n}.
More precisely, this is the area spectrum for a surface
punctured by a single spin network edge. In general the
area is a sum of numbers of this form, one for each
spin network edge puncturing the surface.
>For large n this is approximately
>
> A'_n = n + 1/2
>
>which coincides with the spectrum mentioned above. Is the
>spectrum discussed by Kastrup supposed to be the large n limit
>of the true LQG spectrum?
I don't know about Kastrup, but there are lots of people
who like the answer n + 1/2. Rovelli, Smolin and Ashtekar
did a calculation which gives the answer sqrt(n^2 + n).
There's no way to tell the difference in the large-n
limit. However, the event horizon of a black hole is
probably punctured by a lot of spin network edges labeleld
by rather low spins n. In this case the difference really
matters. We get drastically different thermodynamic behavior
and a different value of the Immirzi parameter depending
on which formula we use! In fact, Ashtekar argues that
the n + 1/2 formula would give noticeable changes in the
spectrum of Hawking radiation even for a large black hole,
which he takes as evidence for the sqrt(n^2 + n) formula.
>On the other hand, I have come across the paper
>
>A. Alekseev, A. Polychronakos & M. Smedbaeck, On Area and
>entropy of a black hole, hep-th/0004036,
>
>which claims to present "evidence" that the area operator of
>LQG, when *properly* quantized (please see there for the
>details) actually does have the spectrum
>
> A'_n = n + 1/2
>
>*exactly*, i.e. also for small n.
Right, these are among the people who like n+1/2. The value
n+1/2 also seems very natural in certain spin foam models -
see my latest paper with Christensen and Egan, where you'll
see 2j+1 all over the place (twice j+1/2). And Bekenstein
and Hod like it for other reasons.
>Now, I am not sure what to think of this paper. On the one
>hand side, the second part of it, which I think I understand
>better, seems dubious to me, on the other hand the authors
>acknowledge discussion with John Baez... :-)
Yes, I talked to them about this. At the time I liked
sqrt(n^2 + n) better. My later work has pushed me
more towards favoring n+1/2. Lots of nice things would
happen if we used this. But Hawking radiation would work
differently, so it's a nontrivial leap of faith.
Urs Schreiber
[...]
> We are able do this because while the Hamiltonian constraint
> is poorly understood in the "bulk" - i.e., in the region outside
> the horizon - we have a good understanding of the geometry of
> the horizon itself. Our very weak hypothesis is that for any
> state of the horizon geometry, there is at least one state of
> the bulk which satisfies the Hamiltonian constraint.
You write "at least" one state. I suppose this is sufficient
since if there were on average n physical bulk states per
boundary state this would merely change the entropy by that
factor n, which can be absorbed into the one parameter freedom
one has anyway. Right?
[...]
From recent papers like gr-qc/0210024 I learn that there is a
surprisingly large industry concerned with *guessing* the BH
energy and area spectrum *without* recourse to any specific
approach to QG but instead based on various plausibility
considerations. So in this context there is the freedom to
"like" one spectrum better than the other. But I would have
thought that once one accepts the premises and results of LQG
that then the determination of the spectrum is a technical
matter, not one of taste. So is there a quantization (operator
ordering?) ambiguity in defining the area operator? From the
previously mentioned paper (hep-th/0004036) I got the
impression that the authors are claiming that the sqrt{n(n+1)}
spectrum is simply not correct and that the correct spectrum
of this operator is n+1/2. Isn't this a purely technical
question which could be decided unambiguously?
Ilja Schmelzer
> The success of general relativity strongly suggests that
> in some sort of classical limit, diffeomorphisms act as gauge
> symmetries. Since diffeomorphisms naturally act as gauge
> symmetries in any background-free theory formulated on a manifold,
> I'd guess the real lesson is this: the laws of physics are
> background-free!
A weak guess because it is not difficult to obtain GR as a limit of
a theory with background. See gr-qc/0205035
Danny Ross Lunsford
> A weak guess because it is not difficult to obtain GR as a limit of
> a theory with background. See gr-qc/0205035
A physics theory is more than numbers and eqns - at some point Occam's razor
must be opened out and wielded. (Having omitted to shave today, I
nevertheless resisted the temptation to say "drawn across the stubble of
existence.") As Mr. Baez has pointed out, there are other ways to see GR as
a background theory, but they don't have the inner simplicity of the usual
approach. That seems to be the case here.
-drl
John Baez
Ilja Schmelzer <schm...@wias-berlin.de> wrote:
>ba...@galaxy.ucr.edu (John Baez) writes:
>> The success of general relativity strongly suggests that
>> in some sort of classical limit, diffeomorphisms act as gauge
>> symmetries. Since diffeomorphisms naturally act as gauge
>> symmetries in any background-free theory formulated on a manifold,
>> I'd guess the real lesson is this: the laws of physics are
>> background-free!
>
>A weak guess because it is not difficult to obtain GR as a limit of
>a theory with background. See gr-qc/0205035
It's always possible to get any beautiful symmetrical theory as
a limit of uglier, less symmetrical theories in the limit
where the ugliness and asymmetry goes away. This should not
be taken as evidence that the beautiful symmetrical theory
really *is* ugly and asymmetrical.
John Baez
In article <3DA86F2B...@uni-essen.de>,
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>John Baez wrote:
>> Our very weak hypothesis is that for any
>> state of the horizon geometry, there is at least one state of
>> the bulk which satisfies the Hamiltonian constraint.
>You write "at least" one state. I suppose this is sufficient
>since if there were on average n physical bulk states per
>boundary state this would merely change the entropy by that
>factor n, which can be absorbed into the one parameter freedom
>one has anyway. Right?
No! The entropy we calculate is completely unaffected by how
many bulk states there are per boundary state, as long as there's
at least one. It doesn't get multiplied by a factor. The reason
is that we're calculating the entropy of the event horizon itself,
not the entropy of the bulk.
Think about an example. Suppose you have a black hole sitting
next to a pot of soup. Should you count the states of the pot
of soup when calculating the entropy of the black hole? No.
That's why we "trace over the bulk degrees of freedom" before
calculating our entropy, and that's why we only need the above
weak hypothesis to do an *exact* calculation.
For more details without too many technicalities, try this
expository paper:
Abhay Ashtekar, Kirill Krasnov
Quantum Geometry and Black Holes
http://xxx.lanl.gov/abs/gr-qc/9804039
especially the middle of page 11 and the footnote on page 13.
For the full story, go here:
Quantum Geometry of Isolated Horizons and Black Hole Entropy
A. Ashtekar, J. Baez, K. Krasnov
Adv. Theor. Math. Phys. 4 (2000) 1-94.
http://xxx.lanl.gov/abs/gr-qc/000512
and see Section VI, "Entropy".
Urs Schreiber
> No! The entropy we calculate is completely unaffected by how
> many bulk states there are per boundary state, as long as there's
> at least one. It doesn't get multiplied by a factor. The reason
> is that we're calculating the entropy of the event horizon itself,
> not the entropy of the bulk.
>
> Think about an example. Suppose you have a black hole sitting
> next to a pot of soup. Should you count the states of the pot
> of soup when calculating the entropy of the black hole? No.
>
> That's why we "trace over the bulk degrees of freedom" before
> calculating our entropy, and that's why we only need the above
>
> For more details without too many technicalities, try this
> expository paper:
>
> Abhay Ashtekar, Kirill Krasnov
> Quantum Geometry and Black Holes
> http://xxx.lanl.gov/abs/gr-qc/9804039
>
> especially the middle of page 11 and the footnote on page 13.
Thanks, I have now taken a look at it. What I did not know before
reading this is that "pots of soup", namely gravitational waves, are
allowed/possible near the black hole when calculating its entropy in
this framework. I certainly understand that one has to get rid of
them.
But by tracing over all bulk states doesn't one also trace out bulk
states that contain no radiation? And shouldn't one keep those states
because otherwise the fact that the BH entropy is proportional to its
area becomes an assumption?
Or is this a quite explicit assumption in the (your) calculations
referred to in the above paper, namely that one takes the entropy of a
black hole a priori to be due to boundary states and then sets out to
calculate these?
Actually, this seems to be what A. Ashtekar and K. Krasnov are saying on
p.11:
"It has often been argued that the degrees of freedom living on the
horizon of [a] black hole are those that account for its entropy.
We take this viewpoint in our approach."
This is related to another question that bothers me: Why is there no
mention of the bulk interior of the BH in this calculation? From a
naive point of view one thing I'd like to learn from a QG calculation
of BH entropy is: Why isn't it due to degrees of freedom residing in
the bulk interior of the BH?
Or is this the wrong question to ask? I am aware that for the
asymptotic observer, looking at the BH and measuring its Hawking
radiation, the notion of the interior of the BH is a subtle one.
On the other hand, other approaches to calculating the BH entropy seem
to be saying something about the interior of the hole:
Ashtekar and Krasnov give a brief comparison with BH entropy
calculations in string theory. They mention that while string theory
(using branes) gets the correct proportionality constant for the
entropy-area relation it does so only for (near-)extremal black holes.
I am wondering why it is seldomly mentioned in this and other reviews
that at the same time string theory gives the
correct-up-to-an-undetermined-factor-of-order-unity entropy for
virtually every type of black hole (e.g. neutral, charged, rotating,
higher dimensional), and in particular the Schwarzschild BH, by means
of the "string/black hole correspondence principle" originally due to
Susskind.
In its more refined form this "principle" amounts to noting that there
is a critical excitation energy where a massive string collapses under
its own gravity to the size of the order of the string scale (becoming
a "string ball") and that precisely at this point its rms radius
coincides with its Schwarzschild radius and furthermore all its
thermodynamical properties (temperature, entropy, radiation, decay
rate) coincide, up to some unknown factors of order unity, with that
of a BH of the same mass (e.g. hep-th/9907030).
(I am not sure how this relates to the D5/D1 brane models, which I
don't know well, but I seem to recall that these brane configurations
are describable, and are described, by "effective long strings", too.)
Anyway, the string/BH correspondence principle gives rise to a neat
mental picture of the BH degrees of freedom which is actually rather
similar to the LQG picture of a "pierced horizon". It is roughly the
following:
A highly excited string with the large mass M >> 1/l_s (l_s =
sqrt{alpha'} is the string scale) is in strikingly good approximation
a random walk of n=sqrt{N} steps of step size l_s, where N is the
level number of the string, i.e. N = M^2 l_s^2. It follows that its
entropy is to leading order
S_s ~ n ~ M l_s .
This is quite unlike the entropy dependence of a black hole, which
goes as
S_BH ~ M^q
with q > 1 . But it so happens that at the above mentioned
correspondence point, which is reached when the mass of the string
becomes the critical value
M_c = 1/g^2 1/l_s
(g is the string coupling) and where the string collapses under its
self-gravity to a ball of diameter ~ l_s, the entropy of the string
and that of a BH of the same size coincide. Still, the entropy of a
random-walk-like string, even in the collapsed form, has a simple
interpretation, it counts the number of decisions one can make while
stepping along the random walk.
Now imagine how that collapsed "random walk" looks like: A chain of n
segments, each of length l_s is restricted to lie within a ball whose
diameter is also about l_s. A typical such state looks somewhat
star-shaped with all the vertices of the random walk on the outside,
forming a sphere. This sphere about coincides with the event horizon
of a BH which has the same mass as our string. The edges of the random
walk cross the interior of this sphere, pierce the horizon, deposit
their vertex there, then return to a point near the corresponding
antipode and so on, thereby covering the sphere with all n vertices,
all about equally spaced (for a typical state). What is the mean area
A_v of the sphere occupied by one such vertex? It is the number of
vertices divided by the area of the horizon, i.e.
A_v ~ n / l_s^(d-1) ~ n g^(d-1) / l_p^(d-1) ~ 1 / l_p^(d-1) .
Here d is the number of spatial dimensions and l_p is the Planck
length, given in string theory by
l_p^(d-1) = g^2 l_s^(d-1) .
It follows that (for instance) in 1+3 dimensions each of the above
vertices occupies an area of about a square Planck length of the event
horizon.The entropy of this system is, due to the nature of a random
walk and by the above formula, proportional to the number of vertices
and hence to the area of the event horizon in Planck units.
This is the picture of black hole microstates at the string/BH
correspondence point, i.e. for BH that are about to decay into a
string state or for strings that are about to become black holes. (In
fact the string ball configuration has been used to predict the
signature of decaying black holes that may be detected in accelerators
one day).
What happens to this crude picture when the mass is increased further?
I am not sure how solid the knowledge obout the answer to this
question is, but there is a lot of literature about "strings on the
stretched horizon". The basic idea is that once the BH description
takes over the above mentioned vertices are somehow frozen on the
event horizon. Since the temperature of the string and the Hawking
temperature agree at the correspondence point and hence the rates of
change of horizon area with mass do, one can show that further quanta
of mass that one throws into a BH at correspondence point correctly
translates into further string "vertices" appearing on the
horizon. But this only holds in the vicinity of the correspondence
point. Farther away one has to take into account the fact that the
energy of an object near a BH horizon is different when measured by an
asymptotically far away observer. When this red-shifting effect is
accounted for one can apparently consistently imagine the BH entropy
being due to a (very) long string which is lying on the "stretched"
event horizon in form of a random walk.
Of course, all this is nowhere near the technical sophistication of
D5/D1 brane-system or LQG calculations. It is rather like a Bohr-atom
model of quantum black holes.
> For the full story, go here:
>
> Quantum Geometry of Isolated Horizons and Black Hole Entropy
> A. Ashtekar, J. Baez, K. Krasnov
> Adv. Theor. Math. Phys. 4 (2000) 1-94.
> http://xxx.lanl.gov/abs/gr-qc/000512
>
> and see Section VI, "Entropy".
I'll have a look at it.
Ilja Schmelzer
The situation is a little bit different in this case. The proposed
theory is not background-free, but is nonetheless quite simple. The
relativistic symmetry (EEP) is derived from axioms and exact. And the
GR limit is also not arbitrary, but roughly speaking the
non-cosmological situation.
If a symmetry appears in a natural way (like the renormgroup if we
consider large distance limits of ugly theories) it is some type of
evidence that the symmetry is only an effective symmetry.
Jay Olson
> >> My
> >> intuition for such matters still isn't up to the point where I feel
> >> comfortable asking, but would something terrible happen if we didn't
> >> mod out by all space-time diffeomorphisms to get the space of states
> >> in the quantum theory? What if these constraints appear as a
> >> manifestation of thermodynamics on the classical level?
>
> I'm not sure what you mean by this, or why you think it might
> happen.
In this analysis, the properties of general relativity in the bulk
(e.g. diff invariance) seem to be derivable from thermodynamic
properties of spacetime on the horizon. Since you really don't need
diffeomorphism invariant states in quantum gravity in order to get the
the right thermodynamics on a horizon (well I suppose that's
debatable), maybe this implies that the fundamental states need not be
diffeomorphism invariant. See where I'm going with this? Is this
simply a wrong-headed notion, or might there be a deep relationship
between spacetime thermodynamics and diffeomorphism invariance?
John Baez
Jay Olson <sjay...@yahoo.com> wrote:
>In this analysis, the properties of general relativity in the bulk
>(e.g. diff invariance) seem to be derivable from thermodynamic
>properties of spacetime on the horizon. Since you really don't need
>diffeomorphism invariant states in quantum gravity in order to get the
>the right thermodynamics on a horizon (well I suppose that's
>debatable), maybe this implies that the fundamental states need not be
>diffeomorphism invariant. See where I'm going with this?
No, but I still don't like it. :-)
Since we can derive the properties of general relativity in the
bulk, e.g., diff invariance, from thermodynamic properties of
spacetime on the horizon, maybe this implies that the fundamental
states need NOT be diffeomorphism invariant??? You could argue
the opposite just as well, or better.
Maybe you mean that you want "fundamental states" to be states of the
horizon rather than the bulk. This would be a very holographic-
hypothesis-ish thing to say. It's interesting, but doesn't have
anything obvious to do with the black hole entropy calculation
we're talking about, since in that calculation we are counting
states of the horizon, not states of the bulk - and we're not
claiming that states of the bulk secretly *are* states of the
horizon (which is the holographic idea).
A.J. Tolland
Sorry this reply took so long to write.
On Sat, 5 Oct 2002, Urs Schreiber wrote:
> So I'd say he does not use diff invariance for his derivation. Unfortunately
> this is not the case for his short remark on the cosmological constant.
I'm bothered by a few things about his derivations. The
biggest is that he restricts his class of spacetimes to those which
(a) admit a Euclidean continuation, and (b) are static. (What the
relation is between (a) and (b), I don't know.) I don't see any way
of extending this approach to generic spacetimes, so I'm not quite
ready to admit that "holography + thermo => diff invariance".
I'm also somewhat bothered by the fact that he uses special
relativity in the local inertial frames. It's not clear to me that
you can write down coordinate dependent actions on general manifolds
which don't violate SR as well.
--A.J.
Urs Schreiber
"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag
news:apnhid$q8b$1...@glue.ucr.edu...
[...]
> since in that calculation we are counting
> states of the horizon, not states of the bulk
But why? Initially one wants to know the entropy of the Schwarzschild
spacetime. From semiclassical calculations one knows that this is purely due to
the horizon. But in LQG one does not want to use and rely on this semiclassical
reasoning, but rather to justify it on deeper grounds. No?
> - and we're not
> claiming that states of the bulk secretly *are* states of the
> horizon (which is the holographic idea).
Too bad that nobody is claiming this! ;-) Seriously, I find (being sufficiently
vague) that the LQG calculation of BH entropy has a rather strong holographic
touch to it. After all, one ends up studying a (non-gravitational,
Chern-Simons) field theory on the boundary of spacetime instead of gravity in
the bulk.
Since one does not know yet how physical bulk states and boundary states are
related in LQG I suppose this still leaves the interesting possibility that
they really are in 1-1 correspondence, which would be holography.
So in case no one has done it before I hereby officially put forward the
following hypothesis:
Hypotheses: "In LQG physical boundary states are in 1-1 correspondence with
physical bulk states."
BTW, another thing about LQG:
Recently T. Thiemann has put his lecture on LQG on the arXive server which he
had held in Bad Honnef. There, in his talk, and now again in the written
version he emphasizes that one/his motivation for studying LQG is to start with
the most conservative approach to QG in order to see where exactly it fails.
The idea is that instead of postulating all the new physics that is used in
other theories of QG one thereby might *derive* the necessary extensions of
known physics by demanding that they make LQG consistent.
Since Thiemann is stressing this point so much I am wondering: Does he have
concrete indications that the LQG program has encountered failure? (Maybe the
classical limit?) Does he have indications as to what modification might be
necessary?
Even though T. Thiemann seems to understand his attitude towards looking for
inconsistencies in a conservative approach to QG as something lacking in other
theories of QG, I wonder if this is correct. I find instead that this attitude
is rather similar to that which seems to have driven much of the development of
string theory: There one starts out with a certain premise (strings instead of
points), studies its implications and finds failures and/or inconsistencies.
These are then used to determine how the theory really "wants" to look like.
(E.g. the need for supersymmetry in string theory.)
Jay Olson
ba...@galaxy.ucr.edu (John Baez) wrote:
> Since we can derive the properties of general relativity in the
> bulk, e.g., diff invariance, from thermodynamic properties of
> spacetime on the horizon, maybe this implies that the fundamental
> states need NOT be diffeomorphism invariant??? You could argue
> the opposite just as well, or better.
Yes, that is what I am saying, although "imply" was probably too
strong a word.
Let me get a little more specific, so you can attack the logic
directly, rather than the vague idea... In your calculation of black
hole entropy, you made the weak assumption that for any state of the
surface, there is at least one bulk state that satisfies the
Hamiltonian constraint. But as a practical matter, the calculation
would have been the same if there was no Hamiltonian constraint at all
-- let's pretend this is true for the fundamental states, that they
aren't required to satisfy the Hamiltonian constraint. Yet if you
take this paper seriously, you will STILL arrive at a classical theory
that has a Hamiltonian constraint, because you have obtained S ~ A for
a horizon in the quantum calculation. Evidently, from this point of
view, the requirements of "local equilibrium" (= equivalence
principle?) of the classical, macroscopic states give you some extra
symmetry. Of course, maybe the fundamental states DO satisfy the
Hamiltonian constraint, but it seems like it would be hard to choose
between the two possibilities without experimental input.
> Maybe you mean that you want "fundamental states" to be states of the
> horizon rather than the bulk. This would be a very holographic-
> hypothesis-ish thing to say. It's interesting, but doesn't have
> anything obvious to do with the black hole entropy calculation
> we're talking about, since in that calculation we are counting
> states of the horizon, not states of the bulk - and we're not
> claiming that states of the bulk secretly *are* states of the
> horizon (which is the holographic idea).
No, this is not what I'm saying, but maybe Urs will have some fun
running with it. It is interesting! :-)
Urs Schreiber
news:<Pine.SOL.4.44.0210292007170.23121-100000@pub-708c-5>...
> On Sat, 5 Oct 2002, Urs Schreiber wrote:
>> So I'd say he does not use diff invariance for his derivation.
>> Unfortunately this is not the case for his short remark on the
>> cosmological constant.
> I'm bothered by a few things about his derivations. The
> biggest is that he restricts his class of spacetimes to those which
> (a) admit a Euclidean continuation, and (b) are static.
But is this a restriction of the result? I'd interpret the use of
static spacetimes in the derivation as showing that: "Requiring that
Rindler horizons in static spacetimes have the desired entropy/area
relation fixes the general action eq. (4) to be the EH action." This
is a weaker assumption than the general one where all horizons,
including those in non-static spacetimes, are required to satisfy some
relation. I am guessing that Padmanabhan tried the easiest case first
(static spacetimes) and since this was already sufficient to fix the
ambiguity he set out to fix he left it at that. (But see below.)
> I don't see any way
> of extending this approach to generic spacetimes, so I'm not quite
> ready to admit that "holography + thermo => diff invariance".
Maybe I do not understand what you are worried about here. I think
that once the ambiguity in eq. (4) is fixed by *some* assumption (in
this case one about Rindler horizons in static spacetimes) we have
obtained the general action. (But, again, see below.)
> I'm also somewhat bothered by the fact that he uses special
> relativity in the local inertial frames. It's not clear to me that
> you can write down coordinate dependent actions on general manifolds
> which don't violate SR as well.
Are you referring to the last sentence on p. 4: "Around any event P
one can construct a local Rindler frame [...]"? After reading your
comment I took a closer look at this seemingly innocent sentence. It
indeed presupposes that we have the freedom to change to a system of
Rindler coordinates. But under the given assumptions we actually don't
have the right to make any coordinate changes at all!
This now looks to me like a problem of Padmanabhans derivation.
However, in the light of what I have said in answer to your previous
comment I am inclined to think that the problem is in the text, not in
the logic. Namely, it really suffices that we require that the area
law (eq. (8), to be explicit) holds for flat Minkowski space written
in any set of Rindler coordinates. That is, we do not need to consider
arbitrary static spacetimes in arbitrary coordinates only to then make
a change of coordinates to a local Rindler system. We can pick a flat
spacetime in Rindler coordinates from the outset and require that the
boundary term of the action (4) has the desired value. Doing this for
all possible Rindler coordinate systems (in particular for any choice
of l(x) in eq. (12)) then still gives the desired result.
(Hm. This might show that thinking in terms of non-diff-invariant
actions for gravity takes a little getting used to.... One is so used
to switching to convenient coordinates when necessary.)
I'll summarize the outline of the proof as I now imagine it, after
thinking about your comments:
Start with a family of theories of a field g_ab, each given by an
element A' of the family of actions eq. (4). Using eq. (6), an element
of this family of actions is parametrized by two real numbers. Fix
these numbers by requiring that a given action A', when evaluated on a
field configuration of the form of eq. (9) (Rindler "metric"),
reproduces the area law eq. (11). The unique element so obtained turns
out to be the EH action which happens to be diff-invariant.
What do you think?
P.S.: I have tried to contact T. Padmanabhan in order to ask him to
comment on this issue, but apparently he is currently on vacation.
Urs Schreiber
Jay Olson wrote:
[...]
> But as a practical matter, the calculation
> would have been the same if there was no Hamiltonian constraint at all
> -- let's pretend this is true for the fundamental states, that they
> aren't required to satisfy the Hamiltonian constraint.
I think there is a subtlety which makes this conclusion problematic: There
is quite a difference between, on the one hand side, not knowing (and not
being required to know) which states satify the Hamiltonian constraint and
on the other hand not requiring any state to satisfy this constraint.
The thermodynamic approach to GR by Padmanabhan and others, just as
holography in general, doesn't do away with GR, but points to a possible
alternative description of it, but ultimately a description of the same
thing. There must be a relation between boundary states and bulk states for
this to be true. You seem to be proposing to remove all conditions on bulk
states while keeping conditions on boundary states. But this would break
the link between them.
Looking closely at the various calculations reveals that this link is always
there: In LQG the field theory on the horizon arises as the unique
extension of the bulk theory to a manifold with boundary. So the boundary
theory and the bulk theory are not independent. In Padmanabhans paper
gr-qc/0209088 the link is essentially in his ansatz eq.(4) for the action.
It consists of two terms, one for the boundary the other for the bulk. But
one determines the other via the relation (14). This should work in both
directions.
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag
>news:apnhid$q8b$1...@glue.ucr.edu...
>> since in that calculation we are counting
>> states of the horizon, not states of the bulk
>But why? Initially one wants to know the entropy of the Schwarzschild
>spacetime.
I guess it depends on who one is. :-)
I want to know the entropy of a black hole, not the
entire spacetime it sits in. For example, if we have
a black hole in our actual universe, the entire spacetime
is very far from the Schwarzschild geometry. Nonetheless,
one can hope that the entropy of the black hole is well-defined
and asymptotically proportional to the area of its event horizon
(in the limit of black holes much larger than the Planck length).
This raises the question of where the entropy actually "lives".
Given that it's proportional the area of the horizon, a natural
guess is: the entropy lives on the horizon.
> From semiclassical calculations one knows that this is purely due to
>the horizon. But in LQG one does not want to use and rely on this
>semiclassical reasoning, but rather to justify it on deeper grounds. No?
It would be nice to understand what happens to the singularity
in loop quantum gravity, but we don't yet. So there are still
mysteries about the precise nature of the interior of a black hole
in loop quantum gravity. Right now, the best we can do is quantize
gravity with boundary conditions asserting the existence of an event
horizon - or more precisely, a marginally outer trapped surface.
>> - and we're not
>> claiming that states of the bulk secretly *are* states of the
>> horizon (which is the holographic idea).
>Too bad that nobody is claiming this! ;-)
I didn't say *nobody* is claiming it. Lee Smolin has claimed it
in one of his papers. I just said "we" aren't claming it - meaning
Ashtekar, Krasnov and myself.
>Seriously, I find (being sufficiently
>vague) that the LQG calculation of BH entropy has a rather strong holographic
>touch to it. After all, one ends up studying a (non-gravitational,
>Chern-Simons) field theory on the boundary of spacetime instead of gravity in
>the bulk.
Right. However, in our calculation this field theory describes
only the geometry of the event horizon, not the geometry of the
bulk.
>Since one does not know yet how physical bulk states and boundary states are
>related in LQG I suppose this still leaves the interesting possibility that
>they really are in 1-1 correspondence, which would be holography.
This would happen if there were really just *one* solution of the
Hamiltonian constraint per state of the horizon.
>So in case no one has done it before I hereby officially put forward the
>following hypothesis:
>
>Hypothesis: "In LQG physical boundary states are in 1-1 correspondence with
>physical bulk states."
Okay - and given the lax standards of refereeing in physics,
maybe you can even publish a one-sentence paper to that effect. :-)
>BTW, another thing about LQG:
>Recently T. Thiemann has put his lecture on LQG on the arXive server which he
>had held in Bad Honnef. There, in his talk, and now again in the written
>version he emphasizes that one/his motivation for studying LQG is to start
>with the most conservative approach to QG in order to see where exactly it
>fails.
I agree that loop quantum gravity is intended to be a "most
conservative approach to quantum gravity", and I think this is
one good reason for studying it. However, it's often very hard
to see "exactly where" a theory of physics fails - especially if
it's right, but even if it's wrong.
>The idea is that instead of postulating all the new physics that is used in
>other theories of QG one thereby might *derive* the necessary extensions of
>known physics by demanding that they make LQG consistent.
That could take a while.
>Since Thiemann is stressing this point so much I am wondering: Does he have
>concrete indications that the LQG program has encountered failure?
I've never heard him say anything of the sort... which is another reason
I'm a bit suspicious of this idea of "seeing exactly where it fails".
Both in loop quantum gravity and string theory, there are very important
and very hard problems that nobody knows how to solve. So far, however,
enthusiasts of these theories take these problems as challenges rather
than "failures". Definitive failures are a bit hard to come by.
>Even though T. Thiemann seems to understand his attitude towards looking
>for inconsistencies in a conservative approach to QG as something lacking
>in other theories of QG, I wonder if this is correct. I find instead that
>this attitude is rather similar to that which seems to have driven much
>of the development of string theory: There one starts out with a certain
>premise (strings instead of points), studies its implications and finds
>failures and/or inconsistencies. These are then used to determine how
>the theory really "wants" to look like. (E.g. the need for supersymmetry
in string theory.)
Right. It seems to me that the main difference is not about whether
one is looking for inconsistencies and seeking to fix them; this
is part of all theory-building. Instead, it seems to be about
whether one is taking a maximally conservative approach. Historically,
some enthusiasm for string theory was due to the fact that unlike
perturbative quantum gravity, it was renormalizable. String theory
probably seemed like the most conservative renormalizable modification
of perturbative quantum gravity. But fans of loop quantum gravity
point out that the whole idea of perturbative quantum gravity is a
strange and unconservative one, so its failure may not mean much.
Jay Olson
> Jay Olson wrote:
> > But as a practical matter, the calculation
> > would have been the same if there was no Hamiltonian constraint at all
> > -- let's pretend this is true for the fundamental states, that they
> > aren't required to satisfy the Hamiltonian constraint.
> I think there is a subtlety which makes this conclusion problematic: There
> is quite a difference between, on the one hand side, not knowing (and not
> being required to know) which states satify the Hamiltonian constraint and
> on the other hand not requiring any state to satisfy this constraint.
I agree that there is quite a difference! That's why it's interesting
to me! If the approach is correct then it could mean that we are
closer than we think to a theory of quantum gravity since the current
"big problem" in finding fundamental LQG physical states (solving the
Hamiltonian constraint) is precisely the thing that you could avoid
having to do. I'm not sure if this idea is "too good to be true"
because it gets around the above problem, "too bad to be true" because
it seems to allow a lot of arbitrarity into the theory, or "just plain
irrelevant" because the idea doesn't hold water for one reason or
another. (or maybe all of the above)
> The thermodynamic approach to GR by Padmanabhan and others, just as
> holography in general, doesn't do away with GR, but points to a possible
> alternative description of it, but ultimately a description of the same
> thing. There must be a relation between boundary states and bulk states for
> this to be true. You seem to be proposing to remove all conditions on bulk
> states while keeping conditions on boundary states. But this would break
> the link between them.
I don't think I'm proposing to remove *all* conditions on bulk states.
Just the ones that aren't needed in order to recover GR in the
classical, thermodynamic limit in the spirit of Padmanabhan's paper
(e.g. the Hamiltonian constraint).
> Looking closely at the various calculations reveals that this link is always
> there: In LQG the field theory on the horizon arises as the unique
> extension of the bulk theory to a manifold with boundary. So the boundary
> theory and the bulk theory are not independent. In Padmanabhans paper
> gr-qc/0209088 the link is essentially in his ansatz eq.(4) for the action.
> It consists of two terms, one for the boundary the other for the bulk. But
> one determines the other via the relation (14). This should work in both
> directions.
So to carry this to conclusion, you're saying that it should be
possible to find a thermodynamic principle that dictates the bulk
term, and then by relations (4) and (14) I can get the boundary term
and thus the full GR action. Presumably, then, by my logic, you could
look at a hypothetical LQG calculation "in the bulk" that predicts the
thermodynamic relation of interest, discover what constraints don't
need to be satisfied by the LQG states in order to make the
calculation work, and then conclude that perhaps one shouldn't require
that the LQG states satisfy these constraints in the first place.
I have no problem with this idea, except that we don't know what the
bulk states are in LQG, and thus it seems like we're not in a position
to actually try it out and confirm that we arrive at the same
conclusions that we get from this first approach (which I agree would
be a neat test of the self-consistency of the idea).
Is there something that makes you think this parallel approach
wouldn't work or would lead you to different conclusions than the
original one?
John Baez
Jay Olson <sjay...@yahoo.com> wrote:
>Let me get a little more specific, so you can attack the logic
>directly, rather than the vague idea... In your calculation of black
>hole entropy, you made the weak assumption that for any state of the
>surface, there is at least one bulk state that satisfies the
>Hamiltonian constraint. But as a practical matter, the calculation
>would have been the same if there was no Hamiltonian constraint at all
>-- let's pretend this is true for the fundamental states, that they
>aren't required to satisfy the Hamiltonian constraint.
So you're having them satisfy the diffeomorphism constraint,
and the gauge-invariance constraint, and the special constraint
which holds at the boundary, but not the Hamiltonian constraint?
The reason I can't stomach this is that *all* these constraints
hold classically. My paper with Ashtekar and Krasnov is the
second part of a 2-part paper - the quantum part. The first
part is purely classical. This part works out all the
constraints on initial data that hold for a classical black
hole - or more precisely, a solution of Einstein's equations
having an "isolated horizon":
Isolated Horizons: the Classical Phase Space
A. Ashtekar, A. Corichi, K. Krasnov
Adv. Theor. Math. Phys. 3 (2000) 419-478.
Also available as gr-qc/9905089
(This was later refined in a number of other papers
by Ashtekar and coauthors.)
In the second part, we carefully quantize the whole story
and get a Hilbert space of states for the horizon of a
quantum black hole.
Throwing out just the Hamiltonian constraint would destroy
the logic of what we're doing. After a careful analysis of
classical general relativity, you'd have us say "Hmm, I don't
like the look of *this* equation" and throw it out before we
quantize the theory! Remember, the Hamiltonian constraint is
just one of Einstein's equations, namely G_{00} = 0, while the
diffeomorphism constraint consists of 3 more, namely G_{0i} = 0.
Throwing one out while keeping the other 3 would not be a
covariant thing to do. Frankly, it would be a very strange
thing to do! And in case you're wondering, we really *do*
need the diffeomorphism constraint to get our calculation to
work.
So, while you can try going down this avenue, it doesn't
seem promising to me. Also:
>Yet if you take this paper seriously, you will STILL arrive
>at a classical theory that has a Hamiltonian constraint,
>because you have obtained S ~ A for a horizon in the quantum
>calculation.
Our paper gets this for an isolated horizon, NOT for an
arbitrary null hypersurface in spacetime. I haven't read
the paper you're talking about, but in Jacobson's derivation
Einstein's equations from thermodynamic considerations, he
needs S ~ A for *all* null hypersurfaces, not just those
measly few that happen to be isolated horizons - i.e., roughly
speaking, horizons of black holes that have nothing falling
into them.
I'll cc this to you because it's taken me so long to reply.
That darn Bogdanov business had me quite distracted.
Please continue on the newsgroup, though!