necessity of degenerate metrics
Nathan Cole
in the unification of GR and QM or possibly QFT. First looking at GR +
QM it seems to me that a degenerate metric should be rigorously
required. Consider the following line of arguments:
1. The action of GR involves the curvature scalar and a matter field
scalar. Looking at the matter field scalar is it not expected that it
should depend on a wave function?
2. Can one expect that the matter field scalar term in the action
integral have a certain form, for instance depend on the metric or
connection coeficients and on the wave function and its first
derivatives in some quadratic way.
3. If 1 and 2 above are both true then when we deal with the action
integral would it not be required that the action integral be
minimized for variations of both the metric AND the wave function?
Now consider what 3 does. Say we have the following action in
accordance with 1, 2 and 3:
S = Integral[(-R/2k + L) (-g)^(-1/2), volume of space]
Here
S = action
R = scalar curvature
k = 8 Pi G / c^2
L = matter field
g = det|g_mn|
This is pretty much GR except that L depends on the wave function.
Still not that unusual though. Now consider what happens if we vary S
for the metric and for the wave function. First if we vary S for the
metric it seems plausible to expect that we get Einstin's field
equations, not a big surprise. What now if we vary the wave function?
Well, since R does not depend on the wave function the term will
become zero. However since L depends quadratically on the wave
function we can expect to get a second order partial differential
equation, i.e. a wave equation. So in summary: vary action with metric
you get GR; vary action with wave function you get QM.
Now to the degeneracy. If we solve the wave function we should get a
set of state functions if the matter scalar is properly defined. These
state functions will depend on the metric and the space-time
variables. Lets enumerate them
|0>, |1>, ...
This is not allways possible but in many cases it is. One can expect
to be able to define T_mn in terms of L (using canonical formalisms).
But what does this mean? Well, since T_mn is a function of a
particular q'th state function |q> and Einstein's field equations give
solutions for g_mn by
G_mn = R_mn - (1/2) g_mn R = -k T_mn(|q>)
Could we not expect that for each |q> there is a possibly new g_mn?
There exists the possibility that even for differing |q> the g_mn
solutions will remain the same but on average it seems that the g_mn
will become quantized if that is the proper word to use. Can this even
be allowed, having multiple metrics in the same space...?
-- NPC
Doug Sweetser
> There exists the possibility that even for differing |q> the g_mn
> solutions will remain the same but on average it seems that the g_mn
> will become quantized if that is the proper word to use. Can this even
> be allowed, having multiple metrics in the same space...?
Two thoughts. First, general relativity like the standard model is a local
theory. It could be that there are multiple indefinite metrics, each in a
different place in spacetime. That is what is seen in Nature.
Second, quanta scale to multiples of hbar, which has units of m L^2/t. A
gravitational field will presumably depend on a mass charge, where the
units for mass come from the mass itself, not hbar. If so, then there are
no quanta that scale to hbar. As you probably know, a big problem with
degenerate metrics is the energy is not bound from below. This problem
might be averted if no quanta can be emited because Planck's constant is
not in the expression for the gravitational field.
doug
quaternions.com
Arkadiusz Jadczyk
>The question I whish to pose is the necessity of a degenerate metric
>in the unification of GR and QM or possibly QFT.
(snip)
Stephen Hawking and others speculated about possible signature
fluctuations - between Lorentzian and Euclidean signatures. A natural
framework for such fluctuations as a dynamical feature should be based
on formulations that are continous across the degenerated configuration.
Theories including scale invariance as a local gauge group also
naturally require "degenerate metrics".
There has been a noticeable progress in understanding these needs
in the last twenty years. For instance Lorentzian General Relativity
naturally splits into "conformal structure" and "scale". We do not need
scale at all to couple massles fields. The scale can be a composite
field, like in the many papers by Pervushin - it can have zeros.
So there are many avenues ro follow in the direction that interests
you.
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
John Baez
Doug Sweetser <swee...@alum.mit.edu> wrote:
>Someone Doug Sweetser referred to solely as "Nathan" wrote:
>> Can this even
>> be allowed, having multiple metrics in the same space...?
>It could be that there are multiple indefinite metrics, each in a
>different place in spacetime. That is what is seen in Nature.
In general relativity there's a different inner product at each
different point in spacetime, but I don't think that's what
Nathan was talking about. Admittedly, I don't understand what
he *was* talking about - but it seemed to involve quantum gravity.
Quantum gravity raises the possibility that the gravitational field
can be in a state that's a *superposition* of states in
which the metric on space takes on different definite values.
This is a more challenging sense of the term "multiple metrics
in the same space" than the one you suggest.
>Second, quanta scale to multiples of hbar, which has units of m L^2/t.
>A gravitational field will presumably depend on a mass charge, where the
>units for mass come from the mass itself, not hbar. If so, then there are
>no quanta that scale to hbar.
I don't understand the logic of this, but in quantum gravity
we expect the existence of quantized excitations of the
gravitational field called "gravitons". These are expected
to satisfy the same relation that photons do:
energy = hbar frequency
We don't expect gravitons to be the *fundamental* description
of quantum gravity, but they should be a decent approximation
in situations where the field is weak and one is probing it
at length scales much larger than the Planck length - about
10^{-35} meters.
>As you probably know, a big problem with
>degenerate metrics is the energy is not bound from below.
>This problem might be averted if no quanta can be emitted
>because Planck's constant is not in the expression for
>the gravitational field.
This seems wrong to me.
Doug Sweetser
Hello John:
Superposition of states with different metrics would be confusing, so I'll
let others remained puzzled on that issue. Let me try to clarify this line
of reasoning:
>>Second, quanta scale to multiples of hbar, which has units of m L^2/t.
>>A gravitational field will presumably depend on a mass charge, where the
>>units for mass come from the mass itself, not hbar. If so, then there are
>>no quanta that scale to hbar.
This trial balloon has nothing to do with the Planck length unlike most
current research in quantum gravity. The energy density of the
electromagnetic field terms has terms of the form:
E^2, B^2, ...
has units of: m/(t^2 L) [or m L^2/(t^2 L^3)]
so you can see it is energy per unit volume. If one thinks about the
energy density for a gravitational field, there should be a contribution
from both source and test masses, contributing an m^2 to the numerator. To
get units of energy density, all it takes is one factor of G:
(mass field)^2 = k (m/R^2)
has units of:
G (m/R^2)^2 = L^3/(m t^2) m^2/L^4 = m/(t^2 L)
Newton's gravitational constant G is required to get energy density, not
hbar. For people who work with natural units where G=hbar=c=1, this point
is not noticed. What happens to the energy density of the squared mass
field as hbar goes to zero? Nothing.
> we expect the existence of quantized excitations of the
> gravitational field called "gravitons". These are expected
> to satisfy the same relation that photons do:
>
> energy = hbar frequency
This is the expected line of logic, no question. It is modeled after EM,
and by extension, the weak and strong forces, the stuff we know well. Yet
the units for a mass field are going to be different from an
electromagnetic one.
No graviton has yet to be detected. No attempt to extend the standard
model in a way to incorporate gravity has work yet. The above dimensional
analysis suggests hbar cannot be used to describe the energy density of a
gravitational field. This may imply that there is no graviton in Nature
because gravity is a change in distance between the known particles of the
standard model.
Doesn't the standard model _have_ to change to deal with gravity? Yes it
does, but not by adding in a new group which would include a new particles.
The standard model already makes statements about the lengths of things.
It is in the "S"'s of U(1)xSU(2)xSU(3). The S stands for special, because
the determinant is exactly equal to one. In curved spacetime, the
determinant might be one plus a delta strictly determined by the amount of
spacetime curvature. All the complex numbers in U(1) have an absolute
value of one in flat spacetime. The absolute value might also change due
entirely to the precise curvature of spacetime.
Please feel free to toss a dart at these trial balloons,
doug
quaternions.com
John Baez
Doug Sweetser <swee...@alum.mit.edu> wrote:
>[...] no graviton has yet to be detected.
And that's good - because if we could detect them with our
current crappy equipment, there'd have to be a *really* strong
source of gravitational radiation out there: something downright
scary and dangerous!
An interesting calculation by Ashtekar shows that all the
handwaving of all the theoretical physicists in the world
has almost surely not generated a single graviton.
To repeat this calculation, just estimate:
1) the total energy emitted in gravitational radiation emitted
by ten thousand guys waving their hands for a thousand years,
2) the energy of a single graviton with frequency equal to that
of this hand-waving - about a hertz, say.
Then notice that energy 2) is much, much more than energy 1)!
The point is, the gravitational radiation will be in a coherent
state where the expectation value of the graviton number is a lot
less than one!
If you get stuck, you can estimate 1) by using the fact that when you've
got a long thin cylinder of length L and mass M, rotating with angular
velocity w, general relativity predicts the power emitted in the form
of gravitational radiation will be
(2/45) G M^2 L^4 w^6
----------------------
c^5
Of course theorists don't wave their hands like this - they
undergo this motion only when they do backflips. Still, this
formula is good enough for a rough order-of-magnitude estimate,
which is all we need here, taking L = 1 meter, M = 1 kilogram and
w = 1/sec to keep the math easy.
You can calculate 2) using
energy = hbar frequency.
the softrat
What is a 'graviton' other than a wave packet of gravitational waves?
What is a gravitational wave other than a (time-dependent) ripple in
the gravitational field (or the curvature of spacetime)? And don't we
get gravitational waves (or ripples) from every binary star, and so
forth?
George D. Freeman IV
the softrat ==> Careful!
I have a hug and I know how to use it!
mailto:sof...@pobox.com
--
Meanwhile, back at the ranch, tension mounted, clicked his spurs
and rode off.
Ray Tomes
>>[...] no graviton has yet to be detected.
John Baez wrote:
> And that's good - because if we could detect them with our
> current crappy equipment, there'd have to be a *really* strong
> source of gravitational radiation out there: something downright
> scary and dangerous!
John, thanks for the story about handwaving not producing a single
graviton - very amusing.
I would like to tell a little story of my own (not so amusing, but
hopefully interesting) and then ask whether I might accidentally have
discovered a gravitational wave detector - the earth. Well, I didn't
discover the earth as such, just the possibility of it being a detector.
;-)
About 8 to 10 years ago I was examining the possibility that earthquakes
might cause a pressure wave (note that sometimes the words "gravity
wave" are used in this context but have an entirely different meaning to
"gravitational wave" in physics) to travel around and through the earth
and after converging on the opposite point to return to the original
point and have an increased probability of an after shock at that
moment. Geophysicists told me that there are a variety of types of waves
which would all take from 44 to 54 minutes to return to the original
location.
So I got a bunch of earthquake data from California (Californians seem
to have the world's greatest obsession with the subject) and made a
histogram of the number of earthquakes occurring at each interval (by
minute) after a previous earthquake. There was no tendency of repetition
after an interval of 44 to 54 minutes. However there was a very clear
increase in probability of earthquake after all multiples of 26.0
minutes up to about a day. The increase was represented by a sharp spike
at every mutiple of 26 minutes, either on the exact minute or one minute
off. Of course when I say there was no increase in probability after
44-54 minutes, there was one after 52 minutes, but because there was
also one after 26 minutes, the point opposite on the earth had to have a
wave depart at exactly the same time as the supposed originating quake.
Eventually I realised that such a possibility could exist if the earth
was oscillating between the shape of a lemon and a grapefruit because
that would produce a period of half the originally expected 44 to 54
minutes - or 22 to 27 minutes. However it seems to me that there is no
easy way to explain such an oscillation mode from any event internal to
the earth, but that it might be what would be expected if gravitational
waves of period 26 minutes were passing through the earth.
Can some physicists knowledgeable in the subject please tell me whether
it is true that gravitational waves of period 26 minutes (or within 2%
of this) would deform the earth in the manner that I described (lemon
and grapefruit alternating) and thereby might lead to the facts that I
observed?
Would this count as gravitational wave detection?
Ray Tomes
Phillip Helbig---remove CLOTHES to reply
In article <bl2bhl$jpd$1...@lust.ihug.co.nz>, Ray Tomes
<rto...@ihug.remove.co.nz> writes:
> I would like to tell a little story of my own (not so amusing, but
> hopefully interesting) and then ask whether I might accidentally have
> discovered a gravitational wave detector - the earth. Well, I didn't
> discover the earth as such, just the possibility of it being a detector.
> ;-)
> Can some physicists knowledgeable in the subject please tell me whether
> it is true that gravitational waves of period 26 minutes (or within 2%
> of this) would deform the earth in the manner that I described (lemon
> and grapefruit alternating) and thereby might lead to the facts that I
> observed?
>
> Would this count as gravitational wave detection?
The amplitude of any gravitational wave would be orders of magnitude too
small to affect geophysics.
Ralph E. Frost
wrote in message news:bl2h0f$oik$2...@online.de...
How about where they have to concentrate, intertwine and equilibrate, say,
like at the center(s) of gravity in something like solar systems?
Squark
Ray Tomes <rto...@ihug.remove.co.nz> wrote in message news:<bl2bhl$jpd$1...@lust.ihug.co.nz>...
> So I got a bunch of earthquake data from California (Californians seem
> to have the world's greatest obsession with the subject) and made a
> histogram of the number of earthquakes occurring at each interval (by
> minute) after a previous earthquake. There was no tendency of repetition
> after an interval of 44 to 54 minutes. However there was a very clear
> increase in probability of earthquake after all multiples of 26.0
> minutes up to about a day. The increase was represented by a sharp spike
> at every mutiple of 26 minutes, either on the exact minute or one minute
> off. Of course when I say there was no increase in probability after
> 44-54 minutes, there was one after 52 minutes, but because there was
> also one after 26 minutes, the point opposite on the earth had to have a
> wave depart at exactly the same time as the supposed originating quake.
Did you try getting statistics on Earthquakes occuring at two
diametrically opposite points (and comparing it to statistics on
earthquakes at two random points)?
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and change the
extension in the obvious way)
John Baez
Ralph E. Frost <ref...@dcwi.com> wrote:
>"Phillip Helbig wrote
>> Ray Tomes wrote:
>> > Can some physicists knowledgeable in the subject please tell me whether
>> > it is true that gravitational waves of period 26 minutes (or within 2%
>> > of this) would deform the earth in the manner that I described (lemon
>> > and grapefruit alternating) and thereby might lead to the facts that I
>> > observed?
>> The amplitude of any gravitational wave would be orders of magnitude too
>> small to affect geophysics.
>How about where they have to concentrate, intertwine and equilibrate, say,
>like at the center(s) of gravity in something like solar systems?
There AREN'T ANY places where gravitational waves
"have to concentrate, intertwine and equilibriate" -
whatever the heck that means. They just zip merrily
along their incredibly weak and wimpy way.
So, the answer to the question is still no.
I think Tomes and Frost are radically underestimating how weak
gravitational waves are. The gravitational waves in our solar
system are probably so weak that they stretch a 4-kilometer-long
object by a distance of at most about 10^{-18} meters - about a
thousandth of the diameter of a proton. Their effect on the earth
as a whole is UTTERLY drowned out by various forms of noise such as
that caused by micro-earthquakes, ocean waves, wind, traffic, etc..
You might as well try to detect the footsteps of an ant when atomic
bombs are going off all over.
This is why LIGO is an ambitious experiment.
Ray Tomes
> There AREN'T ANY places where gravitational waves
> "have to concentrate, intertwine and equilibriate" -
> whatever the heck that means. They just zip merrily
> along their incredibly weak and wimpy way.
Now I didn't write those words, Ralph E Frost did, but it almost makes
me think that he understood something that I was getting at (hey it
happens sometimes!). Of course they don't "have to", but what if the
waves DID concentrate at certain places, like near the centre of solar
systems, because they had large scale standing wave structure related to
the known regularities in super-clusters, galaxies and at other scales?
> I think Tomes and Frost are radically underestimating how weak
> gravitational waves are.
Well, I am not estimating them at all because I have no idea how to :-)
All I am doing is observing that earthquakes occur in the same place at
intervals of time that indicate that the earth is being deformed from
external action not internal (otherwise the period would be twice as
great) and then asking if gravitational waves might be the cause.
> The gravitational waves in our solar
> system are probably so weak that they stretch a 4-kilometer-long
> object by a distance of at most about 10^{-18} meters - about a
> thousandth of the diameter of a proton. Their effect on the earth
> as a whole is UTTERLY drowned out by various forms of noise such as
> that caused by micro-earthquakes, ocean waves, wind, traffic, etc..
> You might as well try to detect the footsteps of an ant when atomic
> bombs are going off all over.
OK, I get the point, thanks. You are saying pretty clearly that
gravitational waves are not a reasonable candidate for earthquakes.
Just one last question on that please John - if we cannot detect
gravitational waves, how do we know that they are this weak (or do we
have an upper limit of measurement)?
The question still remains as to what could be externally deforming the
earth in the manner observed in my original post? If gravitational waves
could not be the cause, could large amplitude e/m waves of period about
26 minutes cause distortions that set off earthquakes? AFAIK no-one has
every tried to measure such low frequency e/m waves, so they seem the
next most likely candidate.
Ray Tomes
John Baez
Ray Tomes <rto...@ihug.remove.co.nz> wrote:
>John Baez wrote:
>> There AREN'T ANY places where gravitational waves
>> "have to concentrate, intertwine and equilibriate" -
>> whatever the heck that means. They just zip merrily
>> along their incredibly weak and wimpy way.
>Now I didn't write those words, Ralph E Frost did, but it almost makes
>me think that he understood something that I was getting at (hey it
>happens sometimes!). Of course they don't "have to", but what if the
>waves DID concentrate at certain places [....]
Instead of asking "what if they did?", it pays to start by asking
"do they?" This is a question that can be answered by general
relativity. I'm pretty sure the answer is "they don't - at least
not enough to care about!"
Of course the usual sort of gravitational lensing caused by galaxies
will apply to gravitational waves just as to light. This is the only
realistic way to focus gravity waves that I can think of! But, this
would just give a little amplification of the puny gravitational waves
caused by very distant sources. It has nothing to do with gravitational
waves causing earthquakes or any other outrageously humongous effects
of the sort you mention.
>> I think Tomes and Frost are radically underestimating how weak
>> gravitational waves are.
>Well, I am not estimating them at all because I have no idea how to :-)
It's not hard - just crack open Misner, Thorne and Wheeler and look
up some of the formulas for power of gravitational waves produced
by various sorts of moving bodies. You don't need to learn general
relativity to use these formulas - just some algebra.
>All I am doing is observing that earthquakes occur in the same place at
>intervals of time that indicate that the earth is being deformed from
>external action not internal (otherwise the period would be twice as
>great) and then asking if gravitational waves might be the cause.
The answer is NO. It's a bit like asking if earthquakes are
caused by people sneezing - only the shock wave caused by a sneeze
is *much* stronger and easier to detect than a gravitational wave.
This is why we don't need a multi-million-dollar project to detect
sneezing.
>Just one last question on that please John - if we cannot detect
>gravitational waves, how do we know that they are this weak (or do we
>have an upper limit of measurement)?
We have experimental upper limits, thanks to LIGO, GEO, VIRGO and many
other detectors. We also have calculations based on general relativity,
which are in accord with these experimental upper limits.
Ray Tomes
Squark wrote:
> Did you try getting statistics on Earthquakes occuring at two
> diametrically opposite points (and comparing it to statistics on
> earthquakes at two random points)?
Good question. The only data source that I found was in California.
Although they had whole Earth statistics as well as Californian
statistics, the whole Earth statistics were for more powerful quakes and
so the data was too sparce. It was the many weak quakes in one region
that allowed me to find the repeating quakes after intervals of 26 minutes.
Ray Tomes
Ralph E. Frost
"John Baez" <ba...@galaxy.ucr.edu> wrote in message
news:bm9h4t$dfv$1...@glue.ucr.edu...
> In article <blg8b0$s4t$1...@lust.ihug.co.nz>,
> Ray Tomes <rto...@ihug.remove.co.nz> wrote:
>
> >John Baez wrote:
>
> >> There AREN'T ANY places where gravitational waves
> >> "have to concentrate, intertwine and equilibriate" -
> >> whatever the heck that means. They just zip merrily
> >> along their incredibly weak and wimpy way.
>
> >Now I didn't write those words, Ralph E Frost did, but it almost makes
> >me think that he understood something that I was getting at (hey it
> >happens sometimes!). Of course they don't "have to", but what if the
> >waves DID concentrate at certain places [....]
>
> Instead of asking "what if they did?", it pays to start by asking
> "do they?" This is a question that can be answered by general
> relativity. I'm pretty sure the answer is "they don't - at least
> not enough to care about!"
Do gravitational waves have origins or do they "just appear"?