What Weinberg Actually Said
Danny Ross Lunsford
...about gravity as geometry (from "Gravitation and Cosmology", preface):
"There was another, more personal reason for my writing this book. In
learning general relativity, and then in teaching it to classes at
Berkeley and MIT, I became dissatisfied with what seemed to be the usual
approach to the subject. I found that in most textbooks geometric ideas
were given a starring role, so that a student who asked why the
gravitational field is represented by a metric tensor, or why freely
falling particles move on geodesics, or why the field equations are
generally covariant would come away with an impression that this had
something to do with the fact that spacetime is a Riemannian manifold.
"Of course, this *was* Einstein's point of view, and his preeminent
genius necessarily shapes our understanding of the theory he created.
However, I believe that the geometrical approach has driven a wedge
between general relativity and the theory of elementary particles. As
long as it could be hoped, as Einstein did hope, that matter would
eventually be understood in geometrical terms, it made sense to give
Riemannian geometry a primary role in describing the theory of
gravitation. But now the passage of time has taught us not to expect
that the strong, weak, and electromagnetic interactions can be
understood in geometrical terms, and that too great an emphasis on
geometry can only obscure the deep connections between gravitation and
the rest of physics.
"In place of Riemannian geometry, I have based the discussion of general
relativity on a principle derived from experiment: the Principle of
Equivalence of Gravitation and Inertia. It will be seen that geometric
objects, such as the metric, the affine connection, and the curvature
tensor naturally find their way into a theory of gravitation based on
the Principle of Equivalence and, of course, one winds up in the end
with Einstein's general theory of relativity. However, I have tried here
to put off the introduction of geometric concepts until they are needed,
so that Riemannian geometry appears only as a mathematical tool for the
explanation of the Principle of Equivalence, and not as the fundamental
basis for the theory of gravitation.
"This approach leads one to ask *why* gravitation should obey the
Principle of Equivalence. In my opinion the answer is not to be found in
the realm of classical physics, and certainly not in Riemannian
geometry, but in the constraints imposed by the quantum theory of
gravitation. It seems to be impossible to construct any Lorentz
invariant quantum theory of particles of mass zero and spin two, unless
the corresponding classical theory obeys the Principle of Equivalence.
Thus the Principle of Equivalence appears to be the best bridge between
the theories of gravitation and of elementary particles. The quantum
basis for the Principle of Equivalence is briefly touched upon here in a
section on the quantum theory of gravitation, but it was not possible to
go far into quantum theory in this book."
==========
Having read these paragraphs again and again, I still do not understand
them as anything more than a wish for a quantum theory of gravitation.
Despite what Weinberg states, the very first chapter is titled "History
of Non-Euclidean Geometry", appearing before the next, "History of the
Theory of Gravitation." He returns to "the geometric analogy" in section
6.9, which he marks as optional on a first reading. He rephrases the
theme of the preface more directly, calling the geometric interpretation
"a mere analogy", and emphasizing that making predictions is the
important object of physics, and that the framework in which those
predictions are actually expressed "simply doesn't matter" - a statement
he immediately acknowledges to be heterodox. Oddly, he then proceeds
immediately to explain the surface geometry behind curvature! This
clearly illustrates that Weinberg is himself rather undecided about his
own attitude, regardless of the words he writes. The actual presentation
in the book is faultlessly geometric.
In section 8.8 he gives an outline of what a quantum theory of
gravitation should look like, without bringing up the "geometric
analogy" at all, and frankly admits that one does not exist. He traces
the failure to lack of proper scaling originating in the dimensionful
gravitational constant, comparing this situation with the Fermi
phenomenology of beta decay (the WSG theory was not yet born). That is
the last we hear about the non-essential nature of geometry. Indeed the
other parts of the book, on cosmology, are explicitly based on an
analysis of symmetric spaces in Riemannian geometry - a tacit admission
that, although he may not believe that gravity is fundamentally
geometrical, he certainly believes that cosmology *is*.
Oddly, Weinberg says nothing at all about the notion of a "background
free" theory, but does elaborate carefully on the analogy with
electrodynamics.
In short - I interpret Weinberg's comments to be more or less a lament
that the ideas that work in field theory, don't really work for gravity,
rather than an actual claim about the nature of gravitation. His acutal
presentation is thoroughly, indeed physically geometric, to the point
that I actually recommend this book above all others for understanding
the *physics* of gravitation.
-drl
Pmb
news:vf3Wb.67$946.51...@newssvr11.news.prodigy.com...
>
>
> ...about gravity as geometry (from "Gravitation and Cosmology", preface):
>
> "There was another, more personal reason for my writing this book. In
> learning general relativity, and then in teaching it to classes at
> Berkeley and MIT, I became dissatisfied with what seemed to be the usual
> approach to the subject. I found that in most textbooks geometric ideas
> were given a starring role, so that a student who asked why the
> gravitational field is represented by a metric tensor, or why freely
> falling particles move on geodesics, or why the field equations are
> generally covariant would come away with an impression that this had
> something to do with the fact that spacetime is a Riemannian manifold.
>
> "Of course, this *was* Einstein's point of view,
To be precise Einstein's view was that view as stated by Einstein himself.
As Einstein said in 1948
------------------------------------------------------------------
I do not agree with the idea that the general theory of relativity is
geometrizing physics or the gravitational field. The concepts of physics
always have been geometrical concepts and I cannot see why the g_jk field
should be called more geometrical than f.i. the electromagnetic field or the
distance of two bodies in Newtonian mechanics. The notion comes probably
from the fact that the mathematical origin of the g_jk field is the
Gauss-Riemann theory of the metrical continuum which we are wont to look at
as part of geometry. I am convinced, however, that the distinction between
geometrical and other kinds of fields is not logically founded.
------------------------------------------------------------------
Pmb
Uncle Al
Danny Ross Lunsford wrote:
>
> ...about gravity as geometry (from "Gravitation and Cosmology", preface):
> "In place of Riemannian geometry, I have based the discussion of general
> relativity on a principle derived from experiment: the Principle of
> Equivalence of Gravitation and Inertia. It will be seen that geometric
> objects, such as the metric, the affine connection, and the curvature
> tensor naturally find their way into a theory of gravitation based on
> the Principle of Equivalence and, of course, one winds up in the end
> with Einstein's general theory of relativity. However, I have tried here
> to put off the introduction of geometric concepts until they are needed,
> so that Riemannian geometry appears only as a mathematical tool for the
> explanation of the Principle of Equivalence, and not as the fundamental
> basis for the theory of gravitation.
>
> "This approach leads one to ask *why* gravitation should obey the
> Principle of Equivalence. In my opinion the answer is not to be found in
> the realm of classical physics, and certainly not in Riemannian
> geometry, but in the constraints imposed by the quantum theory of
> gravitation. It seems to be impossible to construct any Lorentz
> invariant quantum theory of particles of mass zero and spin two, unless
> the corresponding classical theory obeys the Principle of Equivalence.
> Thus the Principle of Equivalence appears to be the best bridge between
> the theories of gravitation and of elementary particles. The quantum
> basis for the Principle of Equivalence is briefly touched upon here in a
> section on the quantum theory of gravitation, but it was not possible to
> go far into quantum theory in this book."
[snip]
This is a tremendously dangerous approach if pursued without stronger
experimental and theoretical support.
1) The Equivalence Principle (EP) is a *postulate.*
Affine/teleparallel gravitation theory can give predictions
indistinguishable from metric theories without recourse to the EP.
The only guaranteed disjoint non-overlap is the EP itself. Do two
local test bodies exist that spontaneously fall along non-parallel
paths in vacuum? Spacetime geometry has never been challenged with
quantitative test mass geometry. It would be easy to look for
testable EP violation in the last possible place it could be hiding,
http://www.mazepath.com/uncleal/qz.pdf
2) The EP is a *postulate.* Rigorous self-consistent theory can be
erected that has no bearing on reality (economics, meteorology).
Euclid made no mistakes. His Fifth (Parallel) Postulate is
incompatible with terrestrial surveying or navigation because the
surface of the Earth is non-Euclidean. There are fully eight
simply-connected geometric 3-manifolds with compact quotients in three
dimensions,
Bull. Amer. Math. Soc. 6 357-381 (1982)
Bull. Lond. Math. Soc. 15(5) 401-487 (1983)
WP Thurston, "Three-dimensional geometry and topology," Vol. 1.
Princeton Mathematical Press, Princeton, NJ, 1997.
Before you put down your bets you had better consider the whole gaming
table.
3) The EP is a *postulate.* We know it could show failure at the
levels of gravitoelectric and gravitomagnetic effects. This is inside
quantum gravitation's purvue. How will the breach be plugged?
> he then proceeds
> immediately to explain the surface geometry behind curvature! This
> clearly illustrates that Weinberg is himself rather undecided about his
> own attitude, regardless of the words he writes. The actual presentation
> in the book is faultlessly geometric.
[snip]
What is the alternative? Is a non-geometric approach calculable? Any
acceptible theory must explicitly allow for the Global Positioning
Satellite system to 11 decimal places,
http://www.eftaylor.com/pub/projecta.pdf
GPS and Relativity
hallmark non-Newtonian perihelion precessions,
http://arXiv.org/abs/astro-ph/0401086
http://arxiv.org/abs/astro-ph/0312071
Deeply relativistic neutron star binaries
and a whole bunch of stuff that gets you published with predictions
vs. observations,
<http://rattler.cameron.edu/EMIS/journals/LRG/Articles/Volume4/2001-4will/index.html>
Experimental constraints on General Relativity.
Essentially nobody uses Weitzenboeck's affine spactime torsion
approach. It is a monster to calculate. It is also unfashionable.
It gains you no advantage.
If M-theory has one singular attribute it is that it makes no testable
predictions. M-theory is an unlimited sheaf of *non-competitive*
sub-theories. Folks can publish as philosophers - pencil, paper, no
wastebasket - and nobody can say "no." Where does it get you
(academic politics aside)?
> Indeed the
> other parts of the book, on cosmology, are explicitly based on an
> analysis of symmetric spaces in Riemannian geometry - a tacit admission
> that, although he may not believe that gravity is fundamentally
> geometrical, he certainly believes that cosmology *is*.
[snip]
What is the alternative? Non-geometric theories are not predictive.
Noether's theorem and other strong correspondences between symmetries
and properties cannot be denied - certainly not locally. If they can
be they certainly haven't been. A quantum approach is specifically
unsuited to global observations. Even if the perfect quantum theory
of gravitation were to be formulated, does one expect it not to look
like General Relativity vis-a-vis Mercury's orbit?
> In short - I interpret Weinberg's comments to be more or less a lament
> that the ideas that work in field theory, don't really work for gravity,
> rather than an actual claim about the nature of gravitation. His acutal
> presentation is thoroughly, indeed physically geometric, to the point
> that I actually recommend this book above all others for understanding
> the *physics* of gravitation.
It is my take on the subject that we do *not* need more beautifully
confirmatory observations. We need a few ugly ones that do not fit
into existing theory. Roses grow from manure,
http://physics.nist.gov/GenInt/Parity/cover.html
--
Uncle Al
http://www.mazepath.com/uncleal/qz.pdf
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)
Arvind Rajaraman
>
> "In place of Riemannian geometry, I have based the discussion of general
> relativity on a principle derived from experiment: the Principle of
> Equivalence of Gravitation and Inertia. However, I have tried here
> to put off the introduction of geometric concepts until they are needed,
> so that Riemannian geometry appears only as a mathematical tool for the
> explanation of the Principle of Equivalence, and not as the fundamental
> basis for the theory of gravitation.
>
> "This approach leads one to ask *why* gravitation should obey the
> Principle of Equivalence. In my opinion the answer is not to be found in
> the realm of classical physics, and certainly not in Riemannian
> geometry, but in the constraints imposed by the quantum theory of
> gravitation. It seems to be impossible to construct any Lorentz
> invariant quantum theory of particles of mass zero and spin two, unless
> the corresponding classical theory obeys the Principle of Equivalence.
> >
> ==========
>
> Having read these paragraphs again and again, I still do not understand
> them as anything more than a wish for a quantum theory of gravitation.
> <snip>
> In short - I interpret Weinberg's comments to be more or less a lament
> that the ideas that work in field theory, don't really work for gravity,
> rather than an actual claim about the nature of gravitation. His acutal
> presentation is thoroughly, indeed physically geometric, to the point
> that I actually recommend this book above all others for understanding
> the *physics* of gravitation.
>
> -drl
Not exactly. Weinberg is asking: suppose a quantum theory of
gravitation exists, then what features will it have so that GR is
reproduced in the classical limit? The answer is: any consistent
quantum theory with a massless spin 2 particle will reproduce GR
classically.
The reasoning is in two steps quoted above: "It seems to be impossible
to construct any Lorentz invariant quantum theory of particles of mass
zero and spin two, unless the corresponding classical theory obeys the
Principle of Equivalence." and " It will be seen that geometric
objects, such as the metric, the affine connection, and the curvature
tensor naturally find their way into a theory of gravitation based on
the Principle of Equivalence and, of course, one winds up in the end
with Einstein's general theory of relativity."
So all we need to find is a consistent QM theory with a spin 2
particle and the rest will follow. This language is much more
consistent with field theory. In this language, geometrical concepts
are secondary, and may not play an important role in the final quantum
theory of gravity. While we're doing classical GR, of course
geometrical concepts are all we need.
mtmtk
>
> What is the alternative? Is a non-geometric approach calculable? Any
> acceptible theory must explicitly allow for the Global Positioning
> Satellite system to 11 decimal places,
>
> http://www.eftaylor.com/pub/projecta.pdf
> GPS and Relativity
>
"we focus only on observable, operationally defined quantities, and
avoid unanswerable questions."
I agree with this and I intrepret the above link according to this
principle. As far as I can understand the article discusses
theoretical considerations using S. metric. There is not one single
equation in it from actual GPS programs.
Then I need to take the writer's authority that by implication the GPS
exactly obeys these theoretical considerations. But other than this
there is nothing as far as I can understand which shows convincingly
that GPS uses General relativistic equations. I am not saying it does
not I am saying that this paper does not show that actual GPS programs
use GR. I want to see a term, as C. Will says the "operationally
defined" general relativistic term which when is taken out it leaves
higher residuals. I don't see this. Again I am not saying that GR
fails or what not, just that this link is not convincing on an
operational level.
Danny Ross Lunsford
> So all we need to find is a consistent QM theory with a spin 2
> particle and the rest will follow. This language is much more
> consistent with field theory.
How are you going to get the non-locality of gravitational energy from
that? The important idea is the Christoffel connection, and that goes
beyond the analysis of spin.
In any case what you actually get from spin-2 is the tetrad formalism.
This is "equivalant" to GR for a particular interpretation of
"equivalent" - but it is not GR.
-drl
Ralph E. Frost
Does the "tetrad formalism" have a simple and short lay-person type of
explanation, or does it require a couple of semesters of grad school and
many chapters of many books of study?
That is, does something like...
http://www-th.phys.rug.nl/~schaar/htmlreport/node52.html translate into a
simple picture?
Also, is there a relevant relationship or connection between the "tetrad
formalism" and "tetrahedral inequalities" as in that discussed by Normal
LaFave in a paper in 1993 on the GR-QC section of the LANL database
(9310036) that showed that 3-d spin networks contained 4-d geometries?
--
Best regards,
Ralph Frost
http://flep.refrost.com
"The essential nature of external reality, Comenius thought,
could be conveyed by education to the simplest intelligence
if all knowledge could be reduced to a basic principle."
- notions ascribed to John Amos Comenius (1592-1670), circa 1640
[Dobbs, Betty Jo Teeter, THE FOUNDATIONS OF NEWTON'S ALCHEMY, Cambridge
University Press, Cambridge 1975 p. 60]
Uncle Al
GPS is an eloquent, extreme precision, multiple platform, and long
time interval vindication of classical gravitational field theory.
Different theory must do as least as well to provide GPS correcitons
vs. the first and non-default corrected satellite's results.
<http://rattler.cameron.edu/EMIS/journals/LRG/Articles/Volume6/2003-1ashby/index.html>
Relativity in the Global Positioning System
http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf
http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm
http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm
http://www.trimble.com/gps/index.html
http://sirius.chinalake.navy.mil/satpred/
http://www.phys.lsu.edu/mog/mog9/node9.html
http://egtphysics.net/GPS/RelGPS.htm
http://www.schriever.af.mil/gps/Current/current.oa1
http://edu-observatory.org/gps/gps_books.html
<http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Unit5/gps.html>
Arvind Rajaraman
> Arvind Rajaraman wrote:
>
> > So all we need to find is a consistent QM theory with a spin 2
> > particle and the rest will follow. This language is much more
> > consistent with field theory.
>
> How are you going to get the non-locality of gravitational energy from
> that? The important idea is the Christoffel connection, and that goes
> beyond the analysis of spin.
I've no idea what the problem is. A spin 2 particle can only couple in
one way, and all the Christoffel symbols etc, will just come out
automatically. Weinberg is clear about this: massless spin 2 implies
the Principle of Equivalence, which implies that the affine connection
etc. will appear correctly.
>
> In any case what you actually get from spin-2 is the tetrad formalism.
> This is "equivalant" to GR for a particular interpretation of
> "equivalent" - but it is not GR.
>
The tetad formalism is necessary to include fermions in GR. Without
fermions, the two formalisms are entirely equivalent. Why do you think
otherwise?
And once again: spin 2 implies the Principle of Equivalence. From this
principle, one can derive GR without ever talking about tetrads, just
as Weinberg does.
> -drl
Italo Vecchi
> As far as I can understand the article
> [GPS and Relativity http://www.eftaylor.com/pub/projecta.pdf ]
> discusses theoretical considerations using S. metric.
> There is not one single equation in it from actual GPS programs.
To my knowledge, in actual GPS operation the potential relativistic
effects (see [3]) are wiped off by continuous clock synchronisation.
Ashby writes in [1]:"At present one cannot easily perform tests of
relativity with the system because the SV clocks are actively steered
to be within 1 microsecond of Universal Coordinated Time (USNO).", but
he also provides some pre-synchronisation GPS results.
In [2] Ashby states that "If such [relativistic] effects are not
accounted for properly, unacceptably large errors in GPS navigation
and time transfer will result.", My naive interpretation of such
statements is that large GR effects could be detected by switching
clock synchronisation off. However in [2] I also read that
"experimental tests of relativity can be performed with GPS, although
generally speaking these are not at a level of precision any better
than previously existing tests." , which I find somewhat vague.
By the way, the NASA Stanford-developed Gravity Probe B ([5]) is going
to be launched soon (April 17, 2004). According to the Stanford
website [4] it will provide " new, very precise tests of Einstein's
general theory of relativity, our fundamental, but very incompletely
tested, theory of the large-scale structure of the Universe. Based on
observations of gyroscopes in a "drag-free" satellite flying in Earth
orbit, the mission will provide (a) by
far the most precise test of general relativity ever attempted, and
(b) the first measurement ever on one of Einstein's most fundamental
predictions, the phenomenon of frame-dragging.". One can also sign up
for weekly updates.
Cheers,
IV
[1] http://www.phys.lsu.edu/mog/mog9/node9.html
[2] http://rattler.cameron.edu/EMIS/journals/LRG/Articles/Volume6/2003-1ashby/index.html
[3] http://groups.google.com/groups?q=GPS+general+relativity&hl=en&lr=&ie=UTF-8&oe=UTF-8&selm=87rle4%
24nl3%241%40mach.thp.univie.ac.at&rnum=2
[4] http://aa.stanford.edu/aeroastro/aalabs.html#gpb
[5] http://einstein.stanford.edu
-------------------------------
"... an experiment is like a very good sword, in that if an agile man
uses it for the defence of prince and realm it achieves many glorious
deeds but if, however, it is used by someone driven by fury , nothing
is to be expected but slaughter and parricide."
Hononrè Fabri, 17th century Jesuit, quoted in Ch. 6 of M.J. Gorman's
thesis http://www.stanford.edu/~mgorman/publications.htm
Arnold Neumaier
> "Danny Ross Lunsford" <antima...@yahoo.NOSE-PAM.com> wrote in message
> news:dnzWb.14545$UE7....@newssvr22.news.prodigy.com...
>
Saying spin 2 is nothing else than saying that there is a
symmetric tensor field, the metric. There is no magic behind
it, and it has nothing to do with tetrads.
The restriction in Weinberg's arguments comes from requiring
that the metric has a representation of mass zero with
nontrivial interactions. These two requirements separately
are easily satisfied in many ways, but their combination
is a very strong restriction.
>>This is "equivalent" to GR for a particular interpretation of
>>"equivalent" - but it is not GR.
The tetrad formalism is completely equivalent to GR, in the
sense that you can translate every true statement in the
standard formalism into a true statement in the other and
conversely.
> Does the "tetrad formalism" have a simple and short lay-person type of
> explanation,
Yes. A tetrad is a set of four linearly independent
vector fields e_0, e_1, e_2, e_3.
Considering them orthonormal in the sense that
g(e_j,e_k)=eta_jk (*)
where eta is the Minkowski metric defines the
metric g uniquely; conversely, for any metric one can
choose (on any chart) such an orthonormal basis.
If the manifold is parallelizable then one can choose
the ONB even globally. In 4 dimensions, any manifold
which allows to define spinors consistently is
parallelizable (by a result of Geroch), hence reality
is most likely described by such a manifold.
Using (*), one can rewrite any formula involving the
metric into one involving instead tetrads, and many
things simplify - using tetrads is closer to the Cartan
formalism than using the metric directly. E.g.,
sqrt(-det g) = det(e).
One has to be slightly careful not to confuse curved
and flat indices, but this is learnt very quickly.
Then one needs much less index shifting.
For gravitation coupled to a (classical) Dirac field,
the tetrad formalism is indispensable, since spinors
cannot be defined without a flat representation.
Arnold Neumaier
Danny Ross Lunsford
>>In any case what you actually get from spin-2 is the tetrad formalism.
>>This is "equivalant" to GR for a particular interpretation of
>>"equivalent" - but it is not GR.
>
> Does the "tetrad formalism" have a simple and short lay-person type of
> explanation, or does it require a couple of semesters of grad school and
> many chapters of many books of study?
>
> That is, does something like...
> http://www-th.phys.rug.nl/~schaar/htmlreport/node52.html translate into a
> simple picture?
Well there's rather more to it than that. Weinberg gives an excellent
presentation in his book, but I would come to it first from the
traditional point of view, the idea of "Ricci rotation coefficients" -
see LP Eisenhart, "Riemannian Geometry". Then understand that the
prototypical spin equation of Dirac is based on the geometry of a
vierbein through the Clifford algebra. Finally follow the presentation
of Weinberg to see how spinors live on curved spacetime. The vielbein
idea is so useful and important that a lot of work to understand it is
justified. Note that the modern word is "frame" and one often hears of
Cartan's "repere mobile", "moving frame".
Strictly speaking the hypothetical graviton should not require a spinor
description, and so also not require description in terms of a frame,
but to link up spin-2 theory with Riemannian geometry, you have to
expand the metric in a Fourier series of elementary polarizations in the
manner of electrodynamics, identify the coefficients as creation and
annihilation operators, form a quantity that is algebraically like the
curvature tensor from these operators etc. The introduction of these
polarization tensors is a tacit introduction of a local frame (vierbein).
> Also, is there a relevant relationship or connection between the "tetrad
> formalism" and "tetrahedral inequalities" as in that discussed by Normal
> LaFave in a paper in 1993 on the GR-QC section of the LANL database
> (9310036) that showed that 3-d spin networks contained 4-d geometries?
An expert in spin nets will have to answer that. There are several who
post here.
-drl
Danny Ross Lunsford
> And once again: spin 2 implies the Principle of Equivalence. From this
> principle, one can derive GR without ever talking about tetrads, just
> as Weinberg does.
One writes down a Hamiltonian with polarization tensors that have as
coefficients creators and annihilators of gravitons. The polarization
tensors are an implicit introduction of a local frame. This is clear in
any case because one is assuming a background to work on.
So spin-2 = frame dynamics on a background, the "equivalent" formulation
of GR in terms of vierbeins. Is this wrong?
-drl
Doug Sweetser
I agree with the start of this sentence...
> Saying spin 2 is nothing else than saying that there is a
> symmetric tensor field, ...
but not the final clause,
> the metric.
Granted it is exceptionally common to work with metric fields, but it
is not a requirement for representing a spin 2 particle. As a counter
example, consider the symmetric tensor A^u;v + A^v;u, the companion to
F^uv of EM. If this appears in an action, one would vary the
4-potential, not the metric. What is kind of odd and tricky to talk
about is the covariant derivative in that tensor, which itself involves
three first-order derivatives of the metric. The symmetric tensor has
first-order derivatives of both the potential and the metric, so there
may be built in ambiguity.
doug
quaternions.com
Arkadiusz Jadczyk
On Sat, 14 Feb 2004 08:13:36 +0000 (UTC), Arnold Neumaier
<Arnold....@univie.ac.at> wrote:
>>>This is "equivalent" to GR for a particular interpretation of
>>>"equivalent" - but it is not GR.
>
>The tetrad formalism is completely equivalent to GR, in the
>sense that you can translate every true statement in the
>standard formalism into a true statement in the other and
>conversely.
This depends on how is "tetrad formalism" defined.
There is a variation of the tetrad formalism, where tetrads ;ive
in an external bundle (that is when you describe gravity as a gauge
theory of the Lorenz group). You need a soldering form in this case, and
the slodering form can happen to be degenerate (for instance on the
Eisnstein-Rosen "bridge"). You can have parallel transport (gauge field)
defined there, but the Levi-Civita connection can not be defined on the
bridge. The statement: there is a non-singular parallel transport across
the bridge is then true - but it is not true in the metric formalism.
Of course the reason is that the two parallel transport concern
different bundles. But, as far as I understand it, "tetrad formalism",
when defined precisely as a gauge theory of the Lorentz group
*needs* an external bundle and a soldering form.
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
Lubos Motl
On Sun, 15 Feb 2004, Danny Ross Lunsford wrote:
> One writes down a Hamiltonian with polarization tensors that have as
> coefficients creators and annihilators of gravitons. The polarization
> tensors are an implicit introduction of a local frame.
A specific creation operator for a graviton is creating a "weak"
gravitational perturbation - a very specific linearized excitation of a
background. In other words, each creation operator is associated with a
concrete symmetric-tensor-valued function of your spacetime (classical
background manifold) that represents an infinitesimal deviation of the
metric; for such an operator, we know how the metric will be perturbed at
each point. The same holds for the annihilation operators. We don't need
any tetrads to define the creation operators. In fact, the tetrad
formalism is strictly needed for the fermions only, as Arvind has
explained.
If you were addressing the fact that the gravitons are usually described
by a momentum and a polarization (e.g. xx-yy and xy-yx oscillating
"linearly polarized" modes or the +-2 helicity modes), yes, that is true,
but this conclusion is a fact of physics, and it is independent on whether
we use the tetrads or not to describe the physical system.
> This is clear in any case because one is assuming a background to work
> on.
In order to talk about the gravitons - which are defined to be quanta of
gravitational waves moving on a background - one needs to have a
background, of course. (The previous sentence is a tautology.) But this
question has nothing to do with the tetrads either.
> So spin-2 = frame dynamics on a background, the "equivalent" formulation
> of GR in terms of vierbeins. Is this wrong?
Spin 2 means that the internal angular momentum of the physical
excitations of a field (e.g. the gravitational field) are equal to +-2
times hbar. The fields whose quanta have this property can usually be
represented as a symmetric tensor - or more generally, a field that
transforms as a Young diagram with two columns, so that the maximal
projection of J_3=J_{12} in the multiplet is always +2.hbar. The
definition and the properties of the spin 2 fields have absolutely nothing
to do with vierbeins. In fact, we have said many times that vierbeins are
an inevitable tool only if we deal with spin 1/2 or spin 3/2 (or higher
half-integer spin) fields.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Only two things are infinite, the Universe and human stupidity,
and I'm not sure about the former. - Albert Einstein
Ilja Schmelzer
> What is the alternative? Is a non-geometric approach calculable? Any
> acceptible theory must explicitly allow for the Global Positioning
> Satellite system to 11 decimal places,
For example, condenses matter interpretations are possible.
See gr-qc/0205035
> What is the alternative? Non-geometric theories are not predictive.
> Noether's theorem and other strong correspondences between symmetries
> and properties cannot be denied - certainly not locally.
Depends on what you name "non-geometric". Of course there is also a lot
of Euclidean geometry in classical Newtonian theory. If Newton counts as
sufficiently non-geometric, then non-geometric theories can be
predictive.
Ilja
Ilja Schmelzer
> The tetad formalism is necessary to include fermions in GR
See hep-th/0310241 for an alternative which needs only
a preferred foliation (ADM decomposition).
Ilja
mtmtk
>
> GPS is an eloquent, extreme precision, multiple platform, and long
> time interval vindication of classical gravitational field theory.
> Different theory must do as least as well to provide GPS correcitons
> vs. the first and non-default corrected satellite's results.
>
> <http://rattler.cameron.edu/EMIS/journals/LRG/Articles/Volume6/2003-1ashby/index.html>
> Relativity in the Global Positioning System
>
Reviews in Relativity. Using the principle that only the operational
terms in the equation will be considered and the rest will be ignored
as non-physical prose, I didn't find what Ashby writes convincing. He
is trying to imply a connection between GPS and GR which is not there
(in this first section I read).
The equation is c^2(t0 - t)^2 = (r0 - r)^2
c = a unit conversion factor
r = position
t = time
I dont see anything General Relativistic in this equation. Ashby first
tries to establish a connection with c and the GR. He claims that the
existence of c in this equation makes it a GRelativistic equation
because "the principle of the constancy of c finds application as the
fundamental concept on which the GPS is based." Not correct. Here c is
simply a unit, and by definition it is constant. We could have chosen
any velocity as unit here and it would work as well, there is nothing
mystical about the number 299792458, we could have used 1mm/s, the
speed of a turtle, and the equation would still work.
Ashby mentions "spacetime" in his prose and implies that his equation
is General Relativistic because "timing signals that are transmitted
from each satellite can be thought of as sequences of events in
spacetime." Again this is his opinion. There is nothing in the
equation about a spacetime. In the equation I see only position and
time expressed separately as position and time. Ashby can mention
"spacetime" as many time as he wishes but his equation does not
contain a term for spacetime, then there is not a one-to-one
correspondence with his prose and his equation, and I take the
authority of the equation not Ashby's authority.
Doug Sweetser
Hello mtmtk:
In classical Newtonian physics, time is separate from space. In
special relativity, a lightlike interval between two events is governed
by this very equation:
> c^2(t0 - t)^2 = (r0 - r)^2
which is more typically written:
ds^2 = dR^2 - c^2 dt^2 = 0 [for lightlight events]
A problem with Newton's law of gravity is that it implies a change in
mass density must change the potential everywhere instantly. This
equation is about how light moves in spacetime, not turtles :-)
doug
quaternions.com
mtmtk
Doug Sweetser <swee...@alum.mit.edu> wrote:
> In classical Newtonian physics, time is separate from space. In
> special relativity, a lightlike interval between two events is governed
> by this very equation:
...
> A problem with Newton's law of gravity is that it implies a change in
> mass density must change the potential everywhere instantly. This
> equation is about how light moves in spacetime, not turtles :-)
Let me understand better what
c^2(t - t0)^2 = (r - r0)^2
says. For simplicity let's place the receiving station at the origin
and let the signal be emitted at time zero, so, r0=0 and t0=0 and the
equation becomes ct = r.
We defined r as position, c is the speed of light and t is some
conventional counting number.
Operationally, this equation says that, if the distance light travels
in unit time is taken as the unit distance, then, any distance r
equals the multiple of the unit time t.
It says nothing about the epistemology of space, time or spacetime. If
we constrain ourselves to operational physics, there is no turtle and
no spacetime in this equation :)
Arnold Neumaier
> Hello Arnold:
>
> I agree with the start of this sentence...
>
>
>>Saying spin 2 is nothing else than saying that there is a
>>symmetric tensor field, ...
>
>
> but not the final clause,
>
>
>>the metric.
>
>
> Granted it is exceptionally common to work with metric fields, but it
> is not a requirement for representing a spin 2 particle. As a counter
> example, consider the symmetric tensor A^u;v + A^v;u, the companion to
> F^uv of EM. If this appears in an action, one would vary the
> 4-potential, not the metric.
This tensor cannot give rise to a long range interaction, hence is
out of the game.
You may wish to read Weinberg's articles which I quoted in some
earlier thread on spin 2 for a deeper understanding of this.
Arnold Neumaier
tes...@tum.bot
relativity is relevant to GPS, but didn't understand what you have read so
far. (By the way, you certainly need to read at least the next few
sections in the review by Ashby!) You should begin by recognizing that
GPS is neither designed nor operated as a test of gtr (or any physical
theory); rather it is designed and operated as a timekeeping system, which
is a far different engineering problem!
Nonetheless it is a fact that before launch the atomic clocks carried by
GPS satellites are set to run -slow- by about 38.5 ms/day wrt to a
standard atomic clock. This correction factor arises from applying gtr to
a very simple but highly idealized model, using "weak-field gtr",
isolating the most relevant relativistic effects. The analysis (see the
derivation below, from a post which appeared in s.p.r. long ago) shows
that an ideal clock carried aboard a satellite in circular orbit with a
period of 12 hours (not 24 hours!) around a nonrotating massive spherical
object E with mass equal to that of the Earth will run -fast- by 38.5
ms/day relative to a similar clock sitting on the surface of E. Thus, the
basic pre-launch correction cancels the major relevant relativistic
effects.
In the actual GPS system, various non-systematic effects such as solar
buffeting are more important than the relativistic effects just mentioned.
Because these effects are unpredictable years in advance, it was
neccessary to design the GPS system so that the operators can occasionally
reset individual on-board clocks by small amounts. The designers wished
to minimize the need for such "active steering" by accounting for all
-systematic effects- in advance. This is the reason for the seemingly
redundant pre-launch correction described in the preceeding paragraph.
You might object to the fact that the basic prelaunch correction arises
from a model in which the Earth is assumed to be -nonrotating-. If we
don't make this assumption, further relativistic effects do arise, but it
turns out these are neglible compared to the minimum error we need to
worry about in accounting adequately for systematic relativistic effects
in the actual GPS system.
A more interesting objection is that in the analysis of the real GPS, as
described in various books, in effect we -ignore gtr- (but take account of
say the Earth's rotation) and simply add in the prelaunch correction at
the end as a sort of unexplained fudge factor. The justification for this
procedure is ultimately that our analysis showed that -weak field gtr-
suffices for our purposes, and the field equation of weak field gtr
happens to be -linear-. This kind of subterfuge, if done properly, is
perfectly valid (and very common in "pertubation analysis"), but in this
case it unfortunately has encouraged some misguided souls to claim that
"gtr is actually irrelevant to GPS" or even that "gtr is falsified by
GPS"!
========= Derivation of the basic pre-launch correction ================
Because the GPS satellites are in nearly perfect circular orbits, it is
unusually easy to compute the frequency shift between a clock on the
surface of the Earth and a clock carried by a GPS satellite. For
convenience I will assume the Earth is nonrotating, that the surface clock
sits on the equator, and that the GPS satellite is in a circular
equatorial orbit with a period of 12 hours.
We need to compute three things:
* the red shift of a static clock sitting on the equator of the Earth
relative to a distant static clock,
* the red shift of a clock carried by a GPS satellite in circular
equatorial orbit (period 12 hours) relative to a distant static clock,
* the difference of these, which gives the blue shift (as it turns out)
of the GPS clock relative to the surface clock.
So here we go:
(1) We compute the gravitational time dilation (red shift) of a static
clock on the surface of the Earth, relative to a distant static observer.
Recall that this is given in the Schwarzschild vacuum by sqrt(1-2m/r);
this is an exact value in our model, not an approximation. We find:
sqrt(1-2m/r) ~ 1 - m/r
~ 1 - 6.96 x 10^(-10)
IOW, a static clock on the equator runs SLOW by about 696 parts per
billion relative to a distant static clock. Or if you prefer, runs SLOW by
about 60.1 microseconds per day relative to a distant static clock. (Not
45.6 microseconds per day.)
Here, I have used for the mass of the Earth and the equatorial radius of
the Earth these values:
m ~ 0.4438 cm
r ~ 6.378 10^8 cm
(2) The simplest way to compute the red shift or time dilation of a clock
carried around on a circular orbit with a distant static clock is to use
the (modelocked or IZAMO--- it doesn't matter which) Hagihara ONB.
(2a) Referring to
http://math.ucr.edu/home/baez/PUB/line
recall that this turns out to be given by the simple and memorable
expression
sqrt(1-3m/R)
where R is the radius of the orbit. This is an exact value in our model,
not an approximation.
(2b) Recall too that the orbital period of a GPS satellite is 12 hours
(more precisely, 12 sidereal hours, but since we are modeling the Earth as
nonrotating, we don't need to worry about this distinction.)
(2c) Recall as well that the orbital frequency as measured by a distant
observer is dv/dt = sqrt(m/R^3). Thus, we can solve to find the R which
gives a period of 12 hours (we don't need to worry about the Earth's
rotation as we ignore it for computing the basic GPS correction). The
required orbital radius turns out to be
R ~ 2.662 x 10^9 cm
Here, to convert from seconds to cm in computing the orbital frequency
(before solving for R), I am using
c ~ 2.998 x 10^10 cm./sec
Exercise: verify the orbital radius given above is correct!
(2d) Thus, we find
sqrt(1-3m/r) ~ (3m)/(2r)
~ 1 - 2.50 x 10^(-10)
IOW, a clock aboard a GPS satellite in equatorial orbit (orbital period 12
hours) runs SLOW by about 250 parts per billion relative to a distant
static clock. Or if you like, runs SLOW by about 21.61 microseconds/day
wrt a distant static clock.
(3) Combining these results, we find that a clock aboard a GPS satellite
in equatorial orbit runs FAST by about 446 parts per billion relative to a
static clock sitting on the equator of the Earth. Or if you like, runs
FAST by about 38.51 microseconds/day wrt a static clock sitting on the
equator of the Earth.
==========================================================================
"T. Essel" (hiding somewhere in cyberspace)
Doug Sweetser
> Let me understand better what
>
> c^2(t - t0)^2 = (r - r0)^2
I'll confess why I don't like the way this is written. Time t and
space r together form a 4-vector. The contraction of this 4-vector
with itself forms a Lorentz invariant scalar. Here are a few ways to
write the same thing:
(r - r0)^2 - c^2(t - t0)^2 = ds^2
dr^2 - c^2(dt)^2 = ds^2
-dR_u dR^u = ds^2 where R^u = (c^2 dt, dR)
-g_uv dR^u dR^v = ds^2 where g_uv is the metric tensor
For a lightlike interval, ds^2 = 0. Only in flat spacetime will
c dt = dR. In curved spacetime, c^2 g_00 dt^2 = g_ii dR^i dR^i. So
for me this contraction looks all about spacetime.
doug
mtmtk
> "mtmtk": if I understand you correctly, you want to understand how
> relativity is relevant to GPS, but didn't understand what you have read so
> far.
No. Imagine someone who is in a dialogue with nature. Nature does not
understand conventional human labels such as GPS and GR, because they
cannot be used in an equation. They are useful only in a dialogue
among professional colleagues.
Ashby writes an equation which simplifies to ct=r, c is the speed of
light, t is a unit counting number, and r is defined as position.
Then, Ashby claims that this equation is about spacetime.
From the point of view of the operational physics Ashby's equation is
not about spacetime because it contains no terms about spacetime.
Doug Sweetser
>> Granted it is exceptionally common to work with metric fields, but it
>> is not a requirement for representing a spin 2 particle. As a
>> counter example, consider the symmetric tensor A^u;v + A^v;u, the
>> companion to
>> F^uv of EM. If this appears in an action, one would vary the
>> 4-potential, not the metric.
>
> This tensor cannot give rise to a long range interaction, hence is
> out of the game.
Why not? If the trace is zero, it should work, no? [I think the trace
may be required to be zero to be long range]. I will bike over to the
library someday.
doug
Italo Vecchi
> Nonetheless it is a fact that before launch the atomic clocks carried by
> GPS satellites are set to run -slow- by about 38.5 ms/day wrt to a
> standard atomic clock.
Very interesting. Where do you have this from? A reference would be
helpful.
Ashby writes that "SV clocks are actively steered to be within 1
microsecond of Universal Coordinated Time (USNO)." . That is not the
same. Is it?
Regards,
IV
Arnold Neumaier
Doug Sweetser wrote:
> Arnold Neumaier replied [>>to me]:
>
>
>>>Granted it is exceptionally common to work with metric fields, but it
>>>is not a requirement for representing a spin 2 particle. As a
>>>counter example, consider the symmetric tensor A^u;v + A^v;u, the
>>>companion to
>>>F^uv of EM. If this appears in an action, one would vary the
>>>4-potential, not the metric.
>>
>>This tensor cannot give rise to a long range interaction, hence is
>>out of the game.
>
>
> Why not? If the trace is zero, it should work, no?
I don't think so; please look at Weiberg's work on 'any spin' to get
familiar with what can one expect. It pays to be at least somewhat
familiar with the things already tried before working on a hard unsolved
problem.
Would you have bet money on the success of a polar expedition
a hundred years ago, if the one undertaking it ignored all that could
be learnt from his predecessors in the attempt to reach the south pole?
Arnold Neumaier
tes...@tum.bot
> > Nonetheless it is a fact that before launch the atomic clocks carried by
> > GPS satellites are set to run -slow- by about 38.5 ms/day wrt to a
> > standard atomic clock.
>
> Very interesting. Where do you have this from? A reference would be
> helpful.
See the middle of page 18 of the review I thought we were discussing:
author = {Ashby, Neil},
title = {Relativity in the Global Positioning System},
note = {http://www.emis.math.ca/EMIS/journals/LRG/Articles/subject.html}}
> Ashby writes that "SV clocks are actively steered to be within 1
> microsecond of Universal Coordinated Time (USNO)."
(plus or minus some whole number of leap seconds)
> That is not the same. Is it?
Certainly not!
To repeat: before launch, the atomic clocks carried by a GPS satellite are
set to run -slow- by a certain factor, the "factory offset". This is a
rate of roughly 446 parts per billion, which works out to about 38.5
microseconds per day, and it compensates for a correction factor which you
can derive from a simple weak-field gtr model. This correction factor
accounts for the most important -systematic- relativistic effects relevant
to GPS (other than the Sagnac effect, which is needed to understand how
one can try to synchronize atomic clocks sitting everywhere on the geoid,
in order to establish TAI). OTH, there are also -unpredictable- or
-nonsystematic- effects, such as solar wind buffeting of individual
satellites, which also significantly affect the performance of the system.
This is why GPS is designed to allow for occasional adjustment, controlled
from the control center in Colorado, of -individual- orbiting clocks.
Looking back, I see you previously said:
> To my knowledge, in actual GPS operation the potential relativistic
> effects (see [3]) are wiped off by continuous clock synchronisation.
Not really. Since Ashby distinguishes between the problem of achieving
synchronization (section 2) and the problem of computing the "factory
offset" (section 5), I'll refer you back to his review.
Italo Vecchi
tes...@tum.bot wrote in message news:<c20cdc$tj9$1...@lfa222122.richmond.edu>...
> On Sun, 29 Feb 2004, Italo Vecchi wrote:
>
> > > Nonetheless it is a fact that before launch the atomic clocks carried by
> > > GPS satellites are set to run -slow- by about 38.5 ms/day wrt to a
> > > standard atomic clock.
> >
> > Very interesting. Where do you have this from? A reference would be
> > helpful.
>
> See the middle of page 18 of the review I thought we were discussing:
>
> author = {Ashby, Neil},
> title = {Relativity in the Global Positioning System},
> note = {http://www.emis.math.ca/EMIS/journals/LRG/Articles/subject.html}}
>
Thanks for the pointer.
> > Ashby writes that "SV clocks are actively steered to be within 1
> > microsecond of Universal Coordinated Time (USNO)."
>
> (plus or minus some whole number of leap seconds)
>
> > That is not the same. Is it?
>
> Certainly not!
>
> To repeat: before launch, the atomic clocks carried by a GPS satellite are
> set to run -slow- by a certain factor, the "factory offset". This is a
> rate of roughly 446 parts per billion, which works out to about 38.5
> microseconds per day, and it compensates for a correction factor which you
> can derive from a simple weak-field gtr model. This correction factor
> accounts for the most important -systematic- relativistic effects relevant
> to GPS (other than the Sagnac effect, which is needed to understand how
> one can try to synchronize atomic clocks sitting everywhere on the geoid,
> in order to establish TAI). OTH, there are also -unpredictable- or
> -nonsystematic- effects, such as solar wind buffeting of individual
> satellites, which also significantly affect the performance of the system.
> This is why GPS is designed to allow for occasional adjustment, controlled
> from the control center in Colorado, of -individual- orbiting clocks.
>
I was wondering how "occasional" this really is. Tom Van Flandern
works on the raw satellite data and concludes that "data shows that
the on-board atomic clock rates do indeed agree with ground clock
rates to the predicted extent" providing strong validation of GR
([1],[2]). This sounds like hard evidence of GR effects in GPS,
complementing the NTS-2 data provided by Ashby.
By the way, in [1] Van Flandern writes that "atomic clocks change
frequencies by small, semi-random amounts (of order 1 ns/day) at
unpredictable times for reasons that are not fully understood.". I
wonder whether this might be related to Unruh&Wald's result that that
any physical clock has a small probability of running backwards.
> Looking back, I see you previously said:
>
> > To my knowledge, in actual GPS operation the potential relativistic
> > effects (see [3]) are wiped off by continuous clock synchronisation.
>
> Not really. Since Ashby distinguishes between the problem of achieving
> synchronization (section 2) and the problem of computing the "factory
> offset" (section 5), I'll refer you back to his review.
>
> "T. Essel" (hiding somewhere in cyberspace)
Thanks again for your feedback.
IV
[1] http://www.metaresearch.org/cosmology/gps-relativity.asp
[2] http://www.metaresearch.org/solar%20system/gps/absolute-gps-1meter.ASP
[3] Unruh & Wald "Time and the interpretation of quantum gravity"
Phys. Rev. D40, 2598--2614 (1989)
---------------------
"Je comprends vite, mais il faut m'expliquer longtemps"
tes...@um.bot
> > GPS is designed to allow for occasional adjustment, controlled from
> > the control center in Colorado, of -individual- orbiting clocks.
>
> I was wondering how "occasional" this really is.
As you no doubt are aware, GPS is operated by the U.S. military, which
tends to be reluctant to reveal details of its various space-based
operations. So people curious about such things probably have to wait for
proposed civilian navigation systems to come on line.
Nice try, but -nooooo- comment :-/
tes...@um.bot
>> "mtmtk": if I understand you correctly, you want to understand how
>> relativity is relevant to GPS, but didn't understand what you have read
>> so far.
mtmtk replied:
> No. Imagine someone who is in a dialogue with nature. Nature does not
> understand conventional human labels such as GPS and GR, because they
> cannot be used in an equation. They are useful only in a dialogue among
> professional colleagues.
Wow!--- I don't have -any- idea what any of this might mean, but perhaps
it doesn't matter.
Hmm... it seems to me that your "No" above could be understood to mean
that you are -not- interested in understanding what Ashby is trying to
explain in his review
author = {Ashby, Neil},
title = {Relativity in the Global Positioning System},
note = {http://www.emis.math.ca/EMIS/journals/LRG/Articles/subject.html}}
Since you have several times quoted (out of context) from this article,
perhaps you will understand why I would find this surprising, but if this
is really the case, I'll stop trying to explain!