On Sep 3, 2011, at 5:27 PM, Paul Zielinski wrote:
Waldyr has been on board with a tensor stress-energy density all along.
Jack Sarfatti replied: Yes, of course. That's partly where I am coming from with the teleparallel-torsion idea I sketched.
It's quite obvious if you localize the global translation group T4 subgroup of special relativity's Poincare group P10 to T4(x), then the compensating spin 1 gauge vector fields that keep all the matter field actions invariant under the larger T4(x) are the Cartan tetrad 1-forms e^I that describe the global LIF geodesic rays of the intrinsic tensor gravity field that you cannot EEP away. Basically, the e^I describe the relative tilts of neighboring light cones in Penrose's sense, i.e. make the linear transform of the e^I to the Rindler-Penrose null tetrads. (see Wikipedia for quick overviews of key terms). The tetrads also have a spinor substratum basically Wheeler's IT FROM BIT with spinors as the BIT and tetrads as the IT.
The torsion 2-form is relative to the spin-connection 1-form covariant exterior derivative D
T^I = De^I = de^I + (spin connection)^IK/\e^K
This is analogous to Yang-Mills theory for SU2(x) and SU3(x).
Maxwell's U1(x) EM uses EM field 2-form
F = dA
A is the EM 4-potential of the Bohm-Aharonov effect etc.
F uses E & B 3-vectors
*F uses D and H 3-vectors
D = (permittivity)E
B = (permeabilty)H
energy densities are ~ E.D and B.H (e.g. Arnold Sommerfeld's Lectures).
No magnetic monopoles and Faraday's induction are
dF = 0
Ampere's law and Gauss's law are
D*F = *J
*J is the electrical current density Cartan 3-form.
D*J = D^2*F = 0
is local conservation of source electrical currents.
The EM field Lagrangian density Cartan 4-form is ~ F/\*F from which you get a local stress-energy tensor of the EM field.
Similarly for the "Yang-Mills" torsion gravity field.
DT^I = 0
D*TI = *J^I
D*J^I = 0
gravity torsion field Lagrangian density has standard "Yang-Mills" form ~ Trace{T^I/\*T^J} giving a local stress-energy tensor for the gravity torsion field.
Now in the tele-parallel theory this gravity torsion field is simply another way of looking at Einstein's curvature theory.
Not only that Hagen Kleinert shows that the curvature and torsion pictures (in the tele-parallel sense) transform into each other as a kind of gauge transformation. It's like Schrodinger wave vs Heisenberg Matrix forms of QM.
The gravity stress-energy tensor is locally zero from the EEP in the curvature picture since it must be zero in an LIF. Hence gravity energy is nonlocal etc when viewed this way. However, when viewed in the torsion picture, there seems to be a well-defined torsion field stress-energy tensor.
That's the idea.
Z: Note that he refers to "the nonsense pseudo tensor of GR". Doesn't that say it all?
On 9/3/2011 10:33 AM, JACK SARFATTI wrote:
On Sep 3, 2011, at 7:24 AM, Waldyr A. Rodrigues Jr. wrote:
Dear Jack,
In my theory of the gravitational field (that has also an interpretation as the teleparallel version of GR the gravitational field) has a legitimate energy-momentum tensor (and not the nonsense pseudo tensor of GR). The explicit and nice tensor for the energy-momentum 1-form fields ta of the gravitational field have been given in my Weimar talk (recall the power point presentation that I sent to you some time ago).
That formula (that according to my best knowledge was not known until last July) is : ta = (∂?∂)ga + 1/2R ga, where the ga are the gravitational potentials (cotetrad fields) and ∂?∂ is the covariant D’Alembertian. The components of the energy-momentum tensor of the gravitational field are tab = ta? gb .
Best regards,
Waldyr
P.S.: I am just ending a paper explaining all that. It will be posted at the arXiv (I hope) this weekend.
On Sep 2, 2011, at 1:53 PM, JACK SARFATTI wrote:
Even in the Newtonian limit of GR where v/c << 1 for test particles
guv ~ nuv(flat) + huv
huv << nuv
and radii of curvature >> measurement scales
the Levi-Civita connection ~ huv,w will still vanish in LIFs via EEP
so if you try to make a Tuv(gravity) quadratic in huv,w it will vanish in an LIF, therefore the gravity energy is still nonlocal even in this Newtonian limit
h00 ~ 2VNewton/c^2 << 1
{LC}^rtt ~ -VNewton,r
i.e.
g = -GM/r^2 etc.
On the other hand the LIF gravity field tetrad Cartan 1-forms e^I are GCT scalar invariants. Their torsion 2-forms
T^I = De^I = de^I + (spin connection)^IKe^K
have local tensor components
Their Yang-Mills Lagrangian density is ~ TraceT^I*T^J yields a local stress-energy gravity field tensor.
Using teleparallelism you can get the Einstein-Hilbert Lagrangian from the torsion tensor.
So this approach seems to solve the problem.
On Sep 2, 2011, at 12:15 PM, Woodward, James wrote:
The difference between Newtonian gravity and GRT is that in Newtonian theory you can localize the gravitational potential energy of objects. In GRT you can't. They aren't the same at all. One is framed in absolute space and time, the other is framed in absolute spacetime. Apples and oranges.
From: Paul Zielinski [
Sent: Friday, September 02, 2011 12:42 PM
To: JACK SARFATTI
Subject: Re: let's be more precise in the use of informal language
On 9/1/2011 10:38 PM, JACK SARFATTI wrote:
On Sep 1, 2011, at 9:50 PM, Paul Zielinski wrote:
Doesn't precisely the same equivalence principle also hold in a generally covariant spacetime formulation of Newtonian theory?
Einstein's GR limits to Newtonian theory in the two limits v/c << 1 and weak curvature i.e.
guv ~ nuv(flat) + huv
huv << nuv
and radii of curvature >> measurement scales
for a spherically symmetric static source
g00(static LNIF) = 1 + 2V/c^2
V = Newton's gravity potential per unit test mass
e.g
V = - c^2rs/r for finite source
r (observer) > rs
rs for Earth is ~ 1 cm
V = - /\r^2 de Sitter space
observer located at r = 0
/\ = Einstein's cosmological constant ~ dark energy density accelerating universe ~ (area-entropy of our future event horizon)^-1
I think the correct answer is "yes".
How can one "locally" distinguish the effects of the dynamical acceleration of a test object observed in its own kinematical rest
frame from the non-tidal effects of a Newtonian gravitational force field acting in a non-accelerating kinematical frame?
Word salad. I have no idea what your babble means without seeing some equations.
It's not a "word salad". In fact it's a very clearly stated question.
The correct answer is that in Newtonian theory you cannot locally distinguish the effects of physically accelerating a test object, as
observed from the object's non-inertial rest frame, from the non-tidal effects of a Newtonian gravitational force observed in an inertial
frame of reference.
The point here is that precisely the same heuristic argument about "local equivalence" holds in Newtonian theory.
Given any object, if an accelerometer clamped to it moves off zero then that object is really accelerating.
http://en.wikipedia.org/wiki/Accelerometer
The whole point of Einstein's "equivalence" argument was that you cannot say that that this is "really" the case, since you could also understand the situation as
being the effect of a gravitational field on the same test object, but observed in a non-accelerating rest frame.
That was Einstein's actual argument for his version of the EP.
"There is nothing to prevent us...."
Acceleration is what accelerometers measure. Anything else is Laputan idiocy in my opinion.
You are now calling Einstein's actual arguments for his version of the equivalence principle "Laputan idiocy"?
As you know I'm also very critical, but I think you should at least acknowledge that they were heuristically fruitful as a matter of historical fact.
They led Einstein to a very successful covariant theory of gravitation.
Accelerometers remain in their null position on timelike geodesics. The local curvature is irrelevant.
Exactly.
Curvature is NOT measured with accelerometers!
Not "locally". But the differential variation of the strength of a non-uniform field from point to point can register curvature.
See MTW to learn how to measure the 4th rank tensor curvature with sets of pairs of test particles each on timelike geodesics with their accelerometers never moving off their null positions.
End of story.
You have completely missed the point of my question Jack. This is about Einstein equivalence. You don't even seem to know what that was.
If so then I guess you basically agree with me about its status in modern GR. But as Synge once remarked, it should be given a decent burial
with all appropriate honors. You seem to want to pretend that it never lived.
If one cannot, does that mean that covariantly formulated Newtonian theory is already an example of "general relativity"?
Newtonian theory is a limiting case of general relativity.
But there is no direct correspondence between Newtonian theory and GR, since the Newtonian field strength has different
transformation properties from the Einstein field strengths (given by the partial derivatives g_uv, w).
The problem is that the Einstein "metric potential" g_uv is a tensor with non-covariant partial derivatives, whereas the Newtonian
potential is a scalar whose partial derivatives are automatically general covariant.
So I think you are sidestepping the question.
General relativity = covariance of local field equations under all kinds of locally coincident frame-detector transformations + equivalence principle connecting coincident LIFs with LNIFs.
But the point of my question above is that you can also give Newtonian theory a covariant spacetime formulation. So general covariance does not
fundamentally distinguish GR from Newtonian theory, does it? Kretschmann argument.
If Kretschmann was right and general covariance is merely a matter of mathematical formulation, and the Einstein EP also holds in Newtonian theory,
then what precisely is so different about GR? What makes GR qualify as "general relativity", while Newtonian theory doesn't qualify?
If you only can manipulate the formalism of GR you still don't understand its physical meaning. That's just playing with symbols.
I completely agree.
But if the Einstein EP holds in Newtonian theory, and Newtonian theory can be given a covariant spacetime formulation, what fundamentally
distinguishes 1916 GR from Newtonian theory such that it qualifies as "general relativity"? That was my question.
OK here's the answer:
The only fundamental difference in this respect between the two theories is that in GR, the inertial trajectories (geodesics in curved spacetime)
depend on the matter distribution, and in Newtonian theory they do not depend on the matter distribution, but are fixed independently.
Thus the gravitational force of Newton's theory is replaced by the objective gravitational deformation of the spacetime geodesics in Einstein's theory.
But since the spacetime of GR geodesics are covariant objects, determined in GR by the covariant geodesic equation, such deformation of
the geodesics has nothing to do with the choice of kinematical frame. Only the coordinate representation of a geodesic depends on the
choice of kinematical reference frame.
So the difference between Newtonian theory and GR has nothing to do with general covariance, and nothing to do with "general relativity". It
has to do only with the matter-dependence of the covariantly determined spacetime geodesics in GR, and the fixity of the spacetime geodesics
in covariantly formulated Newtonian theory. That is the only significant difference in this context.
That's the point.
Z.
How is the situation any different in GR?
On 9/1/2011 6:38 PM, JACK SARFATTI wrote:
Richard Tolman in his 1934 book Relativity, Thermodynamics and Cosmology puts to rest many of the Red Herrings about Einstein's General Relativity.
GR has two organizing principles.
Covariance of the local differential equations of the matter fields under all relevant groups of symmetry transformations. These are the local Euler-Lagrange equations from the invariance of the global action of the matter fields.
Covariance is necessary, but not not sufficient.
In the case of gravity there is also the local equivalence principle that may be expressed in several different ways both formally and informally.
1) the tetrad map between locally coincident LIF and LNIFs
2) vanishing of the Levi-Civita connection at the orgin center of mass of the LIF
3) an accelerating non-inertial frame (LNIF) is locally indistinguishable from Newton's gravity force per unit test mass on the center of mass of the test particle.
the presence or absence of local 4th rank curvature is completely irrelevant here. Bringing it up is a Red Herring from garbled confused thinking - defective logic & inability to related the formal symbols to the operational procedures of experimental physicists and observational astronomers.
Acceleration is absolute - whoever feels "weight" is accelerating in the real sense even if "standing still" in curved spacetime.
As in Hawking's picture
