My problem with General Relativity (GR) is that I learned all my
calculus and subsequent physics based on the idea that Euclid was
right. While it is clear that the completed theory of GR violates
Euclidean geometry, it is unsettling to begin with Reimannian calculus
and the assumption that metric alone governs particle motion. I would
prefer to "see" why Euclid must be abandoned!

Over the years, I developed a "derivation" of GR that permits one to
hold on to Euclidean ideas until we are unequivacably forced to
abandom them. The first step is to assume that the universe is
'flat' and well described by the Minkowskian variables of special
relativity. Let us call these variables (x*, y*, z*, t*). Also
assume that all particles and waves by linear wave theories (such as
Maxwell's equations or relativistic wave equations).

My problem with General Relativity (GR) is that I learned all my
calculus and subsequent physics based on the idea that Euclid was
right.

While it is clear that the completed theory of GR violates
Euclidean geometry, it is unsettling to begin with Reimannian calculus
and the assumption that metric alone governs particle motion. I would
prefer to "see" why Euclid must be abandoned!

Over the years, I developed a "derivation" of GR that permits one to
hold on to Euclidean ideas until we are unequivacably forced to
abandom them.

The first step is to assume that the universe is
'flat' and well described by the Minkowskian variables of special
relativity. Let us call these variables (x*, y*, z*, t*). Also
assume that all particles and waves by linear wave theories (such as
Maxwell's equations or relativistic wave equations).

Key to my "derivation" is the idea that wavepackets of linear waves
obey Hamilton's classical equations of motion, provided one consider
the "eikonal" limit of small wavelength and high frequency. This is
well-known as Ehrenfest's theorem connectiong Schroedinger's equation
to Newtonian motion, but the link between waves and Hamiltonian
equations of motion is valid for virtually all linear waves.

A consequence of this "approximate" isomorphism between Hamiltonian
systems and linear wave equations is that "frequency" is another word
for "Hamiltonian" and "wavenumber" is another word for "canonical
momentum". I STRONGLY feel that the proper notions to use in GR are
"frequency" and "wavenumber" because these involve measurements of
space and time. To me, "wavelength" is physical, while "momentum" is
an abstract mathematical concept.

The first hint that something is pecular about the geometry of GR
comes from the gravitational redshift of light. How can a static
gravitational field induce a shift in frequency?

Homogeneity of space + Noether's theorem gives conservation of linear
momentum. Isotropy of space + Noether's theorem gives conservation of
angular momentum. One fails to see the "abstraction."

General Relativity has been torn apart and rebuilt over time by at
least ten thousand physics graduate students worldwide as part of
their instruction. GR has been torn apart and rebuilt by a tremendous
number of hostile faculty looking for publications. GR is exact to
the limits of observation in all venues at all scales from weak field
GPS to strong field binary pulsars,

On Feb 14, 1:31 am, Uncle Al <Uncle...@hate.spam.net> wrote:
Homogeneity of space + Noether's theorem gives conservation of linear
momentum. Isotropy of space + Noether's theorem gives conservation of
angular momentum. One fails to see the "abstraction."

Noether's theorem assumes that particles follow classical Hamiltonian
(Lagrangian) equations of motion. Herein lies the "abstraction". If
we must question Euclid, why not question the Hamiltonian formalism?
When one "constructs" a new theory, certain concepts are retained
while others are discarded. To me the justification for retaining
classical particle dynamics is that particles are wavepackets of
linear waves, and therefore must obey Hamiltonian equations of motion
(in approximation). (Weinberg proved this for systems of PDEs that
generate wavepackets -- I don't know if anybody proved it for
something as complex as the many-particle system of electrons in a
periodic crystal lattice studied in solid state physics)

Another problem I had with classical momentum versus wavenumber is
that I could not understand see how momentum transforms under
coordinate and/or Lorentzian transformations. As part of an invariant
phase (k_dot_r minus omega*t) I know exactly how to transform the
frequency-wavenumber into any coordinate system.

General Relativity has been torn apart and rebuilt over time by at
least ten thousand physics graduate students worldwide as part of
their instruction. GR has been torn apart and rebuilt by a tremendous
number of hostile faculty looking for publications. GR is exact to
the limits of observation in all venues at all scales from weak field
GPS to strong field binary pulsars,

Good point. I was one of those faculty "looking for publications",
but I would hardly call myself "hostile" to GR. Somebody knowledgible
once told me that GR creates inconsistencies when one attempts to
unify it with other fields of physics (i.e. other forces, quantum
field theories). I lack the knowledge to independently verify this
assertion, but if it is true, then GR is "hostile" to the rest of
physics! And, if it is true, GR needs to be examined from every
possible viewpoint.

One more comment about other attempts to reconstruct GR: The
"massive" text by MTR (Misner, Thorn, Wheeler) - "Gravitation" -
lists about a dozen "paths" to GR. I had great deal of difficulty
following any of them. I also studied two 'derivations' of linearized
GR: One by Ohanian in his textbook on the subject, and one by Feynman
in an obscure printing. In "Gravitation", MTR contend that it is a
bad idea to begin one's study of GR with the linearized version
because it fails to recognize the correct geometry. I agree with
MTR's criticism of all linearized approaches I have seen-- EXCEPT
MINE!

I must concede one point, however. Having "successfully" constructed
a linearized version of GR that correctly gets at the geometry, I
recognize that my approach is not the best pedagogy for a novice.
Every good textbook that I have seen approaches GR by laying the
foundation of Riemannian calculus.

MTR claim that this is essentially
Einstein's original approach

I can't say that I've examined all the "paths" to GR from MTW, but I did
look at one in detail; Schild's argument, Section 7.3, see my recent
post about it:

news:1165525205.367394.113120@n67g2000cwd.googlegroups.com
http://groups.google.ca/group/sci.physics.research/msg/ed7ee90582107b61

Rather than a derivation of GR, this argument starts with a known
empirical fact (gravitational redshift), assumes that space-time has
flat Minkowski geometry, and derives a contradiction. On the other hand,
if we allow non-trivial space-time curvature, as in GR, this assumption
turns out to be consistent with observation. Basically, rather than
deriving GR, the argument shows that special relativity (i.e. flat
space-time) is wrong, while GR is not *necessarily* wrong. I suspect
that other "paths" featured in MTW are of similar character.

But special relativity is more than "flat space time", it is the
assumption that light always travels at c. If we make that assumption
and use it to define distances, then of course the known effects of
gravity will necessarily imply curvature!

We could make an even simpler argument: "Clocks (as viewed by an
observer at rest with respect to the clock) slow down in a
gravitational field, therefore gravity causes curvature, because
according to special relativity it is impossible for such a clock to
slow down."

On the other hand, if we abandon the proposition that light always
travels at c, it seems not at all unlikely that the observable parts of
the universe might be described in terms of flat spacetime and a force
called gravity that affects all particles (including photons).

A black
hole jumping astronaut could determine whether or not GR is correct,
but he would not be able to return to report his findings.

On Feb 14, 1:31 am, Uncle Al <Uncle...@hate.spam.net> wrote:
Homogeneity of space + Noether's theorem gives conservation of linear
momentum. Isotropy of space + Noether's theorem gives conservation of
angular momentum. One fails to see the "abstraction."

Noether's theorem assumes that particles follow classical Hamiltonian
(Lagrangian) equations of motion. Herein lies the "abstraction". If
we must question Euclid, why not question the Hamiltonian formalism?
When one "constructs" a new theory, certain concepts are retained
while others are discarded. To me the justification for retaining
classical particle dynamics is that particles are wavepackets of
linear waves, and therefore must obey Hamiltonian equations of motion
(in approximation). (Weinberg proved this for systems of PDEs that
generate wavepackets -- I don't know if anybody proved it for
something as complex as the many-particle system of electrons in a
periodic crystal lattice studied in solid state physics)

Another problem I had with classical momentum versus wavenumber is
that I could not understand see how momentum transforms under
coordinate and/or Lorentzian transformations. As part of an invariant
phase (k_dot_r minus omega*t) I know exactly how to transform the
frequency-wavenumber into any coordinate system.

General Relativity has been torn apart and rebuilt over time by at
least ten thousand physics graduate students worldwide as part of
their instruction. GR has been torn apart and rebuilt by a tremendous
number of hostile faculty looking for publications. GR is exact to
the limits of observation in all venues at all scales from weak field
GPS to strong field binary pulsars,

Good point. I was one of those faculty "looking for publications",
but I would hardly call myself "hostile" to GR.

Somebody knowledgible
once told me that GR creates inconsistencies when one attempts to
unify it with other fields of physics (i.e. other forces, quantum
field theories). I lack the knowledge to independently verify this
assertion, but if it is true, then GR is "hostile" to the rest of
physics! And, if it is true, GR needs to be examined from every
possible viewpoint.

One more comment about other attempts to reconstruct GR: The
"massive" text by MTR (Misner, Thorn, Wheeler) - "Gravitation" -
lists about a dozen "paths" to GR. I had great deal of difficulty
following any of them. I also studied two 'derivations' of linearized
GR: One by Ohanian in his textbook on the subject, and one by Feynman
in an obscure printing. In "Gravitation", MTR contend that it is a
bad idea to begin one's study of GR with the linearized version
because it fails to recognize the correct geometry. I agree with
MTR's criticism of all linearized approaches I have seen-- EXCEPT
MINE!

I must concede one point, however. Having "successfully" constructed
a linearized version of GR that correctly gets at the geometry, I
recognize that my approach is not the best pedagogy for a novice.
Every good textbook that I have seen approaches GR by laying the
foundation of Riemannian calculus. MTR claim that this is essentially
Einstein's original approach -- and it is simpler, I agree.

I can't say that I've examined all the "paths" to GR from MTW, but I did
look at one in detail; Schild's argument, Section 7.3, see my recent
post about it:

news:1165525205.367394.113120@n67g2000cwd.googlegroups.com
http://groups.google.ca/group/sci.physics.research/msg/ed7ee90582107b61

On Feb 15, 8:36 am, Igor Khavkine <igor...@gmail.com> wrote:

I can't say that I've examined all the "paths" to GR from MTW, but I did
look at one in detail; Schild's argument, Section 7.3, see my recent
post about it:

news:1165525205.367394.113120@n67g2000cwd.googlegroups.com
http://groups.google.ca/group/sci.physics.research/msg/ed7ee90582107b61

I now suspect that Schild's argument, as portrayed by MTW, is flawed.
Here's why:

It is this apparent ease, by which one can construct an incorrect
version of GR set in Euclidean space, that causes me to doubt Schild's
argument (as explained in MTW).

In article <MPG.2042a0ae2517174f98b4a8@news1.eircom.net>, Gerry Quinn wrote:

But special relativity is more than "flat space time", it is the
assumption that light always travels at c. If we make that assumption
and use it to define distances, then of course the known effects of
gravity will necessarily imply curvature!

Actually, it is. Let me quote from Wald's book on GR (Section 4.2):

Thus, the theory of special relativity asserts that *space-time is the
manifold R^4 with a flat metric of Lorentz signature defined on it*.
Conversely, the entire content of special relativity as we have
presented it thus far is contained in this statement, since, given R^4
with a flat Lorentz metric, we can use the geodesics of this metric to
construct global inertial coordinates, etc.

The constancy of the speed of light is an easy consequence (given that
Maxwell's equations hold) of the above concise formulation.

On the other hand, if we abandon the proposition that light always
travels at c, it seems not at all unlikely that the observable parts of
the universe might be described in terms of flat spacetime and a force
called gravity that affects all particles (including photons).

In that case, while you are keeping SR, you are abandoing Maxwell's
equations. You may not have realized this when you made this hypothesis,
probably because your view of the "constancy of the speed of light"
postulate is incorrect or overly spimlistic. See above.

A black
hole jumping astronaut could determine whether or not GR is correct,
but he would not be able to return to report his findings.

That's neither here nor there. And way too vague to make any sense.

I don't understand how you can make any conclusion about Schild's
argument from what you just said. Please explain.

GR's singular attribute is the long reach of its mathematical
simplicity (as these things go). Einstein-Cartan theory is a right
proper pisser to calculate and it must always give the exact same
answers... except it properly handles spin-orbit coupling and predicts
a substantial net signal from a parity calorimetry experiment.
Somebody should look.

GR contains no mistakes - no mistakes in its mathematics, no
mistakes in its predictions compared to observations. If you
wish to falsify GR you can only attack it at its founding
postulates that, by definition, cannot be proven or defended.
That has also been done: Einstein-Cartan and Weitzenb=F6ck; affine,
teleparallel, non-commutative... gravitation theories. These
theories neither postulate the Equivalence Principle (EP) nor
the isotropy of space. GR is then a special case of a more
inclusive treatment.

The singular disjoint non-overlap between metric and other
classes of gravitation theory is angular momentum:
spin-orbit coupling, physical spin (gyroballs

and Gravity Probe-B),

polarized quantum spins and
orbits (magnets, Phys. Rev. Lett. 97 021603 (2006)),

mass distribution geometric parity divergence (J. Math. Phys.
40(9) 4587 (1999)).

No EP or GR violations have been observed in all but the last.