ajiko wrote:
"Tom Roberts" wrote in message
...
ajiko wrote:
Start with a curved GR world. [...]
Now take that curved space and flatten it out. [...]
There are two major problems with this:
1. It is only possible for manifolds with topologies consistent
with a flat metric.
2. The GR manifold is spaceTIME, and you are discussing only the
"space part" -- that's a problem as it can depend on how
you foliate spacetime into space and time.

The point of the post was really more that the vector metric is possible.
The question is: What fundamental accepted principle is being tossed out by
GR?

I think you mean "what principle of GR is missing from this theory?" One
answer is: the field equation. Another is: the ability to derive the
theory from a Lagrangian involving geometrical and physical fields (at
least this is not at all obvious).

But see more on this below.


Is GR necessary for any observations that this metric doesn't cover?

I dunno. The way to answer that is, of course, for you to compute values
for the tests of GR and see how far from GR's values your theory gets.
Unless you get values within experimental error for the known
experiments, your theory is valueless.

The difficult one is probably the binary pulsar data -- at a guess I
don't think your theory has anything like gravitational radiation, and
will probably predict the binary pulsar should be stable, rather than
decaying as observed.... Be sure to check the precession of Mercury's
perihelion. And the bending of light by the sun. My guess is you'll get
them all wrong because you are scaling all spatial directions the same
(e.g. in GR the Schwarzschild manifold only "scales" the radial
direction), but you really need to compute them.


Another way is for you to compute the parameterized post-Newtonian (PPN)
parameters for your theory and compare to the experimentally-measured
values (which "just happen" to agree with GR's values (:-)). The PPN
formalism is a parameterized test theory of GR (google "PPN formalism"
and branch out from there)....


The metric in this post is a straight derivation from the listed principles!
These principles:
0) Principle of relativity is valid.
1) Speed of light is measured with the same value in upper and lower
reference frames (locally Lorentzian).
2) Captured energy gravitates (hot object is slightly heavier, compressed
spring is slightly heavier).
3) Energy is conserved.
4) Gravitation potential of V=-GM/R (his analysis is actually independent of
the details of the potential function)
5) Energy of light is proportional to its frequency.
6) A perpetual motion machine of the first kind is impossible.
Imply this metric:
ScaleT = 1 - (-V)
ScaleXYZ = 1 / (1 - (-V))
Where V is the unitless version of the standard Newton gravity potential.
and -V is positive.

That's quite complicated, and counterintuitive....

Compare to the list for GR:
A) spacetime is locally Minkowskian.
B) the geometry and all matter fields obey the principle of least
action, using a scalar Lagrangian which includes all matter
fields, plus a geometrical term involving the Ricci scalar.

Postulate A is unassailable experimentally (cranks around here
notwithstanding (:-)), and B is so elegant that it seems Nature simply
"has" to use it (:-)). Of course the actual justification is agreement
with experiments which is incredibly good (in spite of the fact there
remain unanswered questions...).


I don't see how I was considering only space.

I took you at your word, and when you said "space" I assumed that is
what you meant.


The flattening concept refers
to all dimension. I had been thinking that the +--- local metric might
interfere with normal concepts of bending. But, definitely, I was imagining
the full 4D. Perhaps you would know: Could a ++++ metric always be
straighten out?

OK. My remarks about the inability to "flatten out" a manifold still
apply. Certainly there are ++++ manifolds for which this is impossible
(e.g. any with topology S^4, or S^3xR, or...).


I do still hold that, if GR does satify the listed principles, then the rest
is actually the proof of the initial hand-waving transform. Hence, GR does
not satify them. I believe it is the conservation of energy that is not
maintained.

GR has LOCAL conservation of energy (i.e. div T = 0, where "div" is the
covariant divergence, and T is the energy-momentum tensor). But in
general there is no global conservation of energy, due basically to the
difficulties of integration in curved manifolds.

GR also does not have your item 4. In particular, a scalar gravitational
potential is not sufficently general to describe GR; it requires a
rank-2 tensor.

I also think that your "principles" above are rather vague, and it will
not be easy to put them on a rigorous enough basis to actually derive a
theory from this handwaving....


Tom Roberts