topological equivalence?

Imagine the following:

Start with a curved GR world. Draw all the world-lines of all the
particles, past and future. Mark up those lines with their \tau lifetime
(time axis) tick marks. With the markup, the metric of GR is no longer
needed. We can manipulate the space, topologically, as we please and have
the same world.

Now take that curved space and flatten it out. It is nearly flat to begin
with. For example, gravity is about -10^-9 at the Earth's surface. Compact
the space where it is curving so that it becomes flat. Black holes would
require a deep reach to pull the space back into the small volume that we
imagine it to be in.

The new space would then be described with a different kind of space-time
metric. There would be a space-time scaling field that maps the original
space-time to the manipulated space-time (or vice-versa).

The possibility of a simple scaling field was analyzed in the 1987 paper now
posted as gravity.htm. In that paper a solution was found. It did not
start with GR and its equivalence principle, but rather with the rest of the
accepted principles (including conservation of energy). It then found the
necessary metric. It then analyzed the bending of light around a star, the
precession of Mercury and event horizon properties.

IMHO, the two theories could be equivalent. If they both satisfy the same
basic principles, and their results are the only ones in their respective
forms, then the two developments represent a proof of their equivalence.

Fundamental accepted principles that are used and maintained:

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.

>From these, find a scaling field that makes them consistent.

The paper looks for a units conversion between different elevations
seperated by a delta potential where the conversion makes the principles
consistent. This is similar to the approach for SR.

1) XYZ dimensions are shown to be isotropic.
2) T and XYZ scalings are the inverse of each other.
3) Starting with a standard gravity potential scalar field V:
1) T' = T*(1-deltaV)
2) X' = X/(1-deltaV)

4) For completeness, he includes force as a possible additional fundamental
unit that might need scaling and finds F' = F.

These imply a global scaling field

ScaleT = 1 - (-V)
ScaleXYZ = 1 / (1 - (-V))

Where V is the unitless version of the standard Newton gravity potential.
and -V is positive.

This is an adjustment of the Newton acceleration.

Qualitatively, the fundamental result is that everything slows and shrinks
as it descends toward a gravitating mass. Not by much in the solar system. V
is about -10^-9 at the surface of the Earth, about -10^-6 at the Sun.

Now, you could take this description of space-time. Draw all the world
lines. Add tick marks for the local proper time (tau) along all the lines.
Then bend and stretch the space so that the tic marks are evenly spaced and
the world lines follow null geodesics.

And we are possibly back to the GR description.

Ned Phipps




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