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Re: Gravitational source terms (was: degeneracy pressure)



David Bowman wrote:
[Part A:]
The reason why the needed supporting pressure diverges at a finite
density in general relativity is that
[Part B:]
in GR the sources of
gravitation (i.e. Ricci curvature of spacetime) include not only the
mass (and the energy) of the matter involved, but also the momentum,
the stress and the pressure of that matter.

Part B is true in isolation. Kinetic energy, stress, and
pressure are among the source terms for gravitation.

But that should not be invoked as the "reason" for part A.
To do so is somewhere between highly misleading and wrong.

To illustrate the point, consider the following puzzle:

_______
/ \
/ > \
| |
| |
\ < /
\___A___/

x
_______
/ B \
/ > \
| |
| |
\ < /
\_______/


We have two massive circular hoops, both rotating clockwise.
We have a test particle (x). At point A, the part of the
upper hoop nearest x is moving right-to-left. At point B,
the part of the lower hoop nearest x is moving left-to-right.
It's obvious by symmetry that there is no net gravitational
force on x.

Observer Alice is comoving with point A. She knows the rest
mass of the hoops; everybody agrees on that. She reports
that nearby parts of the upper hoop are not moving. The
nearby parts of the lower hoop (point B) are moving at high
speed. Therefore they have kinetic energy, which is one of
the "source terms" for the gravitational field. Also
she reports that the density of the lower hoop near point
B has an increased density, due to the Lorentz contraction.
Therefore, she is tempted to predict that the test particle
will accelerate downward, toward point B.

Meanwhile, observer Bob is comoving with point B. He goes
through the same steps. He is tempted to predict that the
test particle will accelerate upward, toward point A.

Obviously both temptations are wrong. Why?
I'll leave this as a puzzle for now.

This is not an elementary puzzle. Newtonian physics doesn't
suffice. Special relativity doesn't suffice. You have to do
the GR, and you have to do it right.

Usual jsd puzzle rules apply: Everything I've said here is
true and helpful, to the best of my knowledge. But obviously
I haven't told you everything; for instance I haven't told
you the answer. This isn't a word game; solving the puzzle
requires understanding the physics, not quibbling about words.