On Feb 20, 1:15 am, Koobee Wublee wrote:
On Feb 16, 12:45 pm, Tom Roberts wrote:

Edward Green wrote:
Suppose we had two masses in relative motion, so that at least one
mass must be in motion in any coordinate system (at least in
approximately locally Lorentz coordinates which encompass both
masses).
At least one of the masses therefore has kinetic energy. Must we
include this kinetic energy in the stress energy tensor, or can we
simply use the rest mass of each as the energy, so long as we treat
the masses discretely?

The energy-momentum tensor automatically includes the motion of an
object. One must include both its kinetic energy and its momentum, in
addition to its mass.

Kinetic energy is observer dependent, and so is the momentum. This is
also true for the observed mass. Thus, the energy-momentum tensor
must be observer dependent as well. Since it is the energy-momentum
tensor that equates with the Einstein tensor, the field equations
allow an observer dependent quantity to shape the curvature of
spacetime that is supposed to be observer independent or invariant.
Please give me a date when you personally will be in peace with this
particular self-inconsistency.

Even I can answer that like an establishment pro!

The _components_ of the tensor may be observer dependent, but the
abstract tensor itself is independent of any coordinate system.

That's just how tensors are. Are you saying the whole machinery was
put together wrong?

...

Since both pressure and kinetic energy stem from the same root: that
the particles are in relative motion, are we over-counting if we
include both?

Not if you do it correctly.

It is less contradictory if you allow an invariant quantity to decide
how spacetime should be curved. shrug

I'm still unsatisfied with this one. It doesn't seem to me to be
"contradictory" as much as "insuifficiently specified".

A relativistic ideal gas has, even in its rest frame, a pressure and a
an energy density, the latter including an increment to the rest
energy. Both of these terms happen to involve the motion of the
molecules, but they are observationally distinct, and don't require us
to know anything about any damn molecules. In treating such a gas in
field equations which include energy and pressure, if we can't simply
plug both the observable values in, but invoke extra accounting
principles -- that's cheating.

Either we can blindly plug in both terms, in or there is something
imperfect about the formulation.