That is certainly not what Einstein was thinking.
$$T_{\mu \nu}$$ is merely the source term in the field equations, and describes the various forms of energy which form sources of gravity, and that includes things like density, flux, stress and momentum. This applies to any energy configuration, i.e. solids, fluids, fields etc; the total SEM tensor is simply the sum of all the various contributions. If $$T_{\mu \nu}=0$$ we are dealing with a vacuum ( exterior field equations ), which, by the way, does not imply that there is no curvature.
Space-time itself, in classical GR, possesses only one degree of freedom, and that is curvature. Space-time cannot be put "under pressure" or "under stress" or anything of that nature...
I'm afraid it is, Markus. You're confusing space and spacetime. See my post 120 where I explained the distinction. Let's run through it again.
Spacetime is an abstract mathematical space in which motion does not occur because it models space at all times. You can draw world-lines in it, and you can draw them curved, but that worldline represents the motion of a body through space over time. The body doesn't actually move through spacetime. People tend to talk of "the spacetime around the Earth" and suggest that light moves through it, but that's wrong. Pay careful attention to Einstein's 1920
Leyden address where he said this:
“According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty”.
Space is inhomogeneous, not curved. And because it's inhomogeneous, motion through this space over time is curved, so we say spacetime is curved. Do refer to the
The Foundation of the General Theory of Relativity and note that Einstein refers to the equations of motion. Also note that
metric is to do with measurement. The
metrical qualities of the continuum of space-time differ because
space is neither homogeneous nor isotropic. You can understand this by placing gedanken light-clocks in an equatorial plane through the Earth. The light-clocks run at different rates, and when you plot your measurements your plot is curved just like the wikipedia
plot of gravitational potential, which is in turn like the
bowling-ball-in-the-rubber-sheet pictures. But the light-clocks didn't run at different rates
because your plot was curved, they ran at different rates because the space they're in is inhomogeneous. Google on
inhomogeneous vacuum for more information. One of the papers you come across is
Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime. Relate this back to Einstein's reference to inhomogeneous space.
...saying that Einstein thought of space as something that could be put "under pressure" or "stretched" is simply wrong, unless you can show that these are expressible in terms of curvature only. The geometry of space-time under GR is expressed as curvature only.
That geometry is the geometry of motion through space. It is not the geometry of space. The
Einstein field equations do include pressure terms and shear-stress terms. And note Einstein's 1929
presentation on field theory:
"It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structure of the physical continuum is not completely described by the Riemannian metric."
The Riemannian metric describes
the state of space. And it features pressure and shear stress. These and other terms result in curved motion through that space, to which we apply the label curved spacetime. It's like a car encountering mud at the side of a road. The road isn't curved, nor is the mud. But the car veers left. It's path is curved.
Markus Hanke said:
Just please don't try to sell statements as fact if really you aren't sure what you are talking about; you know very little about differential geometry, so better stay away from the maths of GR.
With respect Markus, I understand this, and you are confusing space and spacetime. It's important that you understand the difference.
