Plot gravitational potential as a curve, then zoom in on an "infinitesimal region", and it's still curved.
There isn't even really any such thing as gravitational potential in GR. It's a relic from Newtonian gravity that's only sometimes used in weak field approximations.
That's a uniform gravitational field, which cannot exist.
No, you've already been corrected about this before: in Riemannian geometry, it's perfectly possible to have homogenous spaces with non-zero curvature. The sphere and saddle space for example. Can you point to a uniquely distinguishable point on the sphere?
3+1 dimensions reflect the real world of space where light has freedom of motion in any combination of three orthogonal directions, whereafter we derive the time dimension from that motion.
No, that's just your own pet theory, and would be 3 dimensional space + motion, not 3+1 dimensional spacetime. It's called that because it's there right from the beginning in GR, before you even start talking about the motion of
anything.
One of cause and effect. People tend to think light curves because it moves through curved spacetime.
This isn't even a correct restatement of the mainstream understanding of GR. In GR, light travels along geodesic paths, which are the closest things to
straight lines in a Riemannian manifold.
It doesn't, it moves through inhomogeneous space where the coordinate speed of light varies, and veers as a result.
I've also corrected you about this before. It's only correct to say you'll end up with a curved
description of the path of light if the
metric components are inhomogenous, meaning that your
coordinate system (
not space) is inhomogenous. Because the metric components are coordinate dependent, you are completely unjustified in talking about them being inhomogenous in any absolute sense.
There's only one coordinate
independent relation I know of relating to the paths of light (or geodesics in general) in GR. It's the
Jacobi equation:
$$
\frac{\mathrm{D}^2}{\mathrm{d}\lambda^2} J \,+\, R(J,\, \dot{\gamma}) \dot{\gamma} \,=\, 0\,.
$$
Intuitively you can think of it as a differential equation describing the convergence or divergence of light rays (among other possible things) travelling very near one another. Unlike your reference to metric components, this equation is coordinate independent, and it relates the behaviour of nearby light rays directly to the Riemann curvature tensor
R.
He didn't. See
his post and note that he said
We've had this discussion before. We spoke about it previously in
Lattices and Lorentz variance. See
this post where prometheus said
I'm rather confused by your remarks about waves moving though space. He also said
Ok, I grant you that "real" space is not flat but it is locally flat, meaning that I can define an observer that experiences no gravitational forces.. That's wrong. Gravity is where spacetime is not flat.
The only thing you could criticise prometheus for here is where he says "[space] is locally flat", and even that's only if you're really being a jargon nazi. It's especially silly because prometheus
explains what he means by that, saying "meaning that I can define an observer that experiences no gravitational forces" which is
true: a freefall observer in GR locally doesn't experience any gravitational pseudo-forces. The curvature only shows up over sufficiently large distances as
tidal forces.
When prometheus said "[space] is locally flat" he was referring to the fact that Riemannian manifolds locally resemble (flat) Euclidean space (or in GR specifically, flat Minkowski spacetime). This is practically a defining property of Riemannian manifolds.
It's a crucial distinction. Spacetime is the all-times block view. There's no motion in this arena.
Spacetime is no more an "all-times" view than it is an "all-space" view. Spacetime doesn't forbid you from being interested in what's happening at a particular time any more than space forbids you from being interested in what's happening at a particular place.
Any equations of motion you derive are equations of motion through space.
Again, this is a linguistic issue, not a fundamental point about motion and spacetime.
It's the real speed of light
No, that's always
c = 299,792,458 m/s. That's what an observer will always measure the speed of light to be locally. By definition since 1983, but even before that, that was the only real speed of light in GR.
or rapidity if you prefer.
No, rapidity is
something different.
The locally-measured constant speed of light isn't, because you use the motion of light to define your seconds and metres
Nope. Only the metres, and only since 1983, and we only use that definition because we think the speed of light would be different anyway. The whole point of invariance of
c in relativity is that it is
independent of the details of how we define our units.
Even the new definition doesn't guarantee by itself that if you do an actual experiment, you'd measure an invariant
c. The reason is that if you even used the SI definition at all, you'd use it
once to calibrate or measure the size of your equipment. So when you measure the speed of light
immediately afterwards it's no surprise that you get back exactly the same speed that you just put in. But that doesn't explain why you still measure the same speed of light with the same apparatus 6 months later, when the Earth is moving the opposite way on the opposite side of the sun.
which you then use to measure the speed of light. Thus you always measure it to be the same.
You're ignoring the experiments that found an invariant
c before 1983.
The space is inhomogeneous, the vacuum impedance Z[sub]0[/sub] = √(μ[sub]0[/sub]/ε[sub]0[/sub]) varies and c = √(1/ε[sub]0[/sub]μ[sub]0[/sub]) varies too. That's what Einstein kept saying, but it's nowadays dismissed.
It doesn't matter what Einstein said. As I pointed out, to the extent those quantities vary, they're coordinate dependent. This is not out of hand dismissal. If Einstein really did place much fundamental importance on those quantities, we have good grounds for disagreeing with him.
