You've seen the Einstein field equations, but you don't understand them. There's no shame in that, because you can't understand the field equations until you understand the mathematics underlying their notation, and that mathematics is fairly advanced. In particular, you can't understand the Einstein field equations until you understand the mathematical notion of a manifold's intrinsic curvature.
Come off it, Clinger. You were bible thumping the other day, now you're trying to hide behind mathematics. Don’t try it on me, I'm too sharp for that.
No one can accuse you of hiding behind mathematics.
When did mathematics, backed by citations to standard references, turn into "bible thumping"?
Trivial. What you've got isn't actually a triangle, it's a portion of a sphere, a spherical triangle.
Okay. Let's be even more general, and call it a closed path.
That discrepancy can be converted into a mathematically well-defined measure of curvature via careful definitions and taking limits of certain ratios as the size of the path goes to zero. It turns out that the surface of a sphere has constant curvature.
Yep. But I can draw lines through the body of the spherical ball, on the equatorial plane. Straight lines. Three of them. Forming a triangle where the angles add up to 180 degrees.
The sphere is a 2-dimensional surface. On that surface, you cannot connect three geodesics to form a closed path with angles summing to 180 degrees.
From your example, I think you're assuming it's okay to leave the sphere itself and form your triangle within the ball you believe to be enclosed by the sphere. If you're hoping to use that as an analogy for spacetime, you'll have to explain why you're postulating a Euclidean space outside of the spacetime manifold.
And Ricci curvature describes the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. So in an expanding universe a ball of space is expanding and changing volume, but the space itself isn't curved. My equatorial triangle angles always add up to 180 degrees.
That isn't true of all expanding universes, and it certainly isn't true of all coordinate systems.
You have a bad habit of arguing as though there's some privileged coordinate system in which your beloved Euclidean space has been carved out of non-Euclidean spacetime, and that your argument automatically generalizes to all other coordinate systems.
Simplifying the Ricci tensor to a scalar via contraction yields the scalar curvature. Although the full Riemann tensor does not appear in Einstein's field equations, the Ricci tensor and scalar curvature do appear.
You need to read
http://arxiv.org/abs/physics/0204044.
That unpublished arXiv paper supports what I wrote: Curvature is measured by the Riemann curvature tensor and its contracted forms (the Ricci tensor and the scalar curvature).
That arXiv paper also agrees with me about the modern view of gravity as "a manifestation of the
curvature of space-time."
Peter M Brown, the author of that paper, appears to be arguing that Einstein thought about it the other way around, viewing curvature of space-time as a manifestation of gravity. That's a minor historical point, which we can discuss if and when you acquire a better understanding of general relativity. For now, the more important point is that nothing in that paper supports your technical errors:
- You were flat-out wrong to deny that curvature is defined by the Riemann curvature tensor and its contractions (the Ricci tensor and scalar curvature).
- You were flat-out wrong to deny the curvature of spacetime in most FLRW solutions, including all "dusty" solutions.
- You were flat-out wrong to deny the existence of gravity in FLRW solutions.
If the scalar curvature is nonzero, then the spacetime manifold is curved by the very definition of curvature. If the Ricci tensor is nonzero, then the spacetime manifold is curved by the very definition of curvature.
Your definition of curvature seems to be elevating a manifold into something real rather than mathematical artefact. I think you’re confusing a spacetime manifold with physical space.
No. You, however, are confusing mathematics with subjective opinion.
The following statements are mathematical:
- A non-zero Ricci tensor implies curvature.
- A non-zero scalar curvature implies a non-zero Ricci tensor, hence curvature.
- Most FLRW solutions, including all "dusty" solutions, imply curvature of the spacetime manifold.
When you deny those statements, you are flat-out wrong.
The following statement is your subjective opinion:
- Curvature of the spacetime manifold does not imply gravity.
You and Mr Peter M Brown are free to believe that, but Mr Brown at least acknowledges he's out of step with modern presentations of general relativity:
In [modern general relativity] gravity is defined implicitly through what Chandrasekhar called the
zeroth law of gravitation which states
The condition for the absence of any gravitational field is the vanishing of the (curvature tensor).
Do Peter M Brown's quotations from standard references make him a bible thumper?
There is no interpretation needed: The Ricci tensor and scalar curvature are mathematically well-defined. You just need to learn those definitions. Only then will you understand what the individual terms actually mean, and what curved spacetime really is.
I know what it really is: inhomogeneous space.
You are elevating your beliefs over the mathematical definition of curvature that appears quite explicitly in Einstein's field equations. By rejecting the mathematics employed by those equations, you are rejecting those equations.
You're like the fellow at the bowling alley who says the bowling lanes can't be curved because you don't see them bending to the left or right. If I suggest they be lengthened to make the curvature more apparent, he says they won't bend to the left or right no matter how long you make them. If you make them long enough, however, the pins will be set up only 3 meters behind him.
Let’s make our bowling alley follow one of the lines in my equatorial triangle. As space expands the alley gets longer. It isn't curved.
Einstein's static universe, in which space is spherical, is one of the FLRW solutions. Extending the lane indefinitely brings it back to the bowler's feet.
Yet you say that space is Euclidean:
Spacetime, however, is not Euclidean.
But space is.
And it’s a mathematical “space”. It isn’t the same as the space in the universe.
True: the spacetime manifold is a mathematical/scientific model of (some aspects of) the universe.
And the Reimann tensor measures the extent to which the metric tensor is not locally isometric to a Euclidean space. But the angles of my equatorial triangle add up to 180 degrees, and always do. The “curvature” isn’t in the space, and instead can only be said to be there because space expands and the properties of flat space change over time.
Once again, you are assuming a privileged coordinate system. If I weren't so afraid of being called a bible-thumper, I'd quote Einstein's statement of what he called "the fundamental idea of the general principle of relativity".
Cosmologically, the curvature/gravity of an FLRW "dusty" solution manifests itself as a reduced rate of expansion. If the matter density is less than or equal to the critical value, the universe will expand forever, but it will expand slower than it would have without the curvature/gravity due to matter. If the matter density is greater than the critical value, the universe will eventually collapse.
This is wrong I’m afraid. Start with a high-energy-density universe. It expands, reducing the energy density if you adhere to conservation of energy. What happens next? It expands, reducing the energy density. What happens next? It expands, reducing the density. At all times the universe is homogeneous, meaning
there’s no gravity in it. The universe didn’t collapse when it was small and dense. It's going to sail right on past the critical value without a care in the world.
Spinning a yarn is no substitute for doing mathematics.
No one believes that a single FLRW solution describes all of spacetime. The "dusty" FLRW solutions are intended only to model spacetime after radiation and other pressure becomes negligible. If the cosmological constant Λ is zero, then the rate of expansion decreases with time. If the matter density is larger than the critical density, then the rate of expansion eventually goes negative, and the universe begins its collapse into a Big Crunch. If the matter density is less than or equal to the critical density, then the rate of expansion decreases forever but never reaches zero.
In the paragraph above, I am discussing the FLRW models of the universe. The things I said in that paragraph are theorems of mathematics. The real universe may behave differently because the universe is more complex than the FLRW models and may be wildly different from the FLRW models in ways we have not even begun to suspect.
Farsight, however, is just denying the mathematics. That's silly.
