This video is ridiculous. There is no content to it, and it is repeating hackneyed things that are obvious to anyone. Further, the idea could be presented in one sentence of text, saving people a lot of time:
"Can dark matter be spacetime curvature with no matter, and can dark energy by a straightening out of the rubber sheet in a rubber sheet conception of GR?"
The first is silly, since any curvature would necessarily behave as matter. The second is doubly-silly, because the rubber sheet is a terrible analogy for GR, in that it is the wrong components that are curved (space and not time), and the rubber sheet geodesics are repulsive, not attractive.
The rubber sheet is actually a good model of Newtonian gravitational interaction between long parallel rods (or point particles in 2d), to the extent that the rubber sheet is flat (not curved) but has height variations which can be used to drive masses toward each other in the Earth's gravitational field. The curvature of the sheet is second order (in the sense of calculus, it vanishes as the height squared), while the height variations are first order, so it is not a contradiction to imagine a flat sheet with height variations.
The rest of this answer is devoted to a discussion of the finer points.
Curvature without matter
The theory of General Relativity is not just some made up stuff that you can modify willy nilly. You need to be consistent with the basic general principles of physics.
Suppose you have a space region which is curved, and you put it in a big constant slowly varying gravitational field, by bringing a big black hole close, say, what happens? The region of curvature must accelerate toward the black hole, by the equivalence principle. It must fall into the black hole by the black hole horizon property, and it must increase the mass of the black hole, by the horizon area theorem, which is the law of entropy increase.
So you have an object which responds to gravity just like any other matter, and it is matter by definition, if you like, whether you see something there or not. The total mass-energy is determined from the curvature.
The inverse problem
There is a cute point of view very close to this idea which is the following
- Einstein equations relate $T_{\mu\nu}$ to $G_{\mu\nu}$. Solving for the metric is hard. But what if you just take any old metric and solve for $T$ (this is trivial), can't you then find infinitely many trivial solutions to GR?
The issue with this idea is that if you specify the curvature arbitrarily, the matter you get will be grossly unphysical, in that it will have negative energy, it will have matter flow faster than the speed of light, and it will have speed of sound greater than the speed of light in many cases. The restriction on the inverse problem give rise to the energy conditions, which, in addition to the field equation, form the physical content of GR. Here are two of them:
- Null energy condition/Weak energy condition: The (borderline) energy component of T along any null-null direction is nonnegative.
- Strong energy condition: The energy component of T along any timelike or null direction exceeds the sum of the pressure components along the diagonal (in a local orthonormal frame).
The weak energy condition can be colloquially restated as follows
- Gravity always focuses light
And heuristically, perhaps precisely, as the condition
The strong energy condition can be colloquially restated as follows:
- The pressure in matter as a function of density, when integrated from zero density, never has the speed of sound exceed the speed of light.
This condition has an implicit assumption that the pressure be found by classical thermodynamics, by going from a vacuum by adding density at a given temperature. It can be violated if you just have coherent particles making a scalar field expectation value in a vacuum, without making a superluminal speed of sound, just because the perturbations away from the vacuum still obey the strong energy condition, although the vacuum itself does not.
These two conditions are notable in that they allow you to describe two types of results. The weak energy condition gives theorems which are universal to GR in any setting, like closed-trapped surface singularity theorems, and area theorem, while the strong energy condition is used for more special situations where there are no scalar fields giving a bulk cosmological constant, like the big-bang singularity theorem (which fails with scalar field driven inflation).
If the warping introduced by hand violates the weak energy condition, it is difficult to see how it could not be used to signal faster than light, or to violate positive energy and make a perpetual motion machine. If it violates the strong energy condition, and it is not a homogenous scalar field, it is difficult to see how little bumps can't be used to propagate sound faster than light.
So it is believed that only homogenous classical fields violate the strong energy condition, and that nothing classical violates the weak energy condition.
Inflation
The theory of inflation postulates that there is a homogeonous scalar field which had a large expectation value near the big bang, and a large energy density. This gives rise to accelerated expansion, which makes the universe equilibrate to a small-horizon distance sphere called a deSitter space.
The deSitter phase lasts a short time, and seeds the modern era, where we are expanding normally. But we still see some residual deSitter like acceleration, and this is almost certainly due to some residual field energy in our vacuum, a residual scalar (or many scalar) which is left behind after inflation ended into our vacuum.
These ideas are very natural in GR, and in fact are predictions of GR. So it is not reasonable to say that accelerated expansion and dark matter point to a violation of GR. This is like saying that the discovery of Neptune invalidates Newton's model of the solar system, because it alters the orbit of Jupiter. That's included in the theory.
But in the special case of vacuum energy, it is philosophicaly possible in classical GR to consider the energy as part of the equations, or part of the matter, and both positions are viable (classically). The name "Dark energy" reflects the philosophical position that it should be considered matter. The name "cosmological constant" reflects the other view that it should be considered part of the Einstein equations.
These two points of view can't really be distinguished from each other classically in any positivist way, so the two positions are classically equivalent. Whether one is true or the other is completely moot. Quantum mechanically, there is the question of whether deSitter space is stable, and if it is unstable to decay to a zero (or perhaps negative) cosmological constant, then this might be intepreted as resolving the question in favor of the "dark energy" point of view.
