3 ms·
> de Sitter space is a vacuum solution for Einstein's field equations. Yes, an exact solution. > What this means is that the stress-energy tensor is basically
by raattgift 8y ago
> de Sitter space is a vacuum solution for Einstein's field equations.
Yes, an exact solution.
> What this means is that the stress-energy tensor is basically canceled out" everywhere by the uniform energy distribution
No, it means that the stress-energy tensor is zero at every point in the whole spacetime. There is no matter to source any curvature; the curvature is specified by the theoretician writing down the vacuum solution.
In Lorentzian 4-dimensional vacuum de Sitter spacetime ("dS" below) there is no useful concept of energy anywhere. The parameter \Lambda is incorporated directly into the Einstein curvature tensor and so is interpreted geometrically. One would have to perturb the dS vacuum in order to have a useful interpretation of \Lambda as a source of gravitational energy.
Vacuum solutions can be useful in understanding physical systems in which gravitation is important. For example, one can add non-zero stress-energy by hand and use perturbation theory to study the consequences rigorously. Alternatively, one can "cut out" parts of a vacuum solution and work with the rest which closely resembles a physical system (such as deep space far from the matter in and around galaxies). As the expansion of the universe dilutes away the matter contributing to a non-zero stress-energy tensor, our universe will resemble the sparser parts of a de Sitter spacetime. However, from our perspective (i.e. with data in our sky) there are much better approximations at various scales than de Sitter spacetime perturbed by matter fields: there is a lot of obviously non-zero stress-energy under our feet, lighting up our sky day and night, and causing heat to flow from one place to another.
> That means no gravity
There can be extremely strong spacetime curvature in de Sitter space!
In general we can time-orient dS: in one direction objects which only interact gravitationally and do not generate significant stress-energy will always converge, in the other they will always diverge, depending on the value of the \Lambda parameter, which is always positive for dS. If \Lambda is large, all such objects leave the others' causal cones much more quickly than they could, even in principle, in flat spacetime.
If we perturb dS with a dense ball of nonrelativistic (i.e., slowly-moving) dust that feels only the gravitational interaction, the dust will tend to collapse if \Lambda is sufficiently small, and will locally approximate a Lemaître-Tolman-Bondi (LTB) collapsing dust metric, which in turn resembles a perturbation of vacuum Schwarzschild or Kerr (which are eternal black hole models; there is no matter to collapse).
A scattering of such collapsing dust-balls with significant gaps between them has been studied by a number of relativists as a family of "swiss-cheese" cosmologies very similar to the one developed by Einstein & Strauss in 1945, who represented the balls of dust as already-collapsed vacuum Schwarzschild solutions, calling them "holes" in the "cheese" of the surrounding expanding spacetime. The Einstein-Strauss-de Sitter model approximated some features of our universe at intermediate ranges, but has since been superseded by the the Friedmann-Lemaître-Robertson-Walker (FLRW) model of the modern standard model of cosmology that among other things has representations of nonzero stress-energy that the vacuum Einstein & Strauss model did not, and which better matches observables in our sky than models that added more realistic matter fields to Einstein & Strauss vacuum.
One feature of Einstein-Strauss-de Sitter style expanding swiss-cheese models is that a small "test" amount of sparse low-mass nonrelativistic dust scattered in the LTB "holes" will stay confined within the hole while the hole separates from everything else. However, an identical test dust scattered in the "cheese" part well outside of holes will never find its way into a hole. Ultimately the former test dust will converge while the latter will diverge.
We can get highly similar results by perturbing the FLRW model. (The observable universe is well approximated by an almost FLRW model in which there are primordial density fluctuations: over-dense areas collapse gravitationally, comparably to the matter in the "holes" of a non-vacuum Einstein-Strauss-de Sitter universe.)
Locally in all these cases, what we have are the tensor fields contributing to the Einstein Field Equations, and general covariance. We can talk about the values of these tensors at any given point (and its neighbourhood), but General Relativity does not lend itself to universal definitions of energy or even energy-density. We can reduce the tensors by using gauge fixing and other techniques, and "demote" the cosmological constant in FLRW or in dS into an energy. (That's really all Dark Energy is in the standard cosmology). In such a gauge we can talk about the work matter interactions (e.g. the electromagnetic interaction) does "against" dark energy. However, the real physics are in the tensors, and applying coordinates and selecting particular observers, while often able to make things much easier to calculate, can be highly misleading too.
The idea that the cosmological constant is "cancelled" out by matter-matter gravitational interactions (and electromagnetic and nuclear interactions) is a statement that the geometrical nature of the cosmological constant is "demoted" into an extremely weak force field, and leads to all sorts of statements that work in that "demotion" context but not generally. Typical statements are that the solar system is expanding, or the galaxy or cluster is expanding, because of the tiny local force on matter by dark energy.
It is better to retreat to "swiss-cheese" and say that the geometry of our solar system is not well-approximated by expanding metrics like dS or expanding Robertson-Walker, and that consequently it is a better fit to observations that in our "hole" (which the matter of our solar system, galaxy and cluster sources) the cosmological constant simply vanishes.
Better still would be to advance to inhomogeneous metrics, but those are a subject of research beyond the scope of this comment.
Finally, your last paragraph can also be read as a request to be directed to the concept of the Jeans Instability in the Expanding Universe. If in the very early universe there was truly uniform and dense distribution of stress-energy, the slightest perturbations would magnify, leading to structure formation. As I wrote above primordial fluctuations of an otherwise FLRW model approximates modern cosmological observations very well. (One could alternatively introduce dissipation, quantum uncertainties, and so forth, as a source of gravitational Jeans Instability causing the collapse or fragmentation of a uniform distribution of stress-energy).