The Plane Symmetric Einstein-Dust System With Positive Cosmological Constant

Abstract

The Einstein equations with a positive cosmological constant are coupled to the pressureless perfect fluid matter in plane symmetry. Under suitable restrictions on the initial data, the resulting Einstein-dust system is proved to have a global classical solution in the future time direction. Some late time asymptotic properties are obtained as well.

Global existence of solutions for the Einstein–Boltzmann system in a Bianchi type I spacetime for arbitrarily large initial data

“…In the case of collisionless matter, the governing system is the Einstein-Vlasov system, in the pure gravitational case, or this system coupled to other field equations, if these fields are involved in the sources of the Einstein equations. In the collisionless case, several authors proved global results, see [17], [21] for reviews, [12], [22] and [10] for scalar matter fields, also see [19], [18] for the EinsteinVlasov system with cosmological constant, which turns out to be a useful tool for the proof of the fact that, the expansion of the universe is accelerating, see [16] for more details on this question. Now, in the case of collisional matter, the Einstein-Vlasov system is replaced by the Einstein-Boltzmann system, which seems to be the best approximation available and that describes the case of instantaneous, binary and elastic collisions.…”

Plane-symmetric spacetimes with positive cosmological constant: The case of stiff fluids

“…This is especially true when a cosmological constant is included, as we do in the present paper. Global-in-time solutions and the existence of future geodesically complete spacetimes can be established under a smallness condition on the initial data, as recognized by Tchapnda [10] for γ = 1 and under the assumption of plane symmetry and, later, without symmetry and for γ ∈ (1, 4/3), by Rodnianski and Speck [5] and Speck [6,7].…”

“…Observe that the first-order principal part of (2.9) is a strictly hyperbolic system of two equations associated with the two distinct speeds ±e η−λ . Introducing the Riemann invariants 10) and the directional derivatives…”

Proof of the cosmic no-hair conjecture in the $\mathbb T^3$-Gowdy symmetric Einstein–Vlasov setting

“…In the end, we shall choose K Vl = ln t/2. As a consequence, ( 221) and (222) imply that (39) holds with a margin (note that the T 2 -symmetric background solution is such that φi , i = 0, 1, vanish), assuming N ≥ 5; note that in order to prove Theorem 35, it is sufficient to apply Theorem 29 with k 0 = 4.…”

“…Turning to the spatially inhomogeneous setting, there are results in the surface symmetric case with a positive cosmological constant; cf. [37,38,39,15], and see [22] for a definition of surface symmetry. In this case, the isometry group (on a suitable covering space) is 3-dimensional.…”