Numerical study of generic, expanding T²-Symmetric vacuum spacetimes

Generic expanding T²-Symmetric vacuum spacetimes contain two gravitational-wave polarizations propagating in a background spacetime with the metric depending on only one spatial variable and time as well as extra off-diagonal metric components described as non-zero twist and non-zero constants of the motion A and B defined as spatial integrals of nonlinear combinations of the metric variables and their first derivatives. The objective is to identify an asymptotic description of the expansion. Rigorous mathematical results exist for subclasses of the generic case characterized by B = 0 [1, 2]. So far, no rigorous results exist for the B ≠ 0 class. A numerical study is presented to show (1) where the mathematical methods used for B = 0 in [2] fail for B ≠ 0, (2) how the power laws describing the expansion of the spatially averaged variables in expanding T²-Symmetric spacetimes may be found numerically (for all values of B), (3) how spatially averaged variables may be constructed in these spacetimes to demonstrate (different) sink-like attractors for both the B = 0 and B ≠ 0 classes, and (4) that the B ≠ 0 class exhibits an interchange of “energy” between the two gravitational-wave polarizations (if B ≠ 0 but not if B = 0).

[1] P. LeFloch, J. Smulevici, Anal. PDE 9, 363 (2016).

[2] B. K. Berger, J. Isenberg, A. Layne, Ann. Henri Poincaré 21, 675 (2020).