Fajman David, Urban Liam
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Lett Math Phys. 2022;112(2):42. doi: 10.1007/s11005-022-01522-5. Epub 2022 Apr 21.
We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes includes Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes with negative sectional curvature that solve the Einstein equations in the presence of a perfect irrotational fluid with . As such, these results are closely related to the still open problem of past nonlinear stability of such FLRW spacetimes within the Einstein scalar field equations. In contrast to earlier works, our results hold for spatial metrics of arbitrary geometry, hence indicating that the matter blow-up in the aforementioned problem is not dependent on spatial geometry. Additionally, we use the energy estimates derived in the proof in order to formulate open conditions on the initial data that ensure a non-trivial blow-up profile, for initial data sufficiently close to the Big Bang singularity and with less harsh assumptions for .
我们研究具有大爆炸奇点的宇宙时空上线性波动方程解的渐近行为，并表明适当重标度的波收敛于一个爆破轮廓。我们的时空类包括具有负截面曲率的弗里德曼 - 勒梅特 - 罗伯逊 - 沃克（FLRW）时空，它们在存在具有 的完美无旋流体的情况下求解爱因斯坦方程。因此，这些结果与爱因斯坦标量场方程中此类FLRW时空过去非线性稳定性这一仍未解决的问题密切相关。与早期工作不同，我们的结果适用于任意几何形状的空间度量，从而表明上述问题中的物质爆破不依赖于空间几何。此外，我们利用证明中导出的能量估计，以便为初始数据制定开放条件，这些条件确保对于足够接近大爆炸奇点且对 假设不太苛刻的初始数据，存在非平凡的爆破轮廓。
