论文标题
n+1尺寸flrw空间的非线性稳定性
The nonlinear stability of n+1 dimensional FLRW spacetimes
论文作者
论文摘要
我们证明了“ $ n+1 $'家族的非线性lyapunov稳定性 - 弗里德曼·莱玛特·罗伯逊·沃克（FLRW）的一般相对论的一般相对论的维度宇宙学模型，包括积极的宇宙常数。 In particular, we show that the perturbed solutions to the Einstein-Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant $n=3$ case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature.为了实现这一结果，我们在恒定平均外部曲率空间谐波仪（CMCSH）中采用了一种能量法来为Einstein-Euler场方程（CMCSH）耦合。为了处理Euler的方程式，我们从Christodoulou \ cite {Christodoulou}衍生的电流中构建能量（并且与Christodoulou在Minkowski空间上的电流相吻合），并表明这种能量控制着自由度的所需范围。流体能电流以及CMCSH仪条件的使用将Einstein-Euler场方程置于耦合的椭圆形纤维系统中。利用从椭圆方程得出的估计值，我们首先表明重力流体能量功能在扩展方向上保持统一。使用这种统一的界限，我们后来获得了尖锐的衰减估计，如果包括正宇宙常数$λ$，这表明由正宇宙常数引起的物理宇宙的加速扩张足以控制小数据案例中的非线性。讨论了这种稳定结果的一些物理后果。
We prove nonlinear Lyapunov stability of a family of `$n+1$'-dimensional cosmological models of general relativity locally isometric to the Friedman Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein-Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant $n=3$ case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein-Euler field equations in constant mean extrinsic curvature spatial harmonic gauge (CMCSH). In order to handle Euler's equations, we construct energy from a current that is similar to the one derived by Christodoulou \cite{christodoulou} (and which coincides with Christodoulou's current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein-Euler field equations into a coupled elliptic-hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant $Λ$ is included, which suggests that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the non-linearities in the case of small data. A few physical consequences of this stability result are discussed.
