Robertson-Walker度量的最优协变拟合与逆反应的小性
Optimal covariant fitting to a Robertson-Walker metric and smallness of backreaction
摘要：
我们定义了一类“最佳”坐标系，要求偏离精确的罗伯逊-沃克度量是“尽可能小”在一个给定的四维体积。优化是通过最小化几个体积积分，这将消失一个确切的罗伯逊-沃克度量。协方差是自动的。时空的叶是优化过程的一部分。这个过程只涉及度规，不需要对能量动量张量的原点作任何假设。比例因子在优化过程中不会出现，最佳比例因子在最后确定。一般公式是非微扰的。显式微扰处理是可能的。导致最佳坐标的移位服从欧拉-拉格朗日方程，该方程在一阶扰动中制定和求解。扩展到二阶的草图，但原来是不必要的。在坐标的选择上，唯一的自由度是保持罗伯逊-沃克度规形式不变的刚性变换，即空间和时间的平移、空间旋转和空间缩放。空间平均变得微不足道。在微扰的第一级，没有反作用。一个简化的二阶处理的结果在一个非常小的影响，排除了模仿暗能量从逆反应的可能性。这证实（以及矛盾）在文献中的陈述。
英文摘要：
We define a class of "optimal" coordinate systems by requiring that the deviation from an exact Robertson-Walker metric is "as small as possible" within a given four dimensional volume. The optimization is performed by minimizing several volume integrals which would vanish for an exact Robertson-Walker metric. Covariance is automatic. Foliation of space-time is part of the optimization procedure. Only the metric is involved in the procedure, no assumptions about the origin of the energy-momentum tensor are needed. A scale factor does not show up during the optimization process, the optimal scale factor is determined at the end. The general formulation is non perturbative. An explicit perturbative treatment is possible. The shifts which lead to the optimal coordinates obey Euler-Lagrange equations which are formulated and solved in first order of the perturbation. The extension to second order is sketched, but turns out to be unnecessary. The only freedom in the choice of coordinates which finally remains are the rigid transformations which keep the form of the Robertson-Walker metric intact, i.e. translations in space and time, spatial rotations, and spatial scaling. Spatial averaging becomes trivial. In first order of the perturbation there is no backreaction. A simplified second order treatment results in a very small effect, excluding the possibility to mimic dark energy from backreaction. This confirms (as well as contradicts) statements in the literature.
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