标题： 加权标量平均在LTB尘埃模型中的应用 第一部分：统计涨落和引力熵 显示英文标题

标题： Weighed scalar averaging in LTB dust models, part I: statistical fluctuations and gravitational entropy

摘要： 我们引入了一种加权标量平均形式（"q-平均"），用于研究球对称Lemaitre-Tolman-Bondi（LTB）尘埃模型的理论性质和动力学。 通过对密度、哈勃膨胀和空间曲率（这些在FLRW模型中是常见的）应用q-平均而产生的"q-标量"，可以直接用曲率和运动学不变量来表示，并且无须包含Buchert平均所具有的反作用项，它们恒定地满足FLRW演化定律。 相对于q-平均的局部和非局部波动和扰动通过曲率和运动学不变量的比值以及径向梯度的大小传达了不均匀性的影响。 表征这些模型的所有曲率和运动学本征张量都可以作为度规和4速度的不可约代数展开式，其系数是q-标量及其线性和二次局部波动。 这些张量的所有不变收缩都是二次波动，其q-平均直接且精确地与密度和哈勃标量的统计相关矩有关。 我们探讨了这种形式在Hosoya等人（2004 Phys. Rev. Lett. 92 141302）提出的引力熵泛函定义中的应用。 我们证明，密度和哈勃标量波动之间的负相关会导致正熵产生，简要概述了其在各种LTB模型和区域中的实现情况。 虽然q-平均形式特别适用于LTB和Szekeres模型，但它可能为一般非均匀时空中的标量平均特性提供有价值的理论见解。 显示英文摘要

摘要： We introduce a weighed scalar average formalism ("q-average") for the study of the theoretical properties and the dynamics of spherically symmetric Lemaitre-Tolman-Bondi (LTB) dust models models. The "q-scalars" that emerge by applying the q-averages to the density, Hubble expansion and spatial curvature (which are common to FLRW models) are directly expressible in terms of curvature and kinematic invariants and identically satisfy FLRW evolution laws without the back-reaction terms that characterize Buchert's average. The local and non-local fluctuations and perturbations with respect to the q-average convey the effects of inhomogeneity through the ratio of curvature and kinematic invariants and the magnitude of radial gradients. All curvature and kinematic proper tensors that characterize the models are expressible as irreducible algebraic expansions on the metric and 4-velocity, whose coefficients are the q-scalars and their linear and quadratic local fluctuations. All invariant contractions of these tensors are quadratic fluctuations, whose q-averages are directly and exactly related to statistical correlation moments of the density and Hubble expansion scalar. We explore the application of this formalism to a definition of a gravitational entropy functional proposed by Hosoya et al (2004 Phys. Rev. Lett. 92 141302). We show that a positive entropy production follows from a negative correlation between fluctuations of the density and Hubble scalar, providing a brief outline on its fulfillment in various LTB models and regions. While the q-average formalism is specially suited for LTB and Szekeres models, it may provide a valuable theoretical insight on the properties of scalar averaging in inhomogeneous spacetimes in general.