arXiv 0903.43812009-03gr-qc

标准引力理想流体范式的局限性

Limitations of the Standard Gravitational Perfect Fluid Paradigm

Philip D. Mannheim, James G. O'Brien, David Eric Cox

备注：: Final version to appear in General Relativity and Gravitation (the final publication is available at http://www.springerlink.com). 29 pages, 1 figure
DOI：: 10.1007/s10714-010-0997-1

摘要：

我们表明，标准的完美流体范式不一定是一个弯曲的空间稳态引力源的有效描述。仅仅由于不是平坦的，弯曲的空间几何形状必须具有内在的长度尺度，并且这种长度尺度可以影响流体结构。对于波长阶或大于这种尺度的模式，不能应用等离子化几何光学，并且射线不是测地线。因此，协变化不仅需要用协变函数代替平坦空间函数，而且还需要引入平坦空间中不存在的内在尺度。因此，原则上，把弯曲空间能量动量张量构造为基于测地线的平坦时空能量动量张量的协变推广是不可靠的。通过构建的配分函数作为一个完整的一组模式的标量场在弯曲的空间背景中传播的非相干平均，我们表明，对于一个静态的，球对称的几何形状的特定情况下，在一般情况下的稳态能量-动量张量，enamelled将是$T_{μν}=(ρ+p)U_μU_ν+pg_{μν}+π_{μν}$的形式，其中各向异性$π_{μν}$是一个对称的，无痕秩2张量服从$U^μπ_{μν}=0$。在没有内禀长度尺度的时空中，这种$π_{μν}$类型的项对于非相干平均的稳态流体是不存在的，因此原则上在平坦时空$T_{μν}$的协变中会被忽略。虽然这些$π_{μν}$型项的意义需要根据具体情况进行评估，但通过使用动力学理论，我们放心地发现，对于通常使用完美流体源的弱重力星，这些$π_{μν}$型项的影响很小。

英文摘要：

We show that the standard perfect fluid paradigm is not necessarily a valid description of a curved space steady state gravitational source. Simply by virtue of not being flat, curved space geometries have to possess intrinsic length scales, and such length scales can affect the fluid structure. For modes of wavelength of order or greater than such scales eikonalized geometrical optics cannot apply and rays are not geodesic. Covariantizing thus entails not only the replacing of flat space functions by covariant ones, but also the introduction of intrinsic scales that were absent in flat space. In principle it is thus unreliable to construct the curved space energy-momentum tensor as the covariant generalization of a geodesic-based flat spacetime energy-momentum tensor. By constructing the partition function as an incoherent average over a complete set of modes of a scalar field propagating in a curved space background, we show that for the specific case of a static, spherically symmetric geometry, the steady state energy-momentum tensor that ensues will in general be of the form $T_{μν}=(ρ+p)U_μU_ν+pg_{μν}+π_{μν}$ where the anisotropic $π_{μν}$ is a symmetric, traceless rank two tensor which obeys $U^μπ_{μν}=0$. Such a $π_{μν}$ type term is absent for an incoherently averaged steady state fluid in a spacetime where there are no intrinsic length scales, and in principle would thus be missed in a covariantizing of a flat spacetime $T_{μν}$. While the significance of such $π_{μν}$ type terms would need to be evaluated on a case by case basis, through the use of kinetic theory we reassuringly find that the effect of such $π_{μν}$ type terms is small for weak gravity stars where perfect fluid sources are commonly used.

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