论文标题
生成所有静态球形对称（AN）各向同性流体溶液的算法
Algorithms for Generating All Static Spherically Symmetric (An)isotropic Fluid Solutions of Einstein's Equations
论文作者
论文摘要
我们研究曲率坐标中静态球形对称流体系统的爱因斯坦方程，以找到生成所有溶液和所有溶液中心的算法。考虑了表征各向异性系统的四个函数的输入函数的所有可能组合，并确定了中心规则性的所有等效条件（对于各向同性和各向异性系统）。我们提供了使用潜在函数和各向异性作为输入的已知算法的第一个规律性分析。对于输入函数对的其他三个选择（潜在函数，密度或径向压力的任何两个），随后是非常简单的算法，这对于生成常规各向异性溶液非常有效。这是因为该算法中这三对的等效性恰恰来自相同的代数关系，这使得不同的规律性条件的不同等效集成为可能。此外，功能的选择使得该算法非常适合找到接受其他理想物理特性的特定解决方案；我们构建了三个例子。尽管所有各向同性溶液都是作为各向异性系统的一部分，但该算法不接受各向同性极限。其余的两个输入函数对选择（具有径向压或密度的各向异性）导致旧屏障在各向同性系统中遇到：Riccati和Abel方程。但是，借助新算法和现有算法生成的任何解决方案，现在可以构造相应的Riccati方程的一般解决方案，以获取每个输入解决方案的几何几何家族家族。我们讨论了最终解决方案的规律性。
We study the Einstein equations of the static spherically symmetric anisotropic fluid system in curvature coordinates to find algorithms that generate all solutions and all solutions that are regular at the center. All possible combinations of input functions from the set of four functions that characterize the anisotropic system are considered and all equivalent conditions for central regularity are determined (for both isotropic and anisotropic systems). We provide the first regularity analysis of the known algorithm that uses the potential function and anisotropy as inputs. For three other choices of input function pairs (any two of the potential function, density, or radial pressure), a remarkably straightforward algorithm follows, which is very efficient in generating regular anisotropic solutions. This is because the equivalency of the three pairs in this algorithm arises precisely from the same algebraic relation that made the different equivalent sets of regularity conditions possible. In addition, the choice of functions makes this algorithm very suitable for finding particular solutions that admit other desirable physical properties; we construct three examples. This algorithm does not admit an isotropic limit although all isotropic solutions are produced as part of the anisotropic system. The remaining two choices of input function pairs (anisotropy with the radial pressure or density) lead to the old barriers one encounters in the isotropic system: Riccati and Abel equations. However, with any solution generated by the new and existing algorithms, one can now construct the general solution of the corresponding Riccati equation to obtain a one-parameter family of geometries for each input solution. We discuss the regularity of the resulting solutions.
