McGehee regularization of general SO(3)-invariant potentials and applications to stationary and spherically symmetric spacetimes
McGehee regularization of general SO(3)-invariant potentials and applications to stationary and spherically symmetric spacetimes
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一般 SO(3) 不变势的 McGehee 正则化及其在静止和球对称时空中的应用
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
M. Mars
中科院分区：
文献类型：
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作者：
P. Galindo;M. Mars
麦基希正则化是一种研究动力学系统原点奇异性的方法，该动力学系统描述的是一个在幂律势作用下运动的点粒子。它被Belbruno和Pretorius（2011级）使用。量子重力28 195007），以执行动力系统正则化的奇点在史瓦西时空中的无质量的测试粒子的运动的中心。本文推广了麦基希变换，使得我们可以正则化描述任意静止球对称Kerr-Schild形式时空中因果测地线（类时或零）运动的动力系统原点处的奇点。我们首先表明，有质量和无质量的粒子的测地线可以描述为全球的克尔-Schild时空作为一个牛顿点粒子在一个合适的径向潜力的运动和研究的条件下，中心奇点可以正则化使用的麦基希方法的扩展。作为一个例子，我们将这些结果应用于史瓦西和Reissner-Nordström时空中的因果测地线。有趣的是，在这两个时空的整个最大扩张中的测地线轨迹可以用一个具有非平凡拓扑的二维相空间来描述。这种拓扑结构产生于相空间中存在的排除区域，该排除区域由测地线的切向量是因果的和未来定向的条件确定。
The McGehee regularization is a method to study the singularity at the origin of the dynamical system describing a point particle in a plane moving under the action of a power-law potential. It was used by Belbruno and Pretorius (2011 Class. Quantum Grav. 28 195007) to perform a dynamical system regularization of the singularity at the center of the motion of massless test particles in the Schwarzschild spacetime. In this paper, we generalize the McGehee transformation so that we can regularize the singularity at the origin of the dynamical system describing the motion of causal geodesics (timelike or null) in any stationary and spherically symmetric spacetime of Kerr–Schild form. We first show that the geodesics for both massive and massless particles can be described globally in the Kerr–Schild spacetime as the motion of a Newtonian point particle in a suitable radial potential and study the conditions under which the central singularity can be regularized using an extension of the McGehee method. As an example, we apply these results to causal geodesics in the Schwarzschild and Reissner–Nordström spacetimes. Interestingly, the geodesic trajectories in the whole maximal extension of both spacetimes can be described by a single two-dimensional phase space with non-trivial topology. This topology arises from the presence of excluded regions in the phase space determined by the condition that the tangent vector of the geodesic be causal and future directed.