嵌入理论中虚构物质的运动方程分析
Analysis of the equations of motion of fictitious matter in embedding theory
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中文总结 AI 辅助
分析嵌入理论中虚构物质的运动方程，得到壁状、弦状、球状三类静态解，弦状解对应刘维尔方程且无穷远密度减小时解具旋转对称性，球状解在球对称假设下有唯一单参数光滑解族。
中文摘要 AI 辅助
嵌入理论是广义相对论的一种可能修正，它提供了一个框架，用于解释通常归因于暗物质的观测效应。该修正的思路是将我们的时空视为十维平直背景空间中的一个四维曲面。嵌入理论中的运动方程可被重新表述为一组爱因斯坦方程，包含某种额外虚构物质的贡献以及描述该物质的方程。我们分析这些方程的静态解，这些解可简化为壁状、弦状和球状的虚构物质构型。弦状情况最终由刘维尔方程描述，球状情况则由其三维类似物描述。对于弦状情况，我们证明，当无穷远处的密度贡献减小时，刘维尔方程的所有解均具有旋转对称性。对于球状情况，我们证明，在球对称性假设下，存在唯一的单参数族在中心处光滑的解。
英文摘要
Embedding theory is a possible modification of general relativity that provides a framework for explaining the observed effects typically attributed to dark matter. The idea of this modification is to consider our spacetime as a four-dimensional surface in a ten-dimensional flat ambient space. The equations of motion in embedding theory can be reformulated as a set of Einstein equations with the contribution of some additional fictitious matter and of equations describing this matter. We analyze static solutions of these equations, which are reduced to fictitious-matter configurations of the wall, string, and ball types. The string case is ultimately described by the Liouville equation, and the ball case is described by its three dimensional analogue. For the string case, we show that as the density contribution decreases at infinity, all solutions to the Liouville equation are rotationally symmetric. For the case of the ball, we show that under the assumption of spherical symmetry, there exists a unique one-parameter family of solutions that are smooth at the center.
