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arXiv 0809.1030gr-qc
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平坦形变定理与时空对称性

Flat deformation theorem and symmetries in spacetime

Josep Llosa, Jaume Carot

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30 pages, 0 figures
发表信息：
Class.Quant.Grav.26:055013,2009
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摘要：

\emph{flat deformation theorem}指出，给定流形上的一个半黎曼解析度量$g$，局部总是存在一个两种形式的$F$，一个标量函数$c$，以及一个依赖于流形上的点$x$和$F$和$c$的任意指定的标量约束，比如$Ψ(c, F, x)=0$，使得\emph{deformed metric} $η= cg -εF^2$是半黎曼平坦的。在本文中，我们首先表明，上述结果意味着，（洛伦兹分析）度量$g$可以写在\emph{extended Kerr-Schild form}中，即$η_{ab} := a g_{ab} - 2 b k_{(a} l_{b)}$，其中$η$是平坦的，并且$k_a, l_a$是两个空余向量，使得$k_a l^a= -1$ ;接下来，我们更精确地示出了$g$的对称性如何与$η$的对称性相连接;我们表明，如果原来的度量$g$承认共形Killing矢量（包括Killing矢量和同素），那么变形可以进行的方式，使平坦变形度量$η$“继承”的对称性。

英文摘要：

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the point $x$ of the manifold and on $F$ and $c$, say $Ψ(c, F, x)=0$, such that the \emph{deformed metric} $η= cg -εF^2$ is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric $g$ may be written in the \emph{extended Kerr-Schild form}, namely $η_{ab} := a g_{ab} - 2 b k_{(a} l_{b)}$ where $η$ is flat and $k_a, l_a$ are two null covectors such that $k_a l^a= -1$; next we show how the symmetries of $g$ are connected to those of $η$, more precisely; we show that if the original metric $g$ admits a Conformal Killing vector (including Killing vectors and homotheties), then the deformation may be carried out in a way such that the flat deformed metric $η$ `inherits' that symmetry.

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