度量的指标

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 7.1 黎曼流形与伪黎曼流形

由于 $(g^{\mu \nu })$ 是对称矩阵，其特征值都是实数。若 $g$ 是黎曼的，则所有特征值都严格为正；若 $g$ 是伪黎曼的，则其中一些特征值可能为负。若有 $i$ 个正特征值和 $j$ 个负特征值，则称数对 $(i,j)$ 为度量的指标(index of the metric)。若 $j=1$，则称该度量为洛伦兹度量(Lorentz metric)。一旦用适当的正交矩阵把度量对角化，就很容易通过用正数对基向量作适当的伸缩，把所有对角元化为 $\pm 1$。若从黎曼度量出发，最终得到欧几里得度量 $\delta =\mathrm{diag}(1,\dots ,1)$；若从洛伦兹度量出发，则得到闵可夫斯基度量 $\eta =\mathrm{diag}(-1,1,\dots ,1)$。

英文原文

Since $(g^{\mu \nu })$ is a symmetric matrix, the eigenvalues are real. If $g$ is Riemannian, all the eigenvalues are strictly positive and if $g$ is pseudo-Riemannian, some of them may be negative. If there are $i$ positive and $j$ negative eigenvalues, the pair $(i,j)$ is called the index of the metric. If $j=1$ , the metric is called a Lorentz metric. Once a metric is diagonalized by an appropriate orthogonal matrix, it is easy to reduce all the diagonal elements to $\pm 1$ by a suitable scaling of the basis vectors with positive numbers. If we start with a Riemannian metric we end up with the Euclidean metric $\delta =\mathrm{diag}(1,\dots ,1)$ and if we start with a Lorentz metric, the Minkowski metric $\eta =\mathrm{diag}(-1,1,\dots ,1)$ .

出处：7.1 黎曼流形与伪黎曼流形 · node a-f1356c5fc0373aee1dbb884b · 源节点 SHA-256 0d93dea7589e555b…