这是一份andrews圣安德鲁斯大学 MT3506作业代写的成功案例
Proof. If $p q$ is null, the result is obvious (and vacuous) from . Let $p q$ be timelike and choose Minkowski normal coordinates $(t, x, y, z)$ for $N$, with origin at some point $r$, in $N$, lying to the past of $p$ along the extension of $p q$. Choose new coordinates for the region $\hat{N}$ given by $t>\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}$ as follows:
$$
T=\left(t^{2}-x^{2}-y^{2}-z^{2}\right)^{1 / 2},
$$
、
$$
X^{1}=\frac{x}{t}, \quad X^{2}=\frac{y}{t}, \quad X^{3}=\frac{z}{t}
$$
Since the curves $X^{1}, X^{2}, X^{3}=$ const. are timelike geodesics through $r$, and $T=$ const. are spacelike hypersurfaces orthogonal to these, where $T\left(={\Phi(r,)}^{1 / 2}\right)$ measures the length (i.e., proper time) on the geodesic from $r$. we have what is known as a synchronous coordinate system for $N$ (i.e., a Gaussian normal coordinate system in which the geodesics are timelike, being orthogonal to a system of spacelike coordinate hypersurfaces). The metric therefore has the form
MT3506 COURSE NOTES ：
$$
l\left(\gamma_{\xi^{\prime}}\right) \leqq l\left(\gamma_{\xi}\right) \text { if } \xi \subset \xi^{\prime}
$$
by repeated application of $7.2$. Also, given $\xi, \xi^{\prime} \in \Xi$ we have
$$
\left.l\left(\gamma_{c^{\prime}}\right) \leq \min \left(l \gamma_{\xi}\right), l\left(\gamma_{\xi}\right)\right) \text {, }
$$
where $\xi^{\prime \prime}=\xi \cup \xi^{\prime}$. Finally, define $1: \mathscr{G}({p},{q}) \rightarrow \mathbb{R}$ by