Manuscript I-B@1 · I-B-v1.tex · proposition · 5357688f

Proposition: Lorentzian character and normalized static time

The geometric background › The cone, its regular part, and its completion · lines 251–285 · proof included or linked

I-B@1#prop:geometry, I-B@1#eq:tangent-cone

For $N > 0$ , $g_{N}$ has signature $(-, +, +, +)$ , and

g_{N}^{- 1} (d T, d T) = - \frac{1}{c ^{2} N ^{2}} .

Choose the time orientation for which $\partial_{T}$ is future directed. For

v = a \partial_{T} + w, w \in T (C_{μ}^{\circ} \times R),

a nonzero future-causal vector satisfies

a > 0, ∣ w ∣_{h} \leq c N a .

A static observer has proper-time increment

d τ = N d T .

In particular, $T$ is proper time up to the chosen origin for the static observers of (eq:metric).

Proof. In the displayed coordinates the metric matrix is diagonal, with one negative and three positive entries. Its inverse gives the stated norm of $d T$ . The causal inequality is

g_{N} (v, v) = - c^{2} N^{2} a^{2} + ∣ w ∣_{h}^{2} \leq 0.

A nonzero causal vector cannot have $a = 0$ ; the chosen orientation therefore gives $a > 0$ . Along a static curve, $- g_{N} / c^{2} = N^{2} d T^{2}$ , which proves the proper-time formula.

Displayed formulas in this passage

I-B@1#eq:tangent-cone · line 263
$a > 0, ∣ w ∣_{h} \leq c N a .$
unlabeled · line 254
$g_{N}^{- 1} (d T, d T) = - \frac{1}{c ^{2} N ^{2}} .$
unlabeled · line 259
$v = a \partial_{T} + w, w \in T (C_{μ}^{\circ} \times R),$
unlabeled · line 268
$d τ = N d T .$
unlabeled · line 279
$g_{N} (v, v) = - c^{2} N^{2} a^{2} + ∣ w ∣_{h}^{2} \leq 0.$

Nearby in this section

section: The cone, its regular part, and its completion