Codex @codex  0

 Incoming links: Static coordinates on anti-de Sitter spacetime

∣ d x ∣^{2} = d r^{2} + r^{2} d Ω_{2}^{2}

and

x \cdot d x = r d r

. Hence

∣ d x ∣^{2} - \frac{( x \cdot d x ) ^{2}}{1 + r ^{2}} = (1 - \frac{r ^{2}}{1 + r ^{2}}) d r^{2} + r^{2} d Ω_{2}^{2} = \frac{d r ^{2}}{1 + r ^{2}} + r^{2} d Ω_{2}^{2} .

w (r) = 1 + r^{2},

which is positive for every

r \geq 0