37A (General Relativity)

Let

f = 1 - r^{2} / b^{2}

. Then

U_{μ} = (1, - 1/ f, 0, 0)

and

g^{μν} U_{μ} U_{ν} = - 1/ f + f / f^{2} = 0

. Since

d t = d u + d r / f

, the metric becomes

d s^{2} = - fd u^{2} - 2, d u, d r + r^{2} d Ω^{2},

which is regular at

r = b

b

Radial nullness factors as

d u (f, d u + 2 d r) = 0

, giving the two families. For

r < b

u =

constant has

d r / d t = + f

and is outgoing;

d u / d r = - 2/ f

has

d r / d t = - f

and is ingoing. In the

r, t^{*} = r + u

plane they cross the horizon according to these slopes. The cosmological horizon prevents a future-directed inward light signal sent from

r > b

from reaching the static observer inside.

A radial timelike geodesic has conserved

E = f \dot{t}

and normalization

\overset{r}{˙}^{2} = E^{2} - f

. Choose

E > 1

and the outward sign. Then

Δ τ = \int_{0}^{b} \frac{d r}{E ^{2} - 1 + r ^{2} / b ^{2}} < \infty,

so the observer reaches

r = b

in finite proper time even though static coordinate

t

diverges.

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