An Introduction to Modern Astrophysics · Problem 17.13

Close Binary Star Systems — Carroll & Ostlie

Physics — Astrophysics · black-holes-schwarzschild-metric

Problem

(a) Show that the proper distance from the event horizon to a radial coordinate

r

is given by

\Delta\mathcal{L} = r\sqrt{1 - \frac{R_S}{r}} + \frac{R_S}{2}\ln\left(\frac{1 + \sqrt{1 - R_S/r}}{1 - \sqrt{1 - R_S/r}}\right).

This illustrates the danger of interpreting

r

as a distance instead of a coordinate. Hint: Integrate Eq. (17.23). (b) Make a graph of

\Delta\mathcal{L}

as a function of

r

for values of

r

between

r = R_S

and

r = 10R_S

. (c) Show that, for large values of

r

\Delta\mathcal{L} \simeq r.

Thus, far from the black hole, the radial coordinate

r

can be treated as a distance.