7 The Schwarzschild Black Hole

Chapter 7
The Schwarzschild Black Hole

Addendum: An English translation of the original paper by Karl Schwarzschild is available here. As pointed out in this paper, Karl Schwarzschild did not solve the standard Einstein equations but the earliest version of them in which the coordinate condition = 1 (invariant only under volume-preserving diffeomorphisms) had been imposed. This coordinate choice was motivated by convenience and had no physical meaning and could be replaced by any other.
Page 204, Eq. (7.47): according to the redefinition of the P lanck length Eqs. (4.3) (4.4) the entropy is exactly one quarter of the are of the event horizon:

$1 A S = ---2-----, (7.47) 4 ℓPlanck$
Page 209, Eq. (7.65): a factor of i is missing in the denominator

$X - iT -iτ- -------- = e-2i Arg(X+iT) = e-RS . (7.65 ) X + iT$
(Thanks to Ross O’Connell)
Page 212, Eq. (7.81): V ^(d) should be V ^(d-2) and a factor of c³ is missing

$c3V (d- 2) S = -----(d). 4ℏG N$
(Thanks to Curtis Apslund)