Gravity and Strings

198 The Schwarzschild black hole
14. For Killing horizons one can define, following Boyer [177], the quantity known as
surface gravity κ,given by the formula
κ 2 =− 1
2 (∇µ k ν )(∇µkν) horizon . (7.21)
If κ = 0 the Killing horizon is part of a bifurcate horizon, whereas if κ = 0itisa
degenerate Killing horizon.
In the particular case of static spherically symmetric metrics, which can always be
written like this,
ds 2 = gtt(r)dt 2 + grr(r)dr 2 − r 2 d 2 (2) , (7.22)
the Killing vector k µ is just δ µt and the surface gravity takes the value
κ = 1
∂r gtt
2 √
−gttgrr
which for the Schwarzschild BH is the non-vanishing constant
κ =
c4
c, (7.23)
4G (4) . (7.24)
N M
It can be shown that the surface gravity is also constant over the horizon in more
general cases [85, 217, 509]. This is analogous to the fact that the temperature is the
same at any point of a system in thermodynamical equilibrium and it constitutes the
first analogy between the surface gravity and the BH temperature (and the second
between a BH and a thermodynamical system). Physically, the surface gravity is the
force that must be exerted at ∞ to hold a unit mass in place when r → RS and has
dimensions of acceleration, LT −2 .
15. Another set of coordinates that is useful in some problems is isotropic coordinates
{t, x3} with x3 = (x 1 , x 2 , x 3 ) in which the three-dimensional constant-time slices are
conformally flat and isotropic. The change of coordinates is given by
and the metric takes the form
r =
ρ − ω
4
2
ρ, (7.25)
ds 2
= 1 + ω/4
2 1 −
ρ
ω/4
−2 dt
ρ
2
− 1 − ω/4
4 d x
ρ
2
3 ,
(7.26)