Gravity and Strings

7.1 Schwarzschild’s solution 191
where p α =−mdx α /dτ is the observer’s four-momentum, τ is the observer’s
proper time, and we have set ξ = cτ. Onthe other hand, since the Schwarzschild
metric admits a timelike Killing vector k µ = δ µ0 , the observer’s motion
has an associated conserved momentum p(k) ≡ p 0 given by Eq. (3.266) that
we can identify with the observer’s total energy
E =−p 0 c. (7.10)
To simplify the calculations, we assume that the observer has only radial motion
(i.e. zero angular momentum) so pθ = pϕ = 0. Then, using the conservation of
the energy, the mass-shell constraint becomes a simple equation for pr , which
is a differential equation for r,
2
dr E
=
cdτ mc2 2 − W, (7.11)
that can be integrated to give the total proper time:
where
τ = 1
r2
dr
c r1
R0 =
RS
r
RS
1 − [E/(mc 2 )] 2
1
−
RS
2
− , (7.12)
R0
(7.13)
is the value of the radial coordinate r for which the speed of the observer is zero
and E = mc2 .
We can use the above expression to calculate how long it takes for the freefalling
observer to go from r = R0 > RS to the curvature singularity at r = 0,
going through the surface r = RS. The answer is, surprisingly, finite:
τ = π
2c R0
1
R0
2
RS
. (7.14)
This confirms that nothing unphysical happens at r = RS and that the singularity
is only a problem of Schwarzschild’s coordinates. It should, then, be possible
to find a coordinate system which is not singular there. 10
This is essentially the idea on which the Eddington–Finkelstein coordinates
{v,r,θ,ϕ} are based [345, 394]. In these coordinates the Schwarzschild solution
takes the form
ds 2 = Wdv 2 − 2dvdr − r 2 d 2 (2) , (7.15)
where the coordinate v is related to t and r in the region r > RS by
v = ct + r + RS ln |W |, (7.16)
10 There is, of course, another issue: whether the tidal forces at the horizon are big or small. For big enough
Schwarzschild BHs they are small, but this might not be a universal behavior of BHs [555, 556].