Black Holes 97of the emitted photons is unchanged푐their frequency휈goes to zero, and the energy
퐸=ℎ휈of the signal vanishes. One cannot receive signals from beyond the event hori-
zon because photons cannot have negative energy. Thus the outside observer sees the
spacecraft slowing down and the signals redshifting until they cease completely.
The pilot in the spacecraft using local coordinates sees the passage into the black
hole entirely differently. If he started out at distance푟 0 with velocity d푟∕d푡=0attime
푡 0 , he will have reached position푟at proper time휏, which we can find by integrating
d휏in Equation (5.72) from 0 to휏:
∫
휏0√
d휏^2 =휏=
∫푟푟 0[
1 −푟c∕푟
(d푟∕d푡)^2−^1
푐^2 ( 1 −푟c∕푟)] 1 ∕ 2
d푟. (5.75)The result depends on d푟(푡)∕d푡, which can only be obtained from the equation of
motion. The pilot considers that he can use Newtonian mechanics, so he may take
d푟
d푡=푐
√
푟c
푟.
The result is then (Problem 11.4):
휏∝(푟 0 −푟)^3 ∕^2. (5.76)However, many other expressions for d푟(푡)∕d푡also make the integral in Equation (5.75)
converge.
Thus the singularity at푟cdoes not exist to the pilot, his comoving clock shows finite
time when he reaches the event horizon. The fact that the singularity at푟cdoes not
exist in the local frame of the spaceship indicates that the horizon at푟cis a mathe-
matical singularity and not a physical singularity. The singularity at the horizon arises
because we are using, in a region of extreme curvature, coordinates most appropri-
ate for flat or mildly curved space-time. Alternate coordinates, more appropriate for
the region of a black hole and in which the horizon does not appear as a singular-
ity, were invented by Eddington (1924) and rediscovered by Finkelstein (1958). If we
were able to observe the collapse of a neutron star towards the Schwarzschild radius
into a black hole it would appear to take a very long time. Towards the end of it, the
ever-redshifting light would fade and finally disappear completely.
Note from the metric Equation (5.72) that inside푟cthe time term becomes nega-
tive and the space term positive, thus space becomes timelike and time spacelike. The
implications of this are best understood if one considers the shape of the light cone
of the spacecraft during its voyage (see Figure 5.3). Outside the event horizon the
future light cone contains the outside observer, who receives signals from the space-
craft. Nearer푟cthe light cone narrows and the slope d푟∕d푡steepens because of the
approaching singularity in expression on the the right-hand side of Equation (5.73).
The portion of the future space-time which can receive signals therefore diminishes.
Since the time and space axes have exchanged positions inside the horizon, the
future light cone is turned inwards and no part of the outside space-time is included
in the future light cone. The slope of the light cone is vertical at the horizon. Thus
it defines, at the same time, a cone of zero opening angle around the original time
axis, and a cone of 180
∘
around the final time axis, encompassing the full space-time