98 Cosmological Models
DCE0BASpacecraft
world lineObserver
world linetEtmaxrc rs rototA- t t
Figure 5.3The world line of a spacecraft falling into a Schwarzschild black hole. A, the journey
starts at time푡 0 when the spacecraft is at a radius푟s, far outside the Schwarzschild radius푟c,
and the observer is at푟 0. A light signal from the spacecraft reaches the observer at time푡퐴>푡 0
(read time on the right-hand vertical scale!). B, nearer the black hole the future light cone
of the spacecraft tilts inward. A light signal along the arrow will still reach the observer at
atime푡B≫푡A. C, near the Schwarzschild radius the light cone narrows considerably, and a
light signal along the arrow reaches the observer only in a very distant future. D, inside푟cthe
time and space directions are interchanged, time running from up to down on the left-hand
vertical scale. All light signals will reach the center of the hole at푟=0, and none will reach the
observer. The arrow points in the backward direction, so a light signal will reach the center
after the spacecraft. E, the arrow points in the forward direction of the hole, so that a light
signal will reach the center at time푡E, which is earlier than푡max, when the spacecraft ends its
journey.
of the black hole. As the spacecraft approaches the center, d푡∕d푟decreases, defining a
narrowing opening angle which always contains the center.
When the center is reached the metric [Equation (5.72)] has a physical singular-
ity. the spacecraft no longer has a future. One cannot define field equations there, so
general relativity breaks down, unable to predict what will happen. Once across푟cthe
spacecraft reaches the center of the black hole rapidly. For a hole of mass 10푀⊙this
final passage lasts about 10−^4 s.
Some people have speculated that matter or radiation falling in might ‘tunnel’
through a ‘wormhole’ out into another universe. Needless to say, all such ideas are
purely theoretical speculations with no hope of experimental verification.