m/Matter is lifted out of a
Newtonian black hole with a
bucket. The dashed line rep-
resents the point at which the
escape velocity equals the speed
of light. For a real, relativistic
black hole, this is impossible.
object that that isn’t influenced by anything other than gravity. By
this definition, a free-falling rock defines an inertial frame, but this
book sitting on your desk does not.7.4.3 Black holes
The observations described so far showed only small effects from
curvature. To get a big effect, we should look at regions of space in
which there are strong gravitational fields. The prime example is a
black hole. The best studied examples are two objects in our own
galaxy: Cygnus X-1, which is believed to be a black hole with about
ten times the mass of our sun, and Sagittarius A*, an object near
the center of our galaxy with about four million solar masses.
Although a black hole is a relativistic object, we can gain some
insight into how it works by applying Newtonian physics. A spher-
ical body of massMhas an escape velocityv=√
2 GM/r, which
is the minimum velocity that we would need to give to a projectile
shot from a distancerso that it would never fall back down. Ifr
is small enough, the escape velocity will be greater thanc, so that
even a ray of light can never escape.
We can now make an educated guess as to what this means
without having to study all the mathematics of general relativity.
In relativity,cisn’t really the speed of light, it’s really to be thought
of as a restriction on how fast cause and effect can propagate through
space. This suggests the correct interpretation, which is that for an
object compact enough to be a black hole, there is no way for an
event at a distance closer thanrto have an effect on an event far
away. There is an invisible, spherical boundary with radiusr, called
the event horizon, and the region within that boundary is cut off
from the rest of the universe in terms of cause and effect. If you
wanted to explore that region, you could drop into it while wearing
a space-suit — but it would be a one-way trip, because you could
never get back out to report on what you had seen.
In the Newtonian description of a black hole, matter could be
lifted out of a black hole, m. Would this be possible with a real-
world black hole, which is relativistic rather than Newtonian? No,
because the bucket is causally separated from the outside universe.
No rope would be strong enough for this job (problem 12, p. 460).One misleading aspect of the Newtonian analysis is that it en-
courages us to imagine that a light ray trying to escape from a black
hole will slow down, stop, and then fall back in. This can’t be right,
because we know that any observer who sees a light ray flying by
always measures its speed to bec. This was true in special relativity,
and by the equivalence principle we can be assured that the same is
truelocally in general relativity. Figure n shows what would really
happen.450 Chapter 7 Relativity