文档介绍：Foundations
by Greg Egan
3: Black Holes
Copyright © Greg Egan, 1999. All rights reserved.
The previous article in this series began building the framework of ideas needed for
general relativity by describing the geometry of manifolds — mathematical spaces
without any notion of distance or angle — and then showing how it was possible to add a
metric that defined these things in a very general way. The idea of parallel transport of
a vector was introduced: moving along any path, you can carry a kind of “reference
copy” of a vector from your starting point with you. A path is called a geodesic if it
continues to follow the parallel-transported copy of its initial direction, never swerving
away from its original bearing. Parallel transport of a vector around a closed loop can
produce a reference copy back at the starting point that fails to match the original vector,
and this effect is used to quantify the curvature of space (or spacetime), via the Riemann
curvature tensor.
Einstein's equation links the curvature of spacetime with the presence of matter
and energy. We haven't quite said all that we need to about curvature, but this article will
begin by attacking the other side of the equation. This will give us some insight into why
the equation takes the form it does, before we reach the final goal: examining one
solution of the equation, the Schwarzschild solution, which describes a black hole.
Mass
If we want to quantify the amount of matter and energy in a region of spacetime, a good
place to start is the idea of mass. According to Newtonian physics, when we weigh an
object we're measuring the gravitational force that the Earth exerts upon it, and this force
is taken to be proportional to the object's mass. Mass is usually defined quite differently,
though, through the property of inertia: in the absence plications like friction,
when you apply a certain force to an object its rate of acceleration will be inversely