Notes on Relativity and Cosmology - Physics Department, UCSB

9.3. BEYOND THE HORIZON 249
9.3.5 Visualizing black hole spacetimes
We have now had a fairly thorough discussion about Schwarzschild black holes
including the outside, the horizon, the inside, and the “extra regions” (second
exterior and past interior). One of the things that we emphasized was that the
spacetime at the horizon of a black hole is locally flat, just like everywhere else
in the spacetime. Also, the curvature at the horizon depends on the mass of the
black hole. The result is that, if the black hole is large enough, the spacetime
at the horizon is less curved than it is here on the surface of the earth, and a
person could happily fall through the horizon without any discomfort (yet).
It is useful to provide another perspective on the various issues that we have
discussed. The idea is to draw a few pictures that I hope will be illustrative.
The point is that the black hole horizon is an effect caused by the curvature
of spacetime, and the way that our brains are most used to thinking about
curved spaces is to visualize them inside of a larger flat space. For example,
we typically draw a curved (two-dimensional) sphere) as sitting inside a flat
three-dimensional space.
Now, the r, t plane of the black hole that we have been discussing and drawing
on our spacetime diagrams forms a curved two-dimensional spacetime. It turns
out that this two-dimensional spacetime can also be drawn as a curved surface
inside of a flat three-dimensional spacetime. These are the pictures that we will
draw and explore in this section.
To get an idea of how this works, let me first do something very simple: I
will draw a flat two-dimensional spacetime inside of a flat three-dimensional
spacetime. As usual, time runs up the diagram, and we use units such that light
rays move along lines at 45 o angles to the vertical. Note that any worldline of a
light ray in the 3-D spacetime that happens to lie entirely in the 2-D spacetime
will also be the worldline of a light ray in the 2-D spacetime, since it is clearly
a curve of zero proper time. A pair of such crossed light rays are shown below
where the light cone of the 3-D spacetime intersects the 2-D spacetime.