Notes on Relativity and Cosmology - Physics Department, UCSB

270 CHAPTER 9. BLACK HOLES
The point is that, as we have seen, it is often useful to compare what an observer
very far from the black hole sees to what one sees close to the black hole. We
say that an observer very far from the black hole is “at infinity.” Comparing
infinity with finite positions is even more important for more complicated sorts
of black holes that we have not yet discussed. However, it is difficult to draw
infinity on our diagrams since infinity is after all infinitely far away.
How can we draw a diagram of an infinite spacetime on a finite piece of paper?
Think back to the Escher picture of the Lobachevskian space. By ‘squishing’ the
space, Escher managed to draw the infinitely large Lobachevskian space inside
a finite circle. If you go back and try to count the number of fish that appear
along on a geodesic crossing the entire space, it turns out to be infinite. It’s just
that most of the fish are drawn incredibly small. Escher achieved this trick by
letting the scale vary across his map of the space. In particular, at the edge an
infinite amount of Lobachevskian space is crammed into a very tiny amount of
Escher’s map. In some sense this means that his picture becomes infinitely bad
at the edge, but nevertheless we were able to obtain useful information from it.
We want to do much the same thing for our spacetimes. However, for our case
there is one catch: As usual, we will want all light rays to travel along lines at 45
degrees to the vertical. This will allow us to continue to read useful information
from the diagram. This idea was first put forward by (Sir) Roger Penrose 11 ,
so that the resulting pictures are often called “Penrose Diagrams.” They are
also called “conformal diagrams” – conformal is a technical word related to the
rescaling of size.
Let’s think about how we could draw a Penrose diagram of Minkowski space.
For simplicity, let’s consider our favorite case of 1+1 dimensional Minkowski
space. Would you like to guess what the diagram should look like?
As a first guess, we might try a square or rectangle. However, this guess has a
problem associated with the picture below. To see the point, consider any light
ray moving to the right in 1+1 Minkowski space, and also consider any light
ray moving to the left. Any two such light rays are guaranteed to meet at some
event. The same is in fact true of any pair of leftward and rightward moving
objects since, in 1 space dimension, there is no room for two objects to pass
each other!
Left- and right- moving objects
always collide when space has only
one dimension
However, if the Penrose diagram for a spacetime is a square, then there are in
fact leftward and rightward moving light rays that never meet! Some examples
are shown on the diagram below.
11 Penrose is a mathematician and physicist who is famous for a number of things.