Notes on Relativity and Cosmology - Physics Department, UCSB
Notes on Relativity and Cosmology - Physics Department, UCSB 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.
9.6. BLACK HOLE ODDS AND ENDS 271 These light rays do not meet So, the rectangular Penrose diagram does not represent Minkowski space. What other choices do we have? A circle turns out to have the same problem. After a little thought, one finds that the only thing which behaves differently is a diamond: That is to say that infinity (or at least most of it) is best associated not which a place or a time, but with a set of light rays! In 3+1 dimensions, we can as usual decide to draw just the r, t coordinates. In this case, the Penrose diagram for 3+1 Minkowski space is drawn as a half-diamond: 9.6.3 Penrose Diagrams for Black holes Using the same scheme, we can draw a diagram that shows the entire spacetime for the eternal Schwarzschild black hole. Remember that the distances are no longer represented accurately. As a result, some lines that used to be straight get bent 12 . For example, the constant r curves that we drew as hyperbolae before appear somewhat different on the Penrose diagram. However, all light rays still travel along straight 45 degree lines. The result is: 12 This is the same effect that one finds on flat maps of the earth where lines that are really straight (geodesics) appear curved on the map.
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9.6. BLACK HOLE ODDS AND ENDS 271<br />
These light rays do not meet<br />
So, the rectangular Penrose diagram does not represent Minkowski space. What<br />
other choices do we have? A circle turns out to have the same problem. After<br />
a little thought, <strong>on</strong>e finds that the <strong>on</strong>ly thing which behaves differently is a<br />
diam<strong>on</strong>d:<br />
That is to say that infinity (or at least most of it) is best associated not which<br />
a place or a time, but with a set of light rays! In 3+1 dimensi<strong>on</strong>s, we can as<br />
usual decide to draw just the r, t coordinates. In this case, the Penrose diagram<br />
for 3+1 Minkowski space is drawn as a half-diam<strong>on</strong>d:<br />
9.6.3 Penrose Diagrams for Black holes<br />
Using the same scheme, we can draw a diagram that shows the entire spacetime<br />
for the eternal Schwarzschild black hole. Remember that the distances are no<br />
l<strong>on</strong>ger represented accurately. As a result, some lines that used to be straight<br />
get bent 12 . For example, the c<strong>on</strong>stant r curves that we drew as hyperbolae<br />
before appear somewhat different <strong>on</strong> the Penrose diagram. However, all light<br />
rays still travel al<strong>on</strong>g straight 45 degree lines. The result is:<br />
12 This is the same effect that <strong>on</strong>e finds <strong>on</strong> flat maps of the earth where lines that are really<br />
straight (geodesics) appear curved <strong>on</strong> the map.