Black Holes

By Matthew Kahle 

Matthew Kahle, Member (2010-11) in the School of Mathematics, writes about his interest in thinking about what it might be like inside a black hole. This illustration (Figure 1.), from Kip Thorne's Black Holes and Time Warps: Einstein's Outrageous Legacy (W. W. Norton & Company, Inc., 1994), suggests a few probabilities.

I sometimes like to think about what it might be like inside a black hole. What does that even mean? Is it really “like” anything inside a black hole? Nature keeps us from ever knowing. (Well, what we know for sure is that nature keeps us from knowing and coming back to tell anyone about it.) But mathematics and physics make some predictions.

John Wheeler suggested in the 1960s that inside a black hole the fabric of spacetime might be reduced to a kind of quantum foam. Kip Thorne described the idea in his book Black Holes & Time Warps as follows (see Figure 1).

“This random, probabilistic froth is the thing of which the singularity is made, and the froth is governed by the laws of quantum gravity. In the froth, space does not have any definite shape (that is, any definite curvature, or even any definite topology). Instead, space has various probabilities for this, that, or another curvature and topology. For example, inside the singularity there might be a 0.1 percent probability for the curvature and topology of space to have the form shown in (a), and a 0.4 percent probability for the form in (b), and a 0.02 percent probability for the form in (c), and so on.”

In other words, perhaps we cannot say exactly what the properties of spacetime are in the immediate vicinity of a singularity, but perhaps we could characterize their distribution. By way of analogy, if we know that we are going to flip a fair coin a thousand times, we have no idea whether any particular flip will turn up heads or tails. But we can say that on average, we should expect about five hundred heads. Moreover, if we did the experiment many times we should expect a bell-curve shape (i.e., a normal distribution), so it is very unlikely, for example, that we would see more than six hundred heads.

By Juan Maldacena 

Albert Einstein, pictured at left with J. Robert Oppenheimer at the Institute, tried to disprove the notion of black holes that his theory of general relativity and gravity seemed to predict. Oppenheimer used Einstein's theory to show how black holes could form.

The ancients thought that space and time were preexisting entities on which motion happens. Of course, this is also our naive intuition. According to Einstein’s theory of general relativity, we know that this is not true. Space and time are dynamical objects whose shape is modified by the bodies that move in it. The ordinary force of gravity is due to this deformation of spacetime. Spacetime is a physical entity that affects the motion of particles and, in turn, is affected by the motion of the same particles. For example, the Earth deforms spacetime in such a way that clocks at different altitudes run at different rates. For the Earth, this is a very small (but measurable) effect. For a very massive and very compact object the deformation (or warping) of spacetime can have a big effect. For example, on the surface of a neutron star a clock runs slower, at 70 percent of the speed of a clock far away.

In fact, you can have an object that is so massive that time comes to a complete standstill. These are black holes. General relativity predicts that an object that is very massive and sufficiently compact will collapse into a black hole. A black hole is such a surprising prediction of general relativity that it took many years to be properly recognized as a prediction. Einstein himself thought it was not a true prediction, but a mathematical oversimplification. We now know that they are clear predictions of the theory. Furthermore, there are some objects in the sky that are probably black holes.  

Black holes are big holes in spacetime. They have a surface that is called a “horizon.” It is a surface that marks a point of no return. A person who crosses it will never be able to come back out. However, he will not feel anything special when he crosses the horizon. Only a while later will he feel very uncomfortable when he is crushed into a “singularity,” a region with very high gravitational fields. The horizon is what makes black holes “black”; nothing can escape from the horizon, not even light. Fortunately, if you stay outside the horizon, nothing bad happens to you. The singularity remains hidden behind the horizon.