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Black Holes to Blackboards: God Divided by Zero

Jeffrey F. Lockwood
Sahuaro High School

Laying your hands on a black hole is hard (and dangerous) to do, but there are ways to understand these objects and avoid the pain of dimension-bending.

A couple of weeks ago, I was sitting in the second-floor bathroom at Steward Observatory, not thinking at all about astronomy or writing columns. On the stall door in front of me, written perhaps by a clever astronomy student, was an equation in bold black ink: BLACK HOLES = GOD/0. As I pondered the philosophical significance of the equation, I realized that I have never tried to bring black holes to blackboards as my column title states. One of the most esoteric and fascinating objects for students to ponder, a black hole comes with virtually no lab activities. How do you lay your hands on a black hole and survive?

There are, in fact, a few demonstrations that can bring about some understanding of black holes. It’s pretty easy to describe a non-rotating black hole: a single point of infinite density surrounded by a protective sheath called the event horizon. Throw in rotation, though, and things get more complicated. I have never liked the funnel-like diagrams in most textbooks that try to show a black hole’s distortion of space-time. A sharp student will always question this representation, which shows a 3-dimensional warpage, and ask what happens when you approach from a different direction, like from “underneath.”

A black balloon is a much better model, the singularity now imagined to be at the center of the balloon. The mass of a black hole solely determines the size of its event horizon, the imaginary surface around the black hole that marks the border between our universe and that which is unknowable “beyond the horizon.” Karl Schwarzschild discovered early in this century that the event horizon’s radius is equal to 2GM/c2, where G is the Universal Gravitation Constant, c is the speed of light, and M is the mass of the black hole. With the balloon model, the size of the rubber event horizon becomes real. Students can picture what a balloon with a radius of 30 km (Rsch for a 10 solar-mass black hole) would look like if they approached it in their make-believe spaceship. And the notion of an event horizon becomes more real to them if you ask what they can see beyond the black rubber sheet.

embedding diagram of black hole
Down, up, and through the funnel. An embedding diagram is generally a good representation of a black hole’s warping of nearby space-time. But such 2-dimensional illustrations can also cause conceptual problems.

Formation of black holes in supernovae can be demonstrated by having students cover the balloon with aluminum foil to model the core of a massive star about to die. Then have them squeeze the balloon: This simulates the action of the enormous mass of the star collapsing inward on the core. When the balloon pops (students love this part!), the foil can be compressed to a smaller, more dense sphere to simulate the initial compacting of the star’s core. Ask your students to measure the mass of the aluminum core and to calculate the change in density of their “stellar core”; next, have them calculate the density of the foil core if nature allowed them to physically compress it into a sphere of 1 mm, then 1 micrometer, and finally 10-14 m in radius, the size of an atomic nucleus. Typical densities obtained by students are 0.003 g/cm3 for an inflated foil balloon, 0.5 g/cm3 for a compressed foil sphere, and up to a very impressive 1037 g/cm3 for a foil core squeezed inside of a nucleus!

Students now know that black holes have lots of mass and infinite density and that they have shrunk from our view. What else can they know about black holes? Because of all that matter compressed into such a small volume, nothing escapes from them, and all knowledge of objects falling into a black hole is lost once they cross the event horizon. Theoretical astrophysicists tell us that (theoretically speaking) the only things we can know about black holes are their masses, electric charges, and angular momenta. This simplicity is embodied in the No-Hair Theorem: “Black holes have no hair.”

Discussing black holes usually gives students their first chance to confront the fact that scientists don’t and can’t know everything about the universe. We can’t know what craziness happens inside an event horizon because physics and mathematics can’t deal with singularities. Scientists created a law to make sure the universe will never be exposed to a singularity, shielded as it is by the event horizon. The Law of Cosmic Censorship drolly states, “Thou shalt not have naked singularities.”

Fun concepts can be discussed once the basic structure and nature of a black hole is introduced. The warping of space-time by gravity, the existence of wormholes, the existence and nature of a Supreme Being, all become accessible. Student imaginations will run wild with such ideas. And for this reason, the study of black holes can be a link to other areas of the curriculum and provide opportunities for student creativity. They can write stories or read classic ones like Larry Niven’s “The Hole Man.” They can construct or draw their own models of black holes in binary star systems or at the centers of quasars or galaxies. They can even debate whether BLACK HOLES = GOD/0 is a good description of these out-of-our-universe objects!

JEFFREY F. LOCKWOOD is a high-school and college astronomy and physics teacher at Sahuaro High School and Pima Community College in Tucson, Ariz. While always bothered by space-time singularities, he confesses that black holes are really pretty neat. His email address is