pi-day post: in honor of Steven Hawking

March 14 is commonly written as “3-14” or “3 / 14” in the U.S., and because of that, has been dubbed “pi day” to celebrate the mathematical constant π = 3.14159265…

The number π is celebrated for good reasons, for one, it’s an important geometric ratio that is both useful and ubiquitous. It is also easy to define, it’s the ratio of the circumferance of a circle to it’s diameter.

That means:

(length of the Earth’s equator) / (diameter of Earth) = (circumferance of a dime) / (diameter of a dime) = π

Yesterday the great physicist Steven Hawking died, Hawking needs no introduction. His work on black holes is a good topic to dive into on π-day. The simplest type of black hole is a Schwarzschild black hole, that is a spherical object (usually a star) whose mass has collapsed to the Schwarzschild radius. For example, our Sun would have to collapse to about 6 km in diameter to turn into a black hole.

sun black hole

What’s fascinating about General Relativity is that black holes have a “surface of no return” called the Event Horizon. If you imagine the sun collapsed to 6km wide and became a black hole, the orbits of all the planets would stay the same, but the light would disappear. Also, if Matthew McConaughey traveled up to the edge of the black hole and passed the event horizon, he would never be able to get out. Not even light could escape, in fact, the center of a black hole (a singularity) is similar to the future, in that there is no escaping it. Once you fall past the horizon, you will not escape the singularity.

What Hawking discovered is incredible, he used general relativity to study what would happen near an event horizon with virtual particles. Virtual particles are pairs of particles and antiparticles. For example, an electron has a negative charge and a positron has a positive charge, and the same mass as an electron. If an electron and a positron meet, they annihilate into pure energy with no mass.

virtual particles

Physicists have known since before Hawking that empty space is teeming with these particle-antiparticle pairs that appear out of nothing and then annhiliate again. Physicists have tested the existence of these things, one famous consequence of their existence is the Casimir effect.

Back to Hawking, if you imagine a small region of space near an event horizon, then sometimes a pair of virtual particles will appear where one falls into the black hole, and the other escapes. This would appear to outside observers as a new particle radiating away from the black hole. But the conservation of energy appears to be violated. Hawking discovered that the energy would in fact be conserved by the black hole losing mass. So over time, black holes will evaporate! This is called Hawking radiation, after it’s discoverer.

hawking radiation

So the diameter of a Schwarzschild black hole shrinks until all the mass is gone. At all times during this process, it’s circumferance and diameter will still have the same ratio, 3.14159… until finally, after a finite amount of time has passed, it will cease to exist. That brilliant flash of Hawking radiation filling the cosmos will never be quite as bright as the mind that discovered it.