Right Reason

Pi Day 2014

 

In celebration of both Pi Day and Albert Einstein's 135th birthday, Sean Carroll writes about the inclusion of the celebrated constant in Einstein's field equation for gravity:

Each year, the 14th of March is celebrated by scientifically-minded folks for two good reasons. First, it’s Einstein’s birthday (happy 135th, Albert!). Second, it’s Pi Day, because 3/14 is the closest calendrical approximation we have to the decimal expansion of pi, π =3.1415927….

Both of these features — Einstein and pi — are loosely related by playing important roles in science and mathematics. But is there any closer connection?

Of course there is. We need look no further than Einstein’s equation. I mean Einstein’sreal equation — not E=mc2, which is perfectly fine as far as it goes, but a pretty straightforward consequence of special relativity rather than a world-foundational relationship in it’s own right. Einstein’s real equation is what you would find if you looked up “Einstein’s equation” in the index of any good GR textbook: the field equation relating the curvature of spacetime to energy sources, which serves as the bedrock principle of general relativity. It looks like this:

                                 

It can look intimidating if the notation is unfamiliar, but conceptually it’s quite simple; if you don’t know all the symbols, think of it as a little poem in a foreign language. In words it is saying this:

(gravity) = 8 π G × (energy and momentum).

Not so scary, is it? The amount of gravity is proportional to the amount of energy and momentum, with the constant of proportionality given by 8πG, where G is a numerical constant.

Hey, what is π doing there? It seems a bit gratuitous, actually. Einstein could easily have defined a new constant H simply be setting H=8πG. Then he wouldn’t have needed that superfluous 8π cluttering up his equation. Did he just have a special love for π, perhaps based on his birthday?

The real story is less whimsical, but more interesting. Einstein didn’t feel like inventing a new constant because G was already in existence: it’s Newton’s constant of gravitation, which makes perfect sense. General relativity (GR) is the theory that replaces Newton’s version of gravitation, but at the end of the day it’s still gravity, and it has the same strength that it always did.

So the real question is, why does π make an appearance when we make the transition from Newtonian gravity to general relativity?

 

I once had an econometrics professor who noted that the repetition of mathematical constants like pi and euler's number across different disciplines was the best argument for the world being created by intelligent design. Just as pi has a place in Einstein's equation, so too it has a place in the normal probability density function:

Is it more impressive to find a fully operational watch in a field or pi in equations describing seemingly unrelated phenomena?

 

Black Hole Jet

 

Wired's space photo of the day, a jet from a supermassive black hole:

 

Hawking on Apparent Horizons

 

A new paper, a new thing to talk about. In the paper "Information Preservation and Weather Forecasting for Black Holes," Stephen Hawking, who had previously reversed his belief that information is lost in a black hole in 2004 after a bet with John Preskill on the topic, now argues that there is no such thing as what it commonly thought of as a black hole, that is a ravenous maw in space which will consume everything within its reach and whatever is unfortunate enough to fall in it is lost to the universe forever. Instead, there are only objects which can capture matter for so long before eventually that matter, in one form or another, escapes with its information preserved. No event horizons, only apparant horizons.

 National Geographic has a story on the topic: 

A new study from Hawking also says that black holes may not possess "firewalls," destructive belts of radiation that some researchers have proposed would incinerate anything that passes through them but others scientists deem an impossibility.

The conventional view of black holes posits that their gravitational pull is so powerful that nothing can escape from them—not even light, which is why they're called black holes. The boundary past which there is supposedly no return is known as the event horizon.

In this conception, all information about anything that ventures past a black hole's event horizon is destroyed. On the other hand, quantum physics, the best description so far of how the universe behaves on a subatomic level, suggests that information cannot ever be destroyed, leading to a fundamental conflict in theory.

No Event Horizons

Now Hawking is suggesting a resolution to the paradox: Black holes do not possess event horizons after all, so they do not destroy information.

"The absence of event horizons means that there are no black holes, in the sense of regimes from which light can't escape," Hawking wrote in a paper he posted online on January 22. The paper was based on a talk he gave last August at a workshop at the Kavli Institute for Theoretical Physics in Santa Barbara, California.

 

Why the black hole information paradox matters so much for our understanding of the universe is best put by Hawking himself in a lecture he in 2008, entitled "Into a Black Hole":

As particles escape from a black hole  the hole will lose mass, and shrink.This will increase the rate of emission of particles. Eventually, the black hole will lose all its mass, and disappear. What then happens to all the particles and unlucky astronauts,  that fell into the black hole. They can't just re-emerge when the black hole disappears. The particles that come out of a black hole, seem to be completely random, and to bear no relation to what fell in. It appears that the information about what fell in, is lost, apart from the total amount of mass, and the amount of rotation. But if information is lost, this raises a serious problem that strikes at the heart of our understanding of science. For more than 200 years, we have believed in Scientific determinism, that is, that the laws of science, determine the evolution of the universe. This was formulated by La~plass as, If we know the state of the universe at one time, the laws of science will determine it at all future and past times. Napoleon is said to have asked La~plass how God fitted into this picture. La~plass replied, Sire, I have not needed that hypothesis. I don't think that La~plass was claiming that God didn't exist. It is just that He doesn't intervene, to break the laws of Science. That must be the position of every scientist. A scientific law, is not a scientific law, if it only holds when some supernatural being, decides to let things run, and not intervene.  

In La~plass's determinism, one needed to know the positions and speeds of all particles at one time in order to predict the future. But according to the uncertainty relation, the more accurately you know the positions, the less accurately you can know the speeds, and vice versa. In other words, you can't know ~both the positions, ~and the speeds, accurately. How then can you predict the future accurately? The answer is, that although one can't predict the positions and speeds separately, one ~can predict what is called, the quantum state. This is something from which both positions and speeds can be calculated, to a certain degree of accuracy. We would still expect the universe to be deterministic, in the sense that if we knew the quantum state of the universe at one time, the laws of science should enable us predict it at any other time. 

If information were lost in black holes, we wouldn't be able to predict the future, because a black hole could emit any collection of particles. It could emit a working television set, or a leather bound volume of the complete works of Shakespeare, though the chance of such exotic emissions is very low. It is much more likely to be thermal Radiation, like the glow from red hot metal.  It might seem that it wouldn't matter very much if we couldn't predict what comes out of black holes.There aren't any black holes near us. But it is a matter of principle. If determinism breaks down with black  holes, it could break down in other situations. There could be virtual black holes that appear as fluctuations out of the vacuum, absorb one set of particles, emit another, and disappear into the vacuum again. Even worse, if determinism breaks down, we can't be sure of our past history either.The history books and our memories could just be illusions. It is the past that tells us who we are. Without it, we lose our identity.