Sunday, February 14, 2016

The Equivalence Post

About twenty years ago--maybe right around the time LIGO was finally getting funding, when the gravitational waves it just detected were still a couple dozen star systems away--my elementary school class did a living wax museum. We researched a historical figure, dressed up as our subject, and, when a "visitor" to the museum pressed a red dot on our hand, recited a first-person speech based on our research. Unrepentant early nerd that I was, I chose Albert Einstein.

I don't really remember anything about the contents of my monologue. I probably gave a brief biographical sketch, but likely left out the part where Einstein bribed his first wife into divorce with Nobel money he'd yet to receive. I probably talked about the theory of relativity and how it merged space and time, but likely didn't include anything about Riemannian geometry and metric tensors.

My knowledge of the scientist and his science was patchy, to be sure, but that didn't stop me from admiring him. Einstein is the model of the lone genius working tirelessly, using nothing more than the power of his mind to change the world. For a long time, I imagined he and I were equivalent. I imagined that I alone knew the secrets of the universe and that my solitude represented nothing more than the gap in intellect between myself and others.

Before the inevitable deconstruction of that paragraph, let's talk a bit about Einstein the genius. While E=mc2 is his most famous equation, it's not the equation that made him famous. Physicists will tell you that general relativity was his crowning achievement.

GR grew out of Einstein's attempt to extend his special theory of relativity to gravity. SR and electromagnetism fit together perfectly, but gravity did not behave. According to Newton, gravity acts instantaneously, and that didn't sit well with light speed being the ultimate limit. To reconcile gravity with relativity, Einstein looked at a subtle difference between the electrostatic force and the force of gravity.

When two charged particles are sitting next to each other, the electrostatic force that one feels is proportional to the product of their charges divided by the square of the distance between them--simple enough. When two masses are sitting next to each other, the gravitational force on one is proportional to the product of their masses divided by the square of the distance between them. The forces are nearly identical, just swapping charge for mass.

But when a particle feels a force, it follows Newton's second law and accelerates by an amount inversely proportional to its mass, which is what inertia is all about. This means the mass term from gravity and the mass term from inertia cancel out and bodies under the force of gravity experience the same acceleration regardless of their masses. We know this; it's just the idea that a hammer and a feather (ignoring air resistance) fall at the same rate.

Thank you, NASA.
This quirk of gravity gets called the equivalence principle, because it seems to show that "gravitating" mass and "inertial" mass are equivalent, even though there's no particular reason why they need to be.

As Einstein thought about this peculiarity of gravity, he was struck with what he called "the happiest thought" of his life. He postulated a modification to the equivalence principle, which is that being in a gravitational field is equivalent to be in an accelerated reference frame. What he meant was that gravity is not a real force but an effect we observe, so there's no difference between your car seat pushing up against you when you hit the gas and the Earth holding you down.

The link to the other equivalence principle is that, in free fall, any object falling with you moves at the same rate, and the same thing is true in an accelerated reference frame, because the acceleration you feel is a result of the frame (your car, a rocket) and not your mass.

This happiest thought led Einstein to the conclusion that being in free fall in a gravitational field is just as "natural" as being at rest. When you do feel a force (your car seat, the ground), that's just an object getting in the way of your natural path through spacetime. As usual for Einstein, his next step was to imagine what this meant for light.

Assuming his principle is true, weird things happen in gravity. Say you're in a rocket ship at rest in space. If a beam of light comes in one window, it will trace a straight line through the rocket ship and out another window. If you're moving at a constant speed, you observe the exact same thing, because special relativity says you can't tell the difference between different inertial frames.

If you're accelerating, the light will trace out a parabolic curve, because you're moving faster when the light leaves the rocket than when the light enters it. The equivalence principle says you can't tell the difference between gravity and acceleration, so the same thing should happen if you're in a gravitational field. Light passing near the Sun, for example, will curve.

Now it's all well and good to say this happens because of the equivalence principle, but that's not a mechanism. If there isn't a force causing the light to curve, what's doing it? Einstein says this is the wrong question to ask and that what looks like a force is just light taking the only path available.

Here's an imperfect analogy: imagine you're driving up a mountain, maneuvering through twisting switchbacks. If you veer one way, you fall off the mountain. If you veer the other way, you crash into the side of it. So you stick to one narrow path. To the GPS satellites monitoring the position of your phone (but not the mountain or the road), it looks as if your phone, you, and the car are being pushed around by some mysterious force, but in reality you are simply following the only path available.

Except you might think, well that works for light zooming around at 300,000 km/s, but what if there's nothing propelling me? Why am I following any path at all? And the answer is that we are all following a path constantly through spacetime. We're moving forward through time. But in the presence of a gravitational field, spacetime gets warped, and your straight path through it moves a little bit out of time and into space. The "speed" you had going through time gets converted into speed in space, which is why clocks slow down close to a black hole.

Figuring out the specifics of how mass could warp spacetime took Einstein about a decade, but he finally succeeded in 1915, giving the world general relativity. With it came a number of predictions, including the bending of starlight, the correct shape of Mercury's orbit, and the fact that accelerating masses will send out gravitational waves that stretch and shrink spacetime as they pass by. Finally detecting those waves reaffirmed Einstein's genius one more time a century after he first proposed them. And all of that came from Einstein tinkering around with the fact that all objects fall at the same speed.

I said earlier that I equated myself to Einstein, but the truth is I'm no Einstein. I'm a pretty smart guy, but not a genius, and certainly not one of the greatest scientific minds in history, capable of deducing fundamental and quantitative physical truths about the universe from simple thought experiments. What can I possibly hope to achieve compared to that?

But there is an equivalence between me and Einstein, because in reality he was no Einstein, either. It took him a decade to complete general relativity because, talented though he was at math, he was not a mathematician and had to learn an entirely foreign branch of it to make his theory work. He got help from a mathematician friend of his, Marcel Grossmann, who was familiar with Riemannian geometry. That branch of math was invented in the 19th century by a couple of guys, including Bernhard Riemann.

The idea of looking at space and time as a unified thing was partly inspired by Hermann Minkowski, who applied geometrical concepts to Einstein's special relativity. Before Einstein even got to special relativity, which was critical for getting to GR, he frequently discussed difficult subjects with a group of likeminded friends that maybe ironically called themselves the Olymipa Academy. And most of the pieces for SR were put in place by earlier physicists, such as Hendrik Lorentz and George FitzGerald.

Black holes were first theorized about by Karl Schwarzschild, who found one of the simplest solutions to Einstein's field equations while fighting in the trenches during WWI. Roy Kerr figured out how rotating black holes behave. And many others over the ensuing decades contributed to the theory.

As far as gravitational waves are concerned, Einstein himself waffled as far as whether they even existed. But even so, he originally showed only that they could exist and radiate away energy. Solving general relativity for the shape of gravitational waves emitted by two inspiraling, merging black holes took until the 90s. In fact, it was only accomplished with the help of supercomputers using numerical techniques.

And even ignoring the many contributions from theorists not named Einstein, his prediction about gravitational waves would have meant nothing if we did not have the means to detect them. The feat accomplished by LIGO this past week involved scientists who are experts in interferometry, optics, vacuum chambers, thermodynamics, seismology, statistics, etc. The effort required theorists, as well as experimentalists, engineers, and technicians.

I don't mean to imply that Einstein's work would be for naught without the janitors who cleaned his office, that he couldn't have done it without all the little people supporting him. I mean that Einstein's contribution to the discovery was only one part of a vast web of contributions by a host of extremely talented people, alive and dead, who did things Einstein couldn't have done.

On Thursday, we all learned the magnitude of what they had accomplished. Rumors of the discovery had been swirling around for awhile before it was announced. By the time I arrived at school on Thursday to watch the LIGO press conference, I had a pretty good idea of what they were going to say.

Yet that didn't detract from the occasion. Packed into a lounge in the physics department, students, TAs, professors, and I--maybe a hundred altogether--watched the press conference webcast on a giant screen. We all cheered when the discovery was confirmed and cheered again when we heard the primary paper had already been peer reviewed. Half an hour in, I had to leave to go to my theoretical astrophysics course. There, the professor and TA set up a projector and we all continued to watch the press conference. When the webcast ended, the professor took questions about gravitational waves.

Being a part of that, in the minutest and most indirect way, was thrilling. It was a day when Einstein's greatest theory was confirmed yet again, when a new field of astronomy began, and when a thousand scientists got to tell the whole world about the amazing thing they had discovered.

There's a certain--possibly strained--equivalence to my wax museum Einstein moment from 20 years earlier. School was involved, as well as a story about Einstein. But this time I was listening to that story. My passion for science and learning has remained constant, but the attitude has changed. Back then, and for a very long time after that, I took joy in knowing more than others, in being the smartest guy in the room.

Now I know that's not the case. But I also know it doesn't matter. We just don't learn about the universe by sitting alone and thinking brilliant thoughts. That is, at most, one part of the process. So I don’t have to be a mythical genius to contribute. I can be a part of something amazing, of humanity's quest to understand the world around us, just by collaborating with others who are as passionate as I am. I haven't done it yet, obviously, but just as Einstein's magnificent theory has been reaffirmed, so too has my drive to be a scientist.

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