May 24, 2008
Warning: Mach's Principle is not experimentally proven, though it is widely considered to be credible.
Centuries ago, when Galileo was promoting the Copernican model in which the Earth spun on its axis and traveled around the Sun, there was great opposition from those who trusted their common sense:
"How could the Earth be moving? I don't feel it moving! The ground beneath my feet seems perfectly steady!"
And lo, Galileo said: If you were on a ship sailing across a perfectly level sea, and you were in a room in the interior of the ship, you wouldn't know how fast the ship was moving. If you threw a ball in the air, you would still be able to catch it, because the ball would have initially been moving at the same speed as you and the room and the ship. So you can never tell how fast you are moving.
This would turn out to be the beginning of one of the most important ideas in the history of physics. Maybe even the most important idea in all of physics. And I'm not talking about Special Relativity.
Suppose the entire universe was moving. Say, the universe was moving left along the x axis at 10 kilometers per hour.
If you tried to visualize what I just said, it seems like you can imagine it. If the universe is standing still, then you imagine a little swirly cloud of galaxies standing still. If the whole universe is moving left, then you imagine the little swirly cloud moving left across your field of vision until it passes out of sight.
But then, some people think they can imagine philosophical zombies: entities who are identical to humans down to the molecular level, but not conscious. So you can't always trust your imagination.
Forget, for a moment, anything you know about relativity. Pretend you live in a Newtonian universe.
In a Newtonian universe, 3+1 spacetime can be broken down into 3 space dimensions and 1 time dimension, and you can write them out as 4 real numbers, (x, y, z, t). Deciding how to write the numbers involves seemingly arbitrary choices, like which direction to call 'x', and which perpendicular direction to then call 'y', and where in space and time to put your origin (0, 0, 0, 0), and whether to use meters or miles to measure distance. But once you make these arbitrary choices, you can, in a Newtonian universe, use the same system of coordinates to describe the whole universe.
Suppose that you pick an arbitrary but uniform (x, y, z, t) coordinate system. Suppose that you use these coordinates to describe every physical experiment you've ever done—heck, every observation you've ever made.
Next, suppose that you were, in your coordinate system, to shift the origin 10 meters to the left along the x axis. Then if you originally thought that Grandma's House was 400 meters to the right of the origin, you would now think that Grandma's House is 410 meters to the right of the origin. Thus every point (x, y, z, t) would be relabeled as (x' = x + 10, y' = y, z' = z, t' = t).
You can express the idea that "physics does not have an absolute origin", by saying that the observed laws of physics, as you generalize them, should be exactly the same after you perform this coordinate transform. The history may not be written out in exactly the same way, but the laws will be written out the same way. Let's say that in the old coordinate system, Your House is at (100, 10, -20, 7:00am) and you walk to Grandma's House at (400, 10, -20, 7:05am). Then you traveled from Your House to Grandma's House at one meter per second. In the new coordinate system, we would write the history as (110, 10, 20, 7:00am) and (410, 10, -20, 7:05am) but your apparent speed would come out the same, and hence so would your acceleration. The laws governing how fast things moved when you pushed on them—how fast you accelerated forward when your legs pushed on the ground—would be the same.
Now if you were given to jumping to conclusions, and moreover, given to jumping to conclusions that were exactly right, you might say:
"Since there's no way of figuring out where the origin is by looking at the laws of physics, the origin must not really exist! There is no (0, 0, 0, 0) point floating out in space somewhere!"
Which is to say: There is just no fact of the matter as to where the origin "really" is. When we argue about our choice of representation, this fact about the map does not actually correspond to any fact about the territory.
Now this statement, if you interpret it in the natural way, is not necessarily true. We can readily imagine alternative laws of physics, which, written out in their most natural form, would not be insensitive to shifting the "origin". The Aristotelian universe had a crystal sphere of stars rotating around the Earth. But so far as anyone has been able to tell, in our real universe, the laws of physics do not have any natural "origin" written into them. When you write out your observations in the simplest way, the coordinate transform x' = x + 10 does not change any of the laws; you write the same laws over x' as over x.
As Feynman said:
Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong. For example, some philosopher or other said it is fundamental to the scientific effort that if an experiment is performed in, say, Stockholm, and then the same experiment is done in, say, Quito, the same results must occur. That is quite false. It is not necessary that science do that; it may be a fact of experience, but it is not necessary...
What is the fundamental hypothesis of science, the fundamental philosophy? We stated it in the first chapter: the sole test of the validity of any idea is experiment...
If we are told that the same experiment will always produce the same result, that is all very well, but if when we try it, it does not, then it does not. We just have to take what we see, and then formulate all the rest of our ideas in terms of our actual experience.
And so if you regard the universe itself as a sort of Galileo's Ship, it would seem that the notion of the entire universe moving at a particular rate—say, all the objects in the universe, including yourself, moving left along the x axis at 10 meters per second—must also be silly. What is it that moves?
If you believe that everything in a Newtonian universe is moving left along the x axis at an average of 10 meters per second, then that just says that when you write down your observations, you write down an x coordinate that is 10 meters per second to the left, of what you would have written down, if you believed the universe was standing still. If the universe is standing still, you would write that Grandma's House was observed at (400, 10, -20, 7:00am) and then observed again, a minute later, at (400, 10, -20, 7:01am). If you believe that the whole universe is moving to the left at 10 meters per second, you would write that Grandma's House was observed at (400, 10, -20, 7:00am) and then observed again at (-200, 10, -20, 7:01am). Which is just the same as believing that the origin of the universe is moving right at 10 meters per second.
But the universe has no origin! So this notion of the whole universe moving at a particular speed, must be nonsense.
Yet if it makes no sense to talk about speed in an absolute, global sense, then what is speed?
It is simply the movement of one thing relative to a different thing! This is what our laws of physics talk about... right? The law of gravity, for example, talks about how planets pull on each other, and change their velocity relative to each other. Our physics do not talk about a crystal sphere of stars spinning around the objective center of the universe.
And now—it seems—we understand how we have been misled, by trying to visualize "the whole universe moving left", and imagining a little blurry blob of galaxies scurrying from the right to the left of our visual field. When we imagine this sort of thing, it is (probably) articulated in our visual cortex; when we visualize a little blob scurrying to the left, then there is (probably) an activation pattern that proceeds across the columns of our visual cortex. The seeming absolute background, the origin relative to which the universe was moving, was in the underlying neurology we used to visualize it!
But there is no origin! So the whole thing was just a case of the Mind Projection Fallacy—again.
Ah, but now Newton comes along, and he sees the flaw in the whole argument.
From Galileo's Ship we pass to Newton's Bucket. This is a bucket of water, hung by a cord. If you twist up the cord tightly, and then release the bucket, the bucket will spin. The water in the bucket, as the bucket wall begins to accelerate it, will assume a concave shape. Water will climb up the walls of the bucket, from centripetal force.
If you supposed that the whole universe was rotating relative to the origin, the parts would experience a centrifugal force, and fly apart. (No this is not why the universe is expanding, thank you for asking.)
Newton used his Bucket to argue in favor of an absolute space—an absolute background for his physics. There was a testable difference between the whole universe rotating, and the whole universe not rotating. By looking at the parts of the universe, you could determine their rotational velocity—not relative to each other, but relative to absolute space.
This absolute space was a tangible thing, to Newton: it was aether, possibly involved in the transmission of gravity. Newton didn't believe in action-at-a-distance, and so he used his Bucket to argue for the existence of an absolute space, that would be an aether, that could perhaps transmit gravity.
Then the origin-free view of the universe took another hit. Maxwell's Equations showed that, indeed, there seemed to be an absolute speed of light—a standard rate at which the electric and magnetic fields would oscillate and transmit a wave. In which case, you could determine how fast you were going, by seeing in which directions light seemed to be moving quicker and slower.
Along came a stubborn fellow named Ernst Mach, who really didn't like absolute space. Following some earlier ideas of Leibniz, Mach tried to get rid of Newton's Bucket by asserting that inertia was about your relative motion. Mach's Principle asserted that the resistance-to-changing-speed that determined how fast you accelerated under a force, was a resistance to changing your relative speed, compared to other objects. So that if the whole universe was rotating, no one would notice anything, because the inertial frame would also be rotating.
Or to put Mach's Principle more precisely, even if you imagined the whole universe was rotating, the relative motions of all the objects in the universe would be just the same as before, and their inertia—their resistance to changes of relative motion—would be just the same as before.
At the time, there did not seem to be any good reason to suppose this. It seemed like a mere attempt to impose philosophical elegance on a universe that had no particular reason to comply.
The story continues. A couple of guys named Michelson and Morley built an ingenious apparatus that would, via interference patterns in light, detect the absolute motion of Earth—as it spun on its axis, and orbited the Sun, which orbited the Milky Way, which hurtled toward Andromeda. Or, if you preferred, the Michelson-Morley apparatus would detect Earth's motion relative to the luminiferous aether, the medium through which light waves propagated. Just like Maxwell's Equations seemed to say you could do, and just like Newton had always thought you could do.
The Michelson-Morley apparatus said the absolute motion was zero.
This caused a certain amount of consternation.
Enter Albert Einstein.
The first thing Einstein did was repair the problem posed by Maxwell's Equations, which seemed to talk about an absolute speed of light. If you used a different, non-Galilean set of coordinate transforms—the Lorentz transformations—you could show that the speed of light would always look the same, in every direction, no matter how fast you were moving.
I'm not going to talk much about Special Relativity, because that introduction has already been written many times. If you don't get all indignant about "space" and "time" not turning out to work the way you thought they did, the math should be straightforward.
Albeit for the benefit of those who may need to resist postmodernism, I will note that the word "relativity" is a misnomer. What "relativity" really does, is establish new invariant elements of reality. The quantity √(t2 - x2 - y2 - z2) is the same in every frame of reference. The x and y and z, and even t, seem to change with your point of view. But not √(t2 - x2 - y2 - z2). Relativity does not make reality inherently subjective; it just makes it objective in a different way.
Special Relativity was a relatively easy job. Had Einstein never been born, Lorentz, Poincaré, and Minkowski would have taken care of it. Einstein got the Nobel Prize for his work on the photoelectric effect, not for Special Relativity.
General Relativity was the impressive part.
Einstein—explicitly inspired by Mach—and even though there was no experimental evidence for Mach's Principle—reformulated gravitational accelerations as a curvature of spacetime.
If you try to draw a straight line on curved paper, the curvature of the paper may twist your line, so that even as you proceed in a locally straight direction, it seems (standing back from an imaginary global viewpoint) that you have moved in a curve. Like walking "forward" for thousands of miles, and finding that you have circled the Earth.
In curved spacetime, objects under the "influence" of gravity, always seem to themselves—locally—to be proceeding along a strictly inertial pathway.
This meant you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime. You could accelerate the imaginary 'origin' of the universe, while changing a corresponding degree of freedom in the curvature of spacetime, and keep exactly the same laws of physics.
Einstein's theory further had the property that moving matter would generate gravitational waves, propagating curvatures. Einstein suspected that if the whole universe was rotating around you while you stood still, you would feel a centrifugal force from the incoming gravitational waves, corresponding exactly to the centripetal force of spinning your arms while the universe stood still around you. So you could construct the laws of physics in an accelerating or even rotating frame of reference, and end up observing the same laws—again freeing us of the specter of absolute space.
(I do not think this has been verified exactly, in terms of how much matter is out there, what kind of gravitational wave it would generate by rotating around us, et cetera. Einstein did verify that a shell of matter, spinning around a central point, ought to generate a gravitational equivalent of the Coriolis force that would e.g. cause a pendulum to precess. Remember that, by the basic principle of gravity as curved spacetime, this is indistinguishable in principle from a rotating inertial reference frame.)
We come now to the most important idea in all of physics. (Not counting the concept of "describe the universe using math", which I consider as the idea of physics, not an idea in physics.)
The idea is that you can start from "It shouldn't ought to be possible for X and Y to have different values from each other", or "It shouldn't ought to be possible to distinguish different values of Z", and generate new physics that make this fundamentally impossible because X and Y are now the same thing, or because Z no longer exists. And the new physics will often be experimentally verifiable.
We can interpret many of the most important revolutions in physics in these terms:
Whenever you find that two things seem to always be exactly equal—like inertial mass and gravitational charge, or two electrons—it is a hint that the underlying physics are such as to make this a necessary identity, rather than a contingent equality. It is a hint that, when you see through to the underlying elements of reality, inertial mass and gravitational charge will be the same thing, not merely equal. That you will no longer be able to imagine them being different, if your imagination is over the elements of reality in the new theory.
Likewise with the way that quantum physics treats the similarity of two particles of the same species. It is not that "photon A at 1, and photon B at 2" happens to look just like "photon A at 2, and photon B at 1" but that they are the same element of reality.
When you see a seemingly contingent equality—two things that just happen to be equal, all the time, every time—it may be time to reformulate your physics so that there is one thing instead of two. The distinction you imagine is epiphenomenal; it has no experimental consequences. In the right physics, with the right elements of reality, you would no longer be able to imagine it.
The amazing thing is that this is a scientifically productive rule—finding a new representation that gets rid of epiphenomenal distinctions, often means a substantially different theory of physics with experimental consequences!
(Sure, what I just said is logically impossible, but it works.)
Part of The Quantum Physics Sequence
Next post: "Relative Configuration Space"
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