Everything is Relative in the Magic Closet

The dog marches up to my computer as I’m checking my morning email. “What the heck is the deal with relativity?!?”

“Well, good morning to you, too. How are you this fine morning?”

“I’m fine, but I’m confused about relativity.” Sarcasm is totally lost on her.

i-cf62d220b910cb1e754b140a5b2fadd2-sm_office_bone.jpg“What are you confused about?”

“Well, you’ve got Special Relativity, right, and also General Relativity. Special Relativity is all about clocks that run slow when you’re moving, and bunnies that get smaller when you chase them, and General Relativity is all about bowling balls on rubber sheets.”

“Actually, that’s just an analogy for the way that mass distorts space-time in General Relativity. There are no real rubber sheets.”

“Fine, it’s about bending space and bending light and black holes and giant space worms tunneling between planets and stuff. The point is, it’s nothing at all like Special Relativity. How can you possibly call these two things part of the same theory?”

“Well, I admit, they do look pretty different–“

“That’s what I just said!”

“– but at some level, they’re both related to the movement of objects. The usual example people use to explain the similarities is an elevator.”

“An elevator?” She looks puzzled.

“You know, like when we go over to campus? We take the elevator up to my office.”

Recognition dawns. “Oh! The Magic Closet!”

Now I’m the confused one. “Magic Closet?”

“You know. It’s a tiny little room, and when you go inside and close the door, when the door opens again, you’re in a different place.”

“Ah. Yes, well, the point is, an elevator is basically a sealed box that you get in, and it may or may not be moving. Relativity is all about what you can and can’t learn about the motion of the box from inside it.”

“OK, that doesn’t help at all.”

“Well, as I explained before, Special Relativity tells us that there is no absolute frame of reference, and only the relative motion between frames matters. All of the laws of physics behave exactly the same way in an elevator that is moving at constant velocity as they do in an elevator that’s standing still. There’s no experiment you can do from inside the elevator to know whether you’re moving, or standing still.”

“Like that time when we got in the elevator, and you forgot to push any button at all, so we just sat there like idiots?” She wags her tail with exaggerated innocence.

“Yeah. Like that.” We’ll see how many treats she gets tonight. “Anyway, General Relativity extends that to accelerating frames and gravity.”

“But I thought you said that you could detect accelerating motion. Are you changing the story on me?”

“You can distinguish between accelerating motion and motion at constant speed, that’s still true. What General Relativity adds is that you can’t distinguish between accelerating motion and gravity. There’s no detectable difference between being in a stationary elevator in a gravitational field and being in an accelerating elevator in the absence of gravity.”

“So, wait, gravity doesn’t work in the Magic Closet?”

“No, no– gravity still works, it’s just that the sensation of an elevator accelerating is identical to a gravitational force. You know how when the elevator first starts up, you feel heavy for a second?”

“Yeah, I think.”

“When it first starts moving, you feel a little extra weight. That’s the effect of acceleration, and what you feel is just like gravity got stronger for a second.”

“It’s not much of an effect.”

“That’s because our building is old, and the elevators need work. If the elevator accelerated faster, it would be a bigger effect. If you had a fast enough elevator, the effect could be just as big as the gravitational attraction of the Earth, making it seem like your weight doubled. If you had an elevator that could accelerate that fast, and you put it out in space, there would be no way to tell whether you were in space accelerating, or sitting near the surface of the Earth standing still.”

“That would be a mean trick. Don’t do that.” She looks concerned.

“It’s just a thought experiment, to illustrate the idea of the equivalence between acceleration and gravitation.”

“Well, ok. But it’s still a mean gedankenexperiment.” She’s part German Shepherd, and likes to show off. “And anyway, I don’t see how this gets us to bending light.”

“Well, let’s think about what happens to a beam of light shining across the elevator. If the elevator is standing still, or moving at constant speed, the light will pass straight across the elevator.”

“Right. Light always moves in straight lines.”

“Not quite. If the elevator is accelerating, the light will appear to bend, as the floor catches up to it. The elevator would need to be accelerating very rapidly for it to be noticeable, but if it were, you would see that after one tick of a very fast clock, the floor moved up by one unit of distance, and after two ticks, it moved four units, and nine units after three ticks, and so on, catching up to the vertical position of the light. To a person inside the elevator, it would seem like the light was following a bent path, like an object accelerating down.”

“But wouldn’t that allow you to tell that you were moving? Doesn’t that violate relativity?”

“It lets you know that you’re not either standing still or moving with constant velocity, all right, but that’s Special Relativity. What you can’t do in General Relativity is tell the difference between accelerating motion and the effect of gravity.”

“But… doesn’t that mean that light needs to bend due to gravity?”

“Exactly. For relativity to work, light needs to bend in a gravitational field by exactly the amount that it would appear to bend if we were accelerating upward at the acceleration of gravity.”

“So why don’t I notice light bending toward the ground?”

“The effect is really small near the surface of the Earth, because the Earth just doesn’t have that much mass. If we look at really heavy objects, though, we can see the effect of light bending due to gravity. One of the great proofs of General Relativity was the observation that starlight passing close to the Sun bends very slightly. People have also seen ‘gravitational lensing’ where light bends around colossally huge objects like galaxies. The bending of light in a gravitational field is a consequence of General Relativity applied to accelerating motion, in the same way that the slowing of clocks is a consequence of Special Relativity applied to motion at constant velocity. So, you see, the two theories really are similar.”

“I guess so…” she says. She looks thoughtful for a minute, then picks her head up sharply. “Wait a minute. You didn’t mention rubber sheets at all. What happened to bending space?”

“Ah. Well, you remember how you said that light always travels in straight lines a while back?”

“Yeah, but you said I was wrong.”

“It’s not exactly correct, but it’s close. The correct statement is that light always travels along the shortest path between two points.”

She looks really confused. “What’s the difference? I mean, a straight line is always the shortest distance between two points, right?”

“Not necessarily.” I look around the office, and pick up a basketball. “Look, imagine this ball is a globe. We’re here, in Schenectady, on the ‘S’ in ‘Spalding,’ and Paris is here, a bit above the ‘g.’ What’s the shortest path between those two points?”

“A straight line.”

“You might think that, but the shortest path between those two points is actually what’s called a ‘Great Circle’ route, which looks like a curved path on an East-West/ North-South grid of latitude and longitude. If you want to get from here to Paris, the shortest path is actually curved.”

“No it isn’t. The shortest path is a straight line, through the ball.”

“Well, yeah, ok, if you include all three dimensions. In reality, though, we’re confined to moving on or near the surface of the Earth.”

“Maybe you are. I’m a good digger. I can make a tunnel.”

“Not to Paris, you can’t. Anyway, we’ve been over this– only bad dogs dig tunnels.”

“Oh. Yeah.” She looks sad. “I’m a good dog, though, right?”

“You are an excellent dog.” She wags her tail happily. “The point is, the shortest path between two lines is only a straight line if you’re talking about objects on a flat surface. If you’re looking at motion in a curved space, the shortest path between two points can be a curve, not a line.”

“So… since light always takes the shortest path, and light bends in gravity, that means that gravity bends space-time?”

“You got it. It’s probably better to say it the other way around– that is, gravity bends space, and light takes the shortest path, so light bends in a gravitational field– but whichever way you say it, that’s how the world works. Because gravity and acceleration are indistinguishable, if you want to describe the effects of gravity mathematically, you end up describing space as curved due to the presence of mass.”

“Hence the rubber sheets.”

“Exactly.”

“The rubber sheets are a stupid example, anyway. Who has rubber sheets?”

“Well, there are plastic sheets in the baby’s crib. That’s sort of similar.”

“Maybe.” She looks skeptical.

“Anyway, are you satisfied that General and Special Relativity are related, now?”

“Yeah, I guess.” She thinks for a minute, then, “Hey, can I dig a tunnel to Paris?”

“I already said no. Anyway, it’s impossible. And why would you want to do that, anyway?”

“I don’t know. I’ve just got this weird desire to invade France.”

I sigh. “I guess I walked into that, didn’t I?”

“Pretty much.”

Comments

  1. #1 Dean
    September 24, 2008

    Actually, way back in the **s, when I was doubling in math and physics, in my physics courses I was a little creeped by all the older professors repeatedly referring to rubber sheets. I learned that my thoughts and theirs were not on the same topic, or points of imagination.

  2. #2 derek
    September 24, 2008

    I bet she never says it’s mean when it’s a cat in the box.

  3. #3 Rick
    September 24, 2008

    I just happened to watch a show on black holes last night (Nova, I think), and they spent some time talking about “spaghetification” caused by the tidal forces in the vicinity of a black hole. The gravitational strength gradient would appear to be one difference between a gravitational field and an accelerating framework that should be detectable inside the “magic box”. Is this something that’s addressed by general relativity? I have a hard time believing I’m the first person who’s asked this question, so I’m guessing it must be.

  4. #4 ralph
    September 24, 2008

    Let the elevator have windows on opposite walls and be moving at a constant but very high velocity.

    Relative to the floor of the elevator a transverse beam of light entering through one window will exit the other at a lower point because the elevator has moved during the time required for the photons to transit the distance between the windows.

    Therefore the light will appear to bend even were there no acceleration.

  5. #5 Benjamin Geiger
    September 24, 2008

    ralph: No, it won’t bend. At best it’d be diagonal, and that’s because, relative to the elevator, the light source is moving.

  6. #6 Uncle Al
    September 24, 2008

    Are rubber sheets a good example? Compare the sum of the interior angles of a triangle in a mass-deformed rubber sheet (fewer than 180 degrees, hyperbolic) and in mass-deformed space (more than 180 degrees, elliptic). Has that changed?

    Postulated the vacuum is isotropic. A chiral pseudoscalar vacuum background in the massed sector has never been tested. Chemically identical left and right shoes might fall along parallel trajectories with different accelerations. Light, of course, falls with twice the acceleration of matter (and Shapiro delay if you are especially naughty).

    Theory is no better than its founding postulates. Look what happened to Euclid.

  7. #7 Josh
    September 24, 2008

    Rick, the answer is that tidal forces are taken care of in general relativity. The issue is that general relativity has more to it than the “magic box” argument above.

    The “magic box” story above is strictly only for gravitational fields which are constant over length scales comparable to the size of the box. So the box has to be small enough to not notice the tidal forces, then there is a complete equivalence between acceleration and gravity.

    But this caveat doesn’t make the argument useless; indeed it does elucidate the main point of general relativity. The “magic box” argument shows that gravity and acceleration are “locally equivalent” or equivalent over very small distances. so it tells one how point particles behave. This can then be expanded up to objects with real physical extent which is then general enough to address tidal forces.

  8. #8 Tom
    September 24, 2008

    Are you going to also use elevators to teach Emmy about raising and lowering operators perchance?

  9. #9 co
    September 24, 2008

    Ralph, as Benjamin intimated, the light beam will be at some angle to the horizontal. This is a special relativistic effect, and makes things look as though they’re contracting to points in front of the observer. Animations of moving through landscapes close to “c” make it look as though you’re moving through a tunnel, with the images of one’s surroundings getting thrown to the front.

    I’m going on about this, because I’ve just realized that the elevator analogy is a very nice one here (think about making a relativistic pinhole camera!) and will make a nice explanation for my students.

    Thanks for that!

  10. #10 Neil B
    September 24, 2008

    Here’s a fundamental problem I have with thinking of “space” (in effect, the constraint domain of the movement of matter and radiation, right?) as being like a “sheet” (rubber or otherwise) that can be inside another “space” with more dimensions. Here I mean a macro space with more dimensions to “hold” a space with fewer, and not confusing with time either (so: a four-space dimensional manifold to “hold” a curved 3-D space within it.) Sure, in math I can just specify a manifold, a surface or space as part of another space by stating the rule for the locus of points. I can say, “the locus of points equidistant from a given point” which creates by semantic fiat a spherical shell in any space. The shell is literally curved (showing intrinsic non-Euclidean geometry), and has dimensionality one less than the parent space.

    Some physicists and philosophers of science say, there really isn’t (or “doesn’t need to be,” seen as the same point to the empirical minded) some hyperspace that our space has to be “curved into”, it’s just a way of talking about what happens here. Hence you can imagine that space doesn’t really pucker around a mass, but rather that rulers shrink in the radial direction. Some writers literally phrase it that way. That effect would effectively seem to be space curvature (e.g., more rulers can be placed = more distance to travel, when going through the pucker versus around it, etc.) But even if curvature can be “simulated” by distorted rulers “on a flat space,”: don’t you need “real curvature” and not just the equivalent distortion on a flat surface, in order to get closed, finite volumes of space? (Otherwise, the mapping doesn’t work out does it?)

    So as far as physics goes, in what sense is the “space” that holds how matter can move distinguished from the equally empty “space” that holds the first space? I know, there are quantum issues and maybe gravitons work differently, but we still can’t merely draw pictures of surfaces inside another “space” and think we’ve explained anything. Like I said, with mathematics you get to specify loci literally by saying so, but in a natural world there’s “something” that has to keep objects held inside a locus of points defined within a more bountiful (in whatever sense) “space”. IOW, if there’s no difference in “kind” about space and its containing “hyperspace” then it’s just like trying to have water surfaces distinguished inside of water etc. – what makes the difference between them (unless you follow Tegmark’s modal realism.) So, what does that job? What keeps particles etc. penned in when there’s “more room” available in principle? (Please, no circular arguments or semantic tricks.)

  11. #11 ttch
    September 26, 2008

    Why would bowling balls deform rubber sheets in the absence of gravity?

    Using gravity to explain gravity doesn’t explain very much.

  12. #12 Neil B
    September 27, 2008

    Ttch, it’s just an analogy. But it is worth asking, why matter does deform space-time (the “rubber sheet”, but see that I question that metaphor anyway.) We can also ask, why do charges affect other charges etc: at heart it’s the ultimate “Rules”.

  13. #13 Paul
    November 15, 2008

    The sheet was a good enough analogy to be useful. I’m told that people tossed balls on a sheet model of the solar system to figure out good approximate paths for interplanetary craft.

  14. #14 Jonathan Vos Post
    November 16, 2008

    Really? Tell me more, please.

    Re #13: “I’m told that people tossed balls on a sheet model of the solar system to figure out good approximate paths for interplanetary craft.”

    Caltech administers JPL, which for a long time had the US monopoly on interplanetary spacecraft.

    When I arrived at Caltech in 1968, at age 16, on full scholarship, we had an IBM 7090/7094 dual processor that did all the batch computing for JPL and Caltech. In the decade before that, they had smaller computers — but I never heard of the rubber sheets, even from the grizzled veterans with whom I worked on Magellan, Voyager, and Galileo.

    I admit the possibility, plus my own ignorance, and suggest that someone check with the FIRST of all reporters certified by NASA: the bestselling author Ben Bova, who’s also been editor, board member of the National Space Society.

    Or Dr. Thomas D. McDonough, protege of Carl Sagan, science fiction novelist, SETI Director for The Planetary Society, and author of a book on the History of the Space Program.

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