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Space Travel, Part 8 — What's the Deal with the Speed of Light? 
24th-Dec-2015 12:43 am

(Still waffling on whether this should be Space Travel part 8 or Relativity part 1; we'll see...
If you want to go back to part 1, that's here though none of the prior material really matters for this one.)

It's weird to me how everybody knows there's a problem.

How, if you are, say, Issac Asimov or George Lucas or Gene Roddenberry trying to write your Galactic Empire/Federation/Whatever story, you've got to do some kind of handwave about Relativity. We know this because we were taught at some early age that trying to go faster than lightspeed is Somehow Bad, even if nobody ever explains the details.

It's particularly annoying to me because many of the details actually could be explained without going beyond 6th grade math. If you get how right triangles work, that's pretty much all you really need for Special Relativity, except nobody seems to bother trying.

I suppose part of the problem here is the mystique of General Relativity. True story:

Arthur Eddington is at a meeting of Royal Society in 1919 where Sir J.J. Thomson (President) concludes a talk saying nobody's ever really stated in clear language what Relativity is about. The meeting disperses, Ludwig Silberstein (the author of one of the early books on relativity) comes up to Eddington and essentially says, "So you must one of three people in the world who actually get this stuff." Eddington demurs, but Silberstein pushes further, "C'mon, don't be modest," at which point Eddington replies, "Actually, I'm trying to think who #3 is..."

Then again, given the sheer number of people I've encountered who have Misner, Thorne, and Wheeler's Gravitation sitting on their bookshelves, that's clearly not the case anymore.

But if even Albert Fucking Einstein had to take five years off to get up to speed on Differential Geometry — which to some extent is just the Partial Differentiation Chain Rule on Steroids and working through all of the various consequences, but if you didn't get past high school calculus, even that much is going to be a bit hard to take — what hope is there for the rest of us ordinary mortals?

But it so happens that, for Special Relativity, you don't need differential geometry. So let's get started:

Why is the speed of light constant?

How did they even get this idea that the speed of light is always the same? That's the part They never explain. They figure there's no way anybody's going to get that without having the full-ass physics course, so they don't bother trying.

The short answer is that they've done the experiments and that's the way it actually is. Since that's unsatisfying, I'm going to try giving you the half-assed physics course instead:

Really, it's all about electricity and magnetism. 2500 years ago, the ancient Greeks were pretty sure these had nothing to do with each other. Over the last few hundred years various discoveries proved otherwise.


(0) Static Electricity

Rub your feet on the carpet, stuff sticks to you, and your hair stands up. Play around with it a bit and you can come up with some rules:

  • Things can be charged, plus or minus. Like charges repel; opposite charges attract
  • The bigger the charge, the bigger the force
  • The farther apart things are, the smaller the force -- and it's an inverse square thing (twice as far away means four times as weak); that much you can measure.

What is "charge"? Don't know, really. Figure something causes the force and so there's stuff there. Some things have it, other things don't.

You can even have a system of units where the unit of charge is that particular amount of stuff such that when you put it on two distinct objects that are a unit distance apart, you get a unit of force.

(Naturally, the standards committees had to get into the act and mess everything up so that in the MKS system of units that we're stuck with in the real world, the unit of charge is actually Minus Some Completely Arbitrary Number of Electrons and then you have this annoying proportionality constant that relates the force to the charge. Also, they arguably screwed up on which should be plus and which should be minus; I blame Ben Franklin. But it's all bullshit anyway, because the only way we even know the charges are there in the first place is by measuring the forces.)

That's a key point:  measuring the forces is all we can actually do. There's no way to "see" charges in any other sense of the word.

Still, measurement of forces conveys lots of information. We can take a little tiny charged test particle and carry it around with us. Everywhere we go, we measure which direction it gets pushed and how hard. Divide by the charge and we now have a force-per-unit-charge, otherwise known as the "electric field", for every point in space that we've been. Come back later with some other charge that's, say three times as big, and, oddly enough, the force on it there will turn out to be three times as strong in that same direction. You do the experiment, it works, and that's why we have this idea that we're measuring something real.

Where does this electric field come from? Other charged particles located elsewhere. For every charged particle in the universe, divide that charge by the square of how far away it is, and that gives you that particle's contribution to whatever the field is where you are. Because it's all nice and linear, you can figure out the contribution of each source separately. And because it's inverse-square you can ignore all of the crap that's Really Really Far Away because most of the time the effect on your result will be insanely small. Then we just add up all of the contributions that matter and we're done.

That's it, really. This is all of electrostatics. I could go on about Gauss's law and other random math tricks, but you can take the full-assed physics course if you care that much.

Now for the dirty little secret:  Electrostatics plus Relativity is all of classical electrodynamics, including, all of the weird-ass effects I'm about to describe.

(1) Magnets.

They make no sense at all, right?

As it happens, for magnetism, there is, in fact, a vaguely similar story to that for electrostatics, i.e., you can invent a similar notion of North and South "magnetic charges" and come up with similar rules for the forces they exert on each other.

But I'm going to skip all of that because, even though this is the way people thought about magnets from Greek times onwards, it turns out to be a dead end. Unlike with electrical charges, you never, never actually see a bare North by itself. Which is weird, and might possibly be taken as a sign that, under the hood, there's actually something completely different going on.

But there is a "magnetic field" that you can measure using little tiny test magnets, much the same way we measure the electric field using little tiny test charges. That much we can keep. Where does it really come from? Fast forward 2000 years through a bunch of discoveries that I'm probably presenting out of historical order, but who cares:

(2) Ampere's Law:  Moving charges create magnetic fields

If you have current flowing in a wire, even if the wire is electrically neutral, you'll have a magnetic field circling around the wire. This is how electromagnets and electric motors work.

Now, thinking back to Newton's 3rd Law, if moving charges in a wire exert a force on a magnet near the wire, then wouldn't the magnet exert a force on the moving charges? As it happens, yes:

(3) Lorentz Force Law:  Magnets affect moving charges

In particular, if you have a charge moving through a magnetic field, there will be a force that's proportional to the magnetic field strength, the charge, and its velocity. Moreover the force will be at right angles to both the direction of the field and motion of the charge -- which makes it lots of fun to observe physics exams because everybody in the room will be doing stuff with their right thumb and forefinger trying to remember which way things go.

Wait. Velocity?

Yep. Meaning if the charge is not moving, then the magnetic field, no matter how strong it is, exerts no force at all. Weird, eh?

This, by the way, is how the old-style big-glass-tube television works. They take electrons, fling them at the screen (which has phosphorescent stuff on it that lights up when the electron hits), and use magnetic fields to bend the trajectories to make the electrons hit in the right places (doing it all Really Fast).

Hm. It's almost like moving charges exert forces on each other using via magnetic fields, much the way stationary charges exert forces on each other via the electric field.

But wait:  What if you throw a magnet at something that's charged and stationary?

Yes, we just said magnets can't affect stationary stuff, but you'd think it shouldn't make any difference whether the charge is moving at a stationary magnet or the magnet is moving at a stationary charge. How can there be a force in one situation but not the other?

As it turns out, there is a force in the latter situation.

(4) Faraday's Law:  A changing magnetic field creates an electric field.

The idea here is that if the magnet is moving towards you, then the magnetic field where you are will be increasing, and its the increasingness of the field that's doing the job of moving the charge where you are. And since it's something that's causing a stationary charge to move it has to be (at least part of) the electric field -- by definition, really, since, as you'll recall, all we can really do is measure forces.

So, electric fields are produced by both charges and changing magnetic fields.

This, by the way, is how electric generators work.

And now for the tricky bit:

Maxwell, in the course of summarizing the state of the art circa 1860, noticed a bug in Ampere's Law. The short version is that having current flowing in a wire isn't the only way you can have moving charges. Somebody could just throw a single charge at you; there might not be any current right where you are, but the electric field will be increasing because the charge is getting closer, which should have some effect on the magnets where you are.

Well okay, Maxwell actually came at this from a slightly different direction. Imagine a circuit with current flowing, and but also imagine that there's a gap in it — a charging capacitor — where there's no current but there are charges piling up on either side (at least for a short while). He reasoned that the gap shouldn't matter, i.e., it shouludn't make any difference whether there's a current there or an increasing electric field. Somebody did the experiment, a magnetic field measurement between the plates of a charging capacitor, and sure enough:

(5) Maxwell's bugfix:  A changing electric field creates a magnetic field.

In other words, magnetic fields are produced by both currents and changing electric fields.

But then Maxwell noticed something else.

What happens if you have an electric field that's changing at changing rate? That creates a magnetic field that's changing, which then creates an electric field, which then...

Wait a minute, let's try this again.

What happens if you have an electric field that's changing at a rate that's changing at a rate that's changing at a rate that's.... (and so on forever)? And, in case you were wondering, it's fairly easy to come up with functions that behave this way (i.e., that have nonzero derivatives of all orders, if we want to get technical). Here's one such function:

Kind of looks like a wave, doesn't it? If you take the derivative (rate-of-change), you get this

In other words, if the red line is what the electric field is doing then Maxwell's bugfix says that the blue line is what the magnetic field is doing. And if the blue line is what the magnetic field is doing, then Faraday's law says that the red line (which happens to be minus the rate of change of the blue line, because there's a minus sign in there I forgot to tell you about) is what the electric field should be doing.

It all fits together and you have this pair of oscillating electric and magnetic fields generating each other and heading off to infinity in any direction you want. In completely empty space; no charges are needed anywhere once you get the wave going.

Maxwell then asks, how fast does this electric-magneticky-wave-thing go? Crank through the math. Get a number that is the product of the constants that determine the relative strengths of the electric and magnetic fields.

This also just happened to be about the time that the first reasonably accurate measurements of the speed of light start turning up in the journals. Foucalt and Fezier did some clever experiments with rotating mirrors and cogged wheels.

The numbers are the same.

… at which point Maxwell is going, "Fuck me. This is what light is."

(Yes, I'm paraphrasing a bit... Though if I ever get around to doing a translation of Maxwell's thesis from the original Scottish, I'll be sure to use this wording. Um, yeah.)

Sometime later, Heinrich Hertz says, "Gee, I bet we could use electrical circuits to generate really, really, really, really, really, really, really low frequency light...." At which point the whole radio telecommunications industry is off and running.

Now there is this Small Problem:  Usually when you measure the speed of something, it's with respect to something else, e.g., the water, or the air, or whatever medium the wave travels in. But when I say "empty space" above, I meant completely empty. Literally. Nothing. There.

Imagine trying to measure the speed of an air hockey puck and suddenly there's no table anymore.

Never mind that those constants that you multiplied to get the speed of light? They're, well, um, constant. As in, they don't ever change.

Clearly there had to be some kind of otherwise-invisible-and-undetectable medium that light moved with respect to. Problem is, all attempts to measure the velocity of this ineffable medium would come up zero. Earth apparently was always coincidentally in a place where this magic medium was stationary, no matter when anybody tried to do the measurement. (Thank you Michaelson and Morley.)

For the next 20 years, physicists remained baffled by this.

It took Einstein to say, "Screw this. If it looks like a constant and smells like a constant, I'm going to take the hint: It's a fucking constant. Yes, I know this tosses everything Gallileo and Newton had to say about time onto the trash heap. That different observers will necessarily disagree about time measurements and we also have to fuck with measurements made in the direction of motion. But we can make this work. Hell, Lorentz and Poincaré already did the math for me."

(Um, yeah, ... still paraphrasing...)

Which is obviously wacky. But here is what sold it:

Imagine a current in a straight wire and a test charge flying alongside moving in the same direction as the current. The wire is electrically neutral, but once you work through all of the right-hand rules, you find that the magnetic force that the current in the wire exerts on the moving charge attracts that charge to the wire.

Then you look at things from the flying charge's point of view, where now the charge is stationary and it's the wire that's moving. In which case, there can't be any magnetic force, because, well, stationary. But now there's something weird going on with the wire.

In general, having a current in the wire means you've got negative charges with a certain density going one way and positive charges with a certain density going the other way. From the wire's point of view, the densities are the same, the wire is neutral, and thus there's only a magnetic field.

But from the flying charge's point of view, the negative stuff in the wire is moving faster and thus scrunched together, the positive stuff in the wire is moving slower and thus stretched out, the densities are not the same, the wire has a net negative charge, which then gives us an electric field that can pull the positive flying charge in, even though it's stationary.

Same force, completely magnetic from the wire's point of view, completely electric from the flying charge's point of view.

  • Magnetism is entirely an artifact of Relativity.
  • Ampere's Law is entirely an artifact of Relativity.
  • Faraday's Law is entirely an artifact of Relativity.
  • Maxwell's bugfix to Ampere's Law is entirely an artifact of Relativity.

And so on. Electrostatics plus Relativity is Everything Else. (Well okay, there's still quantum mechanics but I'll leave that for another day...).

Coming up with one simple idea that explains all of this weird crap you've been seeing? That is How You Win at Science.

If you want to experience Relativity first hand, there's no need for spaceships that can go near the speed of light, or particle accelerators, or black holes, or any of that shit.

Just go into your kitchen and pull a magnet off of your refrigerator. That's Relativity right there in your hand.

Also, if you want to write a Galactic Empire story, and you want to postulate that Special Relativity doesn't actually work the way we think it does — in order to get around this annoying light-speed limit, which I haven't really explained yet, but we'll get there — then you're also postulating that magnets, electromagnets, CRTs, electric motors, electric generators, radio, ... pretty much everything our current civilization is based on, that none of that stuff works either, or at least doesn't work the way we think it does, and, well,... this is getting close to creationist levels of denial.

Just so that you know.

(to be continued)

26th-Dec-2015 02:32 am (UTC) - Clockwork Rocket
Have you read Greg Egan's Clockwork Rocket? It's set in a universe where the space-time metric has signature (+,+,+,+). And yes, he has done all the math: http://www.gregegan.net/ORTHOGONAL/00/PM.html
18th-Jan-2016 11:33 pm (UTC) - Re: Clockwork Rocket
Haven't yet. I probably should.

I do recall at one point in the time I spent on Usenet decades ago, somebody presenting a proof that given certain reasonable-looking assumptions you could derive what the transformation between observers with different velocities has to look like, and there were 3 possibilities:
  1. Gallileo's (Newton's) version (where the time axis shears sideways, time and spatial measurements are the same for everybody, and the set of events that are 1 hour in your future for all possible velocities forms a line parallel to your current space axis)

  2. Lorentz's (Einstein's) version (where the time and space axes both shear towards the diagonal [light ray trajectory] and stretch out, time slows down in the boosted frame, and the set of events that are 1 hour in your future for all possible velocities forms a hyperbola whose asymptotes are the light ray trajectories)

  3. This weird 3rd possibility where the time and space axes rotate rigidly, time speeds up in the boosted frame, and the set of events that are 1 hour in your future for all possible velocities forms a circle
and then the question was what the hell is #3? Answer: best name for it is the Euclidean transformation; it really is just a rigid rotation of everything. If you go fast enough you're going sideways in time with respect to the original observer (travelling along his/her simultaneity axis). Speed up further and you're going backwards

… which means time travel is trivially easy in this world. That is, if you have enough fuel for your spaceship, you can rotate your frame 180 degrees and coast as far back into the past as you want (well okay, it'll take you 50 years to go 50 years into the past, so you're still limited by your lifetime, but that's still enough to do damage).

And I really hope that Greg Egan considered this because if he didn't, then his story most likely runs afoul of JWZ's rule

Edited at 2016-01-18 11:52 pm (UTC)
14th-Feb-2016 03:15 am (UTC) - Re: Clockwork Rocket
I read The Clockwork Rocket once and as far as I recall, within that book there are no rotations-including-time greater than 90°. Whether they were discussed, I wouldn't remember.

(I like Egan's works' physics. His (non)human elements generally put me off rereading.)
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