Built on Facts

You can break the laws of your local jurisdiction. It might not be a good idea, but if you want to drive 60 in a 45, you can. You’ll pay a hefty fine if you get caught, but you’ll still have been able to do it.

There’s no penalties for breaking the laws of physics because you can’t break the laws of physics. On the other hand, technicalities in both legislative and physical laws can sometimes allow you do to things you wouldn’t expect. Take this one:

You can’t go faster than the speed of light.

Clear enough, until the lawyer chimes in. “What,” he says with a raised eyebrow, “do you mean by speed?” You and I know what speed is, and so we answer that speed is simply the distance covered in a given time. “Yes, but the whole point of relativity is that distance and time aren’t the same in every reference frame. What if I measure distance from the perspective of an observer on the ground but I measure time from the clock in my moving car?”

He’s got a point. You can go faster than the speed of light if you use this definition of speed. Here’s how. In special relativity there’s a formula that tells you how time in your car (or spaceship, or particle in an accelerator, or whatever) is related to time in another coordinate system (say, the ground). If you call time in your car t, time on the ground with the Greek letter τ, and your velocity v, you’ll find that they are related by

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Where

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To you in your vehicle, the ground is going to be length-contracted. From your perspective, you’re going fewer miles per second than the speed of light, but for some reason each miles is squashed and contains more than a mile’s worth of scenery. In fact, the whole external universe seems to be compressed along your direction of travel. Clocks on the outside seem to be going slower as well. But from the perspective of the people on the outside, you seem to be moving in slow motion too. To them you’re going slower than light, but your clock is running so slow that you might cover more than 299,792,458 km by the time one of your slow seconds has gone by.

This seems to suggest that if you get on a suitably powerful spaceship now, you can get to Antares by lunch tomorrow. Sure by the time you get there 600 years will have gone by in the external universe but you nonetheless covered 600 light years (in the external frame) in a few hours (of your frame).

From your perspective you won’t have traveled faster than light because the external distance contracted while you were traveling. And from the external perspective you won’t have been traveling faster than light, but were instead fooling yourself because your clock was slow.

So how fast do you have to be going in the ground frame to be going at the speed of light in this fake hybrid “proper velocity” measurement? We can use those above equations, and if we call the proper velocity w, it will turn out that

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So, we just have to write out gamma explicitly and solve for v. Turns out things end up symmetric:

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So if you want your proper velocity to be c, set w = c and you’ll find that your ground velocity only needs to be v = 0.707c. If you want to make that trip to Antares in 24 hours, you want your proper velocity to be about 219,000c. That’s a pretty fast clip, but it can be done with a ground velocity of 0.99999999998957c. Your local police might give you a ticket for driving that fast, but you’re still under the physics speed limit of the speed of light and so it’s not forbidden.

Drive safely!

Comments

  1. #1 Matt Penfold
    August 21, 2008

    As I understand it should really be nothing can accelerate to faster than the speed of light, or slow to below the speed of light. In other words the speed of light is a barrier that cannot be crossed in either direction.

    Aren’t there some particles that are hypothesised to exist that travel faster than light ?

  2. #2 scotth
    August 21, 2008

    0.707…… I’ve seen that number before. The RMS of a sin wave.

    Interesting to see it pop up here.

    Matt, see tachyons (wiki has a decent page on them).

  3. #3 Uncle Al
    August 21, 2008

    The singular lightspeed naughtiness is being massed and traveling at c. Absolute zero is thermodynamically isolated for the same reason – division by zero is undefined. Negative temps kelvin are trivially accessed: lasing media, NMR and MRI, EPR. The interested reader is invited to propose an analogous finesse of lightspeed.

    A superluminal stardrive is conceptually simple. One need merely go forward in space and backward in time. 3 mph will get you to Antares over an arbitarily small interval. One could do it with a pushcart. Studies for reduction to practice solicit grant funding (SBIR or DARPA).

  4. #4 Mike
    August 21, 2008

    So if you got in a spaceship and left earth and traveled at .99 the speed of light, for 1 day, and returned to earth how many years into the future would it be on earth?

    this stuff is so cool

  5. #5 Eric Lund
    August 21, 2008

    @Scotth: The number in question is the square root of 1/2. It pops up with surprising regularity in physics.

  6. #6 Who Cares
    August 21, 2008

    Now for the interesting question.
    How do you increase the speed of light in a vacuum at a macroscopic scale.

  7. #7 wokka
    August 21, 2008

    I know why I like this blog: well written musings on things you can figure out yourself (but perhaps never bothered to).

    A thought:
    What made me interested in physics in the first place, long ago, was actually the nice feeling of being able to *understand* things combined with the equally nice feeling of coming up with interesting questions.

  8. #8 Matt Springer
    August 21, 2008

    #4, the answer is just over a week. The fact that the traveler turned and came back makes it conceptually harder because now there’s three reference frames: the earth, the outgoing traveler, and the incoming traveler. This leads to the famous twin paradox, which is actually usually explained pretty badly in books not about relativity specifically. The Wikipedia writeup on the topic is ok, and I’ll probably end up writing a post about it eventually myself.

  9. #9 Carl Brannen
    August 22, 2008

    This is a topic that comes up frequently when I rent cars. The guy behind the counter says that I can drive the car for one week with unlimited mileage, and I respond “not according to Albert Einstein”.

    Another good topic would be the speed of gravity. I’ll get you started with my take on the fascinating and controversial subject. Einstein and Newton differ on the speed, so it is one of some interest to students.

  10. #10 Tom
    August 22, 2008

    @Uncle Al (#3) Negative temperatures accessed in lasing media, etc. are non-equilibrium phenomena. You can play similar games with the speed of light if you look at phase velocity vs group velocity in e.g. anomalous dispersion, something which routinely fakes out journalists into stating that relativity has been violated.

  11. #11 IBY
    August 22, 2008

    Relativity physics is just way too cool. I am glad Einstein, Lorentz, and others did the work. :)Hey, and you know another way for the other frame to go “faster” than the speed of light? Just turn around 360 degrees, and see the stars “going” faster than the speed of light. :)Of course, that one doesn’t count.

  12. #12 CCPhysicist
    August 23, 2008

    Read the “relativity faq” before investing your time trying to come up with a better analysis of the twin paradox.

  13. #13 travc
    August 23, 2008

    I can’t remember most of the special relativity paradoxes, because they didn’t seem very paradoxical. The only one that got me confused at all was “the pole and barn paradox”… non simultaneity wasn’t intuitive at first.

    Anyway, with respect to the relativity stuff, communication based paradoxes tend to be more illustrative IMO (less fanciful). Though the really cool communication paradoxes are quantum… several of them (like EPR) actually make perfect sense when you give up Copenhagen and move to a more information theory based framework.

  14. #14 travc
    August 23, 2008

    Oh, I have to share some wisdom from my HS physics teacher regarding speed limits.

    The white speed limit signs are based on the law in a legal sense. The yellow signs are based on the law in a physics sense. You can choose to ignore the white speed limits and maybe not get caught… if you ignore the yellow speed limits you will get always get ‘caught’. (Safety margin yada yada… the point still stands.)

    Very good advice for a bunch of high school students, many of which just got their drivers license.

    PS: I’ve always been annoyed that the safe speed for curves normally isn’t posted if it above the posted legal speed limit.

  15. #15 Benjamin Geiger
    August 23, 2008

    travc @ #14:

    First of all, the yellow signs are advisory, and are generally calibrated for the crappiest cars on the road, and less than ideal circumstances. I may not be able to exceed 35MPH around that curve in a VW Microbus in the rain, but I could probably do 60MPH in my Grand Prix on dry pavement with good tires. I still tend to obey the yellow signs, just because I have this aversion to wrecking.

    Second, the safe speed for curves never needs to be posted when it exceeds the legal speed limit, as nobody should ever (ha ha ha) be going faster.

  16. #16 tomS
    August 28, 2008

    Doesn’t the apparent mass increase as the velocity, so that it takes more and more energy to accelerate closer to c?

  17. #17 tomS
    September 1, 2008

    Doesn’t your mass increase as length shortens? Mass approaches infinity as v approaches c.

  18. #18 Kenny
    December 10, 2008

    I think you can faster than the speed of light.
    Take this example from my friend Kenny, who is smarter than Newton himself.

    If you have a laser and you point it forward, it goes speed of light. BUT if you are in a car going 60mph, then shine that laser, the light is going 60mph + speed of light. This proves that something can go faster than speed of light. Anyone agree?

  19. #19 Matt Springer
    December 11, 2008

    Nope. The light will go forward at the speed of light as seen from the car and as seen from the ground. Time and distance are not the same in each reference frame, and the effect is to keep the speed of light constant no matter what. All relativity follows from that statement.

  20. #20 Stephen Tashiro
    February 28, 2009

    The following problem was posted to physicshelpforum.com:

    A spaceship of proper length 300 m takes 0.75 microseconds to pass an Earth observer. Determine the speed of this spaceship as measured by the Earth observer.

    One might get an answer by solving v/t = L sqrt((1-v^2/c^2)) for v.

    However, this raises the interesting question of how an observer in the theory of special relativity is assumed to measure velocities. Presumably if he observes a “point object” travel a known proper distance in a known proper time, then he must compute the average velocity is the distance divided by the time. But if he observes an object of known proper length move past a fixed point, is it assumed that he uses length contraction in his calculation of its velocity?

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