Built on Facts

Reader Scott writes in with a good question that I’ve heard posed by several people in various places, because it really goes to the core of the counterintuitive nature of relativity.

Okay, let’s assume I’m in a spacecraft moving at a speed of c-1 m/s. That’s one m/s less than the speed of light. Would I be able to move from the back of the ship to the front at 1 m/s? Would I even be able to survive at that speed?

Here’s my reply, extended a bit to include the relevant equations and go into a little more detail. The derivations will all wait until later:

Yes indeed. The question at issue is your speed with respect to a particular reference frame. Let’s pretend that you’re on an airplane flying at 500 mph with respect to the ground. There’s nothing stopping you from walking forward at 1 mph with respect to the plane. And that also happens to mean that you’re walking at 501 mph with respect to an observer on the ground. Assuming the plane ride is nice and smooth and the windows are closed, you wouldn’t even be able to tell the difference between that and walking in a plane parked on the runway. This is fundamental in modern physics: there’s no preferred inertial reference frame, the laws of physics work the same way in all of them.

An inevitable consequence of this are the relativistic effects of time dilation and length contraction. This implies that in fact velocities don’t add in such a simple way. So when you walk at 1 m/s in a ship going 299999999 m/s with respect to the ground (pretending the speed of light is exactly 300000000 m/s for simplicity), you appear to yourself to be walking at 1 m/s inside the ship, just as in any other case. The guy on the ground will (crunching some numbers) see you as walking at about 299999999.00000000667 m/s. This comes from the Einstein velocity addition formula, which for a guy walking at speed v in a plane with speed u means his velocity s with respect to the ground is

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The “why” for all of this is tricky, but within reach for an interested person with a decent grasp of algebra. Of course there’s lots of good websites, but it’s probably hard to beat Einstein’s own book “Relativity: The Special and the General Theory“. And I myself will write up the derivations here one of these days.

Thanks for reading!

Matt

Comments

  1. #1 Fergus Gallagher
    January 7, 2009

    And there’s the interesting case when u=c => s=c. (I suppose not really that interesting as that’s axiomatic, but nice and tidy!)

  2. #2 Kobra
    January 7, 2009

    It defies intuition, but it makes perfect sense.

    I really need to take a physics class. Or two. :P

  3. #3 Jolly Bloger
    January 7, 2009

    The length contraction would also make the ship appear extremely short, would it not? I wonder – would the relativistic changes keep the time it takes to traverse the ship constant?

  4. #4 csrster
    January 7, 2009

    … and of course you’d be fried by the microwave background radiation blue-shifted into bullet-hard gamma rays.

  5. #5 Paul Murray
    January 8, 2009

    Of course, from the POV of you in the spaceship, the spaceship is not moving at 299999999 ms/s – it is *stationary*. And full-length.

  6. #6 Oldfart
    January 8, 2009

    Surely, we can protect ourselves from bullet hard gamma rays! Oth, just what do ordinary bullet hard gamma rays become?

  7. #7 Mike
    January 18, 2009

    To pick up on your aircraft example, do the laws of conservation of momentum not mean that if you were to accelerate yourself forwards to say 1mph, then there would be a corresponding DECREASE in the ground speed of the aircraft?

    Not a big change for an average human in a Jumbo Jet, but possibly far more important in a space craft?

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