# Q & A: Relativistic velocity addition

Q:
So, my friend and I were doing a bit of non-topical theorizing today in physics today, and we came up with an interesting idea. Imagine you are in a rocket ship of some monstrous size... lets say 3000 kilometers for good measure, and inside that there is a 500 kilometer train, and inside of the train there was a 50 kilometer long limousine, and inside that limousine was you. Now, let us assume that the ship is moving 3.00 x 10^7 m/s, and slowly accelerating toward the speed of light. If there as a lare electromagnet at the head of the space ship, and it was turned on, pulling forward the bus, being made of mostly ferrous metals, and then the limousine drove forward at full speed, and you started running though the limousine (assuming that the limousine's floor was flat), we will assume that Σ of all the velocities heading in the same direction as the space ship will pass 3.00 x 10^8 m/s. So in essence, because of Relativity as we know it, you will be able to run faster than the speed of light, within your own relativity, right? The second part of this sort of multifaceted question is, what would happen if you were ejected instantaneously from the limousine, out into space? Discounting most forces, would you not split into at least two identical parts, because one part of you exists within each relativity? And the final part of this question is, is any of this relavent, because when in this sort of ship would you reach the point where you would begin to just turn to energy? -Christopher (15) Maine
- Christopher (age 15)
Falmouth, ME USA
A:
Dear Christopher,

You've done what we've all done when first introduced to special relativity--come up with what seems to be an obvious paradox. In the future, it's a good check of your understanding to make sure what you're saying doesn't imply that you can go faster than the speed of light in any reference frame, but these thoughts mean you're thinking and thinking is always a good thing!

The first thing to know is that velocity addition is different in special relativity. Galilean velocity transformations are as follows:

A man on a train going velocity u with  respect to the earth walking velocity v  with respect to the train's floor will be traveling s = u+v with respect to the earth.

For trains going and people going slowly compared to the speed of light, c, this is approximately correct. Special relativity tells us that, in fact, the person will be traveling

Hence, if you actually carried out this calculation for your individual situation, you would find that the person in the limousine in the train in the rocket ship would indeed be going less than c.

To answer the second part of your question, you would not split into two different people. Remember that relativity only means that people in different inertial frames (a frame of reference that is not accelerating) will witness their own version of events. There is no absolute frame of reference to compare different accounts of events. For instance: if I'm on earth watching you zoom by at, oh, say, 0.8*c on a fancy space ship, I would see you move in very slow motion. You, on the other hand, since you're traveling at a constant velocity, would be perfectly legitimate in saying that I and the entire planet Earth were the ones moving past you at 0.8*c. So you would see me move in very slow motion. Well, who's right? The answer is: we both are. It sounds insane, and obviously paradoxical, but it actually isn't.

Hope that somewhat answers your question. I know it can be confusing. There are plenty of great resources available online and in print.

Those might be a couple of places to start.

Best of luck!

John Hoffman

On your last question, no you wouldn't turn into pure energy. In fact, from your own point of view you're still at rest, even if some things are whizzing past you. /mbw

(published on 02/08/2011)

## Follow-Up #1: Experimental proofs of the relativistic velocity addition formula

Q:
Have there been any experiments that specifically confirm the relativistic velocity addition formula, for matter moving in relation to matter? I know there have been experiments that show light moves at c in matter's rest frame, but that may not confirm the formula, and is not what I'm asking (with light, one of the relative velocities is always c, so this may not specifically confirm the formula). Thank you very much.
- David Martin (age 43)
Arizona
A:

Hello David,

In my field, high energy physics, the most obvious examples are proton accelerators such as the Tevatron at Fermilab or the Large Hadron Collider at CERN in Geneva Switzerland.  Protons at these accelerators are constrained to move in a circular path by means of magnets.   Every time they come around you give them an impulsive swat which increases their energy and momentum.  At very low energies, much smaller than the rest mass of a proton, they pick up speed according to the Galilean non relativistic formula you are used to.   But as they gain more and more energy they eventually reach the terminal velocity of the speed of light.   This can be measured quite precisely and agrees with the relativistic formula.

LeeH

(published on 12/10/2014)