Has the velocity addition formula ever been verified in real world?Is it at all possible to verify it, since observation can only show no object travels faster than $c$, but it's impossible to know what energy corresponds to what speed?

In SR articles it is often quoted the case of a relativistic projectile. Now, consider a gun that normally fires a bullet at 500 m/s: if we accelerate the speed of earth's revolution around the Sun to $c$-100m/s, the hyperphysic calculator says that an observer at rest inthe solar system frame will see the bullet going at a speed less than 1 mm greater than the .99 99 99 c speed of the earth.

Does the observer really see the bullet emerge slowly from the gun in a minute or so?

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    $\begingroup$ We do live at near-c. Just ask someone who is traveling at near-c with respect to us. $\endgroup$
    – mmesser314
    Feb 7, 2017 at 7:24
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    $\begingroup$ The guy who shoots sees the bullet at 500m/s, the one who sees the shooter moving at c-100 will see the bullet moving at c-100+small amount. He will not see a 500m/s speed difference between the bullet and the shooter. $\endgroup$
    – user126422
    Feb 7, 2017 at 7:45
  • $\begingroup$ Is it possible that you run into trouble here when talking about kinetic energy? Check the definition for SR again and everything should be fine. $\endgroup$ Feb 7, 2017 at 8:08
  • $\begingroup$ In the rest frame of a far galaxy which is moving away from us near the speed of light, we are moving at near the speed of light. bullets are fired everyday $\endgroup$
    – anna v
    Feb 7, 2017 at 11:57
  • $\begingroup$ Actually, we are moving fast. See this $\endgroup$ Feb 7, 2017 at 12:51

2 Answers 2


The spectrum of neutrino beams (and for that matter of the electrons produced in the process, but I'm not sure those are measured) is a direct test of the velocity addition formula.

You see, the beam is made by allowing highly relativistic pions to decay to muons which can subsequently decay themselves. The muons can get a gamma of up to about 1.2 in the pions rest frame, which is more than enough to exceed $c$ assuming naive Newtonian velocity addition. And while the decay line is designed so that most of the muons are stopped in the downstream absorber and decay at rest, a fraction of them decay in flight adding an additional neutrinos (with a different spectrum) to the beam.

The fraction that decay is sensitive to their speed and is modeled to produce the assumed neutrino spectrum used as a input to experimental simulation. If this was way off we'd know and it would be a matter of great interest to particle physicists.

  • $\begingroup$ lambert, of course it's not the only example. The acceleration of beam is a very high precision experiment, but it doesn't look like your version unless you understand the differential expression of the problem. I chose this particular case because it involves a impulse high speed projection. You can, in principle make a direct measurement of the muon speeds, but no one bothers. Special relativity is tested to a fare-the-well, and particle physicist and especially accelerator physicists work with the practical results on a daily basis. To the point that this stuff is boring. $\endgroup$ Feb 7, 2017 at 19:01

There is no physical limit that stops you accelerating to any speed less than $c$ relative to your current momentarily comoving frame of reference. By Galileo's relativity principle, a boost from your present frame to one going $500{\rm m\,s^{-1}}$ requires exactly the same effort and physics whether you may be stationary with respect to a second observer, or whether you may be moving at some speed $c-\epsilon$ relative to that observer for any positive $\epsilon$ no matter how small. You gun's muzzle velocity will indeed always be $500{\rm m\,s^{-1}}$ relative to your frame, whatever your speed relative to the second observer.

Suppose you begin stationary relative to observer $A$, then fire your gun and boost to the rest frame relative to the bullet. Now fire your gun again, and boost again to $500{\rm m\,s^{-1}}$ so that you are in the second bullet's rest frame. As long as there is someone in each new frame who can hand you a new shell to fire and some fuel (or a catapult to fling yourself with) to boost to the bullet's rest frame, nothing ever stops you making the same step and the same $500{\rm m\,s^{-1}}$ boost.

However, at each step, your time axis is more and more dilated relative to observer $A$. So this observer sees the speed added by each step as smaller than the first, such that the effect of all the steps asymptotes to $c$ from $A$'s frame. This is the mechanism that prevents greater than $c$ relative motion between observers: from your point of view, there's no barrier ever to your accelerating indefinitely.

I hope this little thought experiment sequence helps.

  • $\begingroup$ @lambertwhite Do you understand that energy is also frame dependent? That there is no absolute measure of a body's energy? Energy is the time component of the momentum four-vector, and the square norm of the latter is $E^2/c^2- p^2$. It is this norm that is Lorentz invariant. $\endgroup$ Feb 8, 2017 at 2:47
  • $\begingroup$ @lambertwhite Yes, from the standpoint of an observer who has been suddenly accelerated by the supernova, because their frame has changed. Look at it this way: if you are propelled towards the Earth at high speed, from your standpoint the Earth is coming towards you at high speed. Any collision you have with it is going to show it has a high kinetic energy. $\endgroup$ Feb 8, 2017 at 7:49

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