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How to deduce the theorem of addition of velocities?

Let's say that you are in a rocket speeding at 90% the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90% the speed of light. According to Newtonian physics, we add both velocities. Thus, the bullet should be going at 180% the speed of light.

But we now know that according to Einstein the sum of these velocities is actually close to 99% the speed of light, because length contracts and time slows down.

But how can you determine mathematically, that bullet is going at 99% the speed of light?
Additionally, can you also calculate, how much time is slowed down, if you know the velocity? Is there any formula?

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kk-dev11
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1 Answers1

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Velocities are strange beasts. Relativity theory tells us rapidities (the sum of the accelerations experienced by an object) are more intuitive quantities. Rapidities are directly observable quantities (external observers relate these to blue/redshifts). And unlike velocities, rapidities do sum up. Velocities depend non-linearly on rapidities, and therefore velocities follow a more complex addition rule.

Imagine three observers $A$, $B$ and $C$ all moving along a railway track. Observer $A$ measures $B$ to have velocity $v_{AB}$. From $B$'s perspective $C$ has a velocity $v_{BC}$. And to close the circle, from $C$'s perspective $A$ has velocity $v_{CA}$.

Common experience (encoded in so-called Galilean relativity) tells us these velocities simply add up to zero:

$v_{AB} + v_{BC} + v_{CA} = 0$

This is wrong. It ignores a non-linear term that becomes important at speeds approaching the speed of light $c$. Lorentzian relativity tells us the correct equation is:

$v_{AB} + v_{BC} + v_{CA} + v_{AB}.v_{BC}.v_{CA} / c^2= 0$

This is all the math you need. Give it a try. Enter $v_{AB}=0.9 c$ and $v_{BC}=0.9 c$ and see what value you get for $v_{AC} = -v_{CA}$.

Johannes
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