# Why is this argument about light speed not true? [duplicate]

Long ago I was watching a scientific video about light speed. At one point, the person said to imagine the following scenario:

You are on a train going from the Solar System to the Alpha Centauri system. The supposed train is going at almost light speed, and you are on top of it. While you are on top of the train, you move forward one step in the direction in which the train is moving. Are you going faster than light speed?

The guy in the video said that the answer is no. When I watched the video I had no idea what the explanation meant because I wasn't into physics at all.

I also can't find the video, but I am curious as to why a scenario like this would be not possible (excluding the limitations of technology, and other flaws in the scenario which have nothing to do with speed.) So, why isn't this possible? :)

## marked as duplicate by Qmechanic♦May 12 '17 at 22:22

• Because velocities add not according to the classical law but according to the relativistic one, and the latter is only close to the classical law for speeds much smaller than the speed of light. Under the relativistic addition the sum of speeds smaller than the speed of light is always smaller than the speed of light, and the sum of the speed of light with anything is itself. Here is also an explanation video. – Conifold May 12 '17 at 21:42
• @Conifold That comment could be converted into a nice answer. – rob May 12 '17 at 21:44

Short answer: when you "add" speeds in relativity, you have to use a special formula. If the train were travelling at 50 mph relative to me and you walked down its carriage at 10 mph relative to seated passengers, I'd see you moving at 60 mph. That's how speeds work, right? Well, relativity reveals it's more complicated than that. "Adding" speeds $u,\,v$ gives $\frac{u+v}{1+uv/c^2}$, $c$ the speed of light. If each speed is much smaller than $c$, the division by $1+uv/c^2$ can be neglected, restoring common sense. If instead $u=c$, the result is $\frac{c+v}{1+v/c}=c$ instead of something greater. You can also show that, if each speed is less than $c$, the final result will be too.

Wouldn't it be great if there were some measure of motion that added straightforwardly? Actually, there is: "rapidity". Speeds $<c$ have finite rapidity; $c$ is infinite rapidity. This makes it easier to prove that less-than-$c$ speeds can't make anything else when you "add" them.

In SR the train could not travel at or greater than the velocity of light in a vacuum, only very close to the the velocity of light. Now consider an observer placed between our solar system and Alpha Centauri who is standing still relative to our solar system and Alpha. As you zipped by, he would measure the train traveling at just less than the velocity of light. But he would also observe time to have virtually stopped flowing on your train. So your step forward would be so slow that it would increase your velocity so minuscular-ly that you would still not be traveling at the velocity of light as measured by the observer.

Now consider your own speedometer. You would experience the time dilation such that you would have arrived at Alpha Centauri before you even moved one cm (or aged 1s). So you would not in fact take a brisk walk to the front of the train during your ride. So you would not measure the observer whizzing by at the velocity of the train plus the velocity of your brisk pace.

This is due to einstein theory of relativity. As far as I know Its based on two principles:

1.The speed of light is the same at any frame of reference.

1. All Physical laws are the same in any inertial system .

Those two principles were proved "inductively" , meaning that the results which we expect on paper still to that day match with what we observe .Of course nobody can do those imaginary experiments you mentioned, however The implications of those principles act perfectly with what we observe. For example einstein discovered those principles because a lot of experiments in the laboratory showed that the speed of light doesn't change no matter with which speed a body is moving . Based on that he tried to give an explanation.