So reading about how it's impossible to accelerate an object to the speed of light relative to another, I thought of this hypothetical situation:

What if you had this row of rocks, floating in space at the speed of 10m/s relative to another row of rocks (let's call the rocks going at 10 m/s row 2, and the ones floating at a relative speed of -10m/s row 1). In this situation, let's say these "rows" are infinite. Then you get another row of rocks and accelerate it to the speed of 10m/s relative to row 2 (row 3). Now row 3 should have a speed of 100 m/s relative to row 1, right? If you continue this, until you have a row going at the speed of 100 000 000m/s relative to row 1, and you get another row and get it to go at a speed of 10m/s relative to that, would it not be moving away from row 1 at a speed greater than the speed of light?. Because relative to row 2, it would just be going at 1/3 of the speed of light, and that should be possible.

Or what about this, you have this row going at a speed of 100 000 000m/s relative to row 1, and then you accelerate row 1 by 10m/s in the opposite direction. What would happen then? Because the speed at which they are moving away from the rock rows with the "in between speeds" is FACT. Isn't it?

• You are aware of how velocities add in special relativity? – mikuszefski May 12 '17 at 5:41
• Where did you get 100 m/s from? In non-relativistic mechanics, row 3 would be moving at (10 m/s + 10 m/s) = 20 m/s relative to row 1. Using Special Relativity, the relative speed would be a tiny bit slower: a shade under 19.999999999999977747 m/s. – PM 2Ring May 12 '17 at 12:34
• Oh right, I think I forgot how to add velocities there, thanks! ...although, on second thought, I'm not sure I agree... If the rocks are situated 1 meter apart that would mean that the row (row 2) going at 10m/s passes 10 rocks in 10 seconds. Then you have row 3 which passes 10 of row 2's rocks every 10 seconds. Wouldn't that make for a relative speed of 100 m/s? – E. Holm May 12 '17 at 22:07

In special relativity velocities don't add linearly. Suppose you have a Person $T$ sitting in a train and a person $S$ standing next to the rails at a train station. Person $S$ will say that the train moves with some velocity $v$ and person $T$ will say that the train station moves with velocity $-v$. Now, if person $T$ stood up and walked along the train (in the direction in which the train is moving), how fast would $T$ move according to $S$?

You assumed the Newtonian law of velocity addition: $v+u=v_T$ , where $u$ is the velocity of person $T$ relative to the train and $v_T$ is the velocity of person $T$ relative to $S$. For example, if the train was moving with $100 km/h$ and the person inside the train was walking with $3km/h$, the person standing at the train station would say that $T$ moves with $103km/h$.

According to special relativity, this law is false. The first postulate of special relativity is that light does always have the same velocity in every reference frame. So if person $T$ switched on a torch and pointed it in the direction of motion of the train, both $T$ and $S$ would say that the light was moving with $c$ (where $c=299,792,458 m/s$). But according to the law of velocity addition from above, the light would have velocity $v+c$ according to $S$ - which is false because it always moves with the same velocity no matter whom you ask.

The Einstein velocity addition formula is the correct way to calculate the velocity. It says that $v_T=\frac {v+u}{1+\frac{vu}{c^2}}$. In everday life, where velocities usually are much smaller compared to the speed of light, this formula reduces to the Newtonian law of velocity addition. In the case of our train $v_T\approx102.999971362km/h$ which is almost equal to $103km/h$. That's why you still can use the normal velocity addition formula when you are dealing with very small velocities - the error is neglegible. The velocity of the light emitted by the torch on the other hand is $c$ according to the formula above - the classical velocity addition formula gives a false result. Also notice that $v_t$ can never be equal to $c$ or be bigger. That's why the speed of light can't be attained by any object that has mass.

Now calculate the different speeds of the stones in your thought experiment using the correct velocity addition formula and you'll see that the speed of light will never be exceeded by your objects (although the involved velocities can become arbitrarily close to $c$).

The thing with special relativity is that the faster an object is moving away from the reference object, the slower the time seems to run for the speedy object. Because of this the speed at which it is flying away from anything else is always below the speed of light. Btw, expansion of space is not the same as moving away because it's not actualy moving trough space. Just the space and the object with it is being moved away.

• Oh, wait, so does that explain why you would need to add infinite energy to accelerate something to c? Because if the impuls on an object is F*t, and one second becomes longer and longer the faster you move, you would need to keep adding F forever eventually. Or am I getting this wrong... – E. Holm May 12 '17 at 22:14
• This is correct. – MaDrung May 18 '17 at 10:45