Why is this idea on addition of velocities in SR wrong?

[nhmllr]

I like to try and derive little things by myself if I think it is manageable. One such thing is the addition of velocities. I gave it a stab, got an answer, and it turned out to be wrong. So tell me where my logic messes up.

There's a spaceship moving at velocity u₁. A man in the spaceship throws a ball, which from his POV is thrown at velocity u₂ in the same direction of the ship. Let's look at the man in the spaceship's POV:
In 1 unit of time, the ball will move u₂ units of distance.

Now, from a man outside of the spaceship, any of distance d₀ in the spaceship is observed to have a distance of d₀√(1-u₁²/c²)

Also, any event that takes a time of t₀ in the spaceship is observed take the time of t₀/√(1-u₁²/c²)

So, in the spaceship the ball travels d₀/t₀

But outside the spaceship, I would think that it would take d₀/t₀ * (1-u₁²/c²)

But that's not it... What's wrong with my reasoning?

 

[jtbell]

I suspect it's because you're not taking into account the third member of the Holy Trinity of relativity (besides length contraction and time dilation): relativity of simultaneity.

I haven't worked out this particular example, but in my experience, almost all apparent paradoxes and inconsistencies that newcomers to SR come up with, turn out to boil down to this.

 

[nhmllr]

I suspect it's because you're not taking into account the third member of the Holy Trinity of relativity (besides length contraction and time dilation): relativity of simultaneity.

I haven't worked out this particular example, but in my experience, almost all apparent paradoxes and inconsistencies that newcomers to SR come up with, turn out to boil down to this.

You might be right, but looking at the example, I don't see how that would be it. There's only one event occurring in one place.

 

[PAllen]

Many things.

1) You define u1 and u2, but use v in formulas, without definition.

2) You try to scale the relative speed of the ball and spaceship, to the outside observer. But this (if it worked as described ... it doesn't) scaled relative speed doesn't factor in u1. Doesn't it seem like its needed? Conceptually closer to the mark (but still wrong) would by u1 + u2 * (scale factor from u1).

3) You don't consider relativity of simultaneity. When you make a statement: at t0, the ball is d0 from the passenger (according to the passenger), you are also implying the statement that passenger's clock reading t0 is simultaneous with the event of ball being d0 away from passenger (all according to passenger). However you ignore that these events are not simultaneous according to the outside observer. Conversely, a pair of simultaneous events for the ball and passenger according to the outside observer, will be non-simultaneous event for the passenger.

[edit: you corrected (1) since I wrote this. Good. The other two comments still apply.]

 

[nhmllr]

3) You don't consider relativity of simultaneity. When you make a statement: at t0, the ball is d0 from the passenger (according to the passenger), you are also implying the statement that passenger's clock reading t0 is simultaneous with the event of ball being d0 away from passenger (all according to passenger). However you ignore that these events are not simultaneous according to the outside observer. Conversely, a pair of simultaneous events for the ball and passenger according to the outside observer, will be non-simultaneous event for the passenger.

Hmm... That's a good point. This seems to be trickier than I thought.