Difficulty understanding invariant 'c' implications

[obliv1]

In our world where 'c' is large, most people intuitively understand Galilean addition of velocities at everyday speeds i.e. if someone stands 40m behind me and rolls a ball towards me at 10m/s (assuming we are not moving relative to each other) it takes 4s to reach me. If we repeat the experiment while I'm walking away from them at 2m/s, the ball would take 5s to reach me by which time I would have moved to be 50m away from the person rolling the ball. In this latter case I am effectively reducing the relative speed at which the ball approaches me meaning it takes longer to reach me.

For simplicity, imagine repeating the above experiment in a universe where 'c', the universal speed limit is 10m/s. If I am standing still, there is no difference to the above case and the ball would still reach me at 10m/s after 4 seconds as before. Using the Relativity equation for addition of velocities S = v+u/1+(vu/c2) I can calculate that if as before I repeat the experiment while I move away from the ball roller at 2m/s, I am unable to change the relative speed of the ball in my direction at all, and it approaches me at 10m/s however fast I travel away from the ball roller.

What I can't quite understand is what this means in practice. i.e. I can grasp what happens in the Galilean view of things where I can reduce the relative velocity of the ball coming towards me (and therefore increase the time it takes to reach me) by moving away from the ball roller, but I can't quite grasp what NOT being able to affect the relative speed of the ball towards me means. Does it still take longer to reach me as in the Galilean view, or does it take 4s no matter how fast I move away from the ball roller?

Although I'm tempted to interpret things to mean that no matter how fast I move away from the ball thrower (in this low-c universe, or from a light source in ours) it always reaches me after the same amount of time, I know this cannot be the case because if in our universe a new star started to shine and during the time it took for it's light to reach Earth I traveled in the opposite direction away from Earth in a spaceship, I would expect to see the new star well after those I left behind on Earth saw it...and yet this seems like a Galilean way of looking at things i.e. it seems I DID then affect the relative velocity of light in my direction by moving away from its source.

I'm confused (clearly!) and would welcome any constructive pointers in the right direction to understand this more clearly. If it helps this all comes from my attempt to understand Relativity of Simultaneity, part of which states that a person on a train sees a lightning flash in the direction their train is traveling before one that happens at an equal distance behind the train, which seems to me as an assumption at odds with invariant 'c' (the relative velocity of the light you are moving towards has been increased and that of the light you are moving away from has been decreased) but clearly I'm missing something.

Regards,
OBL

 

[phinds]

I think what you are missing is that the SPEED of light is invariant but that has nothing to do with the fact that the DISTANCE light travels within one frame of reference IS governed by d=rt [do NOT try to apply this to the expanding universe]

The confusing part of it is that with your ball example, you can change BOTH the time it takes the ball to reach you AND the speed at which it is going relative to you when it reaches you.

With light, you can change the amount of time it takes to reach you, but you can NOT change the fact that when it reaches you, it is moving at c relative to you.What DOES change as you speed up is the amount of red-shifting of the light that reaches you, but it is still traveling at c. Intuitively, this is impossible, but human intuition is pretty much useless on the scales of the very large (cosmology) and the very small (quantum mechanics).

 

[HallsofIvy]

What is confusing is that you have not said what you are measuring things RELATIVE TO. That is, you have not stated a "reference frame" or "coordinate system" in which you are measuring things. As long as you and the person rolling the ball to you are stationary relative to one another, the two "reference frames" are the same and it doesn't matter. But if you are moving at 2 m/s relative to the person who rolls the ball, you have to state which coordinate system you are using.

Relative to the person rolling the ball, you are moving away at 2 m/s and the ball is rolling toward you at 10 m/s. In t seconds, you will be 40+ 2t m away and the ball will be 10t m away. The ball will reach you when 40+ 2t= 10t which gives 5 m just as before- according to his clock.

Relative to you, the other person is moving away from you at 2 m/s. The ball is moving at 10 m/s relative to him so, taking "t" as 10 m/s as you suggest, the same speed as the ball, the ball is moving toward you at (2- 10)/(1+ (2)(-10)/100)= -10. That is the ball is moving toward you at "the speed of light" which is the same in all reference systems. However, you now see the distance between you as "Lorentz contracted". In your frame of reference the distance between you is not 40 m but 40/(sqrt(1+ 2^2/10^2))= 40/sqrt(104/100) which is about 20 meters. Since you still see the ball as moving "at the speed of light", 10 m/s, it will reach you in, by your clock, 20 seconds. However, because he is moving at 2 m/s, relative to you, you will see his time dilated, relative to you, by that same Lorentz factor so you will see his clock, when the ball reaches you, as registering 40 seconds, just as he does.

 

[obliv1]

Thanks folks, really useful replies