Calculating Time Dilation

[coconut]

Hi. I read how to calculate time dilation (here's my version): I am stationary and I shine my torch up and find it takes 1 second to travel 300,000 kms. My co-star is moving to my "left" at speed v, and when they shine their torch up, I will find that it takes longer than 1 second to reach 300,000 kms. This is because light has to travel up 300,000 kms plus it also travels a bit to my left (vt). When I do the calculations I get the usual t' = t / sqrt(1 - v*v / c * c).

 

But.. it doesn't work if we shine our torches sideways, ie. to my "left". When I do those calculations, I get t' = t / (1 - v / c).

 

Anyone know why it only seems to work if we say light is travelling perpendicular to the direction of movement?

 

Thanks,

Coconut

[fafalone]

Are you treating v as a vector or as a scalar? It's a vector.

[coconut]

I'm not too hot with vectors, but I thought it might not matter since the light and the co-star are travelling in same direction (to the left). Here's the calculation I did for shining light sideways to the "left":

 

distance that light travels to reach end of co-stars 300,000 kms (ct') equals distance that light travels to reach end of my 300,000 kms (ct), plus the extra bit due to co-star moving to the left (vt').

 

ct' = ct + vt'

solving for t' gives t' = t / (1 - v/c)

 

and calculation for light travelling "up":

 

same as above, except this time the light is travelling "up" and co-star is travelling "left", so vectors are at right angles:

 

ct' = sqrt ((ct)^2 + (vt')^2)

solving for t' gives t' = t / (1 - v^2/c^2).

 

Maybe the discrepancy has something to do with the length contraction that happens in direction of motion?

[Aeschylus]

Remember:

 

x' = γ(x - u_xt)

 

y' = y

 

z' = z

 

t' = γ(1- βx/c)

 

edited to add: for the love of God turn Html on!!!!!!!!

[Aeschylus]

So in other words the direction most defintely does matter.

[coconut]

I did the calculation for the sideways motion again but this time for a complete trip to the left and back again (like the mirrors example). This time I got the same result as the upwards motion. This seems to imply that overall the time dilation of light averages out the same for left / right motion, as for up / down motion, except that left motion has greater time dilation than right motion. And infact the time dilation for right motion is less than mine!

 

Could this mean that whenever I measure any of my co-stars activities, eg. waving, then the wave to the left is slowed but the wave to the right is sped up, yet overall slower on average according to the usual gamma formula?