# Variable speed of light

1. Jul 3, 2011

### lovetruth

In a frame S, there are two stationary objects A & B at a distance L in the x-direction. Object A is nearer to the observer than object B. Light will take time T to reach object B from A. Therefore in frame S,
L/T = c

In another frame S’, both objects A & B are moving at velocity v in positive x-direction with respect to the observer. Therefore in frame S’, the distance between object A & B will be :
L’ = L/(gamma)
and the time for light to reach object B from A will be:
T’ = T(gamma)

An observer in frame S’ will observe that the total distance travelled by light in reaching object B from A will be:
L’ +v(T’)
Also, the time taken by light to reach object B from A will be: T’

Thus, an observer in frame S’ will calculate the speed of light to be:
[L’ + v(T’)]/T’ = (L’/T’) + v
= [c/(gamma)^2] + v
Thus, in frame S’, the speed of light will not be c. How?

2. Jul 3, 2011

### Staff: Mentor

OK.
No. How did you deduce this?

FYI: You cannot simply apply the 'time dilation' formula here. That formula applies to time measurements made in a moving frame at a single location, but here you are concerned with the time to travel between two locations.

Last edited: Jul 3, 2011
3. Jul 3, 2011

### lovetruth

time dilation

4. Jul 3, 2011

### Staff: Mentor

See the note I added above.

5. Jul 3, 2011

### lovetruth

I dont understand what you mean by 'Time dilation at a single point'. Time is just an interval. Also, I have read about derivation of time dilation by using 'light clock' in which the light travels between two points.

6. Jul 3, 2011

### Staff: Mentor

I spoke of time measurements made at a single point, such as by a clock.
Sure. As long as you take as your 'interval' the round trip time for the light to return to its starting point.

To figure out the time it takes light to travel from A to B in the primed frame, do this:

L/(gamma) + VT' = CT'

7. Jul 3, 2011

### bcrowell

Staff Emeritus
Lovetruth, the problem here is that you're trying to do a calculation that requires the Lorentz transformation, not just gamma factors for time dilation and length contraction. If you write the Lorentz transformation as an equation for (t',x') in terms of (t,x), there are four terms, one of which causes t' to depend on x. (This is all just a different way of saying what Doc Al said.) By finding an inconsistent result, what you've basically proved to yourself is that you need to learn about the Lorentz transformation.

Here are a couple of books that do the Lorentz transformation in detail:
Takeuchi, An Illustrated Guide to Relativity
Mermin, It's About Time: Understanding Einstein's Relativity

8. Jul 3, 2011

### lovetruth

Why the 'interval' for the round trip and not for single trip. I thought time dilation holds for any event (which in this case is the journey of light from object A to B)

9. Jul 3, 2011

### Staff: Mentor

The simple time dilation formula, T = T'*gamma, only applies to time measurements made on a single moving clock (thus at a single location in the moving frame). If you want to apply 'time dilation' to an arbitrary time interval, you need the full Lorentz transformations.

10. Jul 3, 2011

### bcrowell

Staff Emeritus
Yep.

Lovetruth, the amount of time you've spent on this forum discussing relativity indicates an intense interest in the subject. That's great. But I'm frankly extremely shocked that you've spent that amount of time talking, calculating, and thinking about SR without apparently learning what the Lorentz transformation is or why it's necessary. Given the relative sophistication of the topics that you've been trying to discuss, like transformations of electric and magnetic fields between frames, it had never occurred to me that you hadn't built any kind of foundation beyond what one finds in popularizations.

It's a waste of your time and ours for you to keep posting things like this without taking some time to learn the fundamentals of the subject. PF isn't a substitute for reading books at the appropriate level. You cannot learn a nontrivial subject like relativity in the manner you've been doing, by repeatedly attempting calculations, getting them wrong, and then posting them and expecting other people to fix both the calculation and the underlying lack of background.

11. Jul 3, 2011

### lovetruth

Thank you.
I thought that time dilation is valid for any event

12. Jul 3, 2011

### lovetruth

You are right that I only read 'Popularization'. I am just an amateur not a full time physicist. Also, I try to read as much as I can. But you know that all academic books you prescribe to me are long and time-consuming. My lifetime would be too short to read all the books there are in physics, let alone science & math. Even in physics, people have specialized into classical, relativity, quantum mechanics, cosmology.. etc but not whole of physics. So dont construe my efforts to understand physics as insincere.