# Special relativity clocks observed from two frames

• wumple
In summary: This will give you the time that S' records on their clock. However, the time that S actually sees through a telescope will be different due to the effects of relativity and the speed of the two frames. This can be calculated using the Lorentz transformation equations. Essentially, it will be a combination of the time dilation effect and the time delay due to the speed of the frames.
wumple

## Homework Statement

Observers S and S' stand at the origins of their respective frames, which are moving relative to each other with a speed of .6c. Each has a standard clock, which, as usual, they set to zero when the two origins coincide. Observer S keeps the S' clock visually in sight. (a) What time will the S' clock record when the S clock records 5 micro seconds? (b) What time will Observer S actually read on the S' clock when his own clock reads 5 micro seconds?

## Homework Equations

time dilation: t = gamma (proper time)

## The Attempt at a Solution

I can solve part A by using time dilation. My confusion comes in understanding how to interpret the conditions set on part b - how is part b different from part A? I know that for time dilation, the proper time is the time measured when the clock is at rest. But how can I calculate what one observer sees on a clock in another frame?

I assume that part a is in observer S's frame.
I guess they just want to make it clear
that the relativistically calculated time on the S' clock
is different from the time on clock S' as
actually physically observed by S through a telescope.

This is a common mistake by beginners and
they just want to make sure that everyone understands it correctly

but how would I go about calculating what observer S sees on the clock of S' through a telescope? That's the part I don't understand.

work backward

For part B, we need to take into account the effects of length contraction as well. The observer S will see the S' clock as moving at a slower rate due to time dilation, but also as physically shorter due to length contraction. This means that the S' clock will appear to be ticking slower than the S clock, but also moving through space at a slower rate.

To calculate the time that observer S will actually read on the S' clock, we can use the equation for time dilation (t = gamma * t'), where t is the time measured by observer S and t' is the time measured by observer S'. We also need to consider the length contraction factor, which is given by L = L0 * sqrt(1 - v^2/c^2), where L0 is the proper length and v is the relative velocity between the two frames.

So, for part B, the time that observer S will read on the S' clock will be given by:

t = (gamma * t') / (sqrt(1 - v^2/c^2))

where t' is the time recorded by the S' clock when the S clock reads 5 micro seconds.

This shows that the time read by observer S on the S' clock will be less than 5 micro seconds, due to the combined effects of time dilation and length contraction.