Why Do Relativistic Signals Differ Between Observers?

[Scigatt]

I was solving this problem, and I was comparing my answers with a classmate, and we got very different answers. I don't think I made a mistake here, but I may have missed something.

Homework Statement

Tipler - Modern Physics 5e 1-57.
"Two observers agree to test time dilation. They use identical clocks and one observer in frame S' moves with speed v = 0.6c relative to the other observer in frame S. When their origins coincide, they start their clocks. They agree to send a signal when their clocks read 60 min and to send a confirmation signal when each receives the other's signal.

(a) When does the observer in S receive the first signal from the observer in S'

(b) When does he receive the confirmation signal?

(c)Make a table showing the times in S when the observer sent the first signal, received the first signal, and received the confirmation signal. How does this table compare to the one constructed by S?"

$$\beta = 0.6$$

$$\gamma=\frac{1}{\sqrt{1 - \beta^{2}}}$$

$$\gamma\left[\begin{array}{cc}1&-\beta\\-\beta&1\end{array}\right]\left[\begin{array}{cc}d\\ct\end{array}\right]=\left[\begin{array}{cc}d'\\ct'\end{array}\right]$$

The Attempt at a Solution

a) The observer in S' sends their first signal at d' = 0,ct'. To find where this is in S we do an inverse 1D Lorentz transform.

$$\gamma\left[\begin{array}{cc}1&\beta\\\beta&1\end{array}\right]\left[\begin{array}{cc}0\\ct'\end{array}\right]=\left[\begin{array}{cc}\gamma\beta ct'\\\gamma ct'\end{array}\right]$$

This is the point in S when the observer in S' sends the signal. This signal travels at the speed of light in the -d direction until it hits the ct axis.(d = 0)

Thus:

$$ct_{light} = -d_{light} = \gamma\beta ct'$$

The total ct is

$$ct\ =\ ct_{light} + ct_{hour}\ =\ \gamma\beta ct' + \gamma ct'\ =\ \gamma(1 + \beta)ct'$$

entering
$$\gamma\ =\ \frac{1}{\sqrt{1 - 0.6^{2}}}\ =\ 1.25,\ \beta = 0.6,\ t' = 1\ hr.$$

we get t = 2 hrs.

b)First, we need to find our when S' received the first message. Here we want to transform S --> S', so we use the forward Lorentz transform:

$$\gamma\left[\begin{array}{cc}1&-\beta\\-\beta&1\end{array}\right]\left[\begin{array}{cc}0\\ct\end{array}\right]=\left[\begin{array}{cc}-\gamma\beta ct\\\gamma ct\end{array}\right]$$

This is the point in S' when the observer in S sends the signal. This signal travels at the speed of light in the d' direction until it hits the ct' axis.(d' = 0)

Thus:

$$ct'_{light} = d'_{light} = -(-\gamma\beta ct) = \gamma\beta ct$$

The total ct' is

$$ct'\ =\ ct'_{light} + ct'_{hour}\ =\ \gamma\beta ct + \gamma ct\ =\ \gamma(1 + \beta)ct$$

This comes out to t' = 2 hr.

Now we can use the general formula in S for the confirmation pulse:

$$ct\ =\ ct_{light} + ct_{hour}\ =\ \gamma\beta ct' + \gamma ct'\ =\ \gamma(1 + \beta)ct'$$

with t' = 2 hrs. we get t = 4 hrs. as the arrival time of the confirmation pulse.

c)
Time first signal sent: +1 hour.
Time first signal received: +2 hours.
Time confirmation received: +4 hours.

The table should be the same with observer S'.

Attached is a Minkowski diagram of the scenario from the perspective of S which seems to confirm my calculations.

My classmate got a bit over 4000 seconds for a) and just over 2 hours for b).

 

[AvroArrow]

It is possible that your classmate made a mistake in their calculations. It is also possible that they are using different parameters for the problem. For example, if they use a different value for the speed of light, this could lead to a different answer. Additionally, they may have different assumptions about the time at which the clocks were synchronized (e.g., did they start at 0 seconds or 1 hour?). You should compare the parameters and assumptions that you both used to make sure that you are on the same page.