Introductory Special Relativity Problem

[puneeth9b]

Homework Statement

I was reading from Spacetime Physics (2nd Edition) by Taylor and Wheeler when I stumbled upon this
problem I couldn't get right. Here's the gist of the part of the problem that I couldn't solve:

Assume Earth is the rest frame of reference and that the planet Sirius is in the same frame.
Rocket moves at $v = 0.75$.
It starts at Earth at spacetime coordinates $x = 0, t = 0$ in every frame of reference.
It travels at $v$ from Earth to Sirius, at a distance of 8.7 light years, waits there for 7 years and returns back.

The part of the problem that I couldn't solve is this:
[From textbook, James is on the rocket]
As he moves back toward Earth, James is accompanied by a string of incoming lookout stations
along his direction of motion, each one with a clock synchronized to his own. One of these incoming lookout stations, call it Z, passes Earth at the same time (in James's incoming frame) that James leaves Sirius to return home. What time does Z's clock read at this
event of passing? What time does the clock on Earth read at this same event?
[Ends Here]

All distances and time intervals mentioned are measured in the rest frame. All equations use the same units for distance and time and consequently velocity is dimensionless (scaled to c)

2. The attempt at a solution

Since the string of lookout stations have clocks synchronized to the those on the rocket, i assume the clocks are in an inertial frame moving at $v=-0.75$, given I choose the +ve x along motion from Earth to Sirius.
Until now I had used just two equations to solve all my problems:
1. Invariance of interval.
2. Combination of velocities.
This was because the book derived these two from the only two fundamental assumptions I know to exist in special relativity:
1.Light's speed in vacuum is constant in inertial frames.
2.Laws are mathematically equivalent in these frames.

Using the two equations I was able to derive the Lorentz equations but I think I don't fully understand them. Here's what I did:

$t$ = Z's clock reading = time elapsed in Z's frame when James just starts.
Variables followed by '1' are in Z's frame.
Earth to Sirius:
$t = (v*x1 + t1)/√(1 - v^2)$
$x1= 8.7 light years.$
$v = 0.75.$
$t1 = 8.7/0.75$
v is +ve here because in my understanding v is the relative velocity of the frame with x1 and t1 with respect to that in which t is measured.
Stay at Sirius:
$t = (v*x1 + t1)/√(1 - v^2)$
$x1 = 0.$
$t1 = 7 years.$

Adding the two values of time gives me the wrong the answer. Can someone tell me what I'm doing wrong? Thanks!

Note to those with the textbook: It's exercise problem 4-1

 

[TSny]

Variables followed by '1' are in Z's frame.
Earth to Sirius:
$t = (v*x1 + t1)/√(1 - v^2)$
$x1= 8.7 light years.$
$v = 0.75.$
$t1 = 8.7/0.75$

Hello. Note that the 8.7 lt-yr is the distance between Earth and Sirius in the Earth frame. So, what time does 8.7/.75 represent?

 

[puneeth9b]

Hello. Note that the 8.7 lt-yr is the distance between Earth and Sirius in the Earth frame. So, what time does 8.7/.75 represent?

Hey! t1 = 8.7/0.75 years(probably should've mentioned the "years" part) is the time taken by the rocket to reach Sirius in the Earth frame. Thanks

 

[TSny]

Hey! t1 = 8.7/0.75 years(probably should've mentioned the "years" part) is the time taken by the rocket to reach Sirius in the Earth frame. Thanks

OK, That's right. But I'm a bit confused because you said that variables followed by "1" are in Z's frame (not the Earth frame).

 

[puneeth9b]

OK, That's right. But I'm a bit confused because you said that variables followed by "1" are in Z's frame (not the Earth frame).

Oh crap! There's a mistake. All variables followed by '1' are in Earth's frame. I think it should make sense now! Thanks.

Is there a way I can edit the original post?

 

[TSny]

You can edit posts only during the first few hours after making the post (don't know the exact time). But, that's ok.

So, it looks like you've figured out how much Earth time it takes James to reach Sirius and how much additional Earth time that James waits at Sirius before heading back.

Can you figure out how long it takes according to James to travel to Sirius and the addition time according to James that he waits before heading back?

 

[puneeth9b]

You can edit posts only during the first few hours after making the post (don't know the exact time). But, that's ok.

So, it looks like you've figured out how much Earth time it takes James to reach Sirius and how much additional Earth time that James waits at Sirius before heading back.

Can you figure out how long it takes according to James to travel to Sirius and the addition time according to James that he waits before heading back?

Yeah that was a part of the question I got. But Z's frame moves in a direction opposite to that of James' when he travels from Earth to Sirius.
Anyway here's what I did for that part:

Here variables followed by '1' correspond to the Earth frame and those by '2' correspond to James' frame when he travels from Earth to Sirius.

Travel from Earth to Sirius:
$x1^2 - t1^2 = x2^2 -t2^2$
x1 = 8.7;
t1 = 8.7/0.75
x2 = 0 (Rocket is at Earth and Sirius when the events happen);
∴ t2 = 7.67 yrs

Stay at Sirius:
Since he isn't moving t2 = 7 years now.

Total time = 7.67 + 7 = 14.67 years

Thanks!

 

[puneeth9b]

So any idea how I could solve it?

 

[TSny]

Here variables followed by '1' correspond to the Earth frame and those by '2' correspond to James' frame when he travels from Earth to Sirius.

Travel from Earth to Sirius:
$x1^2 - t1^2 = x2^2 -t2^2$
x1 = 8.7;
t1 = 8.7/0.75
x2 = 0 (Rocket is at Earth and Sirius when the events happen);
∴ t2 = 7.67 yrs

Stay at Sirius:
Since he isn't moving t2 = 7 years now.

Total time = 7.67 + 7 = 14.67 years

That looks good. So, at the moment James leaves Sirius to head back to Earth his clock reads 14.67 years. Call that event A. Let event B represent the event of Z passing the earth. What does Z's clock read for event B?

 

[puneeth9b]

That looks good. So, at the moment James leaves Sirius to head back to Earth his clock reads 14.67 years. Call that event A. Let event B represent the event of Z passing the earth. What does Z's clock read for event B?

I'm sorry I don't quite follow how James' frame before his direction change can be related to the frame after the change (i.e Z's frame) ... Also I mentioned my attempt at the solution. Could you spot anything wrong in that? Thanks!

 

[TSny]

I agree with your solution for the time of James' clock at the moment he starts back to earth: 14.67 yrs.

At the event where James starts back to Earth (event A), he suddenly switches from the Earth-Sirius frame to the same frame as Z. The sudden switch in frames does not change the reading of James' clock. So, just after switching to the Z-frame, James' clock will still read 14.67 yrs.

One of your questions is what does the Z clock read at the moment the Z clock passes the Earth (event B). I did not see you give an answer to that question.

Then you have to answer the question: What is the Earth clock time for event B?

 

[puneeth9b]

Ohhh! synchronized means the clocks read the same value! I thought it means the clocks run at the same rate which would mean the velocities of the frames are equal. So 14.67 = time in James' rocket = time in Z's!

 

[TSny]

Ohhh! synchronized means the clocks read the same value! I thought it means the clocks run at the same rate which would mean the velocities of the frames are equal. So 14.67 = time in James' rocket = time in Z's!

Yes, that's right. On the way back, Z and James are in the same frame and their clocks are synchronized.

 

[puneeth9b]

Okay I'm stuck again :P

Variables followed by '1' are in Earth's frame, those followed by '2' are in Z's frame.

$x1^2 - t1^2 = x2^2 - t2^2$
x2 = 0 because the clock is at Earth when it time is recorded and initial value = 0 i.e when the rocket starts.
$x1/t1 = v = -0.75$ because Z's frame moves -0.75 wrt Earth's frame.
$∴ t1^2 (1 - v^2) = t2^2$
t2 =14.67 from before
$t1 = t2/√(1-v^2) = 22.17 years$.. That's wrong :P

 

[TSny]

A spacetime interval is an interval between two events. Consider the two events A and B defined earlier. Then,

Δx₁² - Δt₁² = Δx₂² - Δt₂²

where Δx₁ = x_1B - x_1A, etc.

 

[puneeth9b]

In my solution before these are the two events:
Event 1: Zero of spacetime in all frames i.e when the rocket just starts
Event 2: The clock passing by Earth

Can you tell me what's wrong? Thanks!

 

[TSny]

It will get confusing if you use "1" and "2" to label the events and also use "1" and "2" for the different reference frames. So, let's use "A" and "B" to label the events.

Event "A": James leaves Sirius to head back to earth. So, event A occurs at Sirius.
Event "B": Z-station passes earth. So, event B occurs at Earth.

Let reference frame "1" be the Earth frame and let frame "2" be the Z-station frame.

Let
Δx₁ = the difference in x coordinates of the two events as measured in frame "1".
Δx₂ = the difference in x coordinates of the two events as measured in frame "2".
Δt₁ = the difference in t coordinates of the two events as measured in frame "1".
Δt₂ = the difference in t coordinates of the two events as measured in frame "2".

What are the numerical values (including sign) of Δx₁, Δx₂, and Δt₂? (To determine the signs, you will have to specify how you are setting up the positive direction for the x axes in each frame.)

From the values of these three quantities, you should be able to determine the value of Δt₁ and interpret the result.