# RELATIVITY: Co-ordinates of events

• PeterPeter
In summary: I am supposed to be tutoring someone on relativity. I am failing dismally. Another senior moment. Time to take a break.
PeterPeter

## Homework Statement

A meter ruler moves at velocity u to the right past a stationary observer. The observer is at (0,0) in his rest frame. Give the (x,t) co-ordinates of the following events.
1. The right end of the ruler passes the observer (in the observer's frame)
2. The left end of the ruler passes the observer (in the observer's frame)
3. The observer passes the right end of the ruler (in the ruler's frame)
4. The observer passes the left end of the ruler (in the ruler's frame)

## Homework Equations

t2 = $gamma$*(t2 dash + u * x2 dash / $c^2$)
x2 = $gamma$*(x2 dash + u * t2 dash)

Where the dashed co-ordinates are in the ruler's rest frame.

## The Attempt at a Solution

1. The right end of the ruler passes the observer (in the observer's frame) (0,0)
2. The left end of the ruler passes the observer (in the observer's frame) (0,t2)
3. The observer passes the right end of the ruler (in the ruler's frame) (0,0)
4. The observer passes the left end of the ruler (in the ruler's frame) (-1,t2 dash)

which when I use the above equations I get t2 dash = 1/u

Where have I gone wrong?

PS Not sure how you get subscripts and dashes in LaTex

It would be easier to tell you what goes wrong if you show us how you arrive at your answer.

Regarding the LaTeX part, I have fixed your equations for you:
$$t_2 = \gamma(u) \left(t'_2 + \frac{u x'_2}{c^2}\right)$$
$$x_2 = \gamma(u) (x'_2 + u t'_2)$$
if you want to know how it was done - quoting this message will show you the code.

PeterPeter said:

## Homework Statement

A meter ruler moves at velocity u to the right past a stationary observer. The observer is at (0,0) in his rest frame. Give the (x,t) co-ordinates of the following events.
1. The right end of the ruler passes the observer (in the observer's frame)
2. The left end of the ruler passes the observer (in the observer's frame)
3. The observer passes the right end of the ruler (in the ruler's frame)
4. The observer passes the left end of the ruler (in the ruler's frame)

## Homework Equations

t2 = $gamma$*(t2 dash + u * x2 dash / $c^2$)
x2 = $gamma$*(x2 dash + u * t2 dash)

Where the dashed co-ordinates are in the ruler's rest frame.

## The Attempt at a Solution

1. The right end of the ruler passes the observer (in the observer's frame) (0,0)
2. The left end of the ruler passes the observer (in the observer's frame) (0,t2)
3. The observer passes the right end of the ruler (in the ruler's frame) (0,0)
4. The observer passes the left end of the ruler (in the ruler's frame) (-1,t2 dash)

which when I use the above equations I get t2 dash = 1/u

Where have I gone wrong?

PS Not sure how you get subscripts and dashes in LaTex
Your results for parts 3&4 are correct. You can use the LT to get the results for parts 1 & 2 from the results of parts 3 & 4.

Chet

Chestermiller said:
Your results for parts 3&4 are correct. You can use the LT to get the results for parts 1 & 2 from the results of parts 3 & 4.

Chet

Thanks.

I was expecting to get something like length/gamma ie length contraction!

You will, if you do not remove t2' from your equations and instead use the relation t2' = 1/u to break out a factor t2' from your first equation.

I think it would be more instructive to give the ruler an arbitrary rest length L to be honest. That way units in your expression will make sense and you do not fall into the trap of removing t2'.

PeterPeter said:
Thanks.

I was expecting to get something like length/gamma ie length contraction!
You would only get that if both ends of the stick are observed at the same time in the unprimed frame.

Chet

Last edited:
I have reworked using a ruler length of L instead of 1m. I get:

1. The right end of the ruler passes the observer (in the observer's frame) (0,0)
2. The left end of the ruler passes the observer (in the observer's frame) (0,L/(u*gamma))
3. The observer passes the right end of the ruler (in the ruler's frame) (0,0)
4. The observer passes the left end of the ruler (in the ruler's frame) (-L,L/u)

So the stationary observer would see that the ruler is moving with velocity u and that it takes L/(u*gamma) seconds to pass him.

So I guess he would conclude that the ruler is u*L/(u*gamma) = L/gamma metres long?

PS How does one get the greek letter gamma? I clicked the "Quick symbols" by nothing happened.

Yes, but finding L this way assumes that u is known or measured by the observer. The problem is more well suited to motivate time dilation.

There is a gamma in the quick symbols: γ (between β and δ). If you want the gamma in LaTeX mode: \gamma

Chestermiller said:
You would only get that if both ends of the stick are observed at the same time in the unprimed frame.

As seen in OPs solution, this is not strictly necessary if the velocity u of the object is known. If an object with constant velocity u passes you in time t, you would probably deduce its length as ut and this would be equivalent to using the positions of the end points at the same time in your system.

1 person
Orodruin said:
Yes, but finding L this way assumes that u is known or measured by the observer. The problem is more well suited to motivate time dilation.

There is a gamma in the quick symbols: γ (between β and δ). If you want the gamma in LaTeX mode: \gamma

As seen in OPs solution, this is not strictly necessary if the velocity u of the object is known. If an object with constant velocity u passes you in time t, you would probably deduce its length as ut and this would be equivalent to using the positions of the end points at the same time in your system.
Thanks. I guess I had a mental lapse, because i had known that previously. "Senior moment."

Chet

Unfortunately, the font used by the quick symbols is awfully inadequate. The symbol for gamma "γ" looks too much like an "y". You can always use the latex "##\gamma##" instead.

dauto said:
Unfortunately, the font used by the quick symbols is awfully inadequate. The symbol for gamma "γ" looks too much like an "y". You can always use the latex "##\gamma##" instead.

Equations are always better in ##\LaTeX## :)

Thanks to you all. I think I'm starting to get the basics of SR!

## 1. What is the theory of relativity?

The theory of relativity is a scientific theory developed by Albert Einstein in the early 20th century. It explains the relationship between space and time, and how they are affected by the presence of matter and energy.

## 2. What are co-ordinates of events in relativity?

In relativity, co-ordinates of events refer to the measurements of space and time at a specific location and moment. These co-ordinates are relative to the observer and can be affected by factors such as gravity and velocity.

## 3. How does relativity affect our understanding of time and space?

Relativity has challenged our traditional understanding of time and space. It suggests that time is not absolute and can be affected by factors such as gravity and velocity. It also shows that space is not a fixed, three-dimensional entity, but can be curved and influenced by the presence of matter and energy.

## 4. What is the difference between special relativity and general relativity?

Special relativity deals with the relationship between space and time in the absence of gravity. It explains phenomena such as time dilation and length contraction. General relativity, on the other hand, includes the effects of gravity and explains the curvature of space-time caused by the presence of matter and energy.

## 5. How has the theory of relativity been proven?

The theory of relativity has been extensively tested and has been proven to be accurate in predicting various phenomena, such as the precession of Mercury's orbit and the bending of light around massive objects. It has also been confirmed through experiments such as the Hafele-Keating experiment and the Pound-Rebka experiment.

• Special and General Relativity
Replies
32
Views
2K
• Introductory Physics Homework Help
Replies
7
Views
1K
• Special and General Relativity
Replies
34
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
984
• Introductory Physics Homework Help
Replies
1
Views
845
• Introductory Physics Homework Help
Replies
6
Views
3K
• Introductory Physics Homework Help
Replies
11
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
969
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
962