Distance between two events in a moving reference frame

[nagyn]

Homework Statement

Observer O at the origin of a coordinate system is at rest relative to two equidistant space stations located at x=+3.00×10^6km (station A) and x=−3.00×10^6km (station B) on the x axis. In reference frame O, station A sends out a light pulse at t = 0 (event 1) and station B also sends out a light pulse at t = 0 (event 2). Observer C moves relative to O with velocity 0.650 c0 in the positive x direction.

What is the displacement from event 1 to event 2 according to observer C?

Homework Equations

L = Lproper / gamma

The Attempt at a Solution

If event 1 and event 2 occur at two locations that are at rest relative to observer O, can I say that the measured distance from 1 to 2 in O's reference frame is the proper length? Because that's how I approached the problem.

Lproper = 2*(3.00*10^6 km)*(1000m/1km) = 6*10^9 m
gamma = 1/sqrt(1 - ((0.65c)^2/c^2)) = 1.316

L observer C = L observer O / gamma = (6*10^9 m)/(1.316) = 4.56*10^9 m

Is the assumption I made incorrect?

 

[TSny]

The distance between the two stations A and B according to observer O would be a proper length. That is, if you imagine a rod extending between the two stations, the length of the rod according to O would be the proper length of the rod. The length of this rod according to frame C would be contracted compared to the proper length.

But, the distance between the events 1 and 2 according to frame C is not the same as the length of the rod according to frame C. Can you see why?

 

[nagyn]

The distance between the two stations A and B according to observer O would be a proper length. That is, if you imagine a rod extending between the two stations, the length of the rod according to O would be the proper length of the rod. The length of this rod according to frame C would be contracted compared to the proper length.

But, the distance between the events 1 and 2 according to frame C is not the same as the length of the rod according to frame C. Can you see why?

I'm guessing because the two events are not simultaneous in C's reference frame. Although I'm not sure how to apply that.

I do know that in C's reference frame both pulses will still reach observer O at the same time.

I also know that if observer C was at the location at the instant of event 1 and moving in the positive x direction, he would conclude that event 2 had to have happened first for both pulses to still reach O simultaneously. The question doesn't specify C's location at all so I don't know if I can still apply that.

I also calculated the time it takes in O's reference frame for the pulse to reach them (d/v = (3x10^9)/(3x10^8) = 10 seconds). Is that a good start?

 

[TSny]

I'm guessing because the two events are not simultaneous in C's reference frame.

Yes, that's right.

I do know that in C's reference frame both pulses will still reach observer O at the same time.

Yes.

I also know that if observer C was at the location at the instant of event 1 and moving in the positive x direction, he would conclude that event 2 had to have happened first for both pulses to still reach O simultaneously.

I'm not sure if that's true. Can you convince me with an argument?

From C's point of view, is observer O moving away from the light pulse of event 1 or is observer O moving toward the light pulse of event 1?

The question doesn't specify C's location at all so I don't know if I can still apply that.

The location of C doesn't matter. All observers in C's frame of reference will come to the same conclusions about the events no matter where the observers are located.

I also calculated the time it takes in O's reference frame for the pulse to reach them (d/v = (3x10^9)/(3x10^8) = 10 seconds). Is that a good start?

It might be useful. There are different ways to approach this problem. If you are familiar with the Lorentz transformation equations, then you can use them to get the answer with very little work. Otherwise, you can use kinematics in C's frame of reference along with the fact that the distance between stations A and B is Lorentz contracted in C's frame.

 

[nagyn]

I'm not sure if that's true. Can you convince me with an argument?

I'll try the Lorentz approach, but now I'm curious if my logic for this part is wrong:

Since the speed of light is the same in all reference frames, if C is directly over event 1 (which is a light pulse), then C will see the wavefront emit radially outward in every direction as if event 1 were stationary with him. Observer O and event 2 are moving towards him in his reference frame but event 1 does not move. So when the wavefronts reach observer O, observer C perceives O as having moved nearer to event 1 while remaining the same distance from event 2. In order for both wavefronts to reach O at the same time, if event 2 covered a greater distance it had to have occurred first.

 

[TSny]

I'll try the Lorentz approach, but now I'm curious if my logic for this part is wrong:

Since the speed of light is the same in all reference frames, if C is directly over event 1 (which is a light pulse), then C will see the wavefront emit radially outward in every direction as if event 1 were stationary with him. Observer O and event 2 are moving towards him in his reference frame but event 1 does not move.

Events are neither stationary nor nonstationary with respect to an observer. An event is just a "happening" at a specific spactime point. It doesn't make sense to speak of the motion of an event.

So when the wavefronts reach observer O, observer C perceives O as having moved nearer to event 1 while remaining the same distance from event 2.

You seem to be saying that as the light travels from event 1 towards observer O, observer C sees O move nearer to where event 1 occurred (according to C). I don't think that's right. Just to make sure we're together on the setup, here's a diagram of the situation from the point of view of observer O. Note that C moves to the right according to O.

So, if you switch to the reference frame of C, in which direction do A, B, and O move relative to C?

 

[nagyn]

If A, B and O are all at rest relative to each other then they all move to the left at the same velocity in C's reference frame.

I was able to solve the problem using Lorenz' equations, but I guess I get a little confused about the difference between an event occurring and an event being received/observed. In C's reference frame O is still equidistant from where event 1 and 2 occur, and O still receives both pulses at the same instant. So if the distance both events travel are equal to each other and are received at the same time by O in both frames of reference, how can observer C conclude at all that the events are not also simultaneous in his reference frame?

 

[TSny]

In C's reference frame O is still equidistant from where event 1 and 2 occur,

According to C, observer O is always equidistant between stations A and B. But C would not say that O remains equidistant between the place where event 1 occurred and the place where event 2 occurred. It might help to look at the picture below.

The top figure shows the situation according to C at the instant event 1 occurs. The bottom picture shows the situation according to C at some later time when the light pulse from event 1 is still traveling toward O. The star in the bottom picture marks where event 1 occurred in C's frame of reference.

O still receives both pulses at the same instant.

Yes

So if the distance both events travel are equal to each other

The events don't travel. Only objects travel. So, observer C would say that observer O travels, the space stations A and B travel, and the light pulses travel. But the events don't travel. According to C, the light pulses do not travel equal distances to arrive at O.

and are received at the same time by O in both frames of reference, how can observer C conclude at all that the events are not also simultaneous in his reference frame?

It would be helpful to draw some more figures similar to the one above. Draw a figure for the instant when event 2 occurs. Has event 1 already occurred at this instant according to C? Below this figure, draw another figure for the event where both light pulses arrive at O.

 

[nagyn]

I think I understand now.

First of all it wasn't until I saw your diagram that I realized event 1 was on the positive axis and event 2 was on the negative axis. I had them switched, so when I was arguing that event 2 occurred first in C's frame, I meant event 1.

But to re-approach my argument with what I think is the correct reasoning: events 1 and 2 are stationary in C's frame. The stations themselves and O are all moving away from C at the same velocity. When both pulses reach O, O is closer to event 2 in C's frame. This means, in C's frame, the pulse from event 1 had to travel a greater distance to reach O at the same time as the pulse from event 2, so event 1 must have happened first.

I included a very crude diagram to show this. I hope this is the correct reasoning.

 

[TSny]

I think I understand now.

First of all it wasn't until I saw your diagram that I realized event 1 was on the positive axis and event 2 was on the negative axis. I had them switched, so when I was arguing that event 2 occurred first in C's frame, I meant event 1.

OK. Good.

But to re-approach my argument with what I think is the correct reasoning: events 1 and 2 are stationary in C's frame.

You're probably going to get frustrated with me quibbling about wording. In order to know if something is stationary, that something must exist for some interval of time so that you can see if its position changes over time. But an event occurs at one instant of time. It does not exist for any interval of time. So you cannot say whether or not an event is stationary.

The stations themselves and O are all moving away from C at the same velocity.

Yes

When both pulses reach O, O is closer to event 2 in C's frame.

I would agree that when the pulses arrive at O, O is closer to the place where event 2 occurred than where event 1 occurred (according to C).

This means, in C's frame, the pulse from event 1 had to travel a greater distance to reach O at the same time as the pulse from event 2, so event 1 must have happened first.

Yes!

I included a very crude diagram to show this. I hope this is the correct reasoning.

Your diagram seems to show pulse 2 traveling slower than pulse 1. It also shows event 2 occurring at essentially the same time as event 1. But, as you said, event 1 occurs earlier than event 2.

 

[nagyn]

Your diagram seems to show pulse 2 traveling slower than pulse 1. It also shows event 2 occurring at essentially the same time as event 1. But, as you said, event 1 occurs earlier than event 2.

Ah, the mistake in my diagram is that I tried to show events 1 and 2 simultaneously at the top, as they would be in O's frame. But this is in C's frame.

If I start with the bottom scenario in the diagram and work backwards, the pulse from event 2 will be back at the location of event 2 before the pulse from event 1 makes it back to the location of event 1.

 

[TSny]

Ah, the mistake in my diagram is that I tried to show events 1 and 2 simultaneously at the top, as they would be in O's frame. But this is in C's frame.

If I start with the bottom scenario in the diagram and work backwards, the pulse from event 2 will be back at the location of event 2 before the pulse from event 1 makes it back to the location of event 1.

Yes. That's right.