Measuring Relative Speed of Frames Using Spacetime Interval

[russphelan]

Hi all,

Say two events happen in the same place according to one observer (1). They are separated in time by 3 years.

According to another observer (2) that is moving relative to the first, the events are separated by 5 years. We can calculate, using the invariance of the spacetime interval, that the events are separated in space by 4 lightyears, according to the second observer.

When we want to calculate the speed of observer 2 relative to observer 1, it is necessary to use the time separation of events according to observer 2. So, the relative speed would be

$\frac{4yrs \cdot c}{5yrs} = \frac{4}{5}c$

Why does the time interval according to observer 2 give us relative speed of the frames? Isn't this time interval dilated due to the amount of time it takes light from the events to reach the traveling observer?

Thanks,
Russ

 

[Orodruin]

In order to get the velocity, you need to take a something at rest in one of the system and consider how it moves in the other. In this case, the spatial difference between the events was 4 lightyears and the time difference 5 years in the second frame. In order to get from one event to the other, something moving linearly would need to be moving with 4c/5. This is just the definition of velocity in the second system, difference in space/difference in time. That this is the velocity of the first system relative to the second follows from the assumption that the event ocurs at the same spatial position in the first.

 

[russphelan]

Thank you very much for the reply!

It makes sense that velocity of the second frame is defined in this way. According to the first frame, the events have not moved at all, so their speed is 0. This is the system in which the events are at rest.

 

[Orodruin]

Thank you very much for the reply!

It makes sense that velocity of the second frame is defined in this way. According to the first frame, the events have not moved at all, so their speed is 0. This is the system in which the events are at rest.

This sounds a warning bell for me. Events do not have velocities. The velocity you get is the velocity of an object traveling with constant speed between the events. The events themselves are points in space-time. This might sound picky, but using correct terminology becomes very important and I have seen many people have misconceptions based on misinterpreting the nomenclature.

 

[russphelan]

I meant to say that the speed of the frame in which the events happen in the same place is 0. The speed of the frame in which the events have a spatial separation is given by the distance measured between the events, divided by the time measured between the events, from that frame.

Does that make sense?

This brings up another point though: say a frame is moving with respect to your frame. Is it safe to say that events happening in this moving frame do not have a "speed"? I would guess that the place at which the event actually occurs does not continue to move with the moving frame.

Thanks so much for taking the time to consider these posts.
Russ

 

[Orodruin]

I meant to say that the speed of the frame in which the events happen in the same place is 0. The speed of the frame in which the events have a spatial separation is given by the distance measured between the events, divided by the time measured between the events, from that frame.

To be more precise: The speed relative to the frame where the events occur at the same spatial point is given by this procedure.

Is it safe to say that events happening in this moving frame do not have a "speed"? I would guess that the place at which the event actually occurs does not continue to move with the moving frame.

Events do not happen in a frame. All events occur in all frames, they just have different coordinates. No events have speed, they are just points in space-time. There is no such thing as "a place where the event actually occurs".

There is also no distinction whatsoever between frames. You cannot say that a frame is at absolute rest and all others are moving. Any inertial observers can consider themselves at rest, even if they are moving relative to each other. This is just Galilean relativity.