Relativity along an axis in an inertial frame

[ZanyCat]

Suppose that two events occur on the x-axis of an inertial frame, Δx apart with a time interval between the events of Δt.
a) the proper time interval between the events is...?
b) the proper distance between the events is...?

I think I'm just getting confused by the wording. I imagined that I was in the same frame of reference, and therefore the answers are Δt and Δx. But evidently, I'm wrong. Do I need to set the speed of the frame to 'v' and do something with simultaneous equations to remove that variable?

Thanks!

 

[ZanyCat]

OK, so I defined a stationary reference frame as S, and defined the frame in the question as S'. S' is moving wrt S at a velocity v.
So the proper time is the Δt observed in S', and the proper length is the Δx observed in S'.

I think I've worked out a, but struggling with b. I'm using the length contraction formula as one equation, and the Lorentz coordinate transformation as the second equation, but when I solve them simultaneously I can only achieve v=0.

 

[voko]

Start with definitions. What are proper time and distance?

 

[ZanyCat]

Proper length is measured distance in the FOR where the objects are at rest, i.e. in frame S'.

I'm using the equations L' = L/gamma and x' = gamma(x-vt) and trying to solve these simultaneously, am I on the right track?
I can't determine whether L' = x' and L = x, or L' = x and L = x', though...

 

[voko]

What are the "objects" in the case? Are they at rest as stated? In what reference frame are they at rest?

 

[ZanyCat]

The objects are two arbitrary points situated along the x-axis of S', and are at rest in frame S', thus always separated by delta x.

 

[voko]

If you measure distance between two arbitrary points, you get arbitrary results. I do not think this is what the problem is about. Connect "objects" with the description of the problem.

 

[HallsofIvy]

Suppose that two events occur on the x-axis of an inertial frame, Δx apart with a time interval between the events of Δt.
a) the proper time interval between the events is...?
b) the proper distance between the events is...?

I think I'm just getting confused by the wording. I imagined that I was in the same frame of reference, and therefore the answers are Δt and Δx. But evidently, I'm wrong. Do I need to set the speed of the frame to 'v' and do something with simultaneous equations to remove that variable?

Thanks!

Given the information as stated, with everything motionless in an inertial frame, why would the "proper time interval" not be $\Delta t$ and the "proper distance" $\Delta x$.