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Let us say we have two observers observing the same two events in space time. Relative to one another they are travelling at .5c and are initially separated by 1 lightsecond distance.

The first event happens at the location of observer 1. The second event happens 1 lightsecond away in the direction of observer 2 and happens 1 second after the first. What is the correct way for each observer to calculate the proper time between the two events?

I tried this many ways trying to figure this out but this is the method that made the most sense to me. I am using units such that c=1.

observer 1 witnesses event 1 at time 0 by its clock and location 0. It sees event 2 at time 2 seconds and distance 1.

Δτ=√[(2)

^{2}-(1)

^{2}]=√(3)

intuitively this doesn't feel right. I believe the proper time should be 1 as it would be for someone travelling between the two events at the speed of light so they both occur at the same location.

observer 2 i had trouble determining. I have two methods. one assumes the speed of light is constant and independent of inertial reference frame and the other doesn't. I believe the former is the correct view.

c is constant:

event 1 happens at location 1 relative to observer coordinate system. observer 2 sees event 1 second later at time 1 and location 1. event 2 happens at location .5 and is seen at time 1.5.

Δτ=√[(.5)

^{2}-(.5)

^{2}]=0

c not constant:

event 1 happens at location 1 relative to observer coordinate system. observer 2 sees event 2 second later at time 2 and location 2. event 2 is seen at location 1 and at time 2.

Δτ=√[(0)

^{2}-(1)

^{2}]=i

My understanding is that proper time should be constant between inertial coordinate systems so it shouldn't matter what the speed of the observer is. Suffice it to say I have gotten myself very confused and can go no further on my own. I appreciate any help you all can give.