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## Main Question or Discussion Point

I am refreshing myself on a bit of special relativity and I remember kind of taking the invariance of the spacetime interval for granted the first time. When thinking about it and running some examples I started to get confused so I decided to come here. All of the books I've read always have events happen at the same location but different times so I tried to extrapolate a slightly more complex senario. Below is what I came up with. Please help me figure out what I am doing wrong or misinterpreting. Thanks.

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)

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)

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)

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.

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}]=0c 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}]=iMy 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.