Calculate Proper Time for Two Events in Special Relativity

[electronneutrino]

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 traveling 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 traveling 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.

 

[Orodruin]

It sees event 2 at time 2 seconds and distance 1.
Δτ=√[(2)2-(1)2]=√(3)

This is not correct. Even if the signal does not arrive at the observer until 2 seconds after, what goes into the spacetime interval is the actual time in that reference frame, i.e., the time corrected for the light travel time.

I believe the proper time should be 1 as it would be for someone traveling between the two events at the speed of light so they both occur at the same location.

This is also incorrect. The events are at light-like separation so the spacetime interval between them should be zero.

 

[electronneutrino]

This is not correct. Even if the signal does not arrive at the observer until 2 seconds after, what goes into the spacetime interval is the actual time in that reference frame, i.e., the time corrected for the light travel time.

So this would lead to a proper time of 0 seconds.

This is also incorrect. The events are at light-like separation so the spacetime interval between them should be zero.

ok that makes sense. Is it not also true that proper time can be interpreted as the time measured on a clock by an observer moving such that they are present at both events? In which case wouldn't such an observer measure the distance between them at 0 and the time as 1 second leading to proper time 1?

Further, wouldn't that also make the equation for observer 2
c is constant:
event 1 happens at location 1 relative to observer coordinate system. observer 2 sees event 1 second later at time 0 and location 1. event 2 happens at location .5 and is seen at time 1.

Δτ=√[(1)²-(.5)²]≠0

I do appreciate your help. Thank you.

 

[Orodruin]

So this would lead to a proper time of 0 seconds.

Careful here. Observers in relativity are described by time-like world lines. It is not that meaningful to talk about the "proper time" of a light signal. Light pulses can still be affinely parametrised.

event 1 happens at location 1 relative to observer coordinate system. observer 2 sees event 1 second later at time 0 and location 1. event 2 happens at location .5 and is seen at time 1.

I am sorry, but this is just word sallad. I suggest you write down the coordinates and do the proper Lorentz transformation to find out how the events appear in the other reference frame.

 

[pervect]

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 traveling 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?

The description of the two events is not clear, because you specified it happened "1 second after the first" event, but "one second after the first event is ambiguous without a description of which frame of reference you're using. The position of the event is also ambiguous- presumably the second event is in "the same place" as the first event, but "at the same place" according to whom?

If you could specify the coordinates of the events according to observer 1 (or observer 2), it'd be much easier to understand exactly what question you're asking is.

On a more general note, the proper interval between two events is not necessarily timelike. You have two events, and you draw a straight worldine between them on a space-time diagram. If said worldline is timlike, you have the case that your textbook describe, which gives the timelike interval between two events.

It basically seems to me like you are not fully understanding what a proper time is. The reason that you see two events happening along a timelike worldline at different times is that is one of the simplest, and easiest to grasp, definitions of what a proper time is. At a guess, you are probably conflating "proper time", which is Lorentz invaraint, with some other sort of time interval, but it's not really clear to me what you're asking, so I could be wrong.

 

[pervect]

Let me try an overview which might allow you to answer your own question.

Start off with the idea that events are points on a space-time diagram. One draws two events on said diagram. This is where I'm currently getting stuck, because I don't understand the verbal description of the problem, it's ambiguously worded.

Then one draws a straight line that passes through both events.

If the interval between the two events is timelike, then line on the space-time diagram connecting the two events is the woldline of some observer, a worldline moving at some physically possible velocity. THen the invariant interval (length of the line) is just the amount of time that a clock traversing the worldline would record in moving from one event to the other along the straight line.

Sometimes, though, the worldline won't be timelike separated. This happens when the events are outside of the "light cones" of each other. I'm not sure if the phrase "light cones" is familiar or not, if not I can explain more, I suppose. It's a very useful concept in special relativity, though, and quite common, so I hope it's familar to you. Noting in special relativity can move faster than light, so if the two events are far enough apart in space, there is no light ray that could connect them. We say that the events are "space-like separated" in that case. There is a third case, where a light beam can just reach from one event to the other. In that case, the invariant interval between the two points is always zero.

In the case where the two events are space-liek separated, there will be some frame of reference where both events happen at the same time. This may not be obvious, I can discuss this point in more detail if there is some interest. For this idea to make sense, one needs to realize that "at the same time" in special relativity is observer-dependent, there is no universal notion of "at the same time". This is called the relativity of simultaneity, and it's a common source of confusion about special relativity.

Then the space-like interval between the two events that are space-like separated is given by the distance between the two events in a frame of reference where the two events happen at the same time.

A space-time diagram would help here too, to demonstrate that this is possible. I have stated that there is an obsever who regards both events as simultaneous this without proof at the moment, I am uncertain if it'd be helpful to explain this point in more detail. My inclination is to wait for a question about this issue, because perhaps I'm not answering the question you really wanted to ask. I will say that if you can draw light cones of both events on the space-time diagram, which requires familiarity with both space-time diagrams and light-cones, it's not hard to construct the worldline of the observer who regards the events as simutaneous.