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Firstly, we know that if you have a parameterized curve ξ(s), then you can find the proper time between two events at points s

_{1}and s

_{2}by using this formula (assuming that the curve is timelike):

dτ = ∫ sqrt( -g

_{ab}(dx

^{a}/ds)(dx

^{b}/ds) ) ds

where the bounds of the integral are from s

_{1}to s

_{2}

However, what if you do not have a parameterized curve, but you instead just have a simple vector, or you are simply given two events in a spacetime and then are told to find the proper time between the events?

For example, let's say we were working with Minkowski space and I pointed out two events to you. The first event would be

(t, x, y, z) = (1, 0, 0, 0).

The second event could be:

(t, x, y, z) = (5, 0, 0, 0).

Now I really don't think that the integral formula above would be applicable to this case here (mainly due to derivatives of constants being 0 and whatnot).

Instead, here is what I think that the proper steps to solve this problem are:

First, I would subtract the first event from the second event to get a vector [4, 0, 0, 0]. I believe that I would treat this vector as dx

^{a}when I applied the spacetime interval to it. In other words, dx

^{0}= 4 and the other components equal 0.

Next, I should apply dτ

^{2}= g

_{ab}dx

^{a}dx

^{b}

In the case of this example, dτ

^{2}= -16

Finally, the proper time between the two events would just be sqrt( -dτ

^{2}), which would in this case just be 4.

Is that correct?

If so, I know that this was probably an easy and obvious example. It is just that the most recent textbook that I read from only focused on what to do to get the proper time between events on the ends of a parameterized curve, but it did not talk about instances where you do not have a parameterized curve, but just two events. This was bothering me.

Thanks