Applying the spacetime interval to regular vectors instead of curves

[space-time]

I have some questions. Let us assume for these questions that I am using the (- + + +) sign convention.

Firstly, we know that if you have a parameterized curve ξ(s), then you can find the proper time between two events at points s₁ and s₂ 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 s₁ to s₂

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, dx⁰ = 4 and the other components equal 0.

Next, I should apply dτ² = g_abdx^adx^b

In the case of this example, dτ² = -16

Finally, the proper time between the two events would just be sqrt( -dτ² ), 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

 

[Pencilvester]

The integral formula is still applicable here, you just have to connect the two events by some curve. In SR, the spacetime interval between two events takes the integral over the straight line connecting those two events, which essentially reduces to what you did.

 

[PeterDonis]

First, I would subtract the first event from the second event to get a vector [4, 0, 0, 0].

This only works in flat spacetime. In curved spacetime, displacements from one event to another (i.e., differences in their coordinates) are not vectors.

In general, "proper time between events" is not well-defined unless you specify a curve between them. In flat spacetime many sources gloss over this and do the "subtracting one event from another" thing you did, which, as @Pencilvester said, is equivalent to taking the straight line curve between the events and computing the proper time along that curve. But this does not generalize to curved spacetime; in curved spacetime, there is not in general one unique geodesic between a pair of events, and, as above, events can't be "subtracted" to get vectors anyway.