# Applying the spacetime interval to regular vectors instead of curves

• I
• space-time
In summary, the proper time between two events in flat spacetime can be calculated using the integral formula dτ = ∫ sqrt( -gab(dxa/ds)(dxb/ds) ) ds, where the bounds of the integral are from the first event to the second event. However, in curved spacetime, this formula may not be applicable and a specific curve connecting the events must be specified.
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( -gab(dxa/ds)(dxb/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 dxa when I applied the spacetime interval to it. In other words, dx0 = 4 and the other components equal 0.

Next, I should apply dτ2 = gabdxadxb

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

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.

space-time said:
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.

Pencilvester

## 1. What is the spacetime interval?

The spacetime interval is a concept in physics that measures the distance between two events in the four-dimensional spacetime continuum. It takes into account both spatial and temporal components, allowing for a unified measurement of distance and time.

## 2. How is the spacetime interval applied to regular vectors?

The spacetime interval can be applied to regular vectors by using the Minkowski metric, which transforms the vector components into a four-dimensional spacetime vector. This allows for the calculation of the spacetime interval between two points in space and time.

## 3. What is the benefit of applying the spacetime interval to regular vectors instead of curves?

By applying the spacetime interval to regular vectors, we can calculate the distance between two events in a more precise and unified manner. This allows for a better understanding of the relationship between space and time, and is essential in the study of relativity and other areas of physics.

## 4. Can the spacetime interval be used in everyday situations?

While the concept of spacetime interval is primarily used in the field of physics, it can also be applied to everyday situations. For example, it can be used to calculate the travel time between two points, taking into account both the distance and the time it takes to travel.

## 5. Are there any limitations to applying the spacetime interval to regular vectors?

One limitation to applying the spacetime interval to regular vectors is that it only applies to flat spacetime. In cases where spacetime is curved, such as in the presence of a massive object, the spacetime interval may not accurately reflect the true distance between two events. In these cases, more complex mathematical tools are needed to calculate the distance between events.

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