The spacetime interval, proper time, and time dilation are related concepts in the theory of special relativity, but they are not exactly the same thing.
The spacetime interval is a measure of the distance between two events in spacetime. It is defined as:
Δs^2 = Δt^2 - Δx^2 - Δy^2 - Δz^2
where Δt is the time difference between the events, and Δx, Δy, and Δz are the differences in the spatial coordinates. The spacetime interval is an invariant quantity, meaning that it has the same value in all inertial reference frames. It allows us to calculate the proper time between two events.
The proper time, often denoted as Δτ (delta tau), is the time interval measured by an observer who is at rest relative to the events being considered. It is the time experienced by a clock that is at the same location as the events. The proper time is calculated using the spacetime interval:
Δτ = √(Δs^2)
The proper time is the "true" time experienced by the object or observer between two events, as measured in its own reference frame.
Time dilation, on the other hand, refers to the difference in the passage of time between two observers who are in relative motion. It occurs when two observers have different relative velocities. The faster an observer is moving relative to another, the more time dilation they will experience. Time dilation is calculated by comparing the proper time of one observer to the time measured by another observer.
So, while the spacetime interval is used to calculate the proper time, it is not the same as time dilation. The spacetime interval gives the proper time between events in a specific reference frame, while time dilation refers to the difference in elapsed time between two different observers or reference frames.