Proper time
[1] The interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line.
It is expressed as an integral over the world line (analogous to arc length in Euclidean space).
The formal definition of proper time involves describing the path through spacetime that represents a clock, observer, or test particle, and the metric structure of that spacetime.
Proper time is the pseudo-Riemannian arc length of world lines in four-dimensional spacetime.
Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks.
For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is zero.
Instead, an arbitrary and physically irrelevant affine parameter unrelated to time must be introduced.
In any such frame an infinitesimal interval, here assumed timelike, between two events is expressed as and separates points on a trajectory of a particle (think clock{?}).
The same interval can be expressed in coordinates such that at each moment, the particle is at rest.
Due to the invariance of the interval (instantaneous rest frames taken at different times are related by Lorentz transformations) one may write
Given this differential expression for τ, the proper time interval is defined as
where Δ means the change in coordinates between the initial and final events.
Proper time is defined in general relativity as follows: Given a pseudo-Riemannian manifold with a local coordinates xμ and equipped with a metric tensor gμν, the proper time interval Δτ between two events along a timelike path P is given by the line integral[12] This expression is, as it should be, invariant under coordinate changes.
It reduces (in appropriate coordinates) to the expression of special relativity in flat spacetime.
Then using invariance of the interval, equation (4) follows from it in the same way (3) follows from (2), except that here arbitrary coordinate changes are allowed.
For a twin paradox scenario, let there be an observer A who moves between the A-coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially.
For each leg of the trip, the proper time interval can be calculated using A-coordinates, and is given by
In fact, for an object in a SR (special relativity) spacetime traveling with velocity
) form of the proper time equation is needed, along with a parameterized description of the path being taken, as shown below.
Let there be an observer C on a disk rotating in the xy plane at a coordinate angular rate of
This result is the same as for the linear motion example, and shows the general application of the integral form of the proper time formula.
Because inertial motion in curved spacetimes lacks the simple expression it has in SR, the line integral form of the proper time equation must always be used.
In this new coordinate system, the incremental proper time equation is
So for the inertial at-rest observer, coordinate time and proper time are once again found to pass at the same rate, as expected and required for the internal self-consistency of relativity theory.
[14] The Schwarzschild solution has an incremental proper time equation of
where To demonstrate the use of the proper time relationship, several sub-examples involving the Earth will be used here.
For the Earth, M = 5.9742×1024 kg, meaning that m = 4.4354×10−3 m. When standing on the north pole, we can assume
of 2π divided by the sidereal period of the Earth's rotation, 86162.4 seconds.
This example demonstrates how the proper time equation is used, even though the Earth rotates and hence is not spherically symmetric as assumed by the Schwarzschild solution.
To describe the effects of rotation more accurately the Kerr metric may be used.
