# Proper length

For the cosmological notion of proper distance, see Comoving distance.

Proper length[1] or rest length[2] refers to the length of an object in the object's rest frame.

The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of simultaneity is dependent on the observer.

A different term, proper distance, provides an invariant measure whose value is the same for all observers.

Proper distance is analogous to proper time. The difference is that a proper distance is the square root of (the absolute value of) the spacetime interval of two spacelike-separated events (or the integral of that along a spacelike path), while a proper time is the square root of (the absolute value of) the spacetime interval of two timelike-separated events (or the integral of that along a timelike path).

## Proper length or rest length

The proper length[1] or rest length[2] of an object is the length of the object measured by an observer which is at rest relative to it, by applying standard measuring rods on the object. The measurement of the object's endpoints doesn't have to be simultaneous, since the endpoints are constantly at rest at the same positions in the object's rest frame, so it is independent of Δt. This length is thus given by:

$L_{0}=\Delta x$.

However, in relatively moving frames the object's endpoints have to be measured simultaneously, since they are constantly changing their position. The resulting length is shorter than the rest length, and is given by the formula for length contraction (with γ being the Lorentz factor):

$L=L_{0}/\gamma$.

In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:

$\Delta\sigma=\sqrt{\Delta x^{2}-c^{2}\Delta t^{2}}$.

So Δσ depends on Δt, whereas (as exlained above) the object's rest length L0 can be measured independently of Δt. It follows that Δσ and L0, measured at the endpoints of the same object, only agree with each other when the measurement events were simultaneous in the object's rest frame so that Δt is zero. As explained by Fayngold:[1]

p. 407: "Note that the proper distance between two events is generally not the same as the proper length of an object whose end points happen to be respectively coincident with these events. Consider a solid rod of constant proper length l0. If you are in the rest frame K0 of the rod, and you want to measure its length, you can do it by first marking its endpoints. And it is not necessary that you mark them simultaneously in K0. You can mark one end now (at a moment t1) and the other end later (at a moment t2) in K0, and then quietly measure the distance between the marks. We can even consider such measurement as a possible operational definition of proper length. From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in K0, the distance between the marks is the proper length of the rod regardless of the time lapse between the two markings. On the other hand, it is not the proper distance between the marking events if the marks are not made simultaneously in K0."

## Proper distance between two events in flat space

In special relativity, the proper distance between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous. It is based on the invariant spacetime interval and is in any inertial frame of reference given by

$\Delta\sigma=\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2}$,

where

Two events are spacelike-separated if and only if the above formula gives a real, non-zero value for Δσ.

## Proper distance along a path

The above formula for the proper distance between two events assumes that the spacetime in which the two events occur is flat. Hence, the above formula cannot in general be used in general relativity, in which curved spacetimes are considered. It is, however, possible to define the proper distance along a path in any spacetime, curved or flat. In a flat spacetime, the proper distance between two events is the proper distance along a straight path between the two events. In a curved spacetime, there may be more than one straight path (geodesic) between two events, so the proper distance along a straight path between two events would not uniquely define the proper distance between the two events.

Along an arbitrary spacelike path P, the proper distance is given in tensor syntax by the line integral

$L = c \int_P \sqrt{-g_{\mu\nu} dx^\mu dx^\nu}$,

where

In the equation above, the metric tensor is assumed to use the +−−− metric signature, and is assumed to be normalized to return a time instead of a distance. The − sign in the equation should be dropped with a metric tensor that instead uses the −+++ metric signature. Also, the $c$ should be dropped with a metric tensor that is normalized to use a distance, or that uses geometrized units.