Proper length or rest length is 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 the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events (or along a timelike path).
Proper length or rest length
The proper length In such a specific frame, the distance is given by
<math display="block">\Delta\sigma=\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2} ,</math>
where
- Δx, Δy, and Δz are differences in the linear, orthogonal, spatial coordinates of the two events.
The definition can be given equivalently with respect to any inertial frame of reference (without requiring the events to be simultaneous in that frame) by
<math display="block">\Delta\sigma = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2},</math>
where
- Δt is the difference in the temporal coordinates of the two events, and
- c is the speed of light.
The two formulae are equivalent because of the invariance of spacetime intervals, and since exactly when the events are simultaneous in the given frame.
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
<math display="block">L = c \int_P \sqrt{-g_{\mu\nu} dx^\mu dx^\nu} ,</math>
where
- g<sub>μν</sub> is the metric tensor for the current spacetime and coordinate mapping, and
- dx<sup>μ</sup> is the coordinate separation between neighboring events along the path P.
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 minus 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.
See also
- Invariant interval
- Proper time
- Comoving distance
- Relativity of simultaneity