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In relativistic physics, proper distance is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime. In contrast, proper length or rest length refer to the length of an object in the object's rest frame, which is not necessarily the same as the proper distance between two events.

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. Proper distance provide an invariant measure, whose value is the same for all observers.

Proper distance is analogous to proper time. The difference is that proper distance is the invariant interval of a spacelike path or pair of spacelike-separated events, while proper time is the invariant interval of a timelike path or pair of timelike-separated events.

Proper distance between two events

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 denoted by

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

where

• Δt is the difference in the temporal coordinates of the two events,
• Δx, Δy, and Δz are differences in the linear, orthogonal, spatial coordinates of the two events, and
• c is the speed of light.

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

Proper length or rest length

Also expressions such as proper length or rest length of an object are used. It represents 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:

$L_0=\mathrm{\Delta }x$.

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:

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

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:

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 end-points. 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 of 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 of a path in any spacetime, curved or flat. In a flat spacetime, the proper distance between two events is the proper distance of 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 of 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

$L=c\underset{P}{\int }\sqrt{-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$,

where

• g_μν is the metric tensor for the current spacetime and coordinate mapping, and
• dx^μ 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 - 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

References

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