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| {{For|the cosmological notion of proper distance|Comoving distance}}
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| In relativistic physics, '''proper distance''' is an [[invariant (physics)|invariant]] measure of the [[distance]] between two [[spacelike]]-separated [[Spacetime#Basic concepts|event]]s, or of the length of a spacelike [[Path (topology)|path]] within a [[spacetime]]. In contrast, '''proper length'''<ref name=fayngold>{{cite book |author=Moses Fayngold |title=Special Relativity and How it Works |location=John Wiley & Sons |year=2009 |isbn=3527406077}}</ref> or '''rest length'''<ref name=franklin>{{cite journal |author=Franklin, Jerrold |title=Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity |journal=European Journal of Physics |volume=31 |year=2010 |pages=291–298 |doi=10.1088/0143-0807/31/2/006 |bibcode = 2010EJPh...31..291F |issue=2 |arxiv = 0906.1919 }}</ref> 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.
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| 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 [[Relativity of simultaneity|simultaneity]] is dependent on the observer. Proper distance provide an invariant measure, whose value is the same for all observers.
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| Proper distance is analogous to [[proper time]]. The difference is that proper distance is the invariant [[spacetime interval|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.
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| == Proper distance between two events ==
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| 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
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| <math>\Delta\sigma=\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2}</math>,
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| where
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| * ''Δt'' is the difference in the [[time|temporal]] coordinates of the two events,
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| * ''Δx'', ''Δy'', and ''Δz'' are differences in the [[linear]], [[orthogonal]], [[Three-dimensional space|spatial]] coordinates of the two events, and
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| * ''c'' is the [[speed of light]].
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| Two events are spacelike-separated if and only if the above formula gives a real, non-zero value for ''Δσ''.
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| == Proper length or rest length ==
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| Also expressions such as ''proper length''<ref name=fayngold /> or ''rest length''<ref name=franklin /> 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:
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| :<math>L_{0}=\Delta x</math>.
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| 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]]):
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| :<math>L=L_{0}/\gamma</math>. | |
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| In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:
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| :<math>\Delta\sigma=\sqrt{\Delta x^{2}-c^{2}\Delta t^{2}}</math>.
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| So ''Δσ'' depends on ''Δt'', whereas (as exlained above) the object's rest length ''L''<sub>0</sub> can be measured independently of ''Δt''. It follows that ''Δσ'' and ''L''<sub>0</sub>, 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:<ref name=fayngold />
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| :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 ''l''<sub>0</sub>. If you are in the rest frame ''K''<sub>0</sub> 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 ''K''<sub>0</sub>. You can mark one end now (at a moment ''t''<sub>1</sub>) and the other end later (at a moment ''t''<sub>2</sub>) in ''K''<sub>0</sub>, 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 ''K''<sub>0</sub>, 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 ''K''<sub>0</sub>."
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| == Proper distance of a path ==
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| 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 (topology)|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 (general relativity)|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.
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| Along an arbitrary spacelike path ''P'', the proper distance is given in [[tensor]] syntax by the [[line integral]]
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| <math>L = c \int_P \sqrt{-g_{\mu\nu} dx^\mu dx^\nu} </math>,
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| where
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| * ''g<sub>μν</sub>'' is the [[metric tensor (general relativity)|metric tensor]] for the current [[spacetime]] and [[coordinate]] mapping, and
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| * ''dx<sup>μ</sup>'' is the [[coordinate]] separation between neighboring events along the path ''P''.
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| In the equation above, the metric tensor is assumed to use the '''<tt>+---</tt>''' [[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 '''<tt>-+++</tt>''' metric signature. Also, the <math>c</math> should be dropped with a metric tensor that is normalized to use a distance, or that uses [[Geometrized unit system|geometrized units]].
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| ==See also==
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| *[[Invariant interval]]
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| *[[Proper time]]
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| *[[Comoving distance]]
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| *[[Relativity of simultaneity]]
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| == References ==
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| {{reflist}}
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| [[Category:Theory of relativity]]
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