Editing Proper length (section)

== 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.<ref>{{cite book |title=Gravity: Newtonian, Post-Newtonian, Relativistic |edition=illustrated |first1=Eric |last1=Poisson |first2=Clifford M. |last2=Will |publisher=Cambridge University Press |year=2014 |isbn=978-1-107-03286-6 |page=191 |url=https://books.google.com/books?id=PZ5cAwAAQBAJ}} [https://books.google.com/books?id=PZ5cAwAAQBAJ&pg=PA191 Extract of page 191]</ref><ref>{{cite book |title=Relativistic Celestial Mechanics of the Solar System |first1=Sergei |last1=Kopeikin |first2=Michael |last2=Efroimsky |first3=George |last3=Kaplan |publisher=John Wiley & Sons |year=2011 |isbn=978-3-527-63457-6 |page=136 |url=https://books.google.com/books?id=uN5_DQWSR14C}} [https://books.google.com/books?id=uN5_DQWSR14C&pg=PA136 Extract of page 136]</ref> 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 
* Δ''x'', Δ''y'', and Δ''z'' are differences in the [[linear]], [[orthogonal]], [[Three-dimensional space|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 
* Δ''t'' is the difference in the [[time|temporal]] coordinates of the two events, and
* ''c'' is the [[speed of light]].

The two formulae are equivalent because of the invariance of [[spacetime interval]]s, and since Δ''t'' = 0 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 Δ''σ''.