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See [[Interval and segment|Interval and segment]].
 
See [[Interval and segment|Interval and segment]].
  
 
A space-time interval is a quantity characterizing the relation between two events separated by a spatial distance and a time duration. In special relativity theory the square of an interval is
 
A space-time interval is a quantity characterizing the relation between two events separated by a spatial distance and a time duration. In special relativity theory the square of an interval is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520701.png" /></td> </tr></table>
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$$s^2=c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520702.png" /> is the velocity of light, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520703.png" /> are the space coordinates and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520704.png" /> are the corresponding points in time (for more details, see [[Minkowski space|Minkowski space]]).
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where $c$ is the velocity of light, $x_i,y_i,z_i$ are the space coordinates and $t_i$ are the corresponding points in time (for more details, see [[Minkowski space|Minkowski space]]).
  
 
In general relativity theory one considers the interval between two infinitesimally-close events:
 
In general relativity theory one considers the interval between two infinitesimally-close events:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520705.png" /></td> </tr></table>
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$$ds=\sqrt{-g_{ik}\,dx^i\,dx^k},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520706.png" /> is the infinitesimal difference of the space-time coordinates of these events and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520707.png" /> is the metric tensor.
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where $dx^i$ is the infinitesimal difference of the space-time coordinates of these events and $g_{ik}$ is the metric tensor.
  
  
  
 
====Comments====
 
====Comments====
A space-time interval with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520708.png" /> is called a time-like space-time interval, and one with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052070/i0520709.png" /> is called a space-like space-time interval.
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A space-time interval with $s^2>0$ is called a time-like space-time interval, and one with $s^2<0$ is called a space-like space-time interval.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.F. Lawden,  "An introduction to tensor calculus and relativity" , Methuen  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.K. Sachs,  H. Wu,  "General relativity for mathematicians" , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Tocaci,  "Relativistic mechanics, time, and inertia" , Reidel  (1985)  pp. Sect. A.II.1.4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.F. Lawden,  "An introduction to tensor calculus and relativity" , Methuen  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.K. Sachs,  H. Wu,  "General relativity for mathematicians" , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Tocaci,  "Relativistic mechanics, time, and inertia" , Reidel  (1985)  pp. Sect. A.II.1.4</TD></TR></table>
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[[Category:Geometry]]
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[[Category:Mathematical physics]]

Latest revision as of 14:51, 30 December 2018

See Interval and segment.

A space-time interval is a quantity characterizing the relation between two events separated by a spatial distance and a time duration. In special relativity theory the square of an interval is

$$s^2=c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2,$$

where $c$ is the velocity of light, $x_i,y_i,z_i$ are the space coordinates and $t_i$ are the corresponding points in time (for more details, see Minkowski space).

In general relativity theory one considers the interval between two infinitesimally-close events:

$$ds=\sqrt{-g_{ik}\,dx^i\,dx^k},$$

where $dx^i$ is the infinitesimal difference of the space-time coordinates of these events and $g_{ik}$ is the metric tensor.


Comments

A space-time interval with $s^2>0$ is called a time-like space-time interval, and one with $s^2<0$ is called a space-like space-time interval.

References

[a1] D.F. Lawden, "An introduction to tensor calculus and relativity" , Methuen (1962)
[a2] R.K. Sachs, H. Wu, "General relativity for mathematicians" , Springer (1977)
[a3] E. Tocaci, "Relativistic mechanics, time, and inertia" , Reidel (1985) pp. Sect. A.II.1.4
How to Cite This Entry:
Interval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval&oldid=16796
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article