# Interval

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://www.encyclopediaofmath.org/index.php?title=Interval&oldid=43587