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
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 |
Interval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval&oldid=43587