time coordinates), and indeed four-dimensional spacetime. Line elements in Euclidean space Following are examples of how the line elements are found from the metric. Cartesian coordinates The simplest line element is in Cartesian coordinates - in which case the metric is just the Kronecker delta: (here i, j = 1, 2, 3 for space) or in matrix form (i denotes row, j denotes column): The general curvilinear coordinates reduce to Cartesian coordinates: so Orthogonal curvilinear coordinates For all orthogonal coordinates the metric is given by: where for i = 1, 2, 3 are scale factors, so the square of the line element is: Some examples of line elements in these coordinates are below. General curvilinear coordinates Given an arbitrary basis of a space of dimension , the metric is defined as the inner product of the basis vectors. Where and the inner product is with respect to the ambient space (usually its ) In a coordinate basis The coordinate basis is a special type of basis that is regularly used in differential geometry. Line elements in 4d spacetime Minkowskian spacetime The Minkowski metric is: where one sign or the other is chosen, both conventions are used. This applies only for flat spacetime. The coordinates are given by the 4-position: so the line element is: Schwarzschild coordinates In Schwarzschild coordinates coordinates are , being the general metric of the form: (note the similitudes with the metric in 3D spherical polar coordinates). so the line element is: General spacetime The coordinate-independent definition of the square of the line element ds in spacetime is: In terms of coordinates: where for this case the indices α and β run over 0, 1, 2, 3 for spacetime. This is the spacetime interval - the measure of separation between two arbitrarily close events in spacetime. In special relativity it is invariant under Lorentz transformations. In general relativity it is invariant under arbitrary invertible differentiable coordinate transformations. See also Covariance and contravariance of vectors First fundamental form List of integration and measure theory topics Metric tensor Ricci calculus Raising and lowering indices References Affine geometry Riemannian geometry Special relativity General relativity da:Linjeelement de:Linienelement