One of the simplest non-trivial examples of a four-tensor is the four-displacement
a four-tensor with contravariant rank 1 and covariant rank 0. Four-tensors of this kind are usually known as four-vectors. Here the component x0 = ct gives the displacement of a body in time (coordinate time t is multiplied by the speed of lightc so that x0 has dimensions of length). The remaining components of the four-displacement form the spatial displacement vector x = (x1, x2, x3).
used for calculating the line element and raising and lowering indices. The above applies to Cartesian coordinates. In general relativity, the metric tensor is given by much more general expressions for curvilinear coordinates.
The stress–energy tensor of a continuum or field generally takes the form of a second order tensor, and usually denoted by T. The timelike component corresponds to energy density (energy per unit volume), the mixed spacetime components to momentum density (momentum per unit volume), and the purely spacelike parts to 3d stress tensors.
^ abcdLambourne, Robert J A. Relativity, Gravitation and Cosmology. Cambridge University Press. 2010.
^R. Penrose (2005). The Road to Reality. vintage books. pp. 437–438, 566–569. ISBN978-0-09-944068-0.Note: Some authors, including Penrose, use Latin letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.
^Barut, A.O. Electrodynamics and the Classical theory of particles and fields. Dover. p. 96. ISBN978-0-486-64038-9.
^Barut, A.O. Electrodynamics and the Classical theory of particles and fields. Dover. p. 73. ISBN978-0-486-64038-9. No factor of c appears in the tensor in this book because different conventions for the EM field tensor.