Lorentz Tensor

Definition: An object, P, is called a (contravariant) Lorentz 4-vector if its components, P^a, transforms under a homogeneous Lorentz transformation from one set of Lorentz coordinates (ct, x, y, z) to another set (ct’, x’, y’, z’) as

where, for example

represents the Lorentz transform from S to S’ which are in standard configuration. Therefore the event X = (ct, x, y, z) is a Lorentz 4-vector. In fact X serves as the template of the Lorentz 4-vector. This 4-vector is called the coordinate 4-vector.

Definition: A (contravariant) Lorentz Tensor is an object that transforms under a homogeneous Lorentz transformation from one set of Lorentz coordinates (ct, x, y, z) to another set (ct’, x’, y’, z’) as