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The most general proper Lorentz transformation Λ(v, θ) includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, Λ(0, θ) = R(θ) and Λ(v, 0) = B(v). An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ: ′ =. In index notation, the contravariant and covariant components transform according to, respectively: ′ =, ′ = in which the matrix Λ has components Λ μ ν in row μ and column ν, and the matrix (Λ −1) T has components Λ ...
A derivation for the transformation of the Lorentz force for the particular case u = 0 is given here. [4] A more general one can be seen here. [5] The transformations in this form can be made more compact by introducing the electromagnetic tensor (defined below), which is a covariant tensor.
Examples are listed next. In special relativity, the vector basis can be restricted to being orthonormal, in which case all four-tensors transform under Lorentz transformations. In general relativity, more general coordinate transformations are necessary since such a restriction is not in general possible.
In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.
In matrix notation, the first expression can be written as ... An example is the metric tensor ... , which is the identity map, or a Lorentz transformation . Example ...
Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector ^ = (,,), and a rotation angle θ about a three-dimensional unit vector ^ = (,,) defining an axis, so ^ = (,,) and ^ = (,,) are together six parameters of the Lorentz group (three for rotations and three for boosts). The ...