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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.
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.
(The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1, 3). The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO + (1, 3). [a]
The above relativistic transformations suggest the electric and magnetic fields are coupled together, in a mathematical object with 6 components: an antisymmetric second-rank tensor, or a bivector. This is called the electromagnetic field tensor, usually written as F μν. In matrix form: [13]
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 ...
In the special relativity, Lorentz transformations exhibit the symmetry of Minkowski spacetime by using a constant c as the speed of light, and a parameter v as the relative velocity between two inertial reference frames. Using the above conditions, the Lorentz transformation in 3+1 dimensions assume the form:
Derivation of Lorentz transformation using time dilation and length contraction Now substituting the length contraction result into the Galilean transformation (i.e. x = ℓ), we have: ′ = that is: ′ = ()
The angular momentum tensor M is indeed a tensor, the components change according to a Lorentz transformation matrix Λ, as illustrated in the usual way by tensor index notation ′ = ′ ′ ′ ′ = = =, where, for a boost (without rotations) with normalized velocity β = v/c, the Lorentz transformation matrix elements are = = = = + and the ...