DUALITY OF TIME:

Complex-Time Geometry and Perpetual Creation of Space

by Mohamed Haj Yousef

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3.2.4  Lorentz Group

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The group of all Lorentz transformations of Minkowski space-time is called Lorentz group, and it describes the classical and quantum setting for all non-gravitational physical phenomena, including: the kinematical laws of Special Relativity, Maxwell’s field equations in the theory of electromagnetism, the Dirac equation in the theory of the electron, and the Standard model of particle physics.

All physical laws are Lorentz invariant when gravitational variances are negligible. This group, therefore, expresses the fundamental symmetry of space and time of all known fundamental laws of nature. Lorentz transformations can also be considered as a hyperbolic rotation of Minkowski space.

For one-dimensional motion, Lorentz transformations can be expressed in terms of matrices as follows:

(3.5)

Note that a boost along the-direction does not warp the space in the- and-directions.

The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected, but rather doubly connected. The identity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted. The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time. The restricted Lorentz group has often been presented through a facility of biquaternion algebra.

Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski space-time. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group.

Lorentz transformations are examples of linear transformations; general isometries of Minkowski space-time are affine transformations. Mathematically, the Lorentz group may be described as the generalized orthogonal group, the matrix Lie group that preserves the quadratic form on:

(3.6)

This quadratic form is, when put on matrix form, interpreted in physics as the metric tensor of Minkowski space-time.

The Lorentz group is a subgroup of the Poincaré group, the group of all isometries of Minkowski space-time.

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