III.1.1 Symmetry in Special Relativity

In Classical Mechanics, Galilean transformations are used to transform between the coordinates of two reference frames which differ only by constant relative velocity . For simplicity we consider that motion is only on the  axis, so that the other two axes are unaffected (, ), while  and  are related by the following equation: . Together with spatial rotations and translations in space and time, these transformations form the inhomogeneous Galilean group, which includes the set ofnon-relativisticcontinuous space-time transformations, in  space with absolute time. Simple spatial rotations, translations, and transformations, are subgroups of this Galilean group, each of them has three parameters or dimensions: , , and , whereas temporal translation is a subgroup of one dimension: . Therefor, in total, the Galilean group has  real dimensions.

When we combine these transformations with the principle of the constancy and invariance of the speed of light, we get the Lorentzian transformations which led to Special Relativity. Lorentz invariance relates between two inertial coordinates systems by the following relations, again considering for simplicity that motion is only on the  axis:  and , where :  is the ratio of the velocity  over the speed of light , and  is called Lorentz factor and it is given by: . Lorentz transformations can also be considered as a hyperbolic rotation of Minkowski space, as we explained in Chapter V of Volume II.

When the relative speed between the observers  is very small, in relation to the speed of light, as it is the case in most classical situations, the value of  will be effectively considered zero, and the value of  will be equal to one. Under these conditions, Lorentzian invariance reduces back to the Galilean symmetry expressed above. Therefore, Galilean invariance is an approximation of Special Relativity that is valid for low speeds, while Special Relativity is an approximation of General Relativity that is valid for weak gravitational fields, as we shall describe in section III.1.2.

The set of all Lorentz transformations of Minkowski space-time is called Lorentz group, which 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.

Lorentz violations are allowed in String Theory, super-symmetry and Horava-Lifshitz gravity, as well as some approaches to Quantum Gravity. However, there is no experimental evidence of any violation of Lorentz invariance. When gravitational variances are negligible, all physical laws are Lorentz invariant. Lorentz transformations can also be considered as a hyperbolic rotation of Minkowski space.

The Lorentz group is a six-dimensional non-compact 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, and it is often presented through bi-quaternion algebra. Mathematically, the Lorentz group may be described as the generalized orthogonal group , the matrix Lie group that preserves the quadratic form on : .

Lorentz group is a subgroup of the Poincar group, the group of all isometries of Minkowski space-time, because Lorentz transformations are isometries that leave the origin fixed. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincar group is called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations; general isometries of Minkowski space-time are affine transformations.

In classical physics, the Galilean group is a comparable ten-parameter group that acts on absolute time and space. Instead of boosts, it features shear mappings to relate co-moving frames of reference, whereas Poincar symmetry is the full symmetry of Special Relativity. Objects which are invariant under this group are then said to possess Poincar invariance or relativistic invariance.

Similarly, a physical quantity is said to be Lorentzcovariant if it transforms under a given representation of the Lorentz group. As we explained in the Introduction, according to the representation theory of the Lorentz group, these quantities can be built out of scalars, four-vectors, four-tensors, and spinors. In particular, a Lorentz covariant scalar, such as space-time interval, remains the same under Lorentz transformations and is said to be a Lorentz invariant, which means that they transform under the trivial representation. This can also be generalized to equations, which can be Lorentz covariant if they can be written in terms of such quantities, which hold in all inertial frames.