Lorentz Group - General Dimensions

General Dimensions

The concept of the Lorentz group has a natural generalization to any spacetime dimension. Mathematically, the Lorentz group of n+1 dimensional Minkowski space is the group O(n,1) (or O(1,n)) of linear transformations of Rn+1 which preserve the quadratic form

Many of the properties of the Lorentz group in four dimensions (n=3) generalize straightforwardly to arbitrary n. For instance, the Lorentz group O(n,1) has four connected components, and it acts by conformal transformations on the celestial (n−1)-sphere in n+1 dimensional Minkowski space. The identity component SO+(n,1) is an SO(n)-bundle over hyperbolic n-space Hn.

The low dimensional cases n=1 and n=2 are often useful as "toy models" for the physical case n=3, while higher dimensional Lorentz groups are used in physical theories such as string theory which posit the existence of hidden dimensions. The Lorentz group O(n,1) is also the isometry group of n-dimensional de Sitter space dS_n, which may be realized as the homogeneous space O(n,1)/O(n−1,1). In particular O(4,1) is the isometry group of the de Sitter universe dS₄, a cosmological model.