nLab Lorentz group

Definition

The Lorentz group is the orthogonal group for an invariant bilinear form of signature $(-+++\cdots)$, $O(d-1,1)$.

In physics, in the theory of relativity the Lorentz group acts canonically as the group of linear isometries of Minkowski spacetime preserving a chosen basepoint. This is called the action by Lorentz transformations.

The elements in the Lorentz group in the image of the special orthogonal group $SO(d-1) \hookrightarrow O(d-1,1)$ are rotations in space. The further elements in the special Lorentz group $SO(d-1,1)$, which mathematically are “hyperbolic rotations” in a space-time plane, are called boosts in physics.

One distinguishes the following further subgroups of the Lorentz group $O(d-1,1)$:

• the proper Lorentz group $SO(d-1,1)$ is the subgroup of elements which have determinant +1 (as elements $SO(d-1,1)\hookrightarrow GL(d)$ of the general linear group);

• the proper orthochronous (or restricted) Lorentz group $SO^+(d-1,1) \hookrightarrow SO(d-1,1)$ is the further subgroup of elements which do not act by reflection along the timelike axis.

Properties

Connected components

As a smooth manifold, the Lorentz group $O(3,1)$ has four connected components. The connected component of the identity is the the proper orthochronous group $SO^+(3,1)$. The other three components are

1. $SO^+(3,1)\cdot P$

2. $SO^+(3,1)\cdot T$

3. $SO^+(3,1)\cdot P T$,

where, as matrices

$$P \coloneqq diag(1,-1,-1, \cdots, -1)$$

is the operation of point reflection at the origin in space, where

$$T \coloneqq diag(-1,1,1, \cdots, 1)$$

is the operation of reflection in time and hence where

$$P T = T P = diag(-1,-1, \cdots, -1)$$

is point reflection in spacetime.

Spin cover

While the proper orthochronous Lorentz group $SO^+(d-1,1)$ is connected, it is not simply connected. Its universal double cover is the Lorentzian spin group $Spin(d-1,1)$.

Related concepts

• Lorentz Lie algebra

• quantum group version: quantum Lorentz group

• Lorentzian manifold

• rapidity

• Galilean group

• Minkowski spacetime

• de Sitter spacetime

• anti de Sitter spacetime

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$Pin(n)$	Tring group	$Tring(n)$
special orthogonal group	$SO(n)$	Spin group	$Spin(n)$	String group	$String(n)$
Lorentz group	$\mathrm{O}(n,1)$	$\,$	$Spin(n,1)$	$\,$	$\,$
anti de Sitter group	$\mathrm{O}(n,2)$	$\,$	$Spin(n,2)$	$\,$	$\,$
conformal group	$\mathrm{O}(n+1,t+1)$	$\,$
Narain group	$O(n,n)$
Poincaré group	$ISO(n,1)$	Poincaré spin group	$\widehat {ISO}(n,1)$	$\,$	$\,$
super Poincaré group	$sISO(n,1)$	$\,$	$\,$	$\,$	$\,$
superconformal group

References

Named after Hendrik Lorentz. (Not to be confused with Ludvik Lorenz, whose name is attached to the Lorenz gauge.)

• Wikipedia, Lorentz group