Lorentz Group

The Lorentz group is the group L of time-preserving linear isometries of Minkowski space R^((3,1)) with the Minkowski metric


(where the convention c=1 is used). It is also the group of isometries of three-dimensional hyperbolic geometry. It is time-preserving in the sense that the unit time vector (1,0,0,0) is sent to another vector (x^0,x^1,x^2,x^3) such that x^0>0.

A consequence of the definition of the Lorentz group is that the full group of time-preserving isometries of Minkowski space R^((3,1)) is the group direct product of the group of translations of R^((3,1)) (i.e., R^((3,1)) itself, with addition as the group operation), with the Lorentz group, and that the full isometry group of the Minkowski R^((3,1)) is a group extension of Z_2 by the product L tensor R^((3,1)).

The Lorentz group is invariant under space rotations and Lorentz transformations.

See also

Lorentz Tensor, Lorentz Transformation, Minkowski Metric

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Arfken, G. "Homogeneous Lorentz Group." §4.13 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 271-275, 1985.

Referenced on Wolfram|Alpha

Lorentz Group

Cite this as:

Weisstein, Eric W. "Lorentz Group." From MathWorld--A Wolfram Web Resource.

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