## Lorentz Transformation

A 4-D transformation satisfied by all Four-Vectors ,

 (1)

In the theory of special relativity, the Lorentz transformation replaces the Galilean Transformation as the valid transformation law between reference frames moving with respect to one another at constant Velocity. Let be the Position Four-Vector with , and let the relative motion be along the axis with Velocity . Then (1) becomes
 (2)

where the Lorentz Tensor is given by
 (3)

Here,
 (4) (5)

Written explicitly, the transformation between and coordinate is
 (6) (7) (8) (9)

The Determinant of the upper left Matrix in (3) is
 (10)

so

A Lorentz transformation along the -axis can also be written

 (11)

where is called the rapidity,
 (12)

and
 (13) (14) (15)

References

Fraundorf, P. Accel-1D: Frame-Dependent Relativity at UM-StL.'' http://www.umsl.edu/~fraundor/a1toc.html.

Griffiths, D. J. Introduction to Electrodynamics. Englewood Cliffs, NJ: Prentice-Hall, pp. 412-414, 1981.

Morse, P. M. and Feshbach, H. The Lorentz Transformation, Four-Vectors, Spinors.'' §1.7 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 93-107, 1953.