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Lorentz Transformation


A Lorentz transformation is a four-dimensional transformation

 x^('mu)=Lambda^mu_nux^nu,
(1)

satisfied by all four-vectors x^nu, where Lambda^mu_nu is a so-called Lorentz tensor. Lorentz tensors are restricted by the conditions

 Lambda^alpha_gammaLambda^beta_deltaeta_(alphabeta)=eta_(gammadelta),
(2)

with eta_(alphabeta) the Minkowski metric (Weinberg 1972, p. 26; Misner et al. 1973, p. 68).

Here, the tensor indices run over 0, 1, 2, 3, with x^0 being the time coordinate and (x^1,x^2,x^3) being space coordinates, and Einstein summation is used to sum over repeated indices. There are a number of conventions, but a common one used by Weinberg (1972) is to take the speed of light c=1 to simplify computations and allow ct to be written simply as t for x^0. The group of Lorentz transformations in Minkowski space R^((3,1)) is known as the Lorentz group.

An element x in four-space which is invariant under a Lorentz transformation is said to be a Lorentz invariant; examples include scalars, elements of the form x^2-c^2t^2, and the interval s_(12)^2 between two events (Thorn 2012).

Note that while some authors (e.g., Weinberg 1972, p. 26) use the term "Lorentz transformation" to refer to the inhomogeneous transformation

 x^'^mu=Lambda^mu_nux^nu+a^mu,
(3)

where a^mu is a constant tensor, the preferred term for transformations of this form is Poincaré transformation (Misner et al. 1973, p. 68). The corresponding group of Poincaré transformations is known as the Poincaré group.

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. The Lorentz transformation serves this important role by virtue of the fact that it leaves the so-called proper time

dtau^2=dt^2-dx^2
(4)
=-eta_(alphabeta)dx^alphadx^beta
(5)

invariant. (Here, the convention c=1 is used.) To see this, note that

dtau^('2)=-eta_(alphabeta)dx^('alpha)dx^('beta)
(6)
=-eta_(alphabeta)Lambda^alpha_gammaLambda^beta_deltadx^gammadx^delta
(7)
=-eta_(gammadelta)dx^gammadx^delta
(8)
=dtau^2
(9)

(Weinberg 1972, p. 27).

The set of all Lorentz transformations is known as the inhomogeneous Lorentz group or the Poincaré group. Similarly, the set of Lorentz transformations with a^alpha=0 is known as the homogeneous Lorentz group. Restricting the transformations by the additional requirements

 Lambda^0_0>=1
(10)

and

 det(Lambda)=1,
(11)

where det(Lambda) denotes the tensor determinant, give the proper inhomogeneous and proper homogeneous Lorentz groups.

Any proper homogeneous Lorentz transformation can be expressed as a product of a so-called boost and a rotation.


See also

Four-Vector, Hyperbolic Rotation, Lorentz Group, Lorentz Tensor, Poincaré Group, Poincaré Transformation

This entry contributed by Christopher Stover

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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.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco: W. H. Freeman, 1973.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.Thorn, C. B. "Classical Electrodynamics-Lorentz Invariance and Special Relativity." 83-108, 2012. http://www.phys.ufl.edu/~thorn/homepage/emlectures2.pdf.Weinberg, S. "Lorentz Transformations." §2.1 in Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, pp. 25-29, 1972.

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Lorentz Transformation

Cite this as:

Stover, Christopher. "Lorentz Transformation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LorentzTransformation.html

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