Galilean transformation

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The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light due to Einstein's theory of relativity.


Galileo formulated these concepts in his description of uniform motion [1] The topic was motivated by Galileo's description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity, at the surface of the Earth. The descriptions below are another mathematical notation for this concept.

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[edit] Translation (one dimension)

The Galilean transformation is nothing more than careful addition and subtraction of velocity vectors.

Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship of two coordinate systems (x′ and x) in constant relative motion (velocity u) in the x-direction. All other parameters (t, y, z) are unchanged in the transformation from x′ to x coordinates.

t'=t \,\!
x'=x-ut \,\!
y'=y \,\!
z'=z \,\!
Diagram 1. Views of spacetime along the world line of a slowly accelerating observer.Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower half of the diagram shows the events visible to the observer. Upper half shows those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. This caption probably should be rewritten, to be more relevant for the Galilean transform.
Diagram 1. Views of spacetime along the world line of a slowly accelerating observer.

Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower half of the diagram shows the events visible to the observer. Upper half shows those that will be able to see the observer. The small dots are arbitrary events in spacetime.

The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. This caption probably should be rewritten, to be more relevant for the Galilean transform.

[edit] Galilean transformations

Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations.

The Galilean symmetries (interpreted as active transformations):

Spatial translations:

t\rightarrow t \,\!
\vec{x}\rightarrow \vec{x}+\vec{a} \,\!

Time translations:

t\rightarrow t+\tau \,\!
\vec{x}\rightarrow \vec{x} \,\!

Boosts:

t\rightarrow t \,\!
\vec{x}\rightarrow \vec{x}+\vec{v}t \,\!

Rotations:

t\rightarrow t \,\!
\vec{x}\rightarrow \mathbf{R}\vec{x} \,\!

where R is an orthogonal matrix.

[edit] Central extension of the Galilean group

The Galilean group: Here, we will only look at its Lie algebra. It's easy to extend the results to the Lie group. The Lie algebra of L is spanned by E, Pi, Ci and Lij (antisymmetric tensor) subject to commutators (operators of the form [,]), where

[E,P_i]=0 \,\!
[P_i,P_j]=0 \,\!
[L_{ij},E]=0 \,\!
[C_i,C_j]=0 \,\!
[L_{ij},L_{kl}]=i\hbar [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}] \,\!
[L_{ij},P_k]=i\hbar[\delta_{ik}P_j-\delta_{jk}P_i] \,\!
[L_{ij},C_k]=i\hbar[\delta_{ik}C_j-\delta_{jk}C_i] \,\!
[C_i,E]=i\hbar P_i \,\!
[C_i,P_j]=0 \,\!

We can now give it a central extension into the Lie algebra spanned by E', P'i, C'i, L'ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and

[E',P'_i]=0 \,\!
[P'_i,P'_j]=0 \,\!
[L'_{ij},E']=0 \,\!
[C'_i,C'_j]=0 \,\!
[L'_{ij},L'_{kl}]=i\hbar [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!
[L'_{ij},P'_k]=i\hbar[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!
[L'_{ij},C'_k]=i\hbar[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!
[C'_i,E']=i\hbar P'_i \,\!
[C'_i,P'_j]=i\hbar M\delta_{ij} \,\!

[edit] Notes

  1. ^ Galileo 1638 Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze 191 - 196, published by Lowys Elzevir (Louis Elsevier), Leiden, or Two New Sciences, English translation by Henry Crew and Alfonso de Salvio 1914, reprinted on pages 515-520 of On the Shoulders of Giants: The Great Works of Physics and Astronomy. Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4

[edit] See also

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