Galilean transformation

In physics, a 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, are untenable at speeds that approach the speed of light. In special relativity the Galilean transformations are replaced by Lorentz transformations; conversely, the c→∞ classical limit of Lorentz transformations yields Galilean transformations.

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 near the surface of the Earth.

Translation

Standard configuration of coordinate systems for Galilean transformations.

Though the transformations are named for Galileo, it is absolute time and space as conceived by Isaac Newton that provides their domain of definition. In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as vectors.

This assumption is abandoned in the Lorentz transformations. These relativistic transformations are applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship under the Galilean transformation between the coordinates (x,y,z,t) and (x′,y′,z′,t′) of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion (velocity v) in their common x and x’ directions, with their spatial origins coinciding at time t=t'=0: [2] [3] [4] [5]

x'=x-vt\,
y'=y \,
z'=z \,
t'=t. \,

Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers.

In the language of linear algebra, this transformation is considered a shear mapping, and is described with a matrix acting on a vector. With motion parallel to the x-axis, the transformation acts on only two components:

\begin{pmatrix} x' \\t' \end{pmatrix} = \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix}\begin{pmatrix} x \\t \end{pmatrix}

Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity.

Galilean transformations

The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of space-time.[6] Let x represent a point in three-dimensional space, and t a point in one-dimensional time. A general point in space-time is given by an ordered pair (x,t). A uniform motion, with velocity v, is given by (\bold{x},t) \mapsto (\bold{x}+t\bold{v},t) where v is in R3. A translation is given by (\bold{x},t) \mapsto (\bold{x}+\bold{a},t+b) where a in R3 and b in R. A rotation is given by (\bold{x},t) \mapsto (G\bold{x},t) where G : R3 → R3 is an orthogonal transformation.[6] As a Lie group, the Galilean transformations have dimensions 10.[6]

Galilean group

Two Galilean transformations compose to form a third Galilean transformation. The set of all Galilean transformations SGal(3) on space forms a group with composition as the group operation. The group is sometimes presented as a matrix group with space-time events (t, x, 1) as vectors where t is real and x in R3 is a position in space. A matrix version of SGal(3) has been suggested:[7]

(t,\ x, \ 1)\begin{pmatrix}1 & v & 0 \\ 0 & R & 0 \\ s & y & 1 \end{pmatrix} = (t + s, t v +x R + y, 1) ,

where s is real and v, x, y are in R3 and R is a rotation matrix. The composition of transformations is then accomplished through matrix multiplication. SGal(3) has named subgroups. Let m represent the transformation matrix with parameters v, R, s, y :

G_1 = \{ m : s = 0, y = 0 \} , uniformly special transformations.
G_2= \{ m : v = 0, R = I_3 \} \cong (R^4 , +) , shifts of origin.
G_3 = \{ m : s = 0, y = 0, v = 0 \} \cong SO(3) , rotations of reference frame (see SO(3)).
G_4= \{ m : s = 0, y = 0, R = I_3 \} \cong (R^3, +) , uniform frame motions.

The parameters s, v, R, y span ten dimensions. Since the transformations depend continuously on s, v, R, y, SGal(3) is a continuous group, also called a topological group. The structure of SGal(3) can be understood by reconstruction from subgroups. The semidirect product combination (A \rtimes B ) of groups is required.

  1. G_2 \triangleleft SGal(3) (G2 is a normal subgroup)
  2. SGal(3) \cong G_2 \rtimes G_1
  3. G_4 \trianglelefteq G_1
  4. G_1 \cong G_4 \rtimes G_3
  5. SGal(3) \cong R^4 \rtimes (R^3 \rtimes SO(3)) .

Origin in group contraction

Here, we only look at the Lie algebra of the Galilean group; it is then easy to extend the results to the Lie group.

The relevant Lie algebra is spanned by H, Pi, Ci and Lij (an antisymmetric tensor), subject to commutation relations, where

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

H is the generator of time translations (Hamiltonian), Pi is the generator of translations (momentum operator), Ci is the generator of Galileian boosts, and Lij stands for a generator of rotations (angular momentum operator). This Lie Algebra is seen to be a special classical limit of the algebra of the Poincaré group, in the limit c→∞. Technically, the Galilean group is a celebrated group contraction of the Poincaré group:[8] renaming the generators of the latter as ϵimn Ji ↦ Lmn ; Pi ↦ Pi ; P0 ↦ H/c ; Ki ↦ c Ci, where c is the speed of light or any function thereof diverging as c→∞, the commutation relations (structure constants) of the latter limit to that of the former.

Note the group invariants LmnLmn , Pi P i.

Central extension of the Galilean group

One could, instead,[9] augment the Galilean group by a central extension into the Lie algebra spanned by H', P'i , C'i , L'ij , M, such that M commutes with everything (i.e. lies in the center), and

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

See also

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
  2. ↑ Mould, Richard A. (2002), Basic relativity, Springer-Verla, ISBN 0-387-95210-1, Chapter 2 §2.6, p. 42
  3. ↑ Lerner, Lawrence S. (1996), Physics for Scientists and Engineers, Volume 2, Jones and Bertlett Publishers, Inc, ISBN 0-7637-0460-1, Chapter 38 §38.2, p. 1046,1047
  4. ↑ Serway, Raymond A.; Jewett, John W. (2006), Principles of Physics: A Calculus-based Text, Fourth Edition, Brooks/Cole - Thomson Learning, ISBN 0-534-49143-X, Chapter 9 §9.1, p. 261
  5. ↑ Hoffmann, Banesh (1983), Relativity and Its Roots, Scientific American Books, ISBN 0-486-40676-8, Chapter 5, p. 83
  6. ↑ 6.0 6.1 6.2 Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics (2 ed.). Springer-Verlag. p. 6. ISBN 0-387-96890-3.
  7. ↑ Mehdi Nadjafikhah & Ahmad-Reza Forough (2007) Galilean Geometry of Motions
  8. ↑ Gilmore, Robert (2006). Lie Groups, Lie Algebras, and Some of Their Applications (Dover Books on Mathematics) ISBN 0486445291
  9. ↑ Bargmann, V. (1954). "On Unitary Ray Representations of Continuous Groups", Annals of Mathematics, Second Series, 59, No. 1 (Jan., 1954), pp. 1-46