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. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group (assumed throughout below). Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean relativity action on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view. The equations below, although apparently obvious, are valid only at speeds much less than the speed of light. In special relativity the Galilean transformations are replaced by Poincaré transformations; conversely, the group contraction in the classical limit c → ∞ of Poincaré transformations yields Galilean transformations.

Galileo formulated these concepts in his description of uniform motion.^[1] The topic was motivated by his 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 Poincaré transformations. These relativistic transformations are applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Poincaré 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 spacetime.^[6] Let x represent a point in three-dimensional space, and t a point in one-dimensional time. A general point in spacetime is given by an ordered pair (x, t).

A uniform motion, with velocity v, is given by

${\displaystyle ({\mathbf {x}},t)\mapsto ({\mathbf {x}}+t{\mathbf {v}},t),}$

where v ∈ ℝ³. A translation is given by

${\displaystyle ({\mathbf {x}},t)\mapsto ({\mathbf {x}}+{\mathbf {a}},t+s),}$

where a ∈ ℝ³ and s ∈ ℝ. A rotation is given by

${\displaystyle ({\mathbf {x}},t)\mapsto (G{\mathbf {x}},t),}$

where G : ℝ³ → ℝ³ is an orthogonal transformation.^[6]

As a Lie group, the Galilean transformations span 10 dimensions,^[6] i.e., comprise 10 generators.

Galilean group

Two Galilean transformations G(R, v, a, s) compose to form a third Galilean transformation, G(R' , v' , a' , s' ) G(R, v, a, s) = G(R' R, R' v+v' , R' a+a' +v' s, s' +s). The set of all Galilean transformations Gal(3) on space forms a group with composition as the group operation.

The group is sometimes represented as a matrix group with spacetime events ( x, t, 1) as vectors where t is real and x ∈ ℝ³ is a position in space. The action is given by^[7]

${\displaystyle {\begin{pmatrix}R&v&a\\0&1&s\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\t\\1\end{pmatrix}}={\begin{pmatrix}Rx+vt+a\\t+s\\1\end{pmatrix}},}$

where s is real and v, x, a ∈ ℝ³ and R is a rotation matrix.

The composition of transformations is then accomplished through matrix multiplication. Gal(3) has named subgroups. The identity component is denoted SGal(3).

Let m represent the transformation matrix with parameters v, R, s, a:

${\displaystyle G_{1}=\{m:s=0,a=0\},}$ uniformly special transformations.

${\displaystyle G_{2}=\{m:v=0,R=I_{3}\}\cong (\mathbb {R} ^{4},+),}$ shifts of origin.

${\displaystyle G_{3}=\{m:s=0,a=0,v=0\}\cong \mathrm {SO} (3),}$ rotations of reference frame (see SO(3)).

${\displaystyle G_{4}=\{m:s=0,a=0,R=I_{3}\}\cong (\mathbb {R} ^{3},+),}$ uniform frame motions.

The parameters s, v, R, a span ten dimensions. Since the transformations depend continuously on s, v, R, a, Gal(3) is a continuous group, also called a topological group.

The structure of Gal(3) can be understood by reconstruction from subgroups. The semidirect product combination ($A \rtimes B$) of groups is required.

1. $G_{2}\triangleleft {\mathrm {SGal}}(3)$ (G₂ is a normal subgroup)
2. ${\mathrm {SGal}}(3)\cong G_{2}\rtimes G_{1}$
3. $G_4 \trianglelefteq G_1$
4. $G_{1}\cong G_{4}\rtimes G_{3}$
5. ${\displaystyle \mathrm {SGal} (3)\cong \mathbb {R} ^{4}\rtimes (\mathbb {R} ^{3}\rtimes \mathrm {SO} (3)).}$

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, P_i, C_i and L_ij (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), P_i is the generator of translations (momentum operator), C_i is the generator of Galileian boosts, and L_ij 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 (which, in turn, is a group contraction of the de Sitter group SO(1,4)).^[8]

Renaming the generators of the latter as ϵ_imn J_i ↦ L_mn ; P_i ↦ P_i ; P₀ ↦ H/c ; K_i ↦ cC_i, 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 L_mnL^mn, P_iPⁱ.

In matrix form, for d=3, one may consider the regular representation (embedded in GL(5;ℝ), from which it could be derived by a single group contraction, bypassing the Poincaré group),

${\displaystyle iH=\left({\begin{array}{ccccc}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&1\\0&0&0&0&0\\\end{array}}\right),\qquad }$ ${\displaystyle i{\vec {a}}\cdot {\vec {P}}=\left({\begin{array}{ccccc}0&0&0&0&a_{1}\\0&0&0&0&a_{2}\\0&0&0&0&a_{3}\\0&0&0&0&0\\0&0&0&0&0\\\end{array}}\right),\qquad }$ ${\displaystyle i{\vec {v}}\cdot {\vec {C}}=\left({\begin{array}{ccccc}0&0&0&v_{1}&0\\0&0&0&v_{2}&0\\0&0&0&v_{3}&0\\0&0&0&0&0\\0&0&0&0&0\\\end{array}}\right),\qquad }$ ${\displaystyle i\theta _{i}\epsilon ^{ijk}L_{jk}=\left({\begin{array}{ccccc}0&\theta _{3}&-\theta _{2}&0&0\\-\theta _{3}&0&\theta _{1}&0&0\\\theta _{2}&-\theta _{1}&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\\end{array}}\right)~.}$

The infinitesimal group element is then

${\displaystyle G(R,{\vec {v}},{\vec {a}},s)=1\!\!1_{5}+\left({\begin{array}{ccccc}0&\theta _{3}&-\theta _{2}&v_{1}&a_{1}\\-\theta _{3}&0&\theta _{1}&v_{1}&a_{2}\\\theta _{2}&-\theta _{1}&0&v_{1}&a_{3}\\0&0&0&0&s\\0&0&0&0&0\\\end{array}}\right)+...~.}$

Central extension of the Galilean group

One could, instead,^[9] augment the Galilean group by a central extension of 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

$[H',P'_i]=0 \,\!$

$[P'_i,P'_j]=0 \,\!$

$[L'_{ij},H']=0 \,\!$

$[C'_i,C'_j]=0 \,\!$

$[L'_{ij},L'_{kl}]=i [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!$

$[L'_{ij},P'_k]=i[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!$

$[L'_{ij},C'_k]=i[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!$

$[C'_i,H']=i P'_i \,\!$

$[C'_i,P'_j]=i M\delta_{ij} ~.$

This algebra is often referred to as the Bargmann algebra.

See also

• Representation theory of the Galilean group
• Lorentz group
• Poincaré group
• Lagrangian and Eulerian coordinates

Notes

1. Galilei 1638I, 191–196 (in Italian)
   Galilei 1638E, (in English)
   Copernicus et al. 2002, pp. 515–520
2. Mould 2002, Chapter 2 §2.6, p. 42
3. Lerner 2996, Chapter 38 §38.2, p. 1046,1047
4. Serway 2006, Chapter 9 §9.1, p. 261
5. Hoffmann 1983, Chapter 5, p. 83
6. Arnold 1989, p. 6
7. Nadjafikhah & Forough 2009
8. Gilmore 2006
9. Bargmann 1954

References

• Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics (2 ed.). Springer-Verlag. p. 6. ISBN 0-387-96890-3.
• Bargmann, V. (1954). "On Unitary Ray Representations of Continuous Groups". Annals of Mathematics. 2. 59 (1): 1–46. doi:10.2307/1969831.
• Copernicus, Nicolaus; Kepler, Johannes; Galalei, Galileo; Newton, Isaac; Einstein, Albert (2002). Hawking, Stephen, ed. On the Shoulders of Giants: The Great Works of Physics and Astronomy. Philadelphia, London: Running Press. pp. 515–520. ISBN 0-7624-1348-4.
• Galilei, Galileo (1638I). Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze (in Italian). Leiden: Elsevier. pp. 191–196.
• Galileo, Galilei (1638E). Discourses and Mathematical Demonstrations Relating to Two New Sciences [Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze]. Translated to English 1914 by Henry Crew and Alfonso de Salvio.
• Gilmore, Robert (2006). Lie Groups, Lie Algebras, and Some of Their Applications. Dover Books on Mathematics. Dover Publications. ISBN 0486445291.
• Hoffmann, Banesh (1983), Relativity and Its Roots, Scientific American Books, ISBN 0-486-40676-8, Chapter 5, p. 83
• Lerner, Lawrence S. (1996), Physics for Scientists and Engineers, 2, Jones and Bertlett Publishers, Inc, ISBN 0-7637-0460-1, Chapter 38 §38.2, p. 1046,1047
• Mould, Richard A. (2002), Basic relativity, Springer-Verlag, ISBN 0-387-95210-1, Chapter 2 §2.6, p. 42
• Nadjafikhah, Mehdi; Forough, Ahmad-Reza (2009). "Galilean Geometry of Motions" (PDF). Applied Sciences. pp. 91–105.
• Serway, Raymond A.; Jewett, John W. (2006), Principles of Physics: A Calculus-based Text (4th ed.), Brooks/Cole - Thomson Learning, ISBN 0-534-49143-X, Chapter 9 §9.1, p. 261