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, while 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

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

where $v \in ℝ 3$ . A translation is given by

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

where $a \in ℝ 3$ and $s \in ℝ$ . A rotation is given by

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

where $G : ℝ 3 \to ℝ 3$ 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 $x \in ℝ 3$ is a position in space. The action is given by^[7]

{\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 $v, x, a \in ℝ 3$ 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 $v, R, s, a$ :

G_{1}=\{m:s=0,a=0\},

uniformly special transformations.

G_{2}=\{m:v=0,R=I_{3}\}\cong (\mathbb {R} ^{4},+),

shifts of origin.

G_{3}=\{m:s=0,a=0,v=0\}\cong \mathrm {SO} (3),

rotations of reference frame (see SO(3)).

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.

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

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 $c \to \infty$ . 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 $ϵ imn J i \mapsto L mn; P i \mapsto P i; P 0 \mapsto H / c; K i \mapsto cC i$ , where $c$ is the speed of light, or any function thereof diverging as $c \to \infty$ , the commutation relations (structure constants) of the latter limit to that of the former.

Note the group invariants $L mn L mn$ , $P i P i$ .

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),

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

The infinitesimal group element is then

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.

Notes

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

Galileo Galilei
Scientific career	Observational astronomy Galileo affair Galileo's escapement Galilean invariance Galilean moons Galilean transformation Leaning Tower of Pisa experiment Phases of Venus Celatone Thermoscope
Works	The Assayer De Motu Antiquiora Dialogue Concerning the Two Chief World Systems "Discourse on the Tides" "Letter to the Grand Duchess Christina" Sidereus Nuncius Two New Sciences
Family	Vincenzo Galilei (father) Michelagnolo Galilei (brother) Vincenzo Gamba (son) Maria Celeste (daughter) Marina Gamba (mistress)
Related	"And yet it moves" Villa Il Gioiello Galileo's paradox Sector Museo Galileo Galileo's telescopes Tribune of Galileo
In popular culture	Life of Galileo (1943 play) Lamp At Midnight (1947 play) Galileo (1968 film) Galileo (1975 film) Starry Messenger (1996 book) Galileo's Daughter: A Historical Memoir of Science, Faith, and Love (1999 book) Galileo Galilei (2002 opera) Galileo's Dream (2009 novel)

Relativity

Special
relativity

Background	Principle of relativity Special relativity
Foundations	Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations
Formulation	Galilean relativity Galilean transformation Lorentz transformation
Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks
Spacetime	Light cone World line Spacetime diagram Biquaternions Minkowski space

General
relativity

Background	Introduction Mathematical formulation
Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Minkowski diagram Penrose diagram
Phenomena	Black hole Event horizon Frame-dragging Geodetic effect Lenses Singularity Waves Ladder paradox Twin paradox Two-body problem
Equations	ADM formalism BSSN formalism Einstein field equations Geodesic equation Friedmann equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation
Advanced theories	Brans–Dicke theory Kaluza–Klein Mach's principle Quantum gravity
Solutions	Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub–NUT Milne Robertson–Walker pp-wave van Stockum dust Weyl−Lewis−Papapetrou

Scientists

Einstein field equations:

G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }

and their analytical solution by Ernst equation:

\displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.

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