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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 acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view. In special relativity the homogenous and inhomogenous Galilean transformations are, respectively, replaced by the Lorentz transformations and Poincaré transformations; conversely, the group contraction in the classical limit c → ∞ of Poincaré transformations yields Galilean transformations.

The equations below are only physically valid in a Newtonian framework, and not applicable to coordinate systems moving relative to each other at speeds approaching the speed of light.

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.

Although the transformations are named for Galileo, it is the 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.

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 holds for all Galilean transformations up to addition of a constant, and 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 group of Galilean transformations has dimension 10.[6]
Galilean group

Two Galilean transformations G(R, v, a, s) and 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) 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. Care must be taken in the discussion whether one restricts oneself to the connected component group of the orthogonal transformations.

Gal(3) has named subgroups. The identity component is denoted SGal(3).

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

\( {\displaystyle \{m:R=I_{3}\},} \) anisotropic transformations.
\( {\displaystyle \{m:s=0\},} \) isochronous transformations.
\( {\displaystyle \{m:s=0,v=0\},} \) spatial Euclidean transformations.
\( {\displaystyle G_{1}=\{m:s=0,a=0\},} \) uniformly special transformations / homogenous transformations, isomorphic to Euclidean transformations.
\( G_{2}=\{m:v=0,R=I_{3}\}\cong ({\mathbf {R}}^{4},+), \) shifts of origin / translation in Newtonian spacetime.
\( {\displaystyle G_{3}=\{m:s=0,a=0,v=0\}\cong \mathrm {SO} (3),} \) rotations (of reference frame) (see SO(3)), a compact group.
\( {\displaystyle G_{4}=\{m:s=0,a=0,R=I_{3}\}\cong (\mathbf {R} ^{3},+),} \) uniform frame motions / boosts.

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)
\( {\mathrm {SGal}}(3)\cong G_{2}\rtimes G_{1} \)
\( G_4 \trianglelefteq G_1 \)
\( G_{1}\cong G_{4}\rtimes G_{3} \)
\( {\mathrm {SGal}}(3)\cong {\mathbf {R}}^{4}\rtimes ({\mathbf {R}}^{3}\rtimes {\mathrm {SO}}(3)). \)

Origin in group contraction

The Lie algebra of the Galilean group 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 rotationless Galilean transformations (Galileian boosts),[8] 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 (which, in turn, is a group contraction of the de Sitter group SO(1,4)).[9] Formally, renaming the generators of momentum and boost of the latter as in

P0 ↦ H / c
Ki ↦ c ⋅ Ci,

where c is the speed of light (or any unbounded function thereof), the commutation relations (structure constants) in the limit c → ∞ take on the relations of the former. Generators of time translations and rotations are identified. Also note the group invariants Lmn Lmn and Pi Pi.

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 may consider[10] a central extension of the Lie algebra of the Galilean group, spanned by H′, P′i, C′i, L′ij and an operator M: The so called Bargmann algebra is obtained by imposing \( {\displaystyle [C'_{i},P'_{j}]=iM\delta _{ij}} \), such that M lies in the center, i.e. commutes with all other operators.

In full, this algebra is given as

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

and finally

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

where the new parameter M shows up. This extension and projective representations that this enables is determined by its group cohomology.
See also

Galilean invariance
Representation theory of the Galilean group
Galilei-covariant tensor formulation
Poincaré group
Lorentz group
Lagrangian and Eulerian coordinates

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 1996, Chapter 38 §38.2, p. 1046,1047
Serway & Jewett 2006, Chapter 9 §9.1, p. 261
Hoffmann 1983, Chapter 5, p. 83
Arnold 1989, p. 6
[1]Nadjafikhah & Forough 2009
Ungar, A. A. (2006). Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (illustrated ed.). Springer Science & Business Media. p. 336. ISBN 978-0-306-47134-6. Extract of page 336
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; Galilei, 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

vte

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

De Motu Antiquiora (1589-1592, pub. 1687) Sidereus Nuncius (1610) Letters on Sunspots (1613) Letter to Benedetto Castelli (1613) "Letter to the Grand Duchess Christina" (1615) "Discourse on the Tides" (1616) Discourse on Comets (1619) The Assayer (1623) Dialogue Concerning the Two Chief World Systems (1632) Two New Sciences (1638)

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 Galileo's objective lens Tribune of Galileo Galileo thermometer Galileo spacecraft Galileo Galilei Airport

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 (Galilean relativity Galilean transformation) Special relativity Doubly special relativity

Fundamental
concepts

Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Relativistic mass

Formulation

Lorentz transformation

Phenomena

Time dilation Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox

Spacetime

Light cone World line Minkowski diagram Biquaternions Minkowski space

General relativity
Background

Introduction Mathematical formulation

Fundamental
concepts

Equivalence principle Riemannian geometry Penrose diagram Geodesics Mach's principle

Formulation

ADM formalism BSSN formalism Einstein field equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation

Phenomena

Black hole Event horizon Singularity Two-body problem

Gravitational waves: astronomy detectors (LIGO and collaboration Virgo LISA Pathfinder GEO) Hulse–Taylor binary

Other tests: precession of Mercury lensing redshift Shapiro delay frame-dragging / geodetic effect (Lense–Thirring precession) pulsar timing arrays

Advanced
theories

Brans–Dicke theory Kaluza–Klein Quantum gravity

Solutions

Cosmological: Friedmann–Lemaître–Robertson–Walker (Friedmann equations) Kasner BKL singularity Gödel Milne

Spherical: Schwarzschild (interior Tolman–Oppenheimer–Volkoff equation) Reissner–Nordström Lemaître–Tolman

Axisymmetric: Kerr (Kerr–Newman) Weyl−Lewis−Papapetrou Taub–NUT van Stockum dust discs

Others: pp-wave Ozsváth–Schücking metric

Scientists

Poincaré Lorentz Einstein Hilbert Schwarzschild de Sitter Weyl Eddington Friedmann Lemaître Milne Robertson Chandrasekhar Zwicky Wheeler Choquet-Bruhat Kerr Zel'dovich Novikov Ehlers Geroch Penrose Hawking Taylor Hulse Bondi Misner Yau Thorne Weiss others

Categories
Theory of relativity

Physics Encyclopedia

World

Index

Hellenica World - Scientific Library

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