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]

${\displaystyle x'=x-vt\,}$
${\displaystyle y'=y\,}$
${\displaystyle z'=z\,}$
${\displaystyle t'=t.\,}$

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:

${\displaystyle (x',t')=(x,t){\begin{pmatrix}1&0\\-v&1\end{pmatrix}}.}$

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 ${\displaystyle ({\mathbf {x}},t)\mapsto ({\mathbf {x}}+t{\mathbf {v}},t)}$ where v is in R3. A translation is given by ${\displaystyle ({\mathbf {x}},t)\mapsto ({\mathbf {x}}+{\mathbf {a}},t+b)}$ where a in R3 and b in R. A rotation is given by ${\displaystyle ({\mathbf {x}},t)\mapsto (G{\mathbf {x}},t)}$ where G : R3R3 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]

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

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 :

${\displaystyle G_{1}=\{m:s=0,y=0\},}$ uniformly special transformations.
${\displaystyle G_{2}=\{m:v=0,R=I_{3}\}\cong (R^{4},+),}$ shifts of origin.
${\displaystyle G_{3}=\{m:s=0,y=0,v=0\}\cong SO(3),}$ rotations of reference frame (see SO(3)).
${\displaystyle G_{4}=\{m:s=0,y=0,R=I_{3}\}\cong (R^{3},+),}$ 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 (${\displaystyle A\rtimes B}$) of groups is required.

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 Template:Mvarij (an antisymmetric tensor), subject to commutation relations, where

${\displaystyle [H,P_{i}]=0\,\!}$
${\displaystyle [P_{i},P_{j}]=0\,\!}$
${\displaystyle [L_{ij},H]=0\,\!}$
${\displaystyle [C_{i},C_{j}]=0\,\!}$
${\displaystyle [L_{ij},L_{kl}]=i[\delta _{ik}L_{jl}-\delta _{il}L_{jk}-\delta _{jk}L_{il}+\delta _{jl}L_{ik}]\,\!}$
${\displaystyle [L_{ij},P_{k}]=i[\delta _{ik}P_{j}-\delta _{jk}P_{i}]\,\!}$
${\displaystyle [L_{ij},C_{k}]=i[\delta _{ik}C_{j}-\delta _{jk}C_{i}]\,\!}$
${\displaystyle [C_{i},H]=iP_{i}\,\!}$
${\displaystyle [C_{i},P_{j}]=0~.}$

Template:Mvar 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 JiLmn ; PiPi ; P0H/c ; Kic 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 Template:Mvar commutes with everything (i.e. lies in the center), and

${\displaystyle [H',P'_{i}]=0\,\!}$
${\displaystyle [P'_{i},P'_{j}]=0\,\!}$
${\displaystyle [L'_{ij},H']=0\,\!}$
${\displaystyle [C'_{i},C'_{j}]=0\,\!}$
${\displaystyle [L'_{ij},L'_{kl}]=i[\delta _{ik}L'_{jl}-\delta _{il}L'_{jk}-\delta _{jk}L'_{il}+\delta _{jl}L'_{ik}]\,\!}$
${\displaystyle [L'_{ij},P'_{k}]=i[\delta _{ik}P'_{j}-\delta _{jk}P'_{i}]\,\!}$
${\displaystyle [L'_{ij},C'_{k}]=i[\delta _{ik}C'_{j}-\delta _{jk}C'_{i}]\,\!}$
${\displaystyle [C'_{i},H']=iP'_{i}\,\!}$
${\displaystyle [C'_{i},P'_{j}]=iM\delta _{ij}~.}$