Galilean transformation

Galilean transformations

The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of spacetime. 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

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

where R : R³ → R³ is an orthogonal transformation.

As a Lie group, the group of Galilean transformations has dimension 10.

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

${\displaystyle \left(\begin{array}{ccc}R& v& a\\ 0& 1& s\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ t\\ 1\end{array}\right)=\left(\begin{array}{c}Rx+vt+a\\ t+s\\ 1\end{array}\right),}$

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 / homogeneous transformations, isomorphic to Euclidean transformations.
• ${\displaystyle G_2=\{m:v=0,R=I_3\}\cong \left({\mathbf{R}}^4,+\right),}$ shifts of origin / translation in Newtonian spacetime.
• ${\displaystyle G_3=\{m:s=0,a=0,v=0\}\cong \mathrm{S}\mathrm{O}(3),}$ rotations (of reference frame) (see SO(3)), a compact group.
• ${\displaystyle G_4=\{m:s=0,a=0,R=I_3\}\cong \left({\mathbf{R}}^3,+\right),}$ 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 (${\displaystyle A⋊B}$) of groups is required.

1. ${\displaystyle G_2◃\mathrm{S}\mathrm{G}\mathrm{a}\mathrm{l}(3)}$ (G₂ is a normal subgroup)
2. ${\displaystyle \mathrm{S}\mathrm{G}\mathrm{a}\mathrm{l}(3)\cong G_2⋊G_1}$
3. ${\displaystyle G_4⊴G_1}$
4. ${\displaystyle G_1\cong G_4⋊G_3}$
5. ${\displaystyle \mathrm{S}\mathrm{G}\mathrm{a}\mathrm{l}(3)\cong {\mathbf{R}}^4⋊({\mathbf{R}}^3⋊\mathrm{S}\mathrm{O}(3)).}$

Origin in group contraction

The Lie algebra of the Galilean group is spanned by H, P_i, C_i and L_ij (an antisymmetric tensor), subject to commutation relations, where

${\displaystyle [H,P_i]=0}$

${\displaystyle [P_i,P_j]=0}$

${\displaystyle [L_{ij},H]=0}$

${\displaystyle [C_i,C_j]=0}$

${\displaystyle [L_{ij},L_{kl}]=i[{\delta }_{ik}{L}_{jl}-{\delta }_{il}{L}_{jk}-{\delta }_{jk}{L}_{il}+{\delta }_{jl}{L}_{ik}]}$

${\displaystyle [L_{ij},P_k]=i[{\delta }_{ik}{P}_j-{\delta }_{jk}{P}_i]}$

${\displaystyle [L_{ij},C_k]=i[{\delta }_{ik}{C}_j-{\delta }_{jk}{C}_i]}$

${\displaystyle [C_i,H]=i{P}_i}$

${\displaystyle [C_i,P_j]=0.}$

H is the generator of time translations (Hamiltonian), P_i is the generator of translations (momentum operator), C_i is the generator of rotationless Galilean transformations (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)). Formally, renaming the generators of momentum and boost of the latter as in

P₀ ↦ H / c

K_i ↦ c ⋅ C_i,

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 L_mn L^mn and P_i 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),

${\displaystyle iH=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right),}$ ${\displaystyle i\overrightarrow{a}\cdot \overrightarrow{P}=\left(\begin{array}{ccccc}0& 0& 0& 0& a_1\\ 0& 0& 0& 0& a_2\\ 0& 0& 0& 0& a_3\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),}$ ${\displaystyle i\overrightarrow{v}\cdot \overrightarrow{C}=\left(\begin{array}{ccccc}0& 0& 0& v_1& 0\\ 0& 0& 0& v_2& 0\\ 0& 0& 0& v_3& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),}$ ${\displaystyle i{\theta }_i{ϵ}^{ijk}{L}_{jk}=\left(\begin{array}{ccccc}0& {\theta }_3& -{\theta }_2& 0& 0\\ -{\theta }_3& 0& {\theta }_1& 0& 0\\ {\theta }_2& -{\theta }_1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right).}$

The infinitesimal group element is then

${\displaystyle G(R,\overrightarrow{v},\overrightarrow{a},s)=11_5+\left(\begin{array}{ccccc}0& {\theta }_3& -{\theta }_2& v_1& a_1\\ -{\theta }_3& 0& {\theta }_1& v_2& a_2\\ {\theta }_2& -{\theta }_1& 0& v_3& a_3\\ 0& 0& 0& 0& s\\ 0& 0& 0& 0& 0\end{array}\right)+....}$

Central extension of the Galilean group

One may consider 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 ${\displaystyle [C_i^{\prime },P_j^{\prime }]=iM{\delta }_{ij}}$, such that M lies in the center, i.e. commutes with all other operators.

In full, this algebra is given as

${\displaystyle [H^{\prime },P_i^{\prime }]=0}$

${\displaystyle [P_i^{\prime },P_j^{\prime }]=0}$

${\displaystyle [L_{ij}^{\prime },H^{\prime }]=0}$

${\displaystyle [C_i^{\prime },C_j^{\prime }]=0}$

${\displaystyle [L_{ij}^{\prime },L_{kl}^{\prime }]=i[{\delta }_{ik}{L}_{jl}^{\prime }-{\delta }_{il}{L}_{jk}^{\prime }-{\delta }_{jk}{L}_{il}^{\prime }+{\delta }_{jl}{L}_{ik}^{\prime }]}$

${\displaystyle [L_{ij}^{\prime },P_k^{\prime }]=i[{\delta }_{ik}{P}_j^{\prime }-{\delta }_{jk}{P}_i^{\prime }]}$

${\displaystyle [L_{ij}^{\prime },C_k^{\prime }]=i[{\delta }_{ik}{C}_j^{\prime }-{\delta }_{jk}{C}_i^{\prime }]}$

${\displaystyle [C_i^{\prime },H^{\prime }]=i{P}_i^{\prime }}$

and finally

${\displaystyle [C_i^{\prime },P_j^{\prime }]=iM{\delta }_{ij}.}$

where the new parameter ${\displaystyle M}$ shows up. This extension and projective representations that this enables is determined by its group cohomology.