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. 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. 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.
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.
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:
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:
Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity.
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
where v ∈ ℝ3. A translation is given by
where a ∈ ℝ3 and s ∈ ℝ. A rotation is given by
where G : ℝ3 → ℝ3 is an orthogonal transformation.
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.
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:
- uniformly special transformations.
- shifts of origin.
- rotations of reference frame (see SO(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 () of groups is required.
- (G2 is a normal subgroup)
Origin in group contraction
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), 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)).
Renaming the generators of the latter as ϵimn Ji ↦ Lmn ; Pi ↦ Pi ; P0 ↦ H/c ; Ki ↦ cCi, 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, PiPi.
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),
The infinitesimal group element is then
Central extension of the Galilean group
This algebra is often referred to as the Bargmann algebra.
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