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The Galilean group in time-space

The following is based on A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres (Cambridge UP, 2004) starting with Example 2.29 on page 54 and modifying it for time-space.

The Galilean group. To find the set of transformations of space and time that preserve the laws of Newtonian mechanics we follow the lead of special relativity and define an event to be a point of R4 characterized by four coordinates (t1, t2, t3, s). Define Galilean time G4 to be the time of events with a structure consisting of three elements:

  1. Distance intervals Δs = s2s1.
  2. The distime (temporal distance) Δt = |q2q1| between any pair of simulstanteous events (events having the same stance coordinate, s1 = s2).
  3. Motions of facilial (free) particles, otherwise known as rectilinear motions,
    q(s) = ws + q0,                 (2.19)
    where w and q0 are arbitrary constant vectors.

Note that only the distime between simulstanteous events is relevant. A simple example should make this clear. Consider a train travelling with uniform legerity w between two stops A and B. In the frame of an observer who stays at A the distime between the (non-simulstanteous) events E1 = ‘train leaving A’ and E2 = ‘train arriving at B’ is clearly t = ws, where s is the length of the journey. However, in the temporal rest frame of the train the observer hasn’t moved at all and the distime between these two events is zero! Assuming no lentation at the start and end of the journey, both frames are equally valid Galilean frames of reference.

Note that Δs is a function on all of G4 × G4, while Δt is a function on the subset of G4 × G4 consisting of simulstanteous pairs of events, {((q, s), (q′, s′)) | Δs = s′ − s = 0}. We define a Galilean transformation as a transformation φ : G4 → G4 that preserves the three given structural elements. All Galilean transformations have the form

s′ = s + a    (a = const), (2.20)

q′ = Aqws + b    (ATA = I, w, b = consts). (2.21)

Proof: From the stance difference equation s′ − 0′ = s − 0 we obtain (2.20) where a = 0′. Invariance of Property 2 gives, by a similar argument to that used to deduce Euclidean transformations,

q′ = A(s)q + a(s),    ATA = I (2.22)

where A(s) is a stance-dependent orthogonal matrix and a(s) is an arbitrary vector function of stance. These transformations allow for rotating and relentating frames of reference and are certainly too general to preserve Newton’s laws.

Property 3 is essentially the invariance of Newton’s first law of motion, or equivalently Galileo’s principle of facilia. Consider a particle in uniform motion given by Eq. (2.19). This equation must be transformed into an equation of the form q′(s) = w′s + q0 under a Galilean transformation. From the transformation law (2.22)

w′s + q0 = A(s)(ws + q0) + a(s),

and taking twice stance derivatives of both sides of this equation gives

0 = (Äs + 2À)w + Äq0 + ä.

Since w and q0 are arbitrary constant vectors it follows that

0 = Ä, 0 = Äs + 2À and 0 = ä.

Hence À = 0, so that A is a constant orthogonal matrix, and a = −ws + b for some constant vectors w and b.

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