Two transformations of inertial reference frames are well-known: the Galilean and the Lorentz transformations. There is a third transformation as well, which will be called the dual Galilean transformation. Below is a derivation of all three transformations, closely following the paper Getting the Lorentz transformations without requiring an invariant speed by Andrea Pelissetto and Massimo Testa (American Journal of Physics 83 (2015), p.338-340). Their approach is based on the work of von Ignatowsky in the early 20th century.
We wish to characterize the transformations that relate two different inertial frames. Let us consider two inertial observers K and K′. Let r = (x, x2, x3) and w = (t, t2, t3) be space and time coordinates for K and r´ = (x´, x2´, x3)´ and w´ = (t´, t2´, t3´) be the corresponding quantities for K′.
In order to simplify the argument, we will restrict our considerations to the subgroup of transformations involving x and t only, setting x2´= x2, x3´ = x3, t2´ = t2, and t3´ = t3. This is equivalent to choosing coordinates so that K and K′ are in relative motion along the x and t directions in K and the x′ and t´ directions in K´.
So the displacement magnitude, |Δr| = |Δx|, and the dischronment magnitude, |Δw| = |Δt|, is the time along the t-axis. Then the speed, Δr/|Δw| = Δr/Δt, and the pace, Δw/|Δr| = Δw/Δx,
We assume the validity of the principle of inertia: in an inertial frame free particles undergo a constant rate of rectilinear motion. Therefore, if the trajectory of a particle is a straight line in the frame of observer K and its rate is constant, the trajectory is also a straight line in the frame of observer K′ and also the rate in the new frame is constant. This condition implies that the two inertial frames are related by a linear transformation. We can therefore write
t´ = At + Bx,
x´ = Ct + Dx,
where we have defined the time coordinates t and t´, measured in units of distance, as
t = qt1,
where q is an arbitrary constant with units of speed. The inverse transformation follows immediately:
t = (1/Δ) (Dt´ – Bx´),
x = (1/Δ) (–Ct´ + Ax´),
with
Δ = AD – BC ≠ 0.
It is useful to rewrite the transformation in matrix notation, introducing
and its inverse
Then we have
and
The main ingredient in the proof is the requirement that there are no privileged frames: all inertial frames are equivalent. We wish now to express this hypothesis in a more transparent way that allows us to put constraints on the matrix Λ.
Pelissetto and Testa then show that Δ = 1 so that A = D. These conditions imply that the transformation relating two different inertial frames is of the form
with
A² – BC = 1.
To further constrain the structure of the matrix Λ, we now add the natural requirement that the transformations connecting two inertial frames constitute a group, i.e., that the combination of two such transformations yields a third transformation of the same form.
By multiplying two transformations, sub-scripted 1 and 2, the diagonal elements must again be equal:
B1 C2 = C1 B2.
In order to satisfy this equation for all transformations with equal diagonal elements, we have three different possibilities:
(i) B = αC, where α is a nonzero constant; or
(ii) B = 0 and A = 1; or
(iii) C = 0 and A = 1.
Case (i) corresponds to the Lorentz transformations, as Pelissetto and Testa show. They introduce C´ = C √(|α|) and show that
where A² – C´² = 1. In this case the observer K′ moves with speed v with respect to observer K, where v is determined by
C´/A = v/c.
The condition A² – C´² = 1 implies that
A = γ = 1/√(1 – v²/c²), and
C´ = γv/c,
so that Λ is a generic Lorentz transformation, with the speed of light identified with c.
Case (ii) corresponds to the Galilean transformations:
t´ = t,
x´ = Ct + x,
the parameter C being the relative velocity of the two frames in units of q.
Case (iii) corresponds to the dual Galilean transformations:
t´ = t + Bx,
x´ = x,
the parameter B being the relative lenticity of the two timeframes in units of 1/q.
Thus these three cases lead to three transformations of inertial reference frames.