In a paper titled Nothing but Relativity (Eur. J. Phys. 24 (2003) 315-319) Palash B. Pal derived a formula for transformations between observers that is based on the relativity postulate but not a speed of light postulate. In a paper titled Nothing but Relativity, Redux (Eur. J. Phys. 28 (2007) 1145-1150) Joel W. Gannett presented an alternate derivation with fewer implicit assumptions. Here we’ll use Pal’s approach to derive the time-space version.
Consider two inertial timeframes S and S´, where the second one moves with lenticity u, along the t-axis, with respect to the first one. There are two other time axes. The coordinates and radial distance in the S-timeframe will be denoted by t and x, and in the timeframe S´ will be denoted with a prime. The time-space transformation equations have the form:
t´ = T(t, x, u) and x´ = X(t, x, u),
and out task is to determine these functions. A few properties of these functions can readily be observed. First, the principle of relativity tells us that if we invert the lenticity in these equations, we must obtain the same functional forms:
t = T(t´, x´, –u) and x = X(t´, x´, –u).
Notice that here the third argument of the functions is –u, since that is the lenticity of the timeframe S with respect to S´. Moreover, isotropy of time demands that we could take the t-axis in the reverse direction as well. This is not reversing the order of events but changing the direction of duration. In this case, both t and u change sign, and so does t´. In other words,
T(–t, x, –u) = –T(t, x, u),
X(–t, x, –u) = X(t, x, u).
From Pal’s argument using the homogeneity of time and space, and taking the origins of the two timeframes to coincide at t = x = 0, we can conclude that
T(t, x, u) = At + Bx,
X(t, x, u) = Ct + Dx,
where the additional symbols are functions of the relative lenticity. Given the inverted lenticity equations above, we find that A and D are even functions, while B and C are odd functions of u. From this and the equations above, we obtain the following conditions:
By introducing a third timeframe S´´ we find that the following is constant:
(A² – 1) / (u²A²).
Call this constant K, so that
Thus the most general transformation equations consistent with the principle of relativity are of the form
The lenticity addition law can also be deduced from this:
Specific theories of relativity have to make extra assumptions in order to determine the value of K. In the case of contra-Galilean relativity, this extra assumption shows up in the form of the universality of radial distance, which means x´ = x for any u, which requires K = 0:
t´ = x – ux, x´ = x.
The extra assumption for the Lorentz transformation and Einstein’s theory of relativity is the constancy of the pace of light in a vacuum. From the addition law one can see that K–1/2 is an invariant pace, independent of the timeframe of reference. Thus, K = 1/ç² = c² > 0 in this case.