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The Lorentz transformations relate the space-time coordinates, (which specify the position x,y,z and time of an event) relative to a particular inertial frame of reference (the "rest system"), and the coordinates of the same event relative to another coordinate system moving in the positive x-direction at a constant speed v, relative to the rest system. The coordinates of the event in this "moving sysetm" are denoted and . Before famous Einstein's paper of 1905, the rest system was identified with the "aether", the supposed medium which transmitted electro-magnetic waves, and the moving system as commonly identified with the earth as it moved through this medium.

The transformations were written by different people in slightly different forms before 1905. These were, in order of publication date, Voigt (1887), Hendrik Lorentz (1895, 1899, 1904), Joseph Larmor (1897, 1900), Henri Poincare (1905). We can write them here in a generic form:

Failed to parse (unknown error): t^\prime = \ell \beta \left(t – vx^\prime/c^2\right)

where Failed to parse (unknown error): \beta = 1/\sqrt{1 – v^2/c^2}

 and  is a function of v / c.  

Consider the presentation of the Lorentz transformations given by Joseph Larmor in 1987 (and again in 1900). Larmor presented the transformations in two parts. He was following Lorentz in the first part, with but a small difference which need not concern us. He considered first the transformation from a rest system to a moving system

x ^* = x − vt

y ^* = y

z ^* = z

Failed to parse (unknown error): t^* = t – \epsilon vx^*/c^2

where Failed to parse (unknown error): \epsilon = 1/\left(1 – v^2/c^2\right)

In modern notation ε = γ² where Failed to parse (unknown error): \gamma = 1/\sqrt{1 – v^2/c^2} . This transformation is just the Galilean transformation for the coordinates but contains a form of Lorentz’s "local time". The clocks in the moving frame are unsynchronised (according to the rest system) by varying amounts (εvx ^* / c²) depending on their distance from the origin in the moving frame.

Larmor (1987, 1900) gives no clear interpretation of the origin of this "relativity of simultaneity" (except to show that the transformation makes Maxwell’s equations invariant to first order in v / c. However, Henri Poincaré in 1900 commented on the origin of Lorentz’s “wonderful invention” of local time. He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed c in both directions.

The calculation does not involve length contraction or time dilation. In order to synchronise the clocks here on Earth (the frame) we send a light signal from one clock (at the origin) to another (at x ^* ), and bounce it back. We suppose that the Earth is moving with speed v in the x-direction (= x ^* direction) in some rest system (i.e. the luminiferous aether system for Lorentz and Larmor). All clocks are assumed to run at the same rate and all lengths are assumed idependant of motion. We calculate that the time of flight outwards is

and the time of flight back is

The elapsed time on the clock when the signal is returned is δt_o + δt_b and we ascribed the time to the moment when the light signal reached the distant clock. In the rest frame, of course, the time t = δt_o is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

t ^* = t − εvx ^* / c².

Poincaré gave the result t ^* = t − vx ^* / c², which is the form used by Lorentz before 1899. It seems Poincaré dropped the factor under the assumption that v² / c² < < 1.

Larmor knew that the Michelson-Morley experiment was accurate enough to detect an effect of motion depending on the factor v² / c², and so he sought the transformations which were "accurate to second order" (as he put it). To do this he had to modify the transformations in the first step in two ways:

• he included the Fitzgerald contraction (already known to account for the Michelson-Morley result) and
• he introduced (or predicted) the revolutionary idea of time dilation

Thus he wrote the final transformations as

y' = y ^*

z' = z ^*

from which it can be seen that lengths are shorter by the factor ε^{1 / 2} = γ and time is longer by the factor ε ^{− 1 / 2} = 1 / γ for the moving system. Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v / c", as he put it. His transformations did more than this since only a litte algegra is required to show that the relation between and is

which are the Lorentz transformations, for which we know Maxwell's equations are invariant to any order in v / c. Einstein (1905) and Poincaré (1905) wrote the transformations in this form. It was Poincaré (1905) who named them as the Lorentz Transformations. Lorentz (1899) and (1904) had published the transformations in an identical form to Larmor (as above), and Poincaré was apparently unaware of Larmor's (1987) previous publication. It is worth repeating the first published prediction of time dilation (although I assume Larmor would have thought of it as a dynamical prediction from Maxwell's equations) ... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio ε ^{− 1 / 2} <\quotation>

• Larmor, J. (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc. 190, 205-300 (third and last in a series of papers with the same name).
• Lorentz, H. A. (1899) "Simplified theory of electrical and optical phnomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Lorentz, H. A. (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Macrossan, M. N. (1986) "A note on relativity before Einstein", Brit. J. Phil. Sci., 37, 232-234
• Poincare, H. (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
• Poincare, H. (1905) "Sur la dynamique de l'electron", Comptes Rendues, 140, 1504-8.