History of Lorentz transformations

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The Lorentz transformations relate the space-time coordinates, (which specify the position x, y, z and time t 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. It was devised as a theoretical transformation which makes the velocity of light invariant between different inertial frames. The coordinates of the event in this "moving system" are denoted x′, y′, z′ and t′. Before 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. Early approximations of the transformation were published by Voigt (1887) and Lorentz (1895). They were completed by Larmor (1897, 1900) and Lorentz (1899, 1904) and were brought into their modern form by Poincaré (1905), who gave the transformation the name of Lorentz, and Einstein (1905).

In this article the historical notations are replaced with modern notations, where

\gamma =\frac1{\sqrt{1-v^2/c^2}}

is the Lorentz factor, v is the relative velocity of the bodies, and c is the speed of light.

Contents

[edit] Voigt (1887)

In connection with the Doppler effect and an incompressible medium, Voigt (1887) developed a transformation, which was in modern notation:[1]

\begin{align}x'&=x-vt\\y'&=y/\gamma\\z'&=z/\gamma\\t'&=t-vx/c^2\end{align}

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation. Lorentz (1909) is on record as saying he could have taken these transformations into his theory of electrodynamics, if only he had known of them.[2] Also Hermann Minkowski said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in the 1887.[3]

[edit] Lorentz (1895)

A fundamental concept of Lorentz's ether theory in 1895 was the "theorem of corresponding states" for terms of order v/c.[4] This theorem states that a moving observer (relative to the ether) in his „fictitious“ field makes the same observations as a resting observers in his „real“ field. An important part of the theorem was "local time" (Ortszeit) t′ = t − vx/c², where t is the time coordinate for an observer resting in the ether, and t′ is the time coordinate for an observer moving in the ether. This was the same equation which was already used by Voigt, however, Lorentz later pointed out that at this time he was unaware of Voigt's paper. Lorentz also explained that the dimensions of electrostatic systems rest in the ether and a moving frame are connected by this transformation (where x*  = x − vt):

\begin{align}x'&=\gamma x^* \\y'&=y\\z'&=z\end{align}

And as an additional and independent hypothesis Lorentz claimed (without proof as he admitted), that also intermolecular forces are affected in a similar way and introduced length contraction in his theory. While for Lorentz length contraction was a real physical effect, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation. With the help of local time Lorentz could explain in 1895 the Doppler effect, the aberration of light, and the measurements of the Fresnel drag coefficient by Hippolyte Fizeau in moving and resting liquids as well.

[edit] Larmor (1897, 1900)

Consider the presentation of the Lorentz transformations given by Joseph Larmor in 1897 (and again in 1900).[5] [6] 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 (x, y, z, t) to a moving system (x′, y′, z′, t′)

\begin{align}x'&=x-vt\\y'&=y\\z'&=z\\t'&=t-\gamma^2vx^*/c^2\end{align}

This transformation is just the Galilean transformation for the x, y, z coordinates but contains Lorentz’s "local time". 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). Thus he wrote the final transformations (where x* = x − vt) as:

\begin{align}x'&=\gamma x^*\\y'&=y\\z'&=z\\t'&=t/\gamma-\gamma vx^*/c^2\end{align}

Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v/c", as he put it.

Larmor noted that if it is assumed that the constitution of molecules is electrical than the Fitzgerald-Lorentz contraction is a consequence of this transformation. It's notable that Larmor was the first who recognized that some sort of time dilation is a consequence of this transformation as well, because individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ.

[edit] Lorentz (1899, 1904)

Also Lorentz, by extending his theorem of corresponding states, derived in 1899 the complete transformations. However, he used the undetermined factor ε as an arbitrary function of v. Like Larmor, in 1899 also Lorentz noticed some sort of time dilation effect, and he wrote that for the frequency of oscillating electrons "that in S the time of vibrations be kε times as great as in S0", where S0 is the ether frame,

k=\sqrt{1-v^2/c^2},

and ε is an undetermined factor.[7] The factor l was set to unity in 1904 so Lorentz's equations had the same form as Larmor's (as mentioned above x* must be replaced by x − vt): [8]

\begin{align}x'&=\gamma\ell x^*\\y'&=\ell y\\z'&=\ell z\\t'&=\ell t/\gamma-\gamma\ell vx^*/c^2\end{align}

In connection with this he also derived the correct formulas for the velocity dependence of mass. He concluded, that this transformation must apply to all forces of nature, not only electrical ones and therefore length contraction is a consequence of this transformation.

[edit] Poincaré (1900, 1905)

[edit] Local time

Neither Lorentz or Larmor gave a clear interpretation of the origin of local time. However, Poincaré in 1900 commented on the origin of Lorentz’s “wonderful invention” of local time. [9] 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, which lead to what is nowadays called relativity of simultaneity. Poincaré's calculation does not involve length contraction or time dilation. In order to synchronise the clocks here on Earth (the x*, t* 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 (x,t) (i.e. the luminiferous aether system for Lorentz and Larmor). We calculate that the time of flight outwards is

\delta t_o = \frac{x^*}{\left(c - v\right)}

and the time of flight back is

\delta t_b = \frac{x^*}{\left(c + v\right)}\cdot

The elapsed time on the clock when the signal is returned is δto + δtb and we ascribed the time t* = (δto + δtb)/2 to the moment when the light signal reached the distant clock. In the rest frame, of course, the time t = δto 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 - \frac{\epsilon vx^*}{c^2}\cdot

Poincaré gave the result t* = t − vx*/c2, which is the form used by Lorentz in 1895. Poincaré dropped the factor ε ≅ 1 under the assumption that

\frac{v^2}{c^2}\ll1.\,

[edit] Lorentz transformation

In June 5, 1905 (published June 9) Poincaré simplified the equations (which are algebraically equivalent to those of Larmor and Lorentz) and gave them the modern form (Poincaré set the speed of light to unity):[10]

\begin{align}x'&=\gamma\left(x-vt\right)\\y'&=y\\z'&=z\\t'&=\gamma\left(t-vx\right)\\\gamma&=\frac1{\sqrt{1 - v^2}}\end{align}

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation". He showed that Lorentz's application of the transformation on the equations of electrodynamics didn't fully satisfy the principle of relativity. So by pointing out the group characteristics of the transformation Poincaré demonstrated the Lorentz covariance of the Maxwell-Lorentz equations.

In July 1905 (published in January 1906) Poincaré showed that the transformations are a consequence of the principle of least action;[11] he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group, and he showed that the combination x2 + y2 + z2 − c2t2 is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct√−1 as a fourth imaginary coordinate, and he used an early form of four-vectors.

[edit] Einstein (1905)

In June 30, 1905 (published September 1905) Einstein gave a radical new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light.[12] Contrary to Lorentz, who considered "local time" only as a mathematical stipulation, Einstein showed that the "effective" coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. This was in some respect also done by Poincaré who, however, continued to distinguish between "true" and "apparent" time. Einstein's version was identical to Poincaré's (Einstein didn't set the speed of light to unity):

\begin{align}x'&=\gamma (x - vt)\\y'&= y\\z'&= z\\t'&= \gamma\left(t-\frac{v x}{c^2}\right)\end{align}

[edit] See also

Lorentz ether theory
History of special relativity

[edit] References

Primary sources
  1. ^ Voigt, W. (1887), “Über das Doppler’sche Princip”, Göttinger Nachrichten (no. 2): 41-51 
  2. ^ Lorentz, H.A (1916), The theory of electrons, Leipzig & Berlin: B.G. Teubner 
  3. ^ Bucherer, A. H. (1908), “Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie”, Physikalische Zeitschrift 9 (22): 755-762  See p. 762.
  4. ^ Lorentz, H.A. (1895), Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Kõrpern, Leiden: E.J. Brill 
  5. ^ Larmor, J. (1897), “On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media”, Phil. Trans. Roy. Soc. 190: 205-300 
  6. ^ Larmor, J. (1900), Aether and Matter, Cambridge University Press 
  7. ^ Lorentz, H.A. (1899), “Simplified Theory of Electrical and Optical Phenomena in Moving Systems”, Proc. Roy. Soc. Amst. 1: 427-442 
  8. ^ Lorentz, H.A. (1904), “Electromagnetic phenomena in a system moving with any velocity smaller than that of light”, Proc. Roy. Soc. Amst. 6: 809-831 
  9. ^ Poincaré, H. (1900), “La théorie de Lorentz et le principe de réaction”, Archives néerlandaises des sciences exactes et naturelles 5: 252-278 . Reprinted in Poincaré, Oeuvres, tome IX, pp. 464-488
  10. ^ Poincaré, H. (1905), “Sur la dynamique de l'électron”, Comptes Rendus 140: 1504-1508  Reprinted in Poincaré, Oeuvres, tome IX, S. 489-493.
  11. ^ Poincaré, H. (1906), “Sur la dynamique de l'électron”, Rendiconti del Circolo matematico di Palermo 21: 129-176  Reprinted in Poincaré, Oeuvres, tome IX, pages 494-550. Partial English translation in Dynamics of the electron.
  12. ^ Einstein, A. (1905), “Zur Elektrodynamik bewegter Körper”, Annalen der Physik 17: 891-921  English translation
Secondary sources
  • Darrigol, O. (1994), “The Electron Theories of Larmor and Lorentz: A Comparative Study”, Historical Studies in the Physical and Biological Sciences 24: 265–336 
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