History of Lorentz transformations: Wikis

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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

Voigt (1887)

In connection with the Doppler effect and an incompressible medium, Voigt (1887) developed a transformation, which was in modern notation:[1][2]

x^{\prime}=x-vt,\quad y^{\prime}=\frac{y}{\gamma},\quad z^{\prime}=\frac{z}{\gamma},\quad t^{\prime}=t-x\frac{v}{c^{2}}

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation. In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Maxwell's electrodynamics is both scale invariant and Lorentz invariant, so the combination is invariant too. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a principle of relativity in general. Lorentz acknowledged Voigt's work in 1909 by saying:

In a paper "Über das Doppler'sche Princip", published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (6) (3 of this book) a transformation equivalent to the formulae (287) and (288). The idea of the transformations used above (and in 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper.

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. Voigt responded in the same paper by saying, that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.

Lorentz (1892, 1895)

In 1892 Lorentz developed a model ("Lorentz ether theory") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. To calculate the optics of moving bodies, Lorentz (independently of Voigt) introduced the following quantities to transform from the aether system into a moving system.[3]

x^{\prime}=\gamma x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=t-\gamma^{2} x^{*}\frac{v}{c^{2}}

where x* is the Galilean transformation x-vt. While t is the "true" time for observers resting in the aether, t' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in 2 steps. At first the Galilean transformation - and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. He also (1892b) introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction in his theory (without proof as he admitted). 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.

In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". 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 for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:

x^{\prime}=\gamma x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=t

For solving optical problems Lorentz used the following transformation, whereby for the time variable he used the expression "local time" (Ortszeit):

x^{\prime}=x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=t-x^{*}\frac{v}{c^{2}}

With this concept Lorentz could explain the Doppler effect, the aberration of light, and the Fizeau experiment.[4]

Larmor (1897, 1900)

Larmor in 1897 and 1900 presented the transformations in two parts. Similar to Lorentz, he considered first the transformation from a rest system (xyzt) to a moving system (x′, y′, z′, t′)

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

This transformation is just the Galilean transformation for the xyz 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:

x^{\prime}=\gamma x^{*},\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=\frac{t}{\gamma}-\gamma x^{*}\frac{v}{c^{2}}

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 then 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.[5][6]

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,[7]

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

and ε is an undetermined factor 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):

x^{\prime}=\gamma lx^{*},\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\frac{l}{\gamma}t-\gamma lx^{*}\frac{v}{c^{2}}

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.

Poincaré (1900, 1905)

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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.[8] 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, although 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 + δt b)/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*/c 2, 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.\,

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):[9][10]

x^{\prime}=\gamma(x-vt),\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=\gamma\left(t-vx\right)

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; 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 − c2t 2 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.

Einstein (1905)

In June 30, 1905 (published September 1905) Einstein gave a 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.[11][12][13] 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. For quantities of first order in v/c this was also done by Poincaré in 1900 who, however, continued to distinguish between "true" and "apparent" time, while Einstein derived the complete transformation by this method. Einstein's version of the transformation is identical to Poincaré's except that Einstein didn't set the speed of light to unity:

x^{\prime}=\gamma(x-vt),\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t^{\prime}=\gamma\left(t-x\frac{v}{c^{2}}\right)

See also

Lorentz ether theory
History of special relativity

References

Primary sources
  • Voigt, Woldemar (1887), "Ueber das Doppler’sche Princip", Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen (2): 41–51  
Secondary sources
  1. ^ Miller (1981), 114–115
  2. ^ Pais (1982), Kap. 6b
  3. ^ Miller (1982), 1.4 & 1.5
  4. ^ Janssen (1995), 3.1
  5. ^ Darrigol (2000), Chap. 8.5
  6. ^ Macrossan (1986)
  7. ^ Jannsen (1995), Kap. 3.3
  8. ^ Darrigol (2005), Kap. 4
  9. ^ Pais (1982), Kap. 6c
  10. ^ Katzir (2005), 280–288
  11. ^ Miller (1981), Kap. 6
  12. ^ Pais (1982), Kap. 7
  13. ^ Darrigol (2005), Kap. 6
  • Darrigol, Olivier (2000), Electrodynamics from Ampére to Einstein, Oxford: Oxford Univ. Press, ISBN 0198505949  
  • Katzir, Shaul (2005), "Poincaré’s Relativistic Physics: Its Origins and Nature", Physics in perspective 7: 268–292, doi:10.1007/s00016-004-0234-y  
  • Miller, Arthur I. (1981), Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2  
  • Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: Oxford University Press, ISBN 0-19-520438-7  

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