How did Lorentz transformations get their modern definition?

Question

Historically, Special Relativity was motivated by apparent inconsistencies between Maxwell's Electrodynamics and Newtonian Mechanics. In Einstein's well known paper "On the electrodynamics of moving bodies" he explains quite well his motivations.

Central objects of the theory are the Lorentz transformations. If one forgets motivations, history and intuition the Lorentz transformations are formally defined as the linear transformations $\Lambda:\mathbb{R}^4\to \mathbb{R}^4$ such that

$$\eta(\Lambda v,\Lambda w)=\eta(v,w),$$

where $\eta = \operatorname{diag}(-1,1,1,1)$. Furthermost, it seems that before this definition they were defined as the transformations which keep the speed of light the same in all frames.

My question is: how did Lorentz transformations get this modern definition?

How were they first defined, how did they relate to Einstein's paper, and how did they get the modern definition as "transformations which preserve the spacetime inner product"? Specifically I'm interested in knowing how from the motivations for relativity physicists got to the definition of Lorentz transformations as the transformations $\Lambda$ such that $\eta(\Lambda x,\Lambda y) = \eta(x,y)$

Answer 1

Wikipedia has very adequate and well-sourced article on History of Lorentz transformations. Voigt formulated the not quite the modern ones back in 1887, of which Lorentz was unaware, and had to work out his own version independently. This might have been just as well since he later said he would have used Voigt's if he knew about them, but He presented them partially (without the time dilation) in 1895, the first complete version is due to Larmor (1897). Lorentz was apparently unaware of that either, and supplied his own full version in 1899, see What made Einstein believe (or know) that time was affected by speed and gravity?. None of them viewed the transformations algebraically or kinematically, they were seen as describing dynamic effects on bodies moving at high speeds. Larmor even supplemented a hypothesis that molecular forces are of electromagnetic nature, which would explain the effects. But as Poincare showed in 1905 purely electromagnetic forces could not account for the stability of the electron. He had to conjecture an additional stabilizing non-electromagnetic force that nonetheless obeyed the same transformation laws, which made it ad hoc.

The first algebraic observation, that the transformations form a group, was made by Poincare in his 1904-1906 papers on the dynamics of the electron, but according to Weinstein "Poincaré did not associate this quadratic form with propagation of light in order to define a null interval like Einstein or a metric like Minkowski". This is particularly surprising because groups involved in Kleinian geometries are usually obtained by considering all transformations that preserve a quadratic form, as Poincare well knew. As alluded to by Weinstein, it was Einstein in 1905 who first characterized them kinematically, as the transformations that preserve the speed of light in all frames (i.e. preserve the null interval), and only Minkowsky, inspired by Einstein's paper, gave the modern geometric formulation of them as the transformations that preserve a (pseudo) metric in 1907-1909, see What was the motivation for Minkowski spacetime before special relativity?

Answer 2

Transformations must preserve the structure of Maxwell's equations, which then automatically preserves the speed of light. Voigt, 1887, was the first, but there were several independent derivations. Note that Voight's transformations are not quite the same as those of Lorentz. The latter are consistent with the Principle of Relativity.

Answer 3

Those Wikipedia and Wikiversity pages (History of Lorentz transformations and History of Topics in Special Relativity) have lots of information on the mathematical history of the Lorentz transformations preserving:

$$\pm\left[x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}\right]$$

Starting as early as 1800 by Gauss and many others, quadratic forms equivalent to the spacetime interval and their transformations were studied in the theory of indefinite quadratic forms, elliptic functions, hyperbolic geometry, long before Lorentz, Einstein or Minkowski used them in physics.

Most general Lorentz transformations Lagrange (1773), Gauss (1798–1818), Jacobi (1827, 1833/34), Lebesgue (1837), Bour (1856), Somov (1863), Killing (1878–1893), Poincaré (1881), Cox (1881–1883), Hill (1882), Picard (1882-1884), Callandreau (1885), Lie (1885-1890), Gérard (1892), Hausdorff (1899), Woods (1901-05), Liebmann (1904–05)

LT via imaginary orthogonal transformation Euler (1771), Wessel (1799), Cauchy (1829), Lie (1871), Minkowski (1907–1908), Sommerfeld (1909)

LT via hyperbolic functions Riccati (1757), Lambert (1768–1770), Taurinus (1826), Cayley (1859-84), Beltrami (1868), Klein (1871), Laisant (1874), Escherich (1874), Glaisher (1878), Günther (1880/81), Schur (1885/86, 1900/02), Lindemann (1890–91), Gérard (1892), Killing (1893,97), Woods (1903), Whitehead (1897/98), Liebmann (1904–05), Herglotz (1909/10)

LT via velocity Euler (1735), Beltrami (1868), Schur (1885/86, 1900/02), Lipschitz (1885–86), Voigt (1887), Heaviside (1888), Thomson (1889), Searle (1896), Lorentz (1892, 1895), Larmor (1897, 1900), Lorentz (1899, 1904), Poincaré (1900, 1905), Einstein (1905), Minkowski (1907–1908), Sommerfeld (1909), Herglotz (1909/10), Varićak (1910), Ignatowski (1910), Noether (1910), Klein (1910), Conway (1911), Silberstein (1911), Ignatowski (1910/11), Herglotz (1911), Borel (1913–14), Gruner (1921)

LT via conformal, spherical wave, and Laguerre transformation Lie (1871), Klein & Pockels & Bôcher (1871-91), Laguerre (1882), Stephanos (1883), Darboux (1887), Scheffers (1899), Smith (1900), Bateman and Cunningham (1909–1910)

LT via Cayley–Hermite transformation
Euler (1771), Cayley (1846–1884), Hermite (1853, 1854), Bachmann (1869), Laguerre (1882), Darboux (1887), Smith (1900), Borel (1913–14)

LT via Cayley–Klein parameters, Möbius and spin transformations Lagrange (1773), Gauss (1800), Cayley (1854), Klein (1871–97), Selling (1873–74), Poincaré /1881-86), Bianchi (1888-93), Fricke (1891–97), Woods (1895), Herglotz (1909/10)

LT via quaternions and hyperbolic numbers Euler (1771), Hamilton (1844/45), Cayley (1845), Cockle (1848), Cox (1882), Stephanos (1883), Buchheim (1884–85), Lipschitz (1885/86), Vahlen (1901/02), Noether (1910), Klein (1910), Conway (1911), Silberstein (1911)

LT via trigonometric functions Bianchi (1886–1893), Darboux (1881/94), Scheffers (1899), Eisenhart (1905), Varićak (1910), Gruner (1921)

LT via squeeze mappings Laisant (1874), Lie (1879-84), Günther (1880/81), Laguerre (1882), Darboux (1883–1891), Lipschitz (1885/86), Bianchi (1886–1893), Lindemann (1890/91), Smith (1900), Eisenhart (1905)