Wilhelm Killing — The Killing Form
Publishes foundational work on Lie algebras that, 138 years later, will help prove which universe we live in. Killing's 1888 papers laid the groundwork for what later became known as the Cartan-Killing form.
Lorentz — Transformation Equations
Derives the transformation equations that will eventually bear his name. Gets the mathematics exactly right, but attributes the results to a physical compression of matter moving through an invisible medium called the luminiferous ether.
Poincaré — The Group Structure
Shows that the Lorentz transformations form a group, a deep mathematical insight. In his 1905 work, the Lorentz-group structure becomes explicit. Einstein will build on this foundation the same year, but the algebraic depth of the group structure will take a century more to fully exploit.
Einstein — Special Relativity (Two Postulates)
Publishes "On the Electrodynamics of Moving Bodies." Two postulates, one revolution. Postulate I: the laws of physics are the same in all inertial frames. Postulate II: the speed of light in vacuum is the same in all inertial frames. But why two?
Ignatowski — Relativity from One Postulate
First to show that the Lorentz transformations can be derived without explicitly assuming the constancy of c. Derives the general transformation group from symmetry alone, parameterised by a constant κ. Leaves κ undetermined; neither its sign nor its value can be fixed from the algebra alone. A crucial step forward with one piece missing.
Pauli — "Nothing Can Be Said About the Sign of κ"
In his landmark encyclopaedia article on relativity, Pauli correctly identifies that the sign of κ cannot be determined from symmetry arguments alone, or so he believes. Accepts this as a fundamental limitation of the one-postulate approach. The gap becomes the received wisdom of the field.
Lévy-Leblond — "One More Derivation of the Lorentz Transformation"
Rederives the Lorentz transformations from one postulate in a rigorous and pedagogically clear form. Confirms the existence of the three-case family (κ < 0, κ = 0, κ > 0). Still cannot fix the sign of κ from purely algebraic arguments. The gap identified by Pauli stands. Am. J. Phys. 44, 271 (1976).
Silagadze — "Relativity Without Tears"
A comprehensive pedagogical paper revisiting the one-postulate derivation and making the argument accessible to a wider audience. Reviews the full family of transformation groups. Still κ is left undetermined; the gap persists. The paper's title is prescient: tears are still needed, until the Killing form is applied.
Drory — "The Necessity of the Second Postulate"
Argues, incorrectly as the 2026 paper shows, that the second postulate is genuinely necessary and cannot be derived from the first. Takes Pauli's limitation as definitive. The argument is rigorous but misses the one tool that can resolve the sign: the Killing form of the resulting Lie algebra.
Mostaque — One Postulate
The Killing form, rooted in Killing's 1888 work and central to Cartan's later theory, settles the sign. It distinguishes the three branches by the behaviour of the boost sector: κ = 0 is degenerate on boosts, κ < 0 makes boosts compact like rotations, and κ > 0 alone produces the compact/non-compact split associated with Lorentzian causality. κ > 0 is the only branch that yields a physical universe. Pauli's gap is closed. The second postulate was always redundant.
Nothing can naturally be said about the sign, magnitude and physical meaning of [κ].— Wolfgang Pauli, Theory of Relativity (Encyklopädie der mathematischen Wissenschaften, 1921)
Pauli identified precisely the gap this paper closes. The Killing form distinguishes the three branches by the behaviour of the boost sector: κ = 0 is degenerate on boosts, κ < 0 makes boosts compact like rotations, and κ > 0 alone produces the compact/non-compact split associated with Lorentzian causality.
From 1888 to 2026: 138 years from Killing's discovery to its application.
The tool was always there. The question was always there. Now we have the answer.