The breeding ground for the Special Relativity Theory: let’s make the quabla operator invariant again

In 1632, in his “Dialogo sopra i due massimi sistemi del mondo”, Galileo proposed the Galilean relativity principle, also known as the Galilean invariance, which states that the fundamental laws of physics are invariant in all inertial frames of reference (IFR).

Later, Newton would formalize this principle through the ideas of classical mechanics, which embodied the concepts of absolute, independent time and relative space for inertial observers, where time is universal and spatial coordinates change in accordance to the Galilean addition of velocities.

Although Newton rejected the Cartesian mechanical Aether (more like a plenum in Aristotelian terms), he did believe in absolute space. In Book 1 of the Principia, he states:

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable…

In group theory terms, the Galilean invariance is represented by a Lie Group called Galilean group SGal(3), in which four types of transformations (symmetries) live and combine through composition: spatial translations, rotations, time translations and Galilean Boost. These transformations and their compositions preserve isochronicity (invariance of time intervales), isotropy (invariance under rotation), homogeneity and -for isolated systems- energy conservation.

Let us think of ℝ⁴, with the Euclidean metric as the space in which Newtonian mechanics comes into play along with SGal(3). The Galilean transformations map (x y z t)ᵀ into (x’ y’ z’ t’)ᵀ and viceversa. Let (x y z t)ᵀ be the position vector of an arbitrary event in the IFR of the observer O: S and let (x’ y’ z’ t’)ᵀ be the position vector of the same event but through the perspective of observer O’ in S’. Now suppose that S’ is moving at a constant velocity v, in the x-axis direction from, starting both at t=t’=0 (time in which the origins of both IFR coincide), then:

To go from S to S’, we just need to apply the inverse.

In fact these transformations satisfy the Galilean principle of relativity when it comes to the invariance of Newton’s laws of motion. What about light?

In 1678, in his “Traité de la lumière”, Huygens proposed the undulatory theory of light by modeling light as an object which propagates as spherical wavefronts through a continuous medium: the Luminiferous Aether. This hypothesis will compete with Newton’s corpuscular theory of light (Opticks, 1717).

During the dawn of the electromagnetic theory, in the 19th C., both the undulatory theory of light and the Cartesian Aether (with properties necessary to be the medium of propagation of light waves) would reappear.

In 1801, Thomas Young performed the double-slit experiment, showing interference fringes (a wave pattern) produced by light, thus confirming the undulatory theory of light, perfected a couple of decades later by Fresnel, who incorporated transverse waves to explain polarization (after his diffraction experiments). With this, it was empirically (experimentally) proven that light, indeed, behaves like waves: it diffracts, it reflects and shows interference patterns.

On July 21, 1820, the results of Ørsted observations were published. He noticed that, in presence of a wire in which electric current flows, a magnetic needle deflects, i.e.: an electric current induces a magnetic field. A decade later, Faraday proved the converse: a magnetic field induces an electric current. In addition, Faraday gave birth to the theory of fields by proposing the concept of lines of force, keeping the need for a continuous action between two particles of matter.

Finally, in 1860s, the 19th C. Euclid of electromagnetism: Maxwell, unified the works of Gauss, Ørsted and Faraday, deriving the electromagnetic theory through which the constancy of the speed of light (an electromagnetic wave) was predicted. According to Maxwell’s equations, electromagnetic waves travel at a constant speed of:

Where μ₀ is the permeability of free space and ε₀ is the permittivity of free space (both constants).

Now, since the lines of force of electricity and magnetism are orthogonal to the propagation of the wave, light waves shall be transversal waves, which, in accordance with the undulatory theory, a continuous medium of propagation was needed: the Luminiferous Aether, again.

The undulatory theory of light requires us to admit this kind of elasticity in the luminiferous medium, in order to account for the transverse vibrations. We need not then be surprised if the magneto-electric medium possesses the same property. (Maxwell, 1861)

Now, if light behaves like a wave, then it certainly satisfies the wave equation:

Where ∇² is the Laplacian, ψ is the wave function and c is the speed of propagation of the wave -in this case, it is the speed of light-.

As a law of nature, according to the principle of Galilean relativity, the wave operator (D’Alembertian or quabla operator):

Should be invariant under any Galilean transformation. Let us take the Galilean boost in order to transform the wave operator, noticing that both x and t are both functions of x’, t’.

By the chain rule:

Since x’=x-vt, v constant:

Thus:

Now, we need the second partial derivatives, but first observe that ∇²=∇ ’², so we just have to find the second partial derivative with respect to time:

Since:

then:

And, after applying the Galilean boost, we have obtained:

which means that the quabla operator is not invariant under the Galilean boost.

In 1887, Michelson and Morley made use of the interference of light to assess the state of motion of the all-pervading Aether with respect to Earth’s rotation. The Aether, as an especulative rigid frame, was supposed to generate a “Aether wind” due to the motion of bodies through it. They obtained a negative result by finding no evidence of Aether as from the undeniable (now tested) constancy of the speed of light. The fate of the Luminiferous Aether was sealed.

In 1905, Einstein published his “On the Electrodynamics of moving bodies”, taking the Aether out from the Zeitgeist while killing the Absolute time of Newton. There, he proposed the two principles that govern Special Relativity Theory:

1. The principle of relativity: the laws of physics are the same (invariant) in all inertial reference frames.
2. The constancy of the speed of light: the speed of light in vacuum is constant for all observers, regardless of the motion of the source or the observer.

Now, it is relevant to mention that many great minds contributed to this theory, including Poincaré, Minkowski, Lorentz, FitzGerald, among others.

The time came to make quabla invariant. Grosso modo, in a Minkowski space, the Lorentzian group of transformations (subgroup of a Poincaré group) leaves the D’Alambertian invariant (amongst many other things).

At the end, the electromagnetic theory was right. Why? Because, instead of being a derivation of a set of speculations, it emerged from directly asking to nature (as my relativity professor, Francisco Nettel, in the spirit of Ilya Prigogine, said).

References

Bergmann, P. G. (1976). Introduction to the theory of relativity (Rev. ed.). Dover Publications.

De Andrade, E. M. P., Faber, J., & Rosa, L. P. (2013). A spontaneous physics philosophy on the concept of ether throughout the history of science: Birth, death and revival.Foundations of Science, 18(3), 559 – 577.

Ginoux, J.-M. (2024). Poincaré, Einstein and the discovery of special relativity: An end to the controversy. Springer.

Michelson, A. A., & Morley, E. W. (1887). On the relative motion of the earth and the luminiferous ether. American Journal of Science, 34, 333 – 345.https://doi.org/10.2475/ajs.s3-34.203.333.

Rahaman, F. (2022). The special theory of relativity: A mathematical approach (2nd ed.). Springer.