# Maxwell's equations without special relativity

If Maxwell's equations were around long before special relativity, and yet they don't make sense without special relativity (like if you think of a moving charge next to an electrically neutral wire carrying current, then in the lab frame it experiences magnetic force and in its rest frame, there's no force - unless you take length contraction into account), how were they accepted at the time?

• Because they do make sense without special relativity. – Kyle Kanos Feb 25 at 0:04
• This problem was resolved by Lorentz in his Theory Of Electrons that contained the Lorentz Transformation. He received a Nobel Prize for it in 1902. – safesphere Feb 25 at 9:24

At the time of Maxwell, it was believed that the electric and magnetic fields were separate fields, though related through the latter two of Maxwell's equations (Ampere's laws & the induction eq'n). So there really was no conflict that you somehow perceive there is/was/should be, it all made verifiable sense (cf. the Wikipedia entry on the History of Maxwell's equations, for instance).

What relativity did was enable us to view the electric and magnetic fields as a single entity, the electromagnetic tensor, $$F^{\mu\nu}=\left(\begin{array}{cccc} 0&-E_x&-E_y&-E_z\\ E_x&0&-B_z&B_y\\ E_u&B_z&0&-B_x\\ E_z&-B_y&B_x&0\\ \end{array} \right)$$ The connection between the two fields is discussed in many other places on this site:

All of which would be good reading for you.

• But, ultimately, they don't really make sense without special relativity because of the fact that they spontaneously determine the speed of light in all the frames that they are true in, right? And people realized this and that led to the whole Lorentz-Poincare-Michealson-Morley thingy. – Dvij Mankad Feb 26 at 1:37
• @DvijMankad the individual laws were experimentally determined, so to say they don't make sense is ridiculous. SR added more knowledge about the E&M fields, but it doesn't invalidate any knowledge about them separately. – Kyle Kanos Feb 26 at 1:46

At the time it was a mystery how they could work. One leading theory was the "luminiferous ether". This was a medium which pervaded all things and through which maxwell waves propagated. This ether provided (in theory) the rest frame for the equations.

As the theory was tested by, e.g. the Michaelson Morley, experiment and by the observation of photons it was modified. So, for example, the idea was advanced that the ether could be dragged by bodies (akin to general relativistic frame dragging) and that it was composed of filaments which kept packets of wave energy intact.

The history of Maxwell's equations is a long and hardy story. 1865, James Clerk Maxwell published "A Dynamical Theory of the Electromagnetic Field", unifying all then available results on electricity and magnetism. In this article - which is worthwhile reading - he also postulated the existence of electromagnetic waves, as well as their propagation speed. His theory is presented in the form of 20 equations in 20 unknowns; vector analysis was not yet known at that time.

What we know today as the four "Maxwell's equations" is the work of Oliver Heaviside, who published them in "Electromagnetic Theory", 1893 onwards, 4 volumes. These four equations can be further condensed in 4-space into a single equation, the "Fundamental equation of Electrodyanmics":

((1/c^2)∙∂^2/(∂t^2) - ∂^2/(∂x1^2) - ∂^2/(∂x2^2) - ∂^2/(∂x3^2) )A = μ0∙J

wherein A=(φ/c, (A1, A2, A3)), and J=(ρc, (J1, J2, J3))

(This is the equivalent of Poisson's equation which governs flow processes in 3-space)

James Clerk Maxewll based his theory on the hypothesis of the luminiferous ether, which made its way from there into telecommunication jargon ("sending through the ether" was used synonymous to "wireless"). Heinrich Hertz and Oliver Lodge (the true inventor of radiocommunication; see "The work of Hertz and some of his successors", lectures given in 1894) were firmly adhering to the ether theory.

Michelson and Moorley found no evidence for the existence of a luminiferous ether; however their experiment was interpreted in a 3-dimensional frame; in the 4-dimensional space-time of Special Relativity, the Michelson-Moorley-setup is an inertial system which cannot be used to prove or to disprove the existence of the luminiferous ether. But that our Universe is 4-dimensional was not known at that time.

In 4-dimensional space, physics becomes much simpler, indeed:

Albert Einstein's formulas E=m∙c^2 and the Relativity Invariant E^2/c^2 - p⃗^2= m0^2∙c^2 can be combined (measuring distance in light-seconds; c=1) to E^2 = m^2 = m0^2 + p⃗^2 = m0^2 + p1^2 + p2^2 + p3^2. That means that the rest mass is simply the fourth component of the momentum vector, and the energy is the total momentum.

According to Leonhard Euler's 4-square-identity, any sum of 4 squares can be written as a product of two sums of four squares each. Hence (M0^2 + P1^2 + P2^2 + P3^2) = (r0^2 + r1^2 + r2^2 + r3^2)∙(m0^2 + p1^2 + p2^2 + p3^2).

Herein are

M0 = (r0∙m0 - r1∙p1 - r2∙p2 - r3∙p3)

P1 =(r0∙p1 + r1∙m0 + r2∙p3 - r3∙p2)

P2 =(r0∙p2 - r1∙p3 + r2∙m0 + r3∙p1)

P3 =(r0∙p3 + r1∙p2 - r2∙p1 + r3∙m0)

which represents the quaternion multiplication rule (proof by simple algebraic evaluation)

The squares sums may be interpreted as scalar (inner) products of a vector with itself, hence as a the square of a metric length.

Lets take the vectors as follows:

A⃗ = (m0, p1, p2, p3), a phyical system

R⃗ = (r0, r1, r2, r3), an interaction operator

with (r0^2 + r1^2 + r2^2 + r3^2) = 1; (for energy conservation)

P⃗ = (M0, P1, P2, P3), the resulting physical system

then we can write: P⃗ = R ⃗ * A ⃗ , or also P⃗ = P1⃗ + P2⃗ = R⃗ * (A1⃗ + A2⃗) = R⃗ * A⃗ This is a formula describing the energy conservation in a physical interaction. Energy can only be transmitted, shared, or cumulated between the starting system A⃗ or parts of it (A1⃗ + A2⃗) and the resulting system P⃗ or parts of it P1⃗ + P2⃗, but never be created nor annihilated. The multiplication * sign herein designates the quaternion multiplication rule as given above.

By the way, Leonhard Euler's 4-square-identity in some way prefigures Maxwell's equations, but it was found in a completely different, unrelated context, more than 100 years before. See: https://www.e-periodica.ch/cntmng?pid=fng-001:2017:106::158