Foundations · 1888

Heaviside’s Point Charge Exploration

Length contraction wasn’t invented in 1905 — it fell out of Maxwell’s equations seventeen years earlier. In 1888, Oliver Heaviside solved for the field of a uniformly-moving point charge and found that its equipotentials are squashed into oblate ellipsoids along the direction of motion. FitzGerald (1889) and Lorentz (1892) said: if matter is held together by EM forces, and the EM forces deform this way, the matter must contract by the same factor. Einstein (1905) re-framed the contraction as kinematic — but the geometry was already there.

Charge / rod velocity v / c β = 0.60 γ = 1.250

γ 1.250 E⊥/E₀ 1.250 E∥/E₀ 0.640

① The Heaviside field

Concentric ellipses are surfaces of constant scalar potential. They squash horizontally by 1/γ as β increases. The orange rays show field strength at fixed distance — bunched perpendicular to motion, depleted along it.

The math (one line)

The scalar potential of a uniformly-moving charge is

ϕ (r, θ) = \frac{q}{4 π ϵ _{0} r 1 - β ^{2} sin ^{2} θ}

so equipotentials satisfy $r 1 - β^{2} sin^{2} θ = const$ — an oblate spheroid with x-semi axis $a / γ$ and y-semi $a$ .

The electric field magnitude follows:

E (r, θ) = \frac{q ( 1 - β ^{2} )}{4 π ϵ _{0} r ^{2} ( 1 - β ^{2} sin ^{2} θ ) ^{3/2}}

Two limits: $E_{∥} = E_{0} / γ^{2}$ (forward / backward), $E_{⊥} = γ E_{0}$ (sideways). The field is weaker along motion and stronger perpendicular to it.

② Bonds shrink with the field

Top: dipole at rest with bond length

L_{0}

. Bottom: same dipole moving at β — the EM forces holding it together have deformed, and the new equilibrium spacing along motion is

L_{0} / γ

. Drag β to compare.

Why the rod has to contract

The attractive force between the two charges along the direction of motion scales like $E_{∥} \propto 1/ γ^{2}$ . For the bond to remain in mechanical equilibrium, the charges must move closer along motion by the same factor that restores the force balance. The bookkeeping (and a full calculation including all field components and self-stress) yields exactly:

L = L_{0} / γ

This is the same formula that comes out of the kinematic derivation on the length contraction page — but here it's a consequence of Maxwell + force balance, not of postulates about spacetime.

Dynamic vs. kinematic — what changed in 1905

The Heaviside / FitzGerald / Lorentz reading is dynamic: a moving rod is physically stressed by its own deformed fields and squeezed shorter. Einstein's reading is kinematic: there is no stress, no "real" squeezing — different inertial frames simply slice spacetime into space + time differently, and what one frame calls "the rod at one instant" is geometrically a shorter slice for a moving frame.

Both predict $L = L_{0} / γ$ . The dynamic view is intuitive but privileges one frame as "the truly stationary one" — it doesn't survive the principle of relativity. The kinematic view is more austere but treats all inertial frames symmetrically, which is why it won.

Useful framing: EM was already secretly relativistic. Maxwell's equations are Lorentz-invariant; Heaviside's ellipsoid is the geometric fingerprint of that fact, visible fifteen years before anyone wrote down the transformation that explained it.

Bonus track · separate idea

③ When the charge accelerates: radiation

The Heaviside ellipsoid is for uniform motion. If the charge accelerates, something new happens: a kink in the field lines propagates outward at exactly $c$ — this is electromagnetic radiation. The cause-and-effect speed limit baked into Maxwell's equations is the same $c$ that ends up in $γ$ .

t = 0

the charge gets a brief kick (β = 0.6). Inside the inner ring (

r < c (t - Δ t)

) the field looks like the new uniformly-moving field, radial from the current position. Outside the outer ring (

r > c t

) it still looks like the field of the original at-rest charge. Between the two rings: the radiation pulse — the orange kinks — moving outward at

c

t = 0.00

This is the classical Purcell construction. Take-away: Maxwell already had a finite signal speed wired into it — special relativity didn't invent that, it just made the consequences geometric.

Next stops: see the kinematic derivation on the length contraction page, or watch how the constancy of $c$ forces Galilean addition to break on Galilean → Lorentz.