Opposite pulses and two energy ledgers

// evaluatePulseEnergies
const g = 1 / Math.sqrt(1 - frameSpeed * frameSpeed);
const p1 = (emittedEnergyRestFrame / 2) * g * (1 - frameSpeed * Math.cos(phi));
const p2 = (emittedEnergyRestFrame / 2) * g * (1 + frameSpeed * Math.cos(phi));
const pulseSumMoving = g * emittedEnergyRestFrame; // invariant under phi!

// evaluateSubtraction
const subtractionDifference = emittedEnergyRestFrame * (g - 1);
const kineticEnergyDifference = premise === "unchanged" ? subtractionDifference : null;

Two equal flashes of light leave a body at rest in opposite directions. Seen by someone moving past, the flashes carry more energy than they do for the body, and the extra can only come from the body's energy of motion, which falls as if the body had lost mass.

The mass–energy paper imports one result from §8 of the relativity paper: light of energy l, seen from a frame moving at v along x, has energy l* = l(1 − (v/V) cos φ)/√(1 − (v/V)²), where φ is the angle between the light's direction and the x-axis. A body at rest in (x, y, z) sends out light of energy L/2 at the angle φ and an equal amount the opposite way, and stays at rest. The energy principle must hold in both frames: in the rest frame E₀ − E₁ = L, and in the moving one H₀ − H₁ = γL, writing γ for 1/√(1 − (v/V)²), because the two direction factors, 1 − (v/V) cos φ and 1 + (v/V) cos φ, add to 2. The body's energies E and H stay symbolic, and the lab never gives them values. Subtracting the two balances removes them: (H₀ − E₀) − (H₁ − E₁) = L{1/√(1 − (v/V)²) − 1}. Each H − E is the body's kinetic energy K in the moving frame plus a constant C, and C does not change during the emission, so K₀ − K₁ = L{1/√(1 − (v/V)²) − 1}. At the lab's default, L = 1 J and v = 0.6c, the moving observer counts 1.25 J of light, and the body's kinetic energy falls by 0.25 J whatever the angle. Drop the premise that C is unchanged and the difference is underdetermined, and the lab says so. To lowest order the drop is (L/V²)(v²/2), the kinetic energy of a mass L/V², and the paper concludes that a body giving off energy L as radiation loses mass L/V².

Start with the factor 1/√(1 − (v/V)²). At v = 0.6c it is 1/√0.64 = 1.25. Take L = 1 J, so each flash carries 0.5 J in the body's frame. Along the x-axis, φ = 0: the forward flash, going the same way as the moving observer, has 0.5 × 1.25 × (1 − 0.6) = 0.25 J in the moving frame, and the backward flash has 0.5 × 1.25 × (1 + 0.6) = 1.0 J, together 1.25 J. At φ = 60° the factors are 1 − 0.3 and 1 + 0.3, giving 0.4375 J and 0.8125 J, again 1.25 J: the angle cancels because the two factors always sum to 2, the flashes going in opposite directions. Now write both energy balances. In the body's frame the body loses exactly what the light carries, E₀ − E₁ = 1 J. In the moving frame it loses what the light carries there, H₀ − H₁ = 1.25 J. Subtract the first from the second: (H₀ − E₀) − (H₁ − E₁) = 1.25 − 1 = 0.25 J. Nothing about the body's internal energy was needed, because E₀, E₁, H₀ and H₁ appear only in differences. What is H − E? It is the same body's energy seen from two frames, one in which it rests and one in which it moves, so it is its kinetic energy K in the moving frame plus a constant C that depends only on where each frame puts its zero of energy. With C unchanged, the constants cancel, and K₀ − K₁ = L(γ − 1) = 0.25 J. If C could change during the emission, the two constants would not cancel, and the subtraction would leave K₀ − K₁ tied to an unknown change in C; that is why the lab reports the result as underdetermined when the premise is dropped. The body moves at the same speed before and after, since it stays at rest in its own frame, so a smaller kinetic energy at the same speed means a smaller mass. For slow motion, 1/√(1 − (v/V)²) − 1 ≈ (v/V)²/2, so K₀ − K₁ ≈ (L/V²)(v²/2), which is ½mv² with m = L/V². At 0.6c the approximation gives 0.18 J against the exact 0.25 J. For L = 1 J the mass lost is 1/(2.998 × 10⁸)² = 1.11 × 10⁻¹⁷ kg.

The paper, received on 27 September 1905, writes the factor in full as 1/√(1 − v²/V²), V for the speed of light, L for the emitted energy and l* for the transformed light energy. It cites §8 of the June paper for that transformation and notes in a footnote that the constancy of the speed of light used there is contained in Maxwell's equations. It states the result as L/V², and in cgs units as a mass change of L/9 · 10²⁰ grams for L ergs, and suggests that radium salts might test it. The mass statement rests on the lowest-order term in v/V and on the source premise that C is unchanged. Planck gave a more general treatment in 1907, and Ives argued in 1952 that the 1905 argument was circular, a reading others have disputed. The paper itself says only that the mass of a body is a measure of its energy content.