Connections among the papers · Modern synthesis
One pulse, three questions.
What is its quantum energy? What does another observer measure? What changes when a body emits light?
One pulse · Three distinct questions
Follow the energy without confusing the claims
Static worked example
Accepted settings: ν = 500000000000000 Hz; E = 1 J per pulse; β = 0.6; source-frame angle = 0°. Changing only β changes the observer, not the emitted pulse.
1. Light quanta: an energy scale, not a proof of relativity
In modern notation the quantum energy is hν. Total pulse energy E and frequency ν are independent settings. E/(hν) is shown without rounding: an energy ratio is not an independently measured integer photon count.
| Quantity | Value | Unit |
|---|---|---|
| Frequency in the source frame | 500000000000000 | Hz |
| Pulse energy in the source frame | 1 | J |
| hν in the source frame | 3.313 × 10⁻¹⁹ | J |
| E/(hν) in the source frame | 3.0184 × 10¹⁸ | 1 |
Read the light-quanta argument, §6 · Compare the entropy coefficients
2. Relativity: energy and frequency change together
For the same pulse, the wave owners evaluate ν′/ν and E′/E. Both follow q = γ(1 − β cos θ), so the modern comparison E′/(hν′) equals E/(hν), apart from numerical rounding. This equality compares two computed consequences; it does not independently establish light quanta.
| Quantity | Value | Unit |
|---|---|---|
| Frequency transformation factor | 0.5 | 1 |
| Energy transformation factor | 0.5 | 1 |
| Frequency in the moving frame | 250000000000000 | Hz |
| Pulse energy in the moving frame | 0.5 | J |
| hν in the moving frame | 1.6565 × 10⁻¹⁹ | J |
| E′/(hν′) in the moving frame | 3.0184 × 10¹⁸ | 1 |
3. Mass–energy: first specify the system
A unidirectional light pulse has zero invariant mass. Its energy divided by c² is not its rest mass. To connect to the September argument, now add a distinct, equal pulse traveling in the opposite direction. Their total source-frame energy is 2E and their momenta cancel.
| Quantity | Value | Unit |
|---|---|---|
| Pulse energy divided by c² (not its rest mass) | 1.1127 × 10⁻¹⁷ | kg |
| Invariant mass of a unidirectional light pulse | 0 | kg |
| Opposite equal pulse: moving-frame energy | 2 | J |
| Balanced pair: source-frame energy | 2 | J |
| Balanced pair: moving-frame energy | 2.5 | J |
| Balanced pair: invariant mass | 2.2253 × 10⁻¹⁷ | kg |
| Body's mass decrease for balanced emission | 2.2253 × 10⁻¹⁷ | kg |
For a body initially at rest that emits this balanced pair without recoil, the mass decrease is 2E/c². The displayed system invariant mass is a later interpretation. The historical argument instead compares two energy ledgers and uses the low-speed kinetic-energy premise. It does not require quanta, and we have not assigned the unknown initial body energy E₀ = Mc².
Read the September paper · Follow the two ledgers · Inspect the low-speed coefficient
This is a modern comparison using the exact modern SI definitions of h and c, not a reconstruction of what was measured in 1905. A programmed consequence is not experimental confirmation.
Not modeled: finite spectral bandwidth, diffraction, media, gravitational shifts, detector response, and the recoil from unbalanced single-pulse emission.
Model limits and the code behind the numbers
The idealization is a unidirectional, monochromatic vacuum pulse between inertial frames. The control bounds are numerical admission limits, not claims of physical impossibility.
The frequency and energy factors come from the site's code for relativistic waves. Every number shown comes from the settings this laboratory last accepted. No FrankenSim result is claimed.
Doppler shift and light energy (relativity §§ 7–8) · The light thread across the papers · How the page gathers these results
Follow one flash of light through three papers. Its energy comes in parts whose size is set by its colour; an observer moving away sees it redder and weaker in exactly the same proportion; and a body that sends out such flashes, equally in opposite directions, becomes lighter by their energy divided by the square of the speed of light.
The light paper says that monochromatic radiation behaves, as far as its energy goes, as if it were made of quanta of size hν. The relativity paper gives in §7 the frequency a moving observer measures, and in §8 finds that the energy of a light complex changes with the observer's motion by the same law. At the defaults, a pulse of 1 J at 500 THz, seen by an observer receding at 0.6c along its direction, has half the frequency, 250 THz, and half the energy, 0.5 J, so the number E/hν = 3.02 × 1018 is the same for both observers, although neither paper assumes the other. A pulse sent the opposite way is seen with twice its energy, so a pair of 1 J pulses, 2 J in the frame of the body that emits them, carries 2.5 J for the moving observer, 1.25 times as much. The mass–energy paper uses exactly that difference to show that the body loses mass L/V2, here 2 J/c2 = 2.23 × 10−17 kg. A single pulse has no rest mass; the pair, taken as a system, has.
For an observer moving at β = v/c along the direction of a light pulse, the frequency is multiplied by (1 − β)/√(1 − β2) = √((1 − β)/(1 + β)). At β = 0.6 that is √(0.4/1.6) = √0.25 = 0.5, so 500 THz becomes 250 THz. Section 8 gives the pulse's energy the same factor, so 1 J becomes 0.5 J. One quantum at 500 THz has hν = 6.626 × 10−34 × 5 × 1014 = 3.313 × 10−19 J, so the pulse holds 1/(3.313 × 10−19) = 3.02 × 1018 such amounts; for the moving observer each quantum is 1.657 × 10−19 J and the pulse 0.5 J, which is again 3.02 × 1018. A pulse travelling the other way gets the factor √((1 + β)/(1 − β)) = 2, so 1 J becomes 2 J. Two pulses of 1 J each, sent in opposite directions, carry 2 J in the body's frame and 0.5 + 2 = 2.5 J for the moving observer. The difference is 0.5 J, and 2.5/2 = 1.25 = 1/√(1 − 0.36). That is the mass–energy paper's step: the body gives up 2 J in its own frame but 2.5 J as the moving observer counts it, and the extra 0.5 J can only come out of its energy of motion. For small speeds that extra is ½(L/c2)v2, the kinetic energy of a mass L/c2, so the body has lost that much mass: with L = 2 J and c = 2.998 × 108 m/s, 2/(8.988 × 1016) = 2.23 × 10−17 kg. One pulse alone, with E = pc, has zero rest mass; the pair's momenta cancel in the body's frame, so there all of its energy is rest energy.
Modern qualifications
The light paper never mentions relativity, and the mass–energy paper uses Maxwell's theory of radiation without light quanta. The link here is a later synthesis: because energy and frequency transform alike, the count of quanta is the same for every observer. Einstein wrote V for the speed of light and L for the emitted energy, and stated the result as the mass changing by L/V2, with 9 × 1020 in centimetre-gram-second units. The photon as a particle with energy hν and momentum hν/c came later, with Einstein's 1916 and 1917 papers on radiation and Compton's scattering experiment of 1923.
The light-quanta paper does not establish relativity. The September mass–energy argument does not require quanta. This laboratory connects their consequences without turning one paper’s conclusion into another’s hidden premise.