Special relativity · Electrodynamics §8
A packet of light does not transform
like a rigid material body.
How do the energy and volume of a bounded light complex transform between frames?
SR-10 · The finite light complex
The finite light complex
A bounded packet of light transforms in energy by q = γ(1 − β cos φ) and in volume by 1/q. Neither factor is the material contraction 1/γ.
Einstein's printed volume of the complex is V′/V = 1/q, not 1/γ. Energy and frequency follow q. For receding light along the axis at β = 0.6, q = 1/2: energy halves and volume doubles. A rigid rod of the same rest volume would contract to 0.8.
The discriminating case is a ray transverse in the unprimed frame (cos φ = 0): the light factor is γ and the material factor is 1/γ, so they differ by γ². A ray transverse in the moving frame (cos φ = β) gives q = 1/γ, which equals the material factor and cannot catch the planted negative.
Section 8 notes that the energy and the frequency of a light complex vary with the observer's motion according to the same law. The later photon identification is not a 1905 premise.
A bounded pulse of light does not transform like a solid rod. Energy follows q = γ(1 − β cos φ). Volume follows 1/q. The planted negative “it contracts like a rod” is tested at φ = 90° in K, where q = γ and 1/γ differ by γ², and along the axis. At cos φ = β the energy factor equals 1/γ and cannot catch that mistake.
If the packet were treated as a rigid rod, both energy and volume would scale by 1/γ = 0.8000 (E′_wrong = 0.800 J). The true energy factor is q = 0.5000 and the true volume factor is 1/q = 2.0000. At φ = 90° in K those differ from 1/γ by γ². At cos φ = β they do not.
Accepted snapshot
| Energy in K (E) | 1 J |
|---|---|
| Physical Energy in k (E′) | 0.5 J |
| Volume in K (V) | 1 m³ |
| Physical Volume in k (V′) | 2 m³ |
| Energy ratio E′/E (q) | 0.5 |
| Volume ratio V′/V (1/q) | 2 |
| Material volume factor 1/γ | 0.8 |
| Lorentz factor γ | 1.25 |
| Transformed angle in k (φ′) | 0 ° |
| Countermodel E′ (1/γ) | 0.8 J (wrong model) |
| Countermodel V′ (1/γ) | 0.8 m³ (wrong model) |
Remarkably, the energy factor E′/E equals the Doppler frequency ratio ν′/ν = q across all angles and speeds. This exact proportionality between light energy and wave frequency holds invariantly for any bounded light packet under Lorentz transformations.
Not modeled: media and dispersion; quantum photon structure; finite pulse dispersion in dielectric; gravitational redshift; boundary diffraction at packet edges.
Worked case (readable without JavaScript)
Consider a spherical light complex of initial volume V = 1.0 m³ and total energy E = 1.0 J propagating along the x-axis (φ = 0°) in the stationary system K. An observer moves along the x-axis at speed v = 0.6c (β = 0.6, γ = 1.25).
The Doppler factor is q = γ(1 − β cos φ) = 1.25(1 − 0.6) = 0.5. Because the moving observer's simultaneous spatial plane cuts across a moving wave front, the volume of the complex in k transforms as:
Meanwhile, the energy density transforms with the square of the amplitude ratio, $u'/u = q^2 = 0.25$. The total energy in the moving frame is:
Notice the contrast with a rigid material body: a solid rod of volume V would undergo ordinary Lorentz contraction to V′rod = V/γ = 0.8 m³. Treating the light complex like that rod would give E′wrong = E/γ = 0.80 J, not 0.50 J. The case that isolates this mistake is a ray transverse in K (φ = 90°): q = γ = 1.25 while 1/γ = 0.80, so they differ by γ². A ray transverse in k (cos φ = β) gives q = 1/γ and cannot catch the mistake.
Einstein observed: “It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.”