Special relativity · Electrodynamics §8

Moving mirror reflection, Doppler shift,
and radiation pressure energy balance.

How do the frequency, angle, amplitude, and radiation pressure of light transform when reflected by a moving mirror, and how does energy balance between the light and the mirror's mechanical work?

Read §8 of the 1905 relativity paper →

SR-11 · Moving mirror reflection and radiation pressure

Moving mirror reflection and radiation pressure

Ideal moving mirror, host calculation

JavaScript is off. This is a complete worked example calculated when the site was built. Frequency ratios, radiation pressure, and energy ledgers remain available; changing the settings requires JavaScript.

Light reflected from a moving mirror undergoes a double Doppler shift and changes its reflection angle according to relativistic wave kinematics, while exerting a radiation pressure that balances energy conservation between the electromagnetic field and the mirror's mechanical work.

A receding mirror red-shifts the reflected wave and reduces its energy density; an approaching mirror blue-shifts the wave and increases its energy. The energy difference between incident and reflected light precisely equals the mechanical work rate P·v·Am done on or by the mirror.

Set the mirror velocity ratio β = v/c and incident angle φ, or select a preset scenario. The reflection modifies frequency, ray direction, and amplitude, while radiation pressure does mechanical work that conserves energy across reference frames.

Accepted snapshot

Relativistic wave reflection quantities and energy conservation ledger across frames.
Frequency ratio ν′′′/ν	0.25
Reflection cosine cos(φ′′′)	-1
Reflection angle φ′′′	180 °
Amplitude ratio A′′′/A	0.25
Radiation pressure P	0.5 Pa
Radiation force F	0.5 N
Incident power P_inc	0.4 W
Reflected power P_refl	0.1 W
Work rate P·v·Am	0.3 W
Energy balance residual	0 W

At normal incidence with a receding mirror (β = 0.6), the incident power is 0.4 IA_m, the reflected power is 0.1 IA_m, and the mirror receives mechanical work rate 0.3 IA_m. Energy conservation holds exactly with zero residual.

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Not modeled: mirror mass and acceleration (infinite mass limit); finite mirror thickness and internal absorption; diffraction at mirror edges; quantum radiation pressure fluctuations; non-monochromatic wave packets.

Worked case (readable without JavaScript)

Consider monochromatic radiation of energy density u encountering a perfectly reflecting mirror of area A_m moving at velocity v = 0.6c (β = 0.6, γ = 1.25) along the surface normal (φ = 0°).

The reflected wave frequency undergoes a double Doppler transformation:

\frac{\nu'''}{\nu} = \frac{1 - 2\beta\cos\varphi + \beta^2}{1 - \beta^2} = \frac{1 - 2(0.6)(1) + 0.36}{1 - 0.36} = \frac{0.16}{0.64} = 0.25

The radiation pressure on the mirror is:

P = 2u\,\frac{(\cos\varphi - \beta)^2}{1 - \beta^2} = 2u\,\frac{(1 - 0.6)^2}{1 - 0.36} = 2u\,\frac{0.16}{0.64} = 0.5\,u

Energy balance per unit time in the laboratory system K:

P_{\text{incident}} = u\,c\,A_{\text{m}}(1 - \beta) = 0.4\,u\,c\,A_{\text{m}}

P_{\text{reflected}} = u\left(\frac{\nu'''}{\nu}\right)^2 c\,A_{\text{m}}(1 + \beta) = u(0.0625)c\,A_{\text{m}}(1.6) = 0.1\,u\,c\,A_{\text{m}}

\dot{W}_{\text{mechanical}} = P\cdot v\cdot A_{\text{m}} = (0.5\,u)(0.6\,c)(A_{\text{m}}) = 0.3\,u\,c\,A_{\text{m}}

P_{\text{incident}} - P_{\text{reflected}} - \dot{W}_{\text{mechanical}} = 0.4 - 0.1 - 0.3 = 0

The energy lost by the electromagnetic radiation upon reflection from a receding mirror is converted into mechanical work done on the mirror, preserving exact energy conservation.

Return to paper section 8 →