Who Proved of Kirchhoff's Law of Radiation?

Kirchhoff's Radiation Law stating that the emissivity of a radiating body is equal to its absorptivity presented in 1859, initiated an intense study of blackbody radiation by Rayleigh, Jeans, Wien and others leading into Planck's proof of his radiation law opening to the quantum mechanics of modern physics.

Kirchhoff's Law can be seen as a triviality stating that emission equals absorption as an expression of energy balance. But Kirchhoff's Law concerns emissivity and absorptivity as emission and absorption per unit time, and in this setting it is not at all trivial. The question is why a body capable of absorbing radiation and emitting radiation, must absorb and emit at the same rate? Is it because emission and absorption are simply the reverse of each other with emission simply absorption backwards in time?

No, it is not so trivial, because emission and absorption are different physical processes both with an arrow of time which cannot be reversed. Emission and absorption are not the reverse of each other.

In a previous post I sketched a proof of Kirchhoff's Law based on a wave model with radiative damping analyzed in more detail in Mathematical Physics of Blackbody Radiation.

Let us trace the history of the proof of Kirchhoff's Law with Experimenting theory: The proofs of Kirchhoff's Radiation Law before and after Planck by A. Schirrmacher presenting the following story:

1. Kirchhoff (1859): Thought experiments with mirrors, basic thermodynamics.
2. Planck (1906); Heat rays, basic thermodynamics.
3. Hilbert (1912-14): Integral equation, axiomatic method.

The debate about the proof was intense and no winner was elected. Further studies were made by Dirac and Heitler based on quantum mechanics.

In the modern textbook Radiative Heat Transfer by M. Modest, Kirchhoff's law is presented as a triviality:

• It is easy to show that a black surface also emits a maximum amount of radiative energy, i.e., more than any other body at the same temperature. To show this, we use one of the many variations of Kirchhoff's law: Consider two identical black-walled enclosures, thermally insulated on the outside, with each containing a small object—one black and the other one not. After a long time, in accordance with the Second Law of Thermodynamics, both entire enclosures and the objects within them will be at a single uniform temperature.
• This characteristic implies that every part of the surface (of enclosure as well as objects) emits precisely as much energy as it absorbs. Both objects in the different enclosures receive exactly the same amount of radiative energy. But since the black object absorbs more energy (i.e., the maximum possible), it must also emit more energy than the nonblack object (i.e., also the maximum possible).
• By the same reasoning it is easy to show that a black surface is a perfect absorber and emitter at every wavelength.

We see here an example of a common feature of modern science: A question which once caused a heated debate between the giants of science of the time without ever being settled including

• interpretation of quantum mechanics
• d'Alembert's paradox
• Loschmidt's paradox
• 2nd Law of Thermodynamics

eventually is being put aside as being trivial or a no-issue of little interest.

But Kirchhoff's law is not a triviality and it is a fundamental part of the theory of radiation, and therefore a proof is of considerable interest.

PS To the students at MIT Kirchhoff's Law is presented as trivial energy balance.