The authors Buchdahl, Callen, and Haase make no mention of the passage of radiation, thermal or coherent, across their adiabatic walls. Carathéodory explicitly discusses problems with respect to thermal radiation, which is incoherent, and he was probably unaware of the practical possibility of laser light, which is coherent. Carathéodory in 1909 says that he leaves such questions unanswered.
For the thermodynamic stream of thinking, the notion of empirical temperature is coordinately presupposed in the notion of heat transfer for the definition of an adiabatic wall.
For the mechanical stream of thinking, the exact way in which the adiabatic wall is defined is important.
In the presentation of Carathéodory, it is essential that the definition of the adiabatic wall should in no way depend upon the notions of heat or temperature. This is achieved by careful wording and reference to transfer of energy only as work. Buchdahl is careful in the same way. Nevertheless, Carathéodory explicitly postulates the existence of walls that are permeable only to heat, that is to say impermeable to work and to matter, but still permeable to energy in some unspecified way. One might be forgiven for inferring from this that heat is energy in transfer across walls permeable only to heat, and that such exist as undefined postulated primitives.
In the widely cited presentation of Callen, the notion of an adiabatic wall is introduced as a limit of a wall that is poorly conductive of heat. Although Callen does not here explicitly mention temperature, he considers the case of an experiment with melting ice, done on a summer's day, when, the reader may speculate, the temperature of the surrounds would be higher. Nevertheless, when it comes to a hard core definition, Callen does not use this introductory account. He eventually defines an adiabatic enclosure as does Carathéodory, that it passes energy only as work, and does not pass matter. Accordingly, he defines heat, therefore, as energy that is transferred across the boundary of a closed system other than by work.
As suggested for example by Carathéodory and used for example by Callen, the favoured instance of an adiabatic wall is that of a Dewar flask. A Dewar flask has rigid walls. Nevertheless, Carathéodory requires that his adiabatic walls shall be imagined to be flexible, and that the pressures on these flexible walls be adjusted and controlled externally so that the walls are not deformed, unless a process is undertaken in which work is transferred across the walls. The work considered by Carathéodory is pressure-volume work. Another text considers asbestos and fiberglass as good examples of materials that constitute a practicable adiabatic wall.
The mechanical stream of thinking thus regards the adiabatic enclosure's property of not allowing the transfer of heat across itself as a deduction from the Carathéodory axioms of thermodynamics.