False-SB Violates the 2nd Law

To understand the difference between the two versions of Stefan-Boltzmann's radiation law (SB) under discussion, True-SB and False-SB, let us inspect the proof of SB from Planck's radiation law in its normalized form presented in Computational Blackbody Radiation:

• $R(f ,T) =\gamma Tf^2$ for $f\le T$
• $R(f,T) = 0$ for $f > T$

where $R(f ,T)$ is the radiance of frequency $f$ from a blackbody of temperature $T$, $\sigma$ is a constant and a simplified high-frequency cut-off is used (as compared to Planck's exponential cut-off).

The total radiative transfer $R_{True}$ to a blackbody 1 of temperature $T_1$ from a blackbody 2 of temperature $T_2>T_1$ is given by integration over frequencies as follows:

• $R_{True} =\int_{T_1}^{T_2}\gamma T_2 f^2 df + \int_0^{T_1}\gamma (T_2 - T_1) f^2df \equiv I_1 + I_2$,

where the first integral $I_1$ is the heating effect from 2 above the cut-off of 1 and the second integral the net heating from 2 below cut-off.

We see that $R_{True}$ expressing True-SB is the sum of two integrals with positive integrands, that is, $R_{True}$ is the sum of many small positive contributions all with transfer of heat from 2 to 1.

We shall now see that False-SB arises by rewriting $I_1$ as follows:

• $I_1 = \int_0^{T_2}\gamma T_2f^2df - \int_0^{T_1}\gamma T_2f^2df =\sigma T_2^4 - \sigma T_2T_1^3$

where $\sigma =\frac{\gamma}{3}$, which since $I_2 = \sigma (T_2-T_1)T_1^3$ gives False-SB on the form

• $R_{False} = \sigma T_2^4 - \sigma T_1^4$

expressing the transfer of energy from 2 to 1 as the difference of two gross flows with different signs. We see that False-SB arises by rewriting an integral with positive integrand as the difference of two integrals of different signs as follows:

• $\int_{T_1}^{T_2} f^2 df = \int_0^{T_2}f^2df - \int_0^{T_1}f^2df$,

where the lower integration limit $0$ could be replaced by any positive number smaller than $T_1$. False-SB arises when giving this formal mathematical manipulation a physical meaning stating that one-way net flow is the difference of two-way gross flows, with the flow from 1 to 2 violating the 2nd law of thermodynamics and involving an arbitrary constant.

False-SB thus arises by an ad hoc translation to physics of a mathematical operation which results in a violation of the 2nd law of thermodynamics.

Accordingly False-SB is not found in physics literature, but has appeared outside physics as an ad hoc free invention by climate scientists for the purpose of selling CO2 alarm.