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My question can be asked in either Einstein GR or Maxwell electromagnetism. Suppose we have a system which is localized in space (enclosed in a sphere of finite radius). For example two point masses(charges) orbiting each other. The system starts to radiate due to the acceleration of masses(charges).

What I know: For the energy, there are positivity theorems showing that the total energy of the system $\cal{E}$ decreases in time, i.e. $d{\cal E}/dt\leq 0$. In electromagnetism, this can be proven easily \begin{align} \dfrac{dE_{rad}}{dt}=\oint n_i T_{0i}=\oint n\cdot (E\times B) \end{align} Far from the source, we the wave becomes planar at each direction and we have $B=\frac{1}{c} n\times E$. Using this in prevous equation implies \begin{align} \dfrac{dE_{rad}}{dt}=\oint (|\vec{E}|^2-(n\cdot \vec{E})^2)\geq0 \end{align} Since $E_{rad}+\cal{E}$ is conserved, the latter should be decreasing. For Einstein GR, the same can be proved using the Bondi formalism and looking at the balance equation for energy.

My question: I intuitively expect that the magnitude of the angular momentum $|\vec{J}|^2$of the system is also a decreasing function of time, i.e. $\frac{d}{dt}|\vec{J}|^2\leq 0$. Is this true? If yes, what is the proof? If no, what is an example of such phenomenon? Intuitively, this is strange, because a system can spin up through radiation without any need for external torque.

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Yes, this can happen. As a relatively simple example, consider a black hole binary with spins $S_1$ and $S_2$, anti-aligned with the orbital angular momentum $L$, and $|S_1+S_2|>|L|$. The system will loose orbital angular momentum to gravitational waves, while the magnitude of the spins changes only very little (there is absorption of angular momentum through the horizon but this is about two orders of magnitude smaller than the angular momentum flux to infinity).

Since all contributions to the angular momentum are (anti)-aligned we can consider them as single numbers. If we take $L>0$, the above implie that we both have that $J = S_1+S_2+L <0$ and $dJ/dt\approx dL/dt <0$, such that $d|J|/dt >0$.

In general, you would expected this to occur in systems where the dominant component to the total angular momentum, in not the component of the angular momentum that directly couples to the generation of radiation.

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