Gravitational waves of spherical body with mass $M$ which collapses to a black hole with mass $m<M$

Question

Consider a body with spherical symmetric mass distribution in asymptotically flat space with a total mass $M$. Assume it collapses to a black hole with mass $m<M$.

Does this body emit gravitational waves while collapsing if it maintains spherical symmetry during the collapse? I know that from Birkhoff's theorem follows that a spherical symmetric object cannot emit gravitational waves. But I'm wondering if a collapsing spherical symmetric body could when losing mass. I would say yes, because when calculating the quadrupole moment using the quadrupole formula, i is not constant in time. Hence, the luminosity formula is also nonzero. Is there a simple argument what the maximum energy, radiated by gravitational waves, an observer far away from this object can observe? The hard way is probably to integrate over the luminosity formula and try to somehow maximize it.

Edit: I tried to calculate the mass quadrupole moment: $$Q_{xx}=\int T^{00}\Big(t,\sqrt{x^2+y^2+z^2}\Big)(x^2-\frac{r^2}{3}\delta_{xx}) dx dy dz=\int\limits_0^\pi\int\limits_0^{2\pi}\int\limits_0^\infty \rho (t,r)r^4sin^3(\theta)cos^2(\phi)-\rho (t,r)\frac{r^4}{3} sin(\theta)drd\phi d\theta$$ Where $\rho$ is the spherical symmetric mass distribution. I get similar for the other component, hence the quadrupole moment doesn't vanish.

Answer 1

A perfectly spherical collapse cannot radiate any gravitational waves. (As you say, this follows from Birkhoff's theorem.)

The quadrupole of a spherically symmetric configuration is zero, hence it does not change as the configuration collapses, and the quadrupole formula does not predict any gravitational wave luminosity.

The same is true for any higher multipole moments that could source the generation of gravitational waves.

(Consequently, this also means that the perfectly spherical collapse must result in a black hole with mass $M$.)