Intrinsic Gravitational Modes Sustained by Black Hole Collapsing Binaries

Abstract

Intrinsic Gravitational Modes (IGM) involving electromagnetic field fluctuations are found that are sustained by the time-dependent tridimensional gravitational field of Black Hole binaries as their collapse is approached. These “disk-rippling” modes, emerging from a plasma disk structure surrounding a binary, have ballooning amplitude profiles in the “vertical” direction (referring to the binary angular momentum vector) and rotate mainly with a frequency of twice the binary rotation frequency in the limit where their phase velocity does not exceed the speed of light. Relevant mode–particle resonances (B. Coppi, Plasma Phys. Rep. 45, 438 (2019)) can provide a means to transfer energy from high to low energy populations (a process evidenced by laboratory experiments) and offer an explanation for the absence of detectable high-energy radiation emission as the observed collapse of Black Hole binaries is approached. When the disk structure is immersed in a (stationary) magnetic field (B. Coppi, Plasma Phys. Reports. 45, 438 (2019)), another class of modes, affected by gravity-sustained disk structures, has to be considered.

APPENDIX

1.1 BINARY EVOLUTION

Referring to the presented analysis, the binary collapse is considered to be approached as a result of the loss of energy and angular momentum due to the relevant emission of gravitational waves. The associated radiated power [12] is, for ${{M}_{1}} = {{M}_{2}} \equiv M$

$$\frac{{d{{\varepsilon }_{b}}}}{{dt}} = - \frac{4}{5}{{\left[ {{{d}_{G}}\frac{{{{\Omega }_{{{\text{orb}}}}}}}{c}} \right]}^{3}}M\left( {{{R}_{G}}{{d}_{G}}\Omega _{{{\text{orb}}}}^{2}} \right){{\Omega }_{{{\text{orb}}}}}$$

(A.1)

and, since $\Omega _{{{\text{orb}}}}^{6} = {{\left( {2GM{\text{/}}d_{G}^{3}} \right)}^{3}}$

$$\frac{{d{{\varepsilon }_{b}}}}{{dt}} = - \frac{{64}}{5}\frac{{{{G}^{4}}}}{{{{c}^{5}}}}\frac{{{{M}^{5}}}}{{{{{\left[ {{{d}_{G}}\left( t \right)} \right]}}^{5}}}} \propto \left( {M{{c}^{2}}} \right)\left( {\frac{{R_{G}^{4}}}{{d_{G}^{4}}}} \right)\frac{c}{{{{d}_{G}}}}.$$

(A.2)

Then

$$\frac{d}{{dt}}\left[ {{{d}_{G}}\left( t \right)} \right] = - \frac{8}{5}{{\left[ {\frac{{{{R}_{G}}}}{{{{d}_{G}}\left( t \right)}}} \right]}^{3}}c$$

(A.3)

and

$$\begin{gathered} \frac{1}{{\Omega _{{{\text{orb}}}}^{2}}}\frac{{d{{\Omega }_{{{\text{orb}}}}}}}{{dt}} = - \frac{3}{2}\frac{1}{{{{\Omega }_{{{\text{orb}}}}}}}\frac{1}{{{{d}_{G}}\left( t \right)}}\frac{d}{{dt}}\left[ {{{d}_{G}}\left( t \right)} \right] \\ = - \frac{{12}}{5}\frac{{R_{G}^{2}}}{{{{d}_{G}}\left( t \right)}}\frac{{{{\Omega }_{{{\text{orb}}}}}\left( t \right)}}{c}. \\ \end{gathered} $$

(A.4)