Dyonic black holes
Table of contents
Abstract
I will here give an overview of the work I did on the inspiral of dyonic black holes in order to model its gravitational waveform. Dyonic black holes carry both electric and magnetic charges. The work resulted in two papers: this one and this. The work is covered in detail in the thesis.Idea
Magnetic monopoles have been around as a theoretical concept for a long time, and there are several motivations to why they should exist, among other: They would bring symmetry to the Maxwell's equations; They would explain why electric charge is quantised, as was shown by Dirac in this paper; They are a generic prediction of supersymmetric theories, see these notes, that historically have provided the leading theoretical framework to extend the standard model of particle physics to contain dark matter; And they have been linked to a solution to the confinement problem of quantum chromodynamics. However, in spite of this, they have never been observed experimentally, which puts their existence under strain. Still, if they were produced in the early universe, while the universe was supersymmetric, they might have been so dilluted by now that we do not expect to see them.Nontheless, in this work, we set out to consider the existence of magnetic monopoles agnostically and phenomenologically and ask the question of what we could observe in the mergers of black holes if they could carry both magnetic and electric charges.
The Poincaré cone
The first step is to set up the force equation for either of the black holes, that we consider here as simple point particles without any structure. The position vector \(r_2^i\) receives accelerations from both the magnetic and electric fields, as $$ \begin{eqnarray} m_2 \ddot r_2^i &=& \mu \ddot R^i = C \frac{R^i}{R^3} - D \epsilon_{ijk}\frac{R^j }{R^3} v^k,\label{dyonforce} \\ C &=& \left(-\mu M + q_1 q_2 + g_1 g_2 \right),\quad D = \left( q_2 g_1 - g_2 q_1\right). \end{eqnarray} $$ We have used \(R^i=r_2^i-r_1^i\) to indicate their separation vector. \(q\) and \(g\) are the electric and magnetic charge respectively, while we use the lower index to indicate which point we they belong to. \(\mu=m_1 m_2/M\) is the reduced mass while \(M=m_1+m_2\) is the total mass. If integrate their accelerations numerically in a stationary center of mass system, we find the orbits
and we notice that the orbit is no longer confined to a
plane, but instead traces a conic shape. We will
understand this below. An interesting consequence of this setup, is that the usual anguar momentum conservation no longer applies, \(\rm d /\rm d t\enspace \tilde L \neq 0\), but instead there is a generalised conserved angular momentum
$$ \begin{eqnarray} L^i &\equiv& \tilde L^i - D\hat{R}^i,\label{genangmem}\\ \dot L^i &=& 0 , \end{eqnarray} $$ whose derivative is zero. Since \(\tilde L^i \perp \hat R^i\) and \(D\) is a constant, the norm of \(\tilde L^i\) must be a constant too. Therefore it is only the direction of the angular momentum that is not conserved. The direction of the angular momentum will rotate so that the orbit traces out a cone. Using our new constants of motion, we can perform a derivation of the radial separation similar to what is done with Keplerian orbits. We find $$ \begin{eqnarray} R &=& \frac{a(1-e^2)}{1 + e \cos (\phi \sin \theta)},\label{magmonR}\\ E &=& \frac{C}{2a},\quad \tilde L^2 = \mu|C| a (1-e^2), \end{eqnarray} $$ where \(a,e\) are the semi-major axis and eccentricity respectively, \(\phi\) is the azimuth and the angle \(\cos\theta=-D/L\) is the opening angle of the cone. We therefore see that there is a precessing motion where the maxmum of the separation occurs every \(2\pi /\sin \theta\) radians. Using the radial expression, we can find the 3D orbit, knowing that the zenith angle \(\theta\) is constant $$ R^i = R \begin{pmatrix} \sin\theta\cos\phi \\ \sin\theta\sin\phi \\ \cos\theta \end{pmatrix} . $$ We then have obtained the closed-form expression for the motion of the particles when the energy and generalised angular momentum are conserved. The acceleration can easily be found by differentiation $$ \begin{eqnarray} \ddot{R}^i &=& \frac{|C|}{\mu R^2} g_E^i,\\ g_E^i &=& e\cos x \begin{pmatrix}(\sin\theta-1/\sin\theta)\cos\phi\\(\sin\theta-1/\sin\theta) \sin\phi\\\cos\theta \end{pmatrix} - \frac{1}{\sin\theta}\begin{pmatrix} \cos\phi\\\sin\phi\\ 0\end{pmatrix} , \end{eqnarray} $$ and we notice that we obtain Newton's gravitational law for \(D=0\).Emission
In order to find the orbital evolution like one does in the slow inspiral modelling of black hole mergers, one needs to include the back-reaction of the radiation fields on the orbit. The binary will radiate in both gravitational waves and electromagnetic waves, that will drain the system of energy and angular momentum. For example, the expression for the leading order flux of emitted energy in the electromagnetic channe, by a binary that is only electrically charged, is given by the charge dipole \(Q^i\) like $$ \begin{eqnarray} \dot E &=& -\frac{2 }{3}\langle \ddot Q^i \ddot Q_i \rangle, \\ Q^{i} &=& \int \rm d^3 x\, j^t x^i \end{eqnarray} $$ where \(j^t=\sum_i q_i\delta (\vec x - \vec x_i)\) is the charge density. In order to extend this expression to calculate the emission for dyonic binaries that also carry magnetic charge, we make use of the duality transform of electromagnetism. According to it, electromagnetism is invariant under the transformation $$ \begin{eqnarray} \mathbf E_2 &=& \mathbf E_1 \cos\alpha - \mathbf B_1 \sin\alpha,\\ \mathbf B_2 &=& \mathbf E_1 \sin\alpha + \mathbf B_1 \cos\alpha,\\ q_e &\rightarrow& q_e \cos\alpha + q_g\sin\alpha,\\ q_g &\rightarrow& q_g\cos\alpha - g_e \sin\alpha . \end{eqnarray} $$ This applies to some transformation parameter \(\alpha\). By considering \(\alpha=\pi/2\), we can consider a purely electrically charged system, and then find the analog expression for a purely magnetically charged system. We then wish to superimpose the two emissions, and we realise that we are able to do so, since \(\vec E_1\perp\vec B_1\) and \(\vec E_1 \parallel \vec B_2\implies E_1\perp E_2\) and so on, and the energy \(E\sim \vec E\cdot \vec E + \vec B\cdot \vec B \). In the end, we therefore find the total emissions $$ \begin{eqnarray} \dot E_{\text{tot}} &=& - \frac{ C^2 \left[\left(\Delta\sigma_e\right)^2 + \left(\Delta\sigma_g\right)^2\right]}{24 a^4 (1-e^2)^{5/2} \sin^2\theta}\left( 16 + 28 e^2 + 3 e^4 + (20+ 3 e^2 ) e^2 \cos (2\theta)\right), \\ \dot J_{\text{tot}} &=& -\frac{2\left[\left(\Delta\sigma_e\right)^2+\left(\Delta\sigma_g\right)^2\right]}{3 }\sqrt{\frac{|C|^3 \mu}{a(1-e^2)}}\left\langle \sqrt{\left(\epsilon^i_{\,\,jk} g_J^j g_E^k\right)^2}\,/R^2\right\rangle, \end{eqnarray} $$ where we have defined \(\Delta \sigma_e = q_1/m_1-q_2/m_2\) and \(q\rightarrow g\) for \(\Delta\sigma_g\). See the full derivation for more details.Conclusion
In the end we obtained the orbital motion and expressions for the energy and angular momentum emissions of binary dyonic black holes. The orbit may now be modelled quasi-statically, meaning that the orbital energy and angular momentum changes sufficiently slowly that the the emission is well described by that of a binary moving in a constant energy orbit like the one we found. Then we can couple the emissions to the expressions for \(E\) and \(\tilde L\) and find how the semi-major axis and eccentricity change with time. Finally the gravitational wave emission may be included similarly, and the waveform can be found now that the orbital evolution is known. This is done in this paper and this one for the eccentric orbit.This is the first step to get templates for this kind of binary. In the future one ought to do the analysis for the merger stage, and the ringdown stage in order to have complete enough template waveforms to extract potential signals from the gravitational detector data.