Charged black holes

Written 30th of July 2023 Last updated 30th of July 2023


Here we will give a short overview of this paper, where we considered the inspiral of two black holes that carry electric charge, either from electromagnetism, or some dark sector \(U(1)\) charge. Usually, all astrophysical black holes are assumed to be electrically neutral, see for example this review. The argument for this is very strong, since the timescale for neutralisation would be very small for significantly charged black holes due to among other Schwinger pair-production, or accretion of oppositely charged matter from the environment. However, there are ways to avoid short neutralisation timescales for astrophysical black holes, for example by use of mini-charged dark matter. See this paper. The following work is covered in detail in my thesis. The field of gravitational waves is moving rapidly, and so it should be kept in mind that a lot of work has been done since the time of this project, and the detector network has grown and increased its sensitvity.


The use of matched filtering statistics in the detection- and parameter estimation pipelines of LIGO/Virgo has the consequence that the detectors will have high sensitivity only to signals that are already properly modelled, such as circularly inspiraling black holes with different masses and spins. Black holes will in principle come with one extra parameter, which is the electric charge they carry. Since this is usually neglected because of astrophysical decharging mechanisms, if there somehow where a binary carrying electric charge merging, we might either miss the signal entirely, or interpret it wrongly in terms of our uncharged templates during parameter estimation. In this work, we set out to explore at lowest order what size of error such a projection effect can have on the mass estimation of the binary, and also how large signal losses we might get. Below we illustrate how the noise and signal blends together in a time-series from the detector. The signal amplitude has for presentation been exagerrated, and the y-axis is normalised to unity. The time is in units of seconds.


The first step towards finding the emission of gravitational waves in the slow inspiral regime of the merger, is to find the quasi-static orbit. We are here assuming that the orbital parameters are changing slow with comparison to the timescale of the orbit, so that the orbit at any moment is well described by constants of motion such as orbital energy and angular momentum. After finding the orbit, we will calculate the expected emission for a particle that follows it, and then couple those emissions to the expressions for orbital energy and energy momentum to solve for the orbital parameters as a function of time. Since the Coulomb potential which is set up by charges we introduce goes like \(1/r\) like the gravitational potential, we can add them together by use of an effective gravitational constant like $$ \Phi = - \frac{\tilde G M \mu}{r^2},\quad \tilde G = G_N \left( 1 - \sigma_1 \sigma_2 \right), $$ where \(\sigma_i = q_i/m_i\) is the black hole's charge to mass ratio, \(m_i\) is its mass, and \(M=m_1+m_2\) and \(\mu=m_1 m_2/m\) are the total mass and reduced mass respectively. From here, for the quasi-static orbit, everything is as with the gravitational case, traded in this gravitational constant, and so we can find the Keplerian orbit $$ \begin{eqnarray} r &=& \frac{a(1-e^2)}{1+e\cos\psi},\\ a &=& \frac{\mu M}{2 |E|},\quad e = \sqrt{1 + \frac{2 E L^2}{M^2\mu^3}}, \end{eqnarray} $$ where \(a\) is the semi-major axis and \(e\) is the eccentricity, while \(E,L\) are the orbital energy and angular momentum respectively. We can find the lowest order (in velocity) emission of electromagnetic energy in terms of the charge dipole $$ \begin{eqnarray} \dot E &=& -\frac{2 }{3}\langle \ddot Q^i \ddot Q_i \rangle, \\ Q^{i} &=& \int \rm d^3 x\, j^t x^i , \end{eqnarray} $$ and \(j^t=\sum_i q_i\delta (\vec x - \vec x_i)\) is the charge density. We can do similarly with angular momentum to find a coupled, first order differential equation system for \(a,e\) in terms of \(E,L\), see the thesis for more details. The resultant equation system, not thinking about gravitational wave emission for the moment, is $$ \begin{eqnarray} - \dot E &=& \frac{\tilde G M \mu}{2 a^2}\dot a = \frac{\left(\tilde G M \mu\right)^2 \left( \Delta\sigma\right)^2}{3 a^4 (1-e^2)^{5/2}}(2+e^2),\\ -\dot J &=& \sqrt{\frac{\tilde G M \mu^2}{4 a(1-e^2)}}\left( \dot a (1-e^2) - 2 a e \dot e\right) = \frac{2 \mu^2 \left(\tilde G M \right)^{3/2}\left(\Delta\sigma\right)^2 }{3 a^{5/2}(1-e^2)}, \end{eqnarray} $$ Which solves to a type of evolution for the orbit shown below. A nice result is that orbits at originally sufficiently large enough separation, tend to circularise by the time its gravitational wave emission would enter into detector sensitivity. This would allow us to at first only consider circular orbits in the remaining. However, we find that although the orbit circularises in both cases, in general it is a smaller degree of circularisation in the case of dominating electromagnetic emission, than for gravitational emission. Below one can see the absolute of the change in eccentricity in response to a small fractional change in semi-major axis for the binary system. In both cases the eccentricity is decreasing with smaller separation.

Matched filter and parameter bias

Equipped with the quasi-static orbital evolution, we can find the gravitational emission in terms of the orbital parameters, and therefore the waveform during the inspiral, to lowest order. The waveform depends on parameters such as distance to the source, orientation of the detector, orientation of the binary, et cetera. We are interested in the projection effects on the black hole parameters, so we consider the case of pure plus polarisation radiation, and the relevant part of the waveform is $$ h = \omega^{2/3}\sin\Psi, $$ where \(\omega\) is the angular frequency of the binary. For the full derivation, see the thesis. Since the detectors are most sensitive to the phase of the signal, we can get a sense of the expected parameter bias by considering it in isolation. It is given $$ \frac{5 \Psi}{16 \tau_0}\left(\frac{8\tau_0}{3 A}\right)^{3/8} \equiv \tilde \Psi = 1 - u^{5/8} - q\left(1- u^{7/8}\right), $$ where \(u=\tau/\tau_0\) is the time-parameter and \(\tau\) is the time until coalescence, \(\tau_0\) being the initial time until coalsescense. We have defined \( A = \frac{5}{96\mathcal{M}_*^{5/3}}\), where \(\mathcal M_*\) is the 'chirp mass'. Below, we show the projection effect we find in the chirp mass along the right hand y-axis. The orange line indicates the expected result from doing a least squares phase matching between charged and neutral waveform templates. The blue crosses are the results after doing a matched filtering matching, with simple LIGO/Virgo noise modelling. On the left hand side y-axis, we see the loss of signal incurred by the projection. The x-axis is \(\Delta \sigma=\sigma_2-\sigma_1\), or the charge-over-mass-difference.


We found the projection effect on the parameter estimation and the signal to be at most \(\sim 5\%\) in the worst case considered here. This analysis serves as a proof of concept and initial consideration, but a more thorough analysis would take into account the ringdown and the merger phases of the merger as well, the merger being where the majority of the signal is found. Furthermore, the inspiral should be performed at higher order, and the effects on the full parameter space should be taken into account, including intrinsic and extrinsic binary parameters. A more general takeaway may be that should be careful to interpret the observed mergers as definite exclusions of more exotic sources, seen as we might simply not have modelled them well enough to be sensitive to them, or they may indeed be degenerate with standard GR black holes and cause parameter biases. However, the graviational detector network is also applying model-independent searches, looking for more general waveforms without matched filtering. Employment of such search strategies, although less sensitive, will help to overcome difficulties like the ones presented here.