Auxiliary Content for “Splitting the (Gravitational) Atom”
Auxiliary content for the paper
Splitting the (Gravitational) Atom
José Ferreira, Carlos Herdeiro, Eugen Radu, Miguel Zilhão
TODO: dar ao AI o paper, pedir para ele fazer um side-article para colocar na web com os videos e estilo informal
In this work we performed time evolutions of a class of hairy black holes known as Q-Hairy BHs, introduced in (Herdeiro and Radu 2020). These solutions were studied in spherical symmetry in (Zhang et al 2023), where they found out numerical evidence to suggest they are stable and also have a viable formation mechanism. In this work, we performed time evolutions of these solutions without any assumptions regarding their symmetries.
In short, these are a class of spherically symmetric, static BHs that are a solution to the Einstein-Maxwell-Scalar model
\[ \mathcal{L} = \frac{R}{16 \pi} - \frac{1}{4} F_{\alpha \beta} F^{\alpha \beta} - (\widetilde{\nabla}_\alpha \phi)^* \widetilde{\nabla}^\alpha \phi - V(\left| \phi \right|^2) \,, \]
where \(R\) is the Ricci scalar, \(F_{\mu \nu} \equiv \nabla_\mu A_\nu - \nabla_\nu A_\mu\) is the Maxwell tensor, \(A_\mu\) the 4-potential, \(\phi\) is the (complex) scalar field and \(V(\left| \phi \right|^2)\) its potential. The scalar field is coupled with the field via the coupled covariant derivative, defined as
\[ \widetilde{\nabla}_\mu \phi \equiv \nabla_\mu \phi + i q A_\mu \phi \,, \]
where \(q\) is the coupling constant between the scalar field and the electromagnetic field. We considered the following potential of the scalar field
\[ V(|\phi|) = \mu^2 |\phi|^2 (1 - 2 \lambda |\phi|^2)^2 \,. \]
Performing time evolutions revealed two distinct outcomes: - Fission: When the BH is expelled by the hair; - Absorption: The BH absorbs its hair.
In the first scenario the end state is a stable boson star plus a hairless charged BH, whereas in the second scenario its only a hairless charged BH. Remarkably, we found no evidence for stable solutions. Below, you will see a representative video of each outcome.
In Video 1 we see that the BH starts slowly moving towards the edge of the hair. The offset is exponential, so it speeds up rather quickly even though it can remain stationary for a few hundreds of units of time. The time it takes for it to start moving depends on the region of the parameter space. After the fission happens, you can see that the boson star loses resolution. This is a numerical artifact that is due to the fact that the grid is tracking the BH, and forces us to terminate the solution prematurely.
In Video 2, we see the BH absorbing the hair around it, leaving behind no trace of it.