Charging up regular black holes

Introduction to the show: ident: General Relativity and Quantum Cosmology. The day in general relativity and quantum cosmology.

Kai: We're looking at the paper: "Charging up regular black holes".

Mira: Here's the gist of the paper. We present a general construction of charged regular black holes as solutions of a generalization of the Einstein–Maxwell field equations in spherical symmetry in which the Einstein tensor is deformed into an identically conserved tensor containing up to second derivatives of the gravitational field General Construction and Field Equations The paper presents a general construction of charged regular black holes as solutions of a generalization of the Einstein–Maxwell field equations in spherical symmetry in which the Einstein tensor is deformed into an identically conserved tensor containing up to second derivatives of the gravitational field The generality of the construction allows us to define the field equations satisfied by generic regular black holes when becoming charged The most general field equations for gµν above, in which the spherically symmetric Einstein tensor is deformed into an identically conserved tensor constructed from up second derivatives of qab(x) and r(x), take the form Regularity Conditions and Singularities The conditions that guarantee regularity of charged solutions are evaluated and shown to be more stringent than the regularity conditions for uncharged solutions This implies, in particular, that the charged versions of the Bardeen and Hayward black holes become singular Improved versions of the Bardeen and Hayward metrics that remain regular when charged are proposed Our results indicate that regularizing the vacuum solutions of general relativity is, in general, not enough to yield regular solutions in other situations of physical interest Vacuum Solutions and Metric Families The study starts with a brief review of vacuum solutions, that is, with vanishing electromagnetic fields The field equations (2) describe a wide range of modifications of general relativity which include, in particular, modifications in which black hole solutions have bounded curvature both in vacuum and in the presence of matter The theory is characterized by the deformation of the spherically symmetric Einstein tensor into an identically conserved tensor Two particular deformations of general relativity are considered, for which specific metrics arise as vacuum solutions These include the Bardeen metric and the Hayward metric Charged Solutions and Regularizations The generalization of the Einstein–Maxwell equations given by Eqs.

Kai: (2), (7), (8) and (9) can be explicitly integrated without specifying the functions α and β as long as they satisfy the integrability condition For theories satisfying γ = ∂χα − ∂rβ = 0, we have h(r) = f(r) This leads to charged versions of any regular black hole by recalling Eqs.

Mira: (15-16) and (17-18) The resulting Reissner–Nordstr¨om–Bardeen and Reissner–Nordstr¨om–Hayward black holes become singular at r = 0 and r = r+ > 0, respectively Implications for Realistic Models The main physical implication of our work is the identification of two avenues along which the construction of regular charged solutions can proceed The first avenue is finding other deformations of general relativity which remain regular in the presence of charge, and result in different families of uncharged regular black holes The second avenue is abandoning the idea of regularizing vacuum or electrovacuum solutions and thinking about regular black holes as describing the exterior to suitable matter distributions This allows for the construction of non-electrovacuum metrics by using charged perfect fluids The paper concludes that regularizing charged and rotating regular black holes is subtler than regularizing simpler situations and requires new ingredients Recent advances have provided a general formalism in which to show that deformations of the gravitational interaction such that the latter becomes weaker at the core of black holes can result in regular vacuum solutions The main technical result is that known families of black holes, such as the Bardeen and Hayward black holes, become singular when charged The paper identifies specific theories in the Ziprick–Kunstatter and Kunstatter–Maeda–Taves classes that remain regular when charged It is reasonable to expect that deformations of the Bardeen and Hayward metrics incorporating angular momentum, and reducing to these static geometries for J = 0, may not be regular Such a metric results from the following choice in the Kunstatter–Maeda–Taves class: omegaH2.0(r, χ) = 2(1 − χ)r 4p r 6 − (1 − χ) 2l 4 The requirement of regularity of electrovacuum solutions might be too stringent Introducing other (static or time-dependent) matter fields aside from electromagnetic fields may still result in regular solutions Previous results show that the introduction of repulsive forces, either in the form of a non-zero charge

40: or of a modification of gravity

41, 42: , allows placing spherically symmetric distribution of a perfect fluid in the interior of regular black holes without adding further (distributional) sources Such solutions are guaranteed in the situations analyzed in this paper Acknowledgments We thank Luis Lehner for valuable discussions R.C-R.

Kai: acknowledges financial support provided by the Spanish Government through the Ram´on y Cajal program (contract RYC2023-045894-I), the Grant No.

Paper summary: Kai: Looking at the full paper, "Charging up regular black holes," by Carballo-Rubio, Coviello, and Vellucci, the authors take this general formalism of deformed Einstein-Maxwell equations and show how it dictates the behavior of charged regular black holes. The main claim is that for known metrics like Bardeen and Hayward ones, adding charge makes them singular at both r = zero and some positive radius > <ref:2607.07831#pg3>

Mira: And they confirm that even though they found improved versions of these metrics that remain regular when charged, it's a very stringent condition to meet compared to the uncharged cases >

Lev: That really puts the engineering side into perspective because if we were trying to build hardware based on these solutions, we'd have a lot of trouble realizing those improved metrics with charge without introducing singularities again >

Kai: The implication for us as researchers is that regularizing vacuum solutions isn't enough to guarantee regular black holes in any physical situation involving charge or other fields >

Mira: They suggest two main avenues: either finding new deformations of general relativity that remain regular in the presence of charge and yield different uncharged families, or abandoning the idea of regularizing electrovacuum solutions entirely >

Lev: And they point out that they found specific theories within the Ziprick–Kunstatter and Kunstatter–Maeda–Taves classes where regularity is still maintained when charged >

Kai: It seems like a reasonable expectation is that deformations of the Bardeen and Hayward metrics with angular momentum, which simplify to these static geometries when angular momentum is zero, might not be regular even with charge >

Mira: That specific metric they mentioned, H2 point 0(r, chi) = two (one - chi) r cubed r two- two(one-chi), with alpha H(r, chi) = two r four - three two r squared (one - chi) (one - chi)

r two- two (one-chi): squared, and beta H(r, chi) = - 2r five

r two- two (one-chi): squared in the Kunstatter–Maeda–Taves class, is an example of a theory that might fail regularity requirements > <ref:2607.07831#pg2>

Lev: That complexity suggests that just introducing repulsive forces, like a non-zero charge or gravity modifications, isn't enough to place a perfect fluid distribution in the interior of regular black holes without adding extra distributional sources >

Kai: So the final point is that regularizing charged and rotating regular black holes is actually quite subtle and requires new ingredients to manage it properly >

Conclusion: Kai: So we’ve been looking at how these modified field equations work for charged black holes, and this paper is titled "Charging up regular black holes." It basically shows how you can charge these things up while still keeping them regular.

Mira: Yeah, it's about taking those general constructions we talked about before and seeing what happens when we introduce electric charge into the mix. The authors are Carballo-Rubio, Coviello, and Vellucci.

Lev: So what they’re doing is looking at these generalized Einstein–Maxwell equations where the gravity term is deformed in a specific way. They're not just looking at standard black holes anymore; they're looking at charged versions of these regular ones.

Kai: And the main result is that for some known metrics, like Bardeen or Hayward, when you add charge, they become singular again right at the center and maybe somewhere else out there too.

Mira: That’s a key finding because it means those specific standard models don't handle charge well in this framework. But they did find ways to improve the metrics so they stay regular even with charge, which is interesting because that’s a tough condition to satisfy.

Lev: From an engineering standpoint, if we tried to build hardware based on these exact charged versions of Bardeen or Hayward, we’d run into trouble because of those singularities they found.

Kai: It really highlights the point that just regularizing the vacuum solutions isn't enough for other physical situations like charge to be handled gracefully.

Mira: Exactly. The paper suggests two paths forward: either find different gravity deformations that stay regular with charge, or we have to stop thinking about electrovacuum solutions entirely and start looking at how these black holes sit outside of matter distributions.

Lev: That means we might need to think about building things in the interior, using charged perfect fluids instead of just vacuum or pure electromagnetism.

Kai: It seems like the paper points toward needing new ingredients when you try to build realistic models for charged and rotating black holes. That opens up a whole new set of possibilities for what a "regular" black hole can actually look like.