How can a dyonic Simpson-Visser black-bounce spacetime be sourced by an exotic Einstein cluster?

A Maxwell field plus an exotic Einstein cluster

The paper asks whether the Simpson-Visser black-bounce (SV-BB) spacetime, which is regular and interpolates between a Schwarzschild black hole and an Ellis-Bronnikov traversable wormhole, can be produced by ordinary Maxwell electrodynamics rather than by nonlinear electrodynamics. The authors show that it can, provided the electromagnetic field is dyonic, carrying both an electric charge q_E and a magnetic charge q_B, and provided an additional anisotropic fluid with vanishing radial pressure is present.

That fluid is identified as an Einstein cluster: a dense system of non-interacting particles on circular geodesics whose angular momentum generates tangential pressure only. In the metric ansatz ds^2 = -A(r) dt^2 + A(r)^-1 dr^2 + chi^2(r) dOmega^2, the cluster is described by energy density rho, radial pressure P = 0, and tangential pressure P-tilde. The resulting metric is the dyonic SV-BB with a^2 = q_E^2 + q_B^2, so the bounce parameter a is fixed by the total charge rather than chosen freely.

Maxwell electrodynamics replaces nonlinear models

Earlier constructions of the SV-BB used general relativity coupled to nonlinear electrodynamics (GR-NLED), sometimes with a self-interacting scalar field. Those models were purely magnetic, with the electromagnetic invariant F ~ B^2, and their Lagrangian L(F) = s_0 |F|^(5/4) does not reduce to Maxwell electrodynamics in the weak-field limit. The paper argues this makes them physically questionable as electromagnetic sources.

The new construction instead uses linear Maxwell electrodynamics, L(F) = F, so the field equations are the standard Einstein-Maxwell equations plus the anisotropic fluid. The electromagnetic field tensor has both F_tr and F_theta-phi components, giving F = (q_B^2 - q_E^2)/(2 chi^4). The electromagnetic stress-energy tensor satisfies rho_ED > 0, rho_ED + P_ED = 0, and rho_ED + P-tilde_ED > 0 for all r, so the Maxwell field alone obeys the null and weak energy conditions.

The Einstein cluster density is not positive definite

For the dyonic SV-BB, the Einstein cluster has rho = (q_E^2 + q_B^2)(2m - sqrt(r^2 + q_E^2 + q_B^2)) / [4 pi (r^2 + q_E^2 + q_B^2)^(5/2)] and P-tilde = -(q_E^2 + q_B^2) m / [8 pi (r^2 + q_E^2 + q_B^2)^(5/2)], with P = 0. Because rho is not positive everywhere, the authors call this an exotic Einstein cluster.

The energy-condition analysis splits by configuration. For the regular black hole case 2m > sqrt(q_E^2 + q_B^2), the inequalities rho > 0 and rho + P-tilde >= 0 hold simultaneously only in the region |r| <= sqrt(2.25 m^2 - q_E^2 - q_B^2), which lies strictly inside the event horizon. For the traversable wormhole case 2m < sqrt(q_E^2 + q_B^2), rho <= 0 and rho + P-tilde <= 0 for all r. In both cases the cluster stress-energy tensor violates the null and weak energy conditions somewhere, which is what allows the Penrose singularity theorem to be evaded.

Uniqueness forces a^2 to equal the total charge squared

The authors solve the general static, spherically symmetric black-bounce ansatz with a Maxwell field and an Einstein cluster. The general solution for A(r) contains a logarithmic term proportional to (q_E^2 + q_B^2 - a^2), and the field equations C^t_t = 0 = C^r_r are satisfied only when c_1 = (q_E^2 + q_B^2 - a^2)/a^3. This general solution has a curvature singularity at r = 0, so it is not itself a black bounce.

Regularity for all r requires a^2 = q_E^2 + q_B^2, which collapses the logarithmic term and yields the dyonic SV-BB metric. The authors present this as a uniqueness result: within this GR-Maxwell-Einstein-cluster setting, the SV-BB is the only regular black-bounce solution. They also note that when a -> 0 the fluid quantities vanish and the metric reduces to Reissner-Nordstrom, but the dyonic SV-BB itself does not reduce to Reissner-Nordstrom because a^2 = q_E^2 + q_B^2 is required. This is offered as an explanation for why the SV-BB, though electromagnetically charged, is not the Reissner-Nordstrom black hole.

Electric charge is added to a previously magnetic solution

The paper's stated novelty is that the SV-BB is realized with both electric and magnetic charge, whereas earlier black-bounce constructions supported only magnetic charge. The dyonic solution is also built inside Maxwell electrodynamics, unlike prior dyonic black-bounce work that used nonlinear electrodynamics models which fail the Maxwell weak-field limit.

The traversable-wormhole configuration still requires the exotic Einstein cluster to support the throat. The authors emphasize that the presence of this exotic cluster is what violates the weak energy condition in the dyonic SV-BB and thereby evades the Penrose singularity theorem. The electromagnetic field alone satisfies the energy conditions; the violation comes from the cluster.

Scope of the exact solution

The result is an exact solution of the Einstein field equations for a static, spherically symmetric, asymptotically flat spacetime, with units G = k_B = c = hbar = 1 and metric signature (- + + +). It applies to the specific source combination of a linear Maxwell field and an anisotropic fluid with vanishing radial pressure representing an Einstein cluster. It is not a statement about astrophysical populations, collapse dynamics, or observational signatures.

The energy-condition conclusions are tied to the two parameter regimes: 2m > sqrt(q_E^2 + q_B^2) for the regular black hole with an event horizon at r = sqrt(4m^2 - a^2), and 2m < sqrt(q_E^2 + q_B^2) for the traversable wormhole. The region where rho > 0 and rho + P-tilde >= 0 in the black hole case is strictly inside the horizon, so the cluster is not a normal matter source in the exterior. The paper does not claim the cluster is physically realized; it is introduced as the exotic matter needed to keep the throat open.

The decisive condition is a^2 = q_E^2 + q_B^2

The most decision-relevant result is the regularity condition a^2 = q_E^2 + q_B^2. Without it, the general dyonic solution has a curvature singularity at r = 0 and is not a black bounce. With it, the metric becomes the dyonic SV-BB, the bounce parameter is fixed by the charges, and the solution cannot be tuned to the Reissner-Nordstrom limit.

This also separates the two sources by role. The Maxwell field carries the electric and magnetic charge and satisfies the energy conditions everywhere. The Einstein cluster carries the exotic matter, violates the null and weak energy conditions, and is what makes the throat possible. Readers comparing this to earlier SV-BB constructions should focus on that split: the new ingredient is not a new electromagnetic Lagrangian but a different matter source, and the price is an energy-condition-violating cluster rather than a nonlinear electromagnetic field.

Sources used in this answer