To appreciate why this matters, one has to understand what ModMax actually is. Maxwell’s equations, formulated in the 1860s, are remarkably special: they are conformally invariant, meaning they look the same at every scale, and they possess a continuous electric–magnetic duality that swaps electric and magnetic fields in an elegant rotation. For decades, theorists wondered whether any nonlinear extension of electromagnetism could preserve both of these prized symmetries simultaneously. The answer, established by Bandos, Lechner, Sorokin and Townsend, is that exactly one such theory exists in four dimensions: ModMax. It depends on a single deformation parameter, conventionally written as γ, and when γ is set to zero the theory collapses smoothly back to Maxwell. For nonzero γ, however, the electromagnetic response of the vacuum becomes genuinely nonlinear, and the familiar solution-generating machinery of Maxwell theory largely breaks down.

That breakdown is precisely what makes exact solutions so valuable and so rare. A substantial catalog of Einstein–ModMax solutions has accumulated in recent years — Reissner–Nordström-type black holes, gravitational waves, Taub–NUT geometries, dyonic extensions with NUT wormholes, accelerating black holes, Melvin–Bonnor electromagnetic universes, and even exact multi-black-hole and black-dihole spacetimes. But there is a catch, and it is a deep one. As Ortaggio has shown, broad classes of non-null Einstein–Maxwell configurations can be converted into solutions of conformally invariant nonlinear electrodynamics by a simple constant rescaling of the electromagnetic field. This means many of the known ModMax solutions are, in a sense, Maxwell solutions in disguise. The truly interesting question is whether the nonlinear theory supports exact solutions that have no Maxwell ancestor at all — and the rotating sector, in particular, has proven notoriously rigid. Even in the slow-rotation limit, the standard Maxwell-inspired ansatz generally fails to remain consistent for nontrivial nonlinear electrodynamics, and Newman–Janis-type transformation tricks, often used to manufacture rotating solutions, generally fail to preserve the full field equations.

The difficulty multiplies when a dilaton — a scalar field coupled to the electromagnetic sector — enters the picture. In Einstein–Maxwell–dilaton theory itself, asymptotically flat, charged, rotating black holes are known in closed form only for very special choices of the dilatonic coupling; generic couplings have historically required perturbative treatments and, only recently, fully numerical constructions. Against this backdrop, Bixano and Matos started from a framework they had developed in earlier work: a generalized Ernst-potential formulation for Einstein–ModMax–scalar-field systems. The Ernst formalism, a classic tool of stationary axisymmetric gravity, reduces the coupled field equations to a more tractable structure, and the authors’ extension of it to the ModMax setting provided the technical key. The Lagrangian they work with is written in the Einstein frame, combining the Ricci scalar, the kinetic term of the scalar field with a sign parameter that distinguishes an ordinary dilaton from a phantom field, and an exponentially coupled ModMax Lagrangian built from the two electromagnetic invariants F and G, the latter being the dual invariant involving the dual Faraday tensor.

The new solution belongs to a specific nonlinear sector in which the ratio of the two electromagnetic invariants remains constant across the spacetime. It is stationary and axisymmetric, carrying a nontrivial gravitomagnetic potential ω that encodes frame dragging, together with both a time component A_t and an azimuthal component A_φ of the electromagnetic four-potential, and a nonconstant scalar field φ. Written in spheroidal coordinates, the family comes in an oblate branch and a prolate branch, and it admits configurations with NUT charge — the gravitational analogue of a magnetic monopole’s twist — as well as a NUT-free, asymptotically flat limit. Crucially, the solution exists only in the dilatonic sector, with the sign parameter fixed to the ordinary, non-phantom value.

The most striking feature emerges when one asks whether the solution can be deformed back to Maxwell. Within the authors’ parametrization, the ModMax deformation parameter η₀ is not free: the field equations force it to satisfy the constraint η₀ = (2α₀² − 5)/(2√2 α₀), where α₀ is the dilatonic coupling. Maxwell theory corresponds to η₀ = 0, which is reached only at the isolated coupling α₀² = 5/2. For every established value of the coupling — the low-energy effective superstring value α₀² = 1, the Kaluza–Klein value α₀² = 3, the Einstein–ModMax value α₀ = 0, and the entanglement-relativity value — the parameter η₀ is unavoidably nonzero. In other words, for all the physically motivated dilatonic theories, this rotating configuration is intrinsically ModMax. A Maxwell description, if it exists at all, is confined to a single isolated point in coupling space corresponding to a theory that is not among the standard models. This is the sense in which the solution has no Maxwell analogue: it is not a rescaled Einstein–Maxwell–dilaton spacetime but a genuinely nonlinear creature.

The authors then confronted the singularities. Curvature invariants reveal a ring singularity at the origin of the spheroidal coordinates, two surface singularities along x = ±y in the prolate branch, and an additional surface singularity whose location depends on the parameters τ₀ and λ₀. A careful analysis shows that when the parameter λ₀ is smaller than τ₀, this third singularity is confined to the region x < 1, outside the physical exterior domain of the prolate branch. This matters enormously, because in the prolate sub-extreme regime the surface x = 1 turns out to be an event horizon. The dragging function evaluated on that surface is constant, which means the Killing vector combining time translation and rotation with the horizon angular velocity Ω_H = f₁/(Lλ₀) becomes null there — the defining signature of a genuine event horizon. In Boyer–Lindquist-type coordinates, the horizon sits at r_H = l₁ + L, where l₁ is the NUT-related offset.

Even more remarkably, the exterior region of this prolate black hole satisfies the null energy condition — the requirement that T_{μν} l^μ l^ν ≥ 0 for every null vector l^μ — throughout the entire domain 0 < λ₀ ≤ τ₀ < 1, x ≥ 1, and −1 ≤ y ≤ 1. The proof is a piece of mathematical craftsmanship: the authors show that each bracketed polynomial factor in the numerator of the energy expression is non-negative over the physical domain, and that the remaining term, viewed as a quadratic polynomial in the variable u = y², is concave, so its minimum occurs at the endpoints where it evaluates to manifestly non-negative quantities. With the denominator strictly positive in the same parameter range, the null energy condition holds. Consequently, the black hole respects Penrose’s weak cosmic censorship conjecture: all curvature singularities remain hidden behind the event horizon. The authors note that in the limiting case λ₀ = τ₀ the energy condition is violated at the poles and some singularities touch the horizon there, so the inequality must be strict for the physically clean sector.

The conserved charges reveal one final, structurally elegant surprise. The NUT parameter, the Komar mass, the Einstein–Maxwell–dilaton-type electric charge, and the ModMax electric and magnetic charges are all proportional to a single parameter τ₀, while the Komar angular momentum J is independently controlled by λ₀. This produces an unusually clean separation between the monopolar and rotational sectors of the solution. Setting τ₀ = 0 simultaneously switches off the mass, the NUT charge, and all the electric and magnetic monopole charges, yet leaves the angular momentum nonzero — yielding an asymptotically flat spacetime that is still nontrivial, describing a pure mass dipole accompanied by ModMax electric and magnetic dipole moments. Moreover, the EMD-type charge and the ModMax electric charge can never be made equal, which the authors take as a strong indication that no analogous solution exists in the Einstein–Maxwell–dilaton framework. A complete linear stability analysis, including coupled gravitational, electromagnetic, and dilatonic perturbations and their dependence on the ModMax parameter, is currently in progress and will be presented separately. For now, theorists have a new benchmark: an exact rotating dilatonic black hole that exists only because Maxwell’s theory was left behind.

Subject of Research: Exact rotating dilatonic black hole solutions in Einstein–ModMax nonlinear electrodynamics without a Maxwell analogue

Article Title: Exact rotating dilatonic branch in ModMax electrodynamics without Maxwell analogue

Article References: Bixano, L., & Matos, T. (2026). Exact rotating dilatonic branch in ModMax electrodynamics without Maxwell analogue. The European Physical Journal C, 86(8), Article 1020. https://doi.org/10.1140/epjc/s10052-026-16268-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16268-w

Keywords: ModMax electrodynamics, black holes, nonlinear electrodynamics, dilaton, Ernst potentials, event horizon, null energy condition, cosmic censorship, Kaluza–Klein theory, superstring theory, NUT charge, general relativity

The stakes of this question are enormous. In 1969, Roger Penrose proposed the cosmic censorship conjecture, arguing that singularities arising from gravitational collapse should always be concealed within event horizons, forever shielded from any external observer. If that conjecture holds, the predictive power of general relativity is preserved, because a naked singularity would be a region where the theory’s equations break down in plain sight, spewing causally connected unpredictability into the observable universe. If it fails, physics as we know it could lose its ability to forecast the future in certain extreme scenarios. Despite decades of effort, no general proof or disproof of cosmic censorship has ever been achieved, and the new study adds a crucial piece to this ongoing investigation.

The central insight of the new research is that the fate of a collapsing matter cloud is governed by the dynamics of the apparent horizon, a surface that marks the boundary of a region where even light cannot escape. Unlike the event horizon, which is defined globally and requires knowledge of the entire future evolution of the spacetime, the apparent horizon is a quasi-local construct: it can be located by examining the geometry of the spacetime at a given moment. Within the apparent horizon lie trapped surfaces, two-dimensional surfaces from which both outgoing and ingoing light rays converge. The authors show that whether the final singularity is visible or hidden depends critically on the relative timing of when these trapped surfaces first form and when the central singularity comes into existence.

Technically, the analysis is carried out for spherically symmetric collapse of what relativists call a type-I matter field, a broad class that includes most physically reasonable forms of matter, such as dust, perfect fluids, and scalar fields, where the energy-momentum tensor has a well-behaved algebraic structure. The authors work with initial data satisfying C-squared regularity, meaning the relevant functions are twice continuously differentiable, and the weak energy condition, which demands that matter has non-negative energy density as measured by any observer. These are the same broad physical assumptions that underpin most rigorous work on collapse in classical general relativity, giving the results considerable generality within the spherical model.

The key diagnostic tool in the study is the behavior of outgoing null geodesics, the trajectories that light rays follow as they emerge from the region near the forming singularity. If families of future-directed outgoing null geodesics can escape from arbitrarily close to the central singularity, then the singularity is locally visible: light signals generated at or near it can reach distant observers, at least for a finite period before the apparent horizon eventually swallows the region. If, on the other hand, the apparent horizon forms early enough in the collapse, before the singularity has a chance to communicate with the outside, then every outgoing light ray is trapped, and the singularity is destined to be hidden within a black hole. The relative timing of these two events, the birth of trapped surfaces and the birth of the singularity, therefore acts as the decisive switch between the two possible end states.

This timing criterion connects the new work to a long lineage of collapse studies stretching back to the foundational models of the 1930s and the numerical analyses of Eardley and Smarr in 1979, which first made explicit how the apparent horizon curve in a collapsing dust cloud determines what external observers can see. Subsequent work by Christodoulou in the 1980s and 1990s demonstrated both violations of cosmic censorship in dust collapse and the instability of certain naked singularities in scalar field collapse, while Joshi and collaborators showed through the 1990s and 2000s that naked singularities arise in inhomogeneous dust, perfect fluid, and scalar field models under wide classes of initial conditions. The new paper synthesizes this tradition by characterizing, in terms of the apparent horizon’s dynamics, precisely when visibility occurs for a general type-I matter field rather than for one specialized equation of state at a time.

Why does the distinction between apparent horizon and event horizon matter so much? The event horizon of a black hole is teleological: its location at any instant depends on the entire future history of the spacetime, including whether collapse will ever end. This makes it useless for real-time diagnostics, both analytically and in numerical simulations. The apparent horizon, by contrast, is determined by the local geometry, specifically by the vanishing of the expansions of both future-directed null congruences emanating from a closed two-surface. Because of this quasi-local character, apparent horizons can be located in numerical relativity codes during dynamical simulations, and they have become the workhorse of modern black hole physics, from modeling binary mergers detected by gravitational-wave observatories to testing general relativity with future detectors such as LISA. Establishing that the apparent horizon’s evolution controls singularity visibility therefore ties an abstract question about censorship to quantities that observers and simulators can actually compute.

The results also carry weight for observational astrophysics. If naked singularities can form in realistic collapse, they would not be mere mathematical curiosities but potential astrophysical objects with distinct observational signatures. Recent theoretical work has explored how accretion flows around naked singularities differ from those around black holes: hydrodynamic and general-relativistic magnetohydrodynamic simulations of accretion onto Reissner-Nordström-type naked singularities and timelike naked singularities reveal outflows and disc properties that differ qualitatively from the black hole case. Earlier studies by Joshi, Malafarina, and Narayan showed that accretion disc profiles can in principle distinguish black holes from naked singularities. A rigorous characterization of when collapse produces visible singularities, grounded in the apparent horizon dynamics, helps identify the parameter regimes in which such exotic objects might plausibly exist and be searched for with X-ray and gravitational-wave facilities.

There are also connections to quantum gravity and to the critical phenomena discovered by Choptuik in 1993, where scalar field collapse exhibits universal scaling behavior near the threshold of black hole formation, with naked singularities appearing exactly at the critical point. The authors’ own recent research program has extended this picture, examining universality and criticality in massless scalar field collapse and the possible formation of primordial naked singularities in the early universe. If tiny naked singularities were produced in the primordial cosmos, their subsequent quantum evaporation, a possibility explored by Goswami, Joshi, and Singh in 2006, could have left observable imprints. The new timing criterion provides a framework for assessing whether such scenarios are dynamically permitted under generic, physically reasonable initial data.

Of course, important caveats remain. The analysis is restricted to spherical symmetry, and the cosmic censorship question in full generality, with rotation, gravitational radiation, and arbitrary asymmetries, remains open. Numerical studies of nonspherical collapse, from Shapiro and Teukolsky’s rotating spheroids to modern binary black hole simulations, continue to probe whether censorship holds beyond the symmetric idealization. Nevertheless, by reducing the visibility question to the concrete, computable dynamics of the apparent horizon and trapped surfaces, and by establishing the result for general type-I matter under standard energy and regularity assumptions, Koushiki and Joshi have converted a famously intractable conjecture into a sharply posed dynamical problem. The race between trapped surfaces and the singularity, it turns out, is the whole story, and knowing the rules of that race brings physicists one step closer to deciding whether the universe’s most violent endings are always hidden from view.

Subject of Research: Visibility of spacetime singularities and apparent horizon dynamics in spherical gravitational collapse

Article Title: Apparent horizon and causal structure of spacetime singularities

Article References: Koushiki, & Joshi, P. S. (2026). Apparent horizon and causal structure of spacetime singularities. General Relativity and Gravitation, 58(10), Article 114. https://doi.org/10.1007/s10714-026-03618-y

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03618-y

Keywords: general relativity, gravitational collapse, spacetime singularities, apparent horizon, trapped surfaces, naked singularity, cosmic censorship, black holes, null geodesics, event horizon, type-I matter field, weak energy condition