Einstein-Maxwell fields as solutions of Einstein gravity coupled to conformally invariant non-linear electrodynamics

This paper establishes a criterion for extending Einstein-Maxwell solutions to conformally invariant non-linear electrodynamics via constant field rescaling, demonstrating that all static or twistfree configurations are extendable and providing explicit examples such as generalized black holes and gravitational waves.

The Gist

Imagine the universe as a giant, complex video game. For decades, physicists have been playing with the "standard rules" of how gravity and electricity interact, known as Einstein-Maxwell theory. These rules work great for most things, but they are a bit like a basic version of the game: they assume electricity behaves in a perfectly straight, predictable line (linear).

However, in the real world, or in extreme environments like the center of a black hole, electricity might get "squishy" or "stretchy." It might not follow those straight lines anymore. Physicists call these more complex, "squishy" rules Non-Linear Electrodynamics (NLE).

The problem? Solving the equations for these "squishy" rules is incredibly hard. It's like trying to solve a Sudoku puzzle where the numbers keep changing their rules every time you look at them.

The Big Discovery: The "Universal Adapter"

This paper, written by Marcello Ortaggio, discovers a special "Universal Adapter" for this video game.

Here is the core idea:
There is a specific group of solutions (patterns of gravity and electricity) in the standard game that are immune to the rules changing. If you take one of these solutions and simply turn a "dial" (rescale the electric field by a constant number), it instantly becomes a valid solution for any of the complex, "squishy" theories, including the newly proposed ModMax theory.

Think of it like this:

• The Standard Game: You have a map of a city with straight roads.
• The Complex Game: The roads are now made of rubber and stretch when cars drive on them.
• The Discovery: Ortaggio found that if you have a specific type of city layout (a "static" one where the roads don't twist and turn dynamically), you can take that exact map, stretch the rubber roads slightly, and it works perfectly in the new game without having to redraw the whole city.

The "Static" Secret

How do you know which maps work? The paper proves a very cool rule: If the universe is "static" (not spinning or twisting in a weird way), it works.

Imagine a spinning top. If the top is wobbling and twisting, it's hard to predict how the rubber roads will stretch. But if the top is perfectly still (static) or spinning in a very simple, straight line, the "stretchiness" of the electricity doesn't break the solution.

The author shows that:

1. Static Universes: Any solution where gravity and electricity are calm and steady (like a black hole sitting still) can be upgraded to these new theories.
2. The "Dial" (Rescaling): You don't need to re-solve the math. You just take the electric field you already have, multiply it by a specific number (the "dial"), and boom—you have a solution for the new, complex theory.

Why This Matters

Before this paper, if a physicist wanted to study a black hole in a "squishy" electricity universe, they would have to start from scratch and do years of difficult math.

Now, they can:

1. Look at the huge library of known solutions for standard gravity and electricity.
2. Pick the "static" ones.
3. Apply the "Universal Adapter" (the rescaling).
4. Instantly have a valid solution for the new, complex theories.

The Examples (The "Showcase")

The paper doesn't just talk theory; it shows off the "Universal Adapter" with some famous examples:

• Ozsváth's Universe: A weird, homogeneous universe that looks the same everywhere.
• The Black Hole in a Rubber Universe: A black hole sitting inside a universe that is already filled with strong magnetic fields (the Levi-Civita/Bertotti-Robinson universe).
• The Accelerating Black Hole: A black hole that is being pulled apart (the C-metric).
• Gravitational Waves: Ripples in space-time moving through a magnetic field.

The Takeaway

In simple terms, this paper is a shortcut. It tells us that for a huge class of calm, non-twisting cosmic scenarios, the universe is surprisingly robust. Whether electricity is "straight" (Maxwell) or "squishy" (Non-Linear), the fundamental shapes of these cosmic structures remain the same; they just need a little adjustment to the strength of the electric field.

It's like realizing that a perfectly still house will stand up just fine whether the wind is a gentle breeze or a hurricane, as long as you just reinforce the walls slightly. You don't need to rebuild the house; you just need to know how to reinforce it.

Technical Summary

1. Problem Statement

The paper addresses the difficulty of finding exact solutions in Non-Linear Electrodynamics (NLE) coupled to Einstein gravity. While exact solutions are well-known for linear Maxwell theory, the non-linear nature of general NLE theories (such as Born-Infeld or ModMax) typically renders the field equations intractable.

The specific problem investigated is: Under what conditions can a known solution of the Einstein-Maxwell (EM) equations be extended to become a solution for any Conformally Invariant Non-Linear Electrodynamics (CINLE) theory?

The author focuses on the subclass of NLE theories that are conformally invariant, characterized by a Lagrangian density $L(I, J) = I W(x)$ where $x = I/J$ and $I, J$ are the electromagnetic invariants. The goal is to identify a class of "extendable" EM solutions that remain valid (up to a constant rescaling of the field) regardless of the specific functional form of the non-linear function $W(x)$.

2. Methodology

The author employs a combination of algebraic analysis of the field equations and geometric arguments involving Killing vector fields.

• Field Equations Analysis: The paper starts with the action for Einstein gravity coupled to CINLE. By varying the action, the author derives the Einstein equations and the generalized Maxwell equations.
  • The Einstein equation differs from the standard Einstein-Maxwell case only by an overall factor $-2(xW)'$ on the energy-momentum tensor side.
  • The generalized Maxwell equation involves a 2-form $H_{ab}$ which is a linear combination of $F_{ab}$ and its dual $\ast F_{ab}$ with coefficients depending on $x$.
• The "Extendability" Criterion: The author posits that if the ratio of the electromagnetic invariants $x = I/J$ is constant throughout the spacetime, the factor $(xW)'$ becomes a constant. This allows the generalized Maxwell equations to be satisfied by a simple constant rescaling of the original Maxwell field strength $F_{ab} \to \Omega^{-1} F_{ab}$, while keeping the metric $g_{ab}$ unchanged.
• Newman-Penrose (NP) Formalism: To characterize these solutions geometrically, the author utilizes the complex self-dual 2-form formalism and the NP tetrad. The condition $x = \text{const}$ is translated into a condition on the phase of the complex scalar $\Phi_1$ (the only non-vanishing Maxwell scalar for non-null fields). Specifically, the phase $\theta$ must be constant.
• Geometric Proofs (Staticity): The paper proves that for static configurations (possessing a timelike, hypersurface-orthogonal Killing vector field), the electric and magnetic field vectors are necessarily parallel (up to a constant factor). This is revisited using Das' alignment condition, proving that the phase $\theta$ is constant, thereby satisfying the extendability criterion. This result is extended to spacelike, twist-free Killing vector fields.

3. Key Contributions

1. Derivation of the Extendability Criterion: The paper establishes a necessary and sufficient condition for an Einstein-Maxwell solution to be extendable to any CINLE: the electromagnetic field must be duality-equivalent to a purely electric or purely magnetic field (i.e., the ratio of invariants $I/J$ is constant, or equivalently, the phase of the self-dual field is constant).
2. Proof for Static and Twist-Free Configurations: It is rigorously proven that all static Einstein-Maxwell solutions (and more generally, those with a non-null, twist-free Killing vector field) satisfy this criterion. This implies that the vast literature of static EM solutions is directly applicable to the entire class of CINLE theories.
3. Duality Invariance in General Theories: The work highlights that the duality invariance of the underlying Einstein-Maxwell theory allows for the construction of dyonic solutions (containing both electric and magnetic charges) in more general CINLE theories, provided the solution is static.
4. Explicit Construction of New Solutions: The paper provides a systematic method to generate exact solutions for specific CINLE models (like ModMax) by taking known Einstein-Maxwell solutions and applying a constant rescaling factor $\Omega$ derived from the specific $W(x)$ of the theory.

4. Results and Examples

The author demonstrates the utility of this method by presenting several explicit examples of Einstein-Maxwell solutions that can be extended to CINLE:

• Ozsváth's Homogeneous Universe: A Petrov type I solution with a negative cosmological constant. Although the timelike Killing vector is twisting, a spacelike twist-free vector exists, making it extendable.
• Black Hole in the LCBR Universe: A Schwarzschild black hole immersed in the Levi-Civita, Bertotti, and Robinson universe. The paper shows how to extend the dyonic version of this solution.
• Interpolating Black Hole: A solution interpolating between the Bonnor-Melvin universe and the LCBR universe, obtained via Harrison transformation.
• Generalized C-Metric: An accelerating charged black hole solution. The paper notes that both the standard charged C-metric and its generalizations (including those with acceleration in a Bonnor-Melvin background) are extendable.
• Non-expanding Gravitational Waves: A time-dependent example of gravitational waves in a Bonnor-Melvin background. While not covered by the staticity proof, it is shown to be extendable due to the specific form of its electromagnetic field.

5. Significance

• Bridging Linear and Non-Linear Theories: The paper provides a powerful "shortcut" for researchers. Instead of solving complex non-linear differential equations for every new NLE model, one can simply take a known Einstein-Maxwell solution, verify the extendability condition (usually satisfied by staticity), and apply a constant rescaling.
• Universality of Static Solutions: It establishes that the "universality" of solutions (being independent of the specific non-linear corrections) is not limited to null fields (as previously known) but extends to a broad class of non-null, static configurations.
• ModMax and Beyond: The results are particularly relevant for the recently proposed ModMax electrodynamics, which is conformally invariant and duality invariant. The paper confirms that all static Einstein-ModMax solutions can be derived directly from Einstein-Maxwell literature.
• Duality and Dyons: The work clarifies that even in theories where duality symmetry might be broken or modified, the existence of static dyonic solutions is guaranteed if the underlying Einstein-Maxwell solution is static, due to the constant phase property.

In summary, Ortaggio's work significantly expands the toolkit for finding exact solutions in modified gravity and electrodynamics by identifying a large, theory-independent class of solutions characterized by the constancy of the electromagnetic invariant ratio.