⚛️ general relativity

Schwarzschild black-hole immersed in an electric or magnetic background in Entangled Relativity

This paper presents the first exact neutral black-hole solutions in Entangled Relativity by describing a Schwarzschild black-hole immersed in a Melvin-like electric or magnetic background, demonstrating that these solutions recover General Relativity's Schwarzschild limit and suggest astrophysical black-holes in this theory are observationally indistinguishable from those in General Relativity due to weak interstellar matter fields.

Original authors: Olivier Minazzoli, Maxime Wavasseur

Published 2026-02-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Olivier Minazzoli, Maxime Wavasseur

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A Universe That Needs "Stuff" to Exist

Imagine General Relativity (Einstein's famous theory of gravity) as a stage play. In Einstein's version, the stage (spacetime) can exist even if there are no actors on it. You can have an empty, silent stage, and the laws of physics still work.

The theory presented in this paper, called Entangled Relativity, says: "No, that's impossible."

In this new theory, the stage and the actors are so tightly linked that the stage cannot exist without the actors. If you try to remove all the matter (the actors) from the universe, the theory says the stage itself disappears or becomes undefined. This is based on a concept called Mach's Principle, which suggests that space and time are defined entirely by the matter inside them.

The Problem: Finding a "Neutral" Black Hole

Scientists wanted to test this theory by looking at black holes.

The Old Way: Previous studies looked at black holes that had an electric charge (like a battery). This worked because the electric field counts as "matter," so the theory could function.
The New Challenge: Real black holes in space are usually "neutral" (they don't have a massive electric charge). If you take away the charge, you are left with a vacuum. According to Entangled Relativity, a vacuum shouldn't exist. So, how can you have a neutral black hole in this theory?

The Solution: The "Background Noise" Trick

The authors solved this puzzle by imagining the black hole isn't in a perfect vacuum, but is instead sitting inside a very faint, invisible "background field" (like a magnetic or electric field filling the whole universe).

Think of it like this:

The Black Hole is a heavy rock in the middle of a pond.
The Background Field is a very gentle, constant breeze blowing across the water.

Even though the rock itself isn't "charged," the breeze (the background field) provides the necessary "stuff" for the theory to work. The authors found exact mathematical solutions for a black hole sitting in a magnetic field and another sitting in an electric field.

The Surprising Result: It Looks Just Like Einstein's

Here is the most important finding: When the background breeze gets weaker and weaker, the solution smoothly turns into the standard black hole we know from Einstein's General Relativity.

The Analogy: Imagine you are wearing noise-canceling headphones. When the background noise (the magnetic field) is loud, the headphones (the theory) work differently than when it's quiet. But as you turn the volume down to zero, the headphones behave exactly like a normal pair of ears.
The Discovery: The authors found that as the background magnetic or electric field approaches zero, their complex new equations simplify perfectly into Einstein's famous Schwarzschild black hole equations.

This is huge because it means that for all practical purposes, black holes in this new theory look exactly the same as black holes in Einstein's theory.

Why This Matters

No More "Vacuum" Paradox: They proved that you can have a neutral black hole in this theory without breaking the rule that "space needs matter to exist." The background field acts as that necessary matter.
Indistinguishable from Reality: Since the magnetic fields in our galaxy are incredibly weak, the black holes we observe in the real world (like the one in the center of our galaxy) would look identical whether you use Einstein's old theory or this new "Entangled" theory.
A Twist on the Melvin Solution: In Einstein's theory, if you remove the black hole from a magnetic field, you get a specific shape of space called the "Melvin solution." In this new theory, if you remove the black hole, you get something slightly different. However, since we don't see black holes without magnetic fields in the real world, this difference is mostly a mathematical curiosity.

The Bottom Line

The authors found a way to describe a neutral black hole in a universe where "empty space" is forbidden. They did this by placing the black hole in a faint cosmic background field. The result is reassuring: our current observations of black holes cannot tell the difference between Einstein's General Relativity and this new Entangled Relativity. The new theory successfully mimics the old one in the conditions we see in our universe.

Technical Summary: Schwarzschild Black-Hole Immersed in an Electric or Magnetic Background in Entangled Relativity

Problem Statement
Entangled Relativity (ER) is a reformulation of General Relativity (GR) characterized by a non-linear coupling between the Ricci scalar curvature ( $R$ ) and matter fields ( $L_m$ ). A defining feature of ER is that it precludes the existence of vacuum solutions ( $L_m = 0$ ), thereby satisfying Einstein's definition of Mach's Principle where the metric field is wholly determined by matter. Consequently, standard vacuum solutions of GR, such as the Schwarzschild black hole, are not exact solutions in ER.

Previous investigations into black holes within ER focused on charged black holes, where the electromagnetic field ensures $L_m$ is non-zero everywhere. These studies demonstrated that the Schwarzschild solution emerges as a limit when the charge (and thus the matter field) vanishes. However, astrophysical black holes are expected to be electrically neutral. This creates a theoretical gap: it was previously unknown whether ER could support exact, neutral black-hole solutions immersed in background fields, which are necessary to model realistic astrophysical scenarios without violating the theory's prohibition of vacuum solutions.

Methodology
The authors sought exact, neutral (spherical) black-hole solutions in ER by immersing a Schwarzschild black hole in external electric or magnetic background fields, analogous to the Schwarzschild-Melvin solution in GR.

Theoretical Framework: The theory is defined by a quantum phase $\Theta$ involving the ratio of the matter Lagrangian density squared to the Ricci scalar ( $L_m^2/R$ ). The classical field equations are derived from the stationary phase condition ( $\delta \Theta = 0$ ).
Field Equations: The authors utilized the specific field equations for ER, which include an additional scalar degree of freedom sourced by the difference between the trace of the stress-energy tensor ( $T$ ) and the matter Lagrangian ( $L_m$ ).
Ansatz and Solution:
- Magnetic Case: The authors constructed a metric and electromagnetic 4-vector ansatz similar to the GR Schwarzschild-Melvin solution but modified by conformal factors dependent on the ratio $L_m/R$ . They verified the solution using the SageManifolds computational package.
- Electric Case: The electric solution was derived from the magnetic solution via a specific duality transformation involving the electromagnetic tensor and a conformal rescaling of the metric.
Verification: The solutions were checked to ensure they satisfy the ER field equations, specifically verifying that the ratio $R/L_m$ remains well-defined even as the background field strength approaches zero.

Key Contributions and Results
The paper presents the first exact, neutral black-hole solutions in Entangled Relativity.

Magnetic Solution: The authors derived a metric for a Schwarzschild black hole immersed in a magnetic field aligned with the $z$ $z$ -axis. The metric includes a conformal factor $\Lambda$ $Λ$ dependent on the magnetic field strength $B$ $B$ and the Schwarzschild radius $r_S$ $r_{S}$ . Crucially, the solution satisfies $T=0$ $T = 0$ (as expected for a pure electromagnetic field) but maintains $L_m \neq 0$ $L_{m} \neq = 0$ everywhere, thus avoiding the vacuum prohibition.
- The ratio $R/L_m$ is found to be $-\Lambda^{-2/13}$ , which remains finite and well-defined even as $B \to 0$ .
- The solution is algebraically general (Petrov classification), similar to the GR counterpart.
Electric Solution: An analogous solution was derived for an electric background field. The metric and vector potential differ from the magnetic case by specific conformal factors and sign changes in the Lagrangian ( $L_m > 0$ $L_{m} > 0$ for electric, $L_m < 0$ $L_{m} < 0$ for magnetic).
- The ratio $R/L_m$ is found to be $-\Lambda^{2/13}$ .
- Like the magnetic case, the solution is well-defined in the limit where the field vanishes.
Limiting Cases:
- Vanishing Field ( $B, E \to 0$ ): As the background field approaches zero, the metric converges exactly to the Schwarzschild metric of General Relativity. The ratio $R/L_m$ remains well-defined in this limit, allowing the scalar degree of freedom to settle to a constant value, recovering GR equations.
- Vanishing Black Hole ( $r_S \to 0$ ): When the black hole size is set to zero, the solution does not reduce to the standard Melvin solution of GR. Instead, it retains a structure distinct from GR due to the sourced scalar degree of freedom.
- Vanishing Both ( $B, r_S \to 0$ ): The solution converges to Minkowski space, with the ratios $R/L_m$ and $L_m/R$ remaining finite throughout the limit.

Significance and Claims
The authors claim these results are significant for several reasons:

Existence of Neutral Solutions: This work confirms that Entangled Relativity can support neutral black-hole solutions, resolving a previous uncertainty in the theory.
Recovery of General Relativity: The Schwarzschild black hole of GR emerges as a valid limit of ER solutions when the background matter field (electric or magnetic) vanishes. This suggests that astrophysical black holes in ER are indistinguishable from those in GR, given the extremely weak interstellar density of matter fields and the negligible charge of astrophysical black holes.
Distinction from GR Limits: The paper highlights a subtle but important difference: while the Schwarzschild limit is recovered, the Melvin solution (pure background field without a black hole) is not recovered when the black hole size vanishes. This implies that the "vacuum" of ER is fundamentally different from the vacuum of GR, even if they appear similar in the presence of a massive object.
Stability of the Coupling Parameter: The authors note that the gravitational coupling parameter $\kappa$ (proportional to $L_m/R$ ) maintains a consistent sign across both electric and magnetic solutions, despite $L_m$ changing sign. This prevents the theory from predicting repulsive gravity in these regimes, maintaining consistency with observations.
Observational Implications: The authors conclude that, based on these findings, it is unlikely that current or near-future observations of black holes (such as shadow imaging or gravitational wave ringdowns) will be able to distinguish between ER and GR. They suggest that neutron stars may be more promising targets for testing the theory's deviations.

The paper does not propose new experimental setups or applications but frames these exact solutions as a necessary step to validate the theory's viability under astrophysical conditions.