Radiating solutions in Entangled Relativity

This paper demonstrates that Mineur--Vaidya radiating solutions can be consistently embedded within Entangled Relativity and other Einstein--Maxwell--dilaton theories as limits of vanishing electromagnetic fields, thereby proving that these frameworks, like General Relativity, permit the dynamical formation of naked singularities.

The Gist

Imagine the universe as a giant, stretchy trampoline. In our current best understanding of physics (General Relativity), massive objects like stars sit on this trampoline, causing it to curve. This curvature is what we feel as gravity.

For decades, physicists have wondered: Is General Relativity the only rulebook for gravity? Or is there a hidden "secret sauce" that changes how gravity works under extreme conditions, like inside a black hole?

This paper explores a new, slightly different rulebook called Entangled Relativity. Here is the story of what the authors found, explained without the heavy math.

1. The "Ghost" Problem

In the new theory (Entangled Relativity), gravity isn't just about bending space; it's also about a mysterious "ghost" variable (a scalar field) that connects matter and the curvature of space. Think of this ghost as a volume knob that adjusts how loud gravity is.

However, there's a catch. This volume knob is defined by a specific ratio: How much matter is there compared to how much the space is curving?

The authors looked at a famous, classic scenario in physics: The Mineur-Vaidya solution. This describes a star that is rapidly losing mass by shooting out light (radiation) in all directions.

• The Problem: In this specific scenario, the math says the "curvature" is zero and the "matter" is zero in a way that makes the ratio undefined. It's like trying to calculate the price of an apple by dividing zero apples by zero dollars. The "volume knob" breaks.
• The Consequence: Because the knob breaks, the classic "radiating star" solution doesn't technically work in this new theory. It's a dead end.

2. The "Flavoring" Solution

To fix this, the authors decided to add a little "flavor" to the soup. They imagined that this radiating star isn't just sitting in empty space; it's sitting inside a massive magnetic or electric field.

Think of it like this:

• Old Scenario: A campfire in a vacuum. (The math breaks).
• New Scenario: A campfire inside a giant, swirling magnetic storm.

By adding this magnetic (or electric) field, the "ghost" volume knob suddenly has something to measure. The ratio becomes defined again. The math works!

3. The "Limit" Trick

Here is the clever part of the discovery. The authors showed that if you take their new solution (the star in the magnetic storm) and slowly turn the magnetic field down to zero, the solution smoothly transforms back into the classic Mineur-Vaidya solution.

It's like a video game where you can toggle a "Magnetic Mode."

1. Magnetic Mode ON: The physics works perfectly in the new theory.
2. Magnetic Mode OFF: The game reverts to the classic General Relativity rules.

This proves that the classic solution is actually a valid part of the new theory; it just lives at the very edge where the magnetic field disappears.

4. The Big Surprise: Naked Singularities

Why does this matter? The main goal was to see if this new theory could prevent the formation of singularities (points of infinite density, like the center of a black hole).

Some scientists had speculated that Entangled Relativity might act like a "repulsive gravity" force, preventing stars from collapsing into singularities, or at least hiding them behind event horizons (the point of no return). This would save the "Cosmic Censorship Conjecture"—the idea that nature always hides its ugly, infinite secrets behind a black hole's veil.

The Verdict: The authors found that no, it doesn't hide them.
Even with this new theory, if you collapse a star in a specific way (like the Mineur-Vaidya scenario), you can create a Naked Singularity.

• Hidden Singularity: A black hole. The infinite point is hidden behind a curtain (the event horizon).
• Naked Singularity: The infinite point is exposed to the rest of the universe. The curtain is ripped off.

The paper shows that Entangled Relativity allows for these "naked" singularities just as easily as standard General Relativity does. The "ghost" volume knob doesn't save the day; the universe can still tear its own fabric in a way that is visible to the outside world.

Summary

• The Theory: Entangled Relativity is a new way to look at gravity that adds a "volume knob" to the equations.
• The Issue: A classic physics solution (a radiating star) broke the math because the knob had nothing to measure.
• The Fix: The authors added a magnetic field to the mix, which fixed the math. They then showed that removing the field brings you back to the classic solution.
• The Result: This proves that Entangled Relativity allows for Naked Singularities. It means that, contrary to some hopes, this new theory doesn't magically prevent the universe from creating "naked" points of infinite chaos. It behaves very much like our current understanding of gravity.

In short: The universe is still capable of ripping its own fabric, even with this new rulebook.

Technical Summary

1. Problem Statement

The paper addresses a specific theoretical gap in Entangled Relativity (ER), a non-linear reformulation of General Relativity (GR) proposed as a parsimonious alternative with fewer fundamental constants.

• The Core Issue: ER introduces an additional scalar degree of freedom defined by the ratio between the Ricci scalar ($R$) and the matter Lagrangian ($L_m$), specifically $\vartheta = -L_m/R$.
• The Singularity Problem: In the classic Mineur–Vaidya radiating solutions (which describe spherically symmetric gravitational collapse with outgoing/ingoing null radiation), the stress-energy tensor is traceless ($T=0$) and conformally invariant. Consequently, $R=0$ and $L_m=0$ everywhere. This renders the scalar field $\vartheta$ ill-defined (division by zero), meaning the standard Mineur–Vaidya solution is strictly not a solution to Entangled Relativity.
• The Question: Can Entangled Relativity support the formation of singularities, including naked singularities (which violate the Cosmic Censorship Conjecture), given that its defining scalar field breaks down in the standard radiating collapse scenarios? Previous speculation suggested ER might resolve singularities via repulsive gravity; this paper tests that hypothesis.

2. Methodology

The authors employ a constructive approach to embed the radiating solutions into a framework where the scalar field is well-defined, then take a limit to recover the original scenario.

1. Embedding in Einstein–Maxwell–Dilaton (EMD) Theories:

   • The authors recognize that ER is a specific case of EMD theories (with coupling constant $\alpha = 1/(2\sqrt{3})$).
   • They utilize the Mineur–Vaidya–Melvin procedure: embedding the standard Vaidya radiating metric into a background magnetic (or electric) field.
   • In this embedded scenario, the presence of the electromagnetic field ensures that $L_m \neq 0$ and $R \neq 0$, making the scalar field $\vartheta$ well-defined.

2. Derivation of Solutions:

   • Magnetic Case: They derive the metric, vector potential, and scalar field for a radiating solution embedded in a magnetic field within the EMD framework.
   • Electric Case: They apply a duality transformation (involving the Levi-Civita tensor and the scalar field) to generate the electric counterpart.
   • Conformal Transformation: Since ER is defined in a specific "entangled frame" (distinct from the Einstein frame used in standard EMD formulations), the authors perform an inverse conformal transformation ($g_{\mu\nu} = e^{4\alpha\phi}\tilde{g}_{\mu\nu}$) to map the EMD solutions back to the ER field equations.

3. Limit Analysis:

   • The authors analyze the behavior of these new solutions as the magnetic (or electric) field strength ($B$) approaches zero.
   • They verify that in the limit $B \to 0$, the metric converges to the standard Mineur–Vaidya solution, and the scalar field behavior remains consistent with the theory's requirements.

4. Verification:

   • The solutions were verified using symbolic computation notebooks (SageManifolds), which are made publicly available by the authors.

3. Key Contributions

• Explicit Solutions in ER: The paper provides the first explicit, exact radiating solutions for Entangled Relativity that are mathematically well-defined. These solutions include a magnetic or electric field to regularize the scalar degree of freedom.
• Resolution of the Ill-Defined Scalar: The work demonstrates that the apparent singularity in the definition of ER's scalar field for Vaidya solutions is an artifact of the vacuum limit. By embedding the solution in a field, the theory remains consistent.
• Generalization to EMD Theories: The authors prove that the Mineur–Vaidya solution is a valid solution for all Einstein–Maxwell–dilaton theories (for any coupling constant $\alpha$) when the dilaton is constant, provided the source is a conformally invariant null field.
• Limit Recovery: They rigorously show that the standard Mineur–Vaidya solution is the limiting case of these new solutions as the external field vanishes.

4. Results

• Existence of Singularities: The derived solutions in Entangled Relativity exhibit the same singularity structure as their General Relativity counterparts. Specifically, the collapse of the radiating matter leads to the formation of a curvature singularity.
• Formation of Naked Singularities: Crucially, the paper demonstrates that naked singularities (singularities not hidden behind an event horizon) can form dynamically in Entangled Relativity.
  • This holds true for both the magnetic and electric embedded cases.
  • The analysis along the axis of symmetry ($\theta=0$) confirms that the lack of spherical symmetry (introduced by the Melvin field) does not prevent the formation of naked singularities.
• Stability: The resulting Cauchy horizons are noted to be stable against non-spherical perturbations, similar to the GR case.
• Universality: The conclusion that naked singularities can form applies to the entire class of Einstein–Maxwell–dilaton theories, not just ER.

5. Significance

• Refutation of "Repulsive Gravity" Resolution: The results directly challenge the speculative argument (from previous literature) that Entangled Relativity might resolve classical singularities through repulsive gravity. The paper proves that ER, like GR, allows for the formation of naked singularities under specific radiating collapse conditions.
• Cosmic Censorship: The findings suggest that Entangled Relativity does not inherently enforce the Cosmic Censorship Conjecture. If naked singularities are physically realizable in GR, they remain so in ER.
• Theoretical Robustness: By showing that ER recovers the pathological features of GR (naked singularities) in the appropriate limits, the paper establishes ER as a viable, albeit non-trivial, alternative to GR that shares its core phenomenological predictions in the observable universe (dust/radiation dominated).
• Clarification of ER's Domain: The work clarifies the mathematical domain of ER, showing that while the scalar field definition is sensitive to the $L_m/R$ ratio, the theory remains consistent and predictive when matter fields are non-trivial (non-zero trace or non-conformal).

In summary, Minazzoli and Wavasseur demonstrate that Entangled Relativity is not a "singularity-free" theory; rather, it inherits the capacity for naked singularity formation from General Relativity, thereby preserving the same fundamental challenges regarding the Cosmic Censorship Conjecture.