⚛️ general relativity

Black hole and wormhole branches in gravitational decoupling

This paper demonstrates that applying Minimal Geometric Deformation to a Schwarzschild black hole seed yields two mutually exclusive global spacetime branches—a black hole or a two-ended wormhole—depending on whether the coupling strength keeps the deformation-induced root inside or outside the original horizon, with the transition between these non-diffeomorphic manifolds dictated by the preservation of Lorentzian signature and distinguished by their second homology groups.

Original authors: Francisco Tello-Ortiz, Y. Gomez-Leyton, Vitalii Vertogradov, Jean Baez Cuevas

Published 2026-09-10

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CC BY 4.0

Original authors: Francisco Tello-Ortiz, Y. Gomez-Leyton, Vitalii Vertogradov, Jean Baez Cuevas

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

Gravity is the force that shapes the universe on the largest scales, bending the path of light and the flow of time itself. In our current understanding of physics, this force is described by the geometry of space and time, a fabric that can be stretched, twisted, and warped by matter. Two of the most extreme predictions of this theory are black holes, regions where gravity is so strong that nothing can escape, and wormholes, theoretical tunnels that could connect distant parts of the universe. For decades, physicists have treated these as separate categories of objects, often requiring different kinds of matter or energy to exist. A black hole is a one-way trap, while a wormhole is a bridge that, in theory, could be crossed. The question of whether a single physical process could naturally lead to one or the other, depending only on a specific setting, has remained a puzzle.

A new study by a team of researchers in Chile, China, Russia, and Azerbaijan explores this very question using a mathematical technique called gravitational decoupling. This method allows scientists to take a known solution to the equations of gravity, such as the description of a simple, non-rotating black hole, and add a new layer of complexity to it. Imagine starting with a clean, perfect sphere and then gently stretching it in a specific direction. The researchers applied this technique to a standard black hole model, introducing a new, invisible component of matter that interacts with gravity but does not exchange energy with the original black hole. They found that this single, consistent addition does not just make the black hole slightly different. Instead, it forces the universe to choose between two completely different realities based on a single number that controls the strength of this new interaction.

The researchers began with the classic description of a black hole, a region of space defined by a specific boundary called the event horizon, beyond which nothing can return. They then introduced a mathematical function that describes how the new matter stretches the space around the black hole. This function is fixed; it does not change. The only variable is a single number, a coupling strength, which determines how intensely this new matter affects the geometry of space. When the researchers adjusted this number, they discovered a sharp, unexpected split. If the number was above a certain critical value, the result was a familiar black hole. The new matter simply modified the space around the horizon, but the black hole remained a black hole, with a single boundary and a singular point at its center.

However, when the researchers turned the number down below that critical threshold, the same mathematical setup produced something entirely different. The point where the new matter stretched the space so much that it vanished did not stay hidden behind the black hole's horizon. Instead, it moved outward, appearing in the space where an observer could theoretically stand. At this new location, the geometry of space broke down in a way that made it impossible to describe the region between the original black hole boundary and this new point using the standard rules of physics. The fabric of space and time lost its ability to distinguish between past and future in that gap. Because this gap could not exist as a normal part of the universe, the only way to make the math work was to cut it out entirely.

When this forbidden region is removed, the remaining space does not look like a black hole at all. Instead, it forms a bridge. The new point where the space vanished becomes the narrowest part of a tunnel, connecting two separate, identical universes. This is a wormhole. The researchers proved that this is not a gradual transformation where a black hole slowly turns into a wormhole. There is no smooth path between the two. If the coupling number is on one side of the critical value, the universe is a black hole. If it is on the other side, the universe is a wormhole. The two outcomes are mutually exclusive; the same physical ingredients cannot produce both at the same time, nor can they produce a hybrid version. The choice is absolute and determined by the value of that single number.

This finding challenges the idea that black holes and wormholes are just different shapes of the same object. The study shows that they are fundamentally different structures, like two different buildings that can be constructed from the same set of blueprints depending on how a single switch is set. One building has a basement that leads to a dead end, while the other has a tunnel that leads to a second building. The switch does not change the materials; it changes the entire layout. The researchers also confirmed that for the wormhole version to exist, the new matter must violate a fundamental rule of physics known as the null energy condition. This means the matter must have properties that are not found in ordinary stars or gas, behaving in a way that allows space to stay open rather than collapsing. This is a known requirement for wormholes, and the study confirms that this new method produces exactly that kind of exotic matter.

The significance of this work lies in its clarity. It demonstrates that the global shape of the universe—whether it is a trap or a bridge—can be dictated by a simple parameter in the equations, without requiring a complex change in the underlying physics. The researchers did not claim to have found a wormhole in the sky or to have built one in a lab. Instead, they showed that the mathematics of gravity itself contains a built-in switch. Depending on how a specific interaction is tuned, the same starting point leads to two distinct, non-interchangeable realities. This provides a new way to think about how the universe might be structured, suggesting that the difference between a black hole and a wormhole might not be a matter of degree, but a matter of a single, decisive choice in the laws of nature.

Technical Summary: Black Hole and Wormhole Branches in Gravitational Decoupling

Problem Statement
The paper addresses a structural ambiguity within the Minimal Geometric Deformation (MGD) method of gravitational decoupling. While MGD is widely used to generate new solutions from known "seed" geometries (such as the Schwarzschild black hole) by introducing an anisotropic matter sector, the global topological nature of the resulting spacetime has not been rigorously classified based on the coupling strength. Specifically, the authors investigate whether a single, fixed decoupler function $h(r)$ , derived from a specific equation of state, can force a static configuration into mutually exclusive classes—either a black hole or a wormhole—solely as a function of the coupling parameter $k$ . The central question is whether the transition between these regimes represents a continuous deformation of a single spacetime or a discontinuous bifurcation into two non-diffeomorphic manifolds.

Methodology
The authors employ the MGD framework applied to a static, spherically symmetric Schwarzschild seed metric. The total energy-momentum tensor is decomposed into a seed sector and a decoupling sector ( $\theta$ -sector) coupled via a strength parameter $\alpha$ (absorbed into $k$ ).

Fixed Decoupler: The $\theta$ -sector is closed using a single equation of state. This yields a unique, fixed radial function $h(r)$ that encodes the deformation, independent of the coupling strength $k$ .
Deformed Metric: The radial metric component is modified by a factor $F(r; k) = 1 + k h(r)$ . The full metric is given by $ds^2 = -A(r)dt^2 + [A(r)F(r; k)]^{-1}dr^2 + r^2d\Omega^2$ .
Root Analysis: The authors analyze the roots of $F(r; k) = 0$ . They distinguish between roots lying inside the seed horizon ( $r < 2M$ ) and those appearing in the exterior ( $r > 2M$ ).
Signature Obstruction: A critical step involves analyzing the Lorentzian signature of the metric in the interval between the seed horizon ( $r=2M$ ) and a root $r^*$ that has moved outside the horizon.
Topological Classification: The authors compute the second homology group ( $H_2$ ) of the spatial hypersurfaces for both potential global completions (black hole exterior vs. doubled wormhole) to establish a discrete invariant distinguishing the branches.

Key Contributions and Results

The Signature Obstruction (Lemma 1): The paper proves that if the function $F(r; k)$ develops a simple root $r^*$ outside the seed horizon ( $r^* > 2M$ ), the metric component $g_{rr}$ becomes negative in the interval $(2M, r^*)$ while $g_{tt}$ remains negative. This results in a signature of $(-,-,+,+)$ , which is non-Lorentzian. Consequently, the standard smooth extension of the exterior metric through the seed horizon is mathematically impossible.
Forced Global Completion (Proposition 1): Due to the signature obstruction, the only admissible smooth Lorentzian completion for the case $r^* > 2M$ is a "doubled" manifold. The region $(2M, r^*)$ is excised, and the geometry is extended by attaching a second copy of the region $r \ge r^*$ across the minimal sphere $T = \{r = r^*\}$ . This creates a two-ended wormhole.
Mutually Exclusive Branches: The study demonstrates that the same fixed function $h(r)$ $h (r)$ generates two distinct, non-diffeomorphic manifolds depending on $k$ $k$ :
- Black Hole Branch ( $k > k_c$ ): The root $r^*$ lies inside the horizon (or does not exist in the exterior). The spacetime retains the Schwarzschild horizon structure.
- Wormhole Branch ( $k < k_c$ ): The root $r^*$ lies outside the horizon. The seed horizon is excised, and the spacetime is a wormhole.
Discrete Invariant: The authors explicitly compute the homology groups to distinguish the branches:
- For the black hole exterior relative to its horizon: $H_2(\Sigma_{BH}, H) = 0$ .
- For the completed wormhole manifold: $H_2(\Sigma_{WH}) \cong \mathbb{Z}$ .
  This discrete change in homology confirms that the transition is not a continuous deformation of a single manifold but a bifurcation between two topologically distinct solutions.
Energy Conditions: The analysis confirms that the wormhole branch necessarily violates the Null Energy Condition (NEC) at the throat, consistent with the Morris-Thorne theorem. In the explicit example provided, the NEC is violated throughout the entire exterior region of the wormhole branch.
Photon Sphere: The paper notes that the MGD deformation does not shift the radius of the photon sphere as long as it lies outside the throat. If the throat moves outside the seed photon sphere, the standard photon sphere is removed from the accessible exterior.

Significance and Claims
The paper claims to identify a structural property intrinsic to the gravitational decoupling method itself, rather than an artifact of a specific deformation profile. The primary significance lies in the demonstration that a single, physically sourced decoupler function can force a bifurcation into two geometrically inequivalent classes of spacetimes based solely on the value of a coupling parameter.

Crucially, the authors emphasize that this is not a claim of a dynamical topology change (which would violate Geroch-Tipler no-go theorems). Instead, for any fixed value of $k$ , there exists exactly one admissible static spacetime. The "switch" between black hole and wormhole is a discontinuous selection between two different static solutions, dictated by the requirement of a smooth Lorentzian metric. The metric itself, via the signature obstruction, dictates which completion is admissible for a given $k$ .

The work clarifies that the global properties of MGD solutions depend not only on the local field equations of the $\theta$ -sector but also on the global requirement of Lorentzian smoothness, which acts as a selection criterion for the manifold's topology. The paper does not claim to have discovered a new wormhole solution but rather provides a classification theorem for the MGD solution space, showing that black holes and wormholes can be forced alternatives of a single deformation sector.

Technical Summary: Black Hole and Wormhole Branches in Gravitational Decoupling

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