The conceptual foundations were laid in 1935, when Albert Einstein and Nathan Rosen found that the Schwarzschild solution, sliced in a particular way, contains a bridge joining distant regions of space-time—a bridge that pinches shut too quickly for anything to cross. The modern era opened in 1988, when Michael Morris and Kip Thorne specified what a wormhole a traveler could actually survive would require. Their analysis produced the field’s defining embarrassment: keeping a throat open violates the classical energy conditions, the inequalities that normally forbid, among other things, a locally negative energy density. The culprit is geometric. The flare-out condition—the requirement that the funnel open back up rather than close—forces matter at the throat into behavior no known substance exhibits. In the decades since, theorists have tried to shrink, localize or disguise this exotic ingredient, embedding wormholes in phantom energy, Casimir vacuum, dark matter halos, loop-quantum-gravity corrections and the effective geometries of modified gravity.

Thin-shell wormholes are the most economical realization of that program. Instead of spreading strange matter through the bulk of space, the cut-and-paste construction takes two identical copies of a well-behaved seed geometry and glues them along a spherical surface—the shell—which becomes the wormhole’s throat. All the exotic material is then concentrated on that two-dimensional interface, like surface tension on a soap bubble. The price of gluing is computed with the Darmois–Israel junction conditions, the relativistic bookkeeping that converts the jump in extrinsic curvature across the shell into a surface stress tensor. From those conditions the authors read off the two numbers characterizing the throat’s supporting fluid: a surface energy density and a tangential pressure. Whether the object survives its own perturbations is decided by the linearized stability analysis introduced for Schwarzschild shells by Eric Poisson and Matt Visser and refined since for charged, cosmological, rotating, higher-dimensional and quantum-corrected backgrounds. Squeeze the throat slightly, and the question becomes brutally simple: does it spring back, or does it run away?

The novelty of the new work is the environment. Cosmic voids are the emptiest regions of the Universe: expanses where the matter density plunges far below the cosmic average, wrapped in walls and filaments of galaxies and together occupying a substantial fraction of the volume of space. Because they are weak-field, weakly screened environments, voids amplify subtle gravitational signatures that are difficult to isolate in dense clusters, which has made them a favorite hunting ground for dark energy and modified-gravity effects. The team anchors its geometry in the universal density profile of Hamaus, Sutter and Wandelt, a phenomenological formula that captures both the underdense core of a typical void and the compensating overdense wall around it. The profile is set by a mean background density, a negative density contrast, a scale radius, a void radius and two shape parameters controlling the inner and outer slopes. Crucially, the void’s contribution imprints a de Sitter-like character on the gravitational field, so the environment behaves on large scales like the exponentially expanding space associated with a positive cosmological constant.

Drop a black hole into that profile and something qualitatively new appears. The lapse function—the quantity that governs how clocks and radial distances are warped—acquires two roots instead of one. The inner root is an ordinary black-hole horizon; the outer root is a cosmological-like horizon generated not by a true cosmological constant but by the de Sitter-like character of the void itself, in close analogy with the Schwarzschild–de Sitter solution. Between the two horizons lies a finite region where the lapse function is positive, and it is precisely there that the authors perform their surgery. Following the cut-and-paste recipe, they take two copies of this black-hole-in-void spacetime, excise everything beyond a chosen radius and sew the remaining pieces throat to throat. The result carries no exotic matter in the bulk at all: whatever strange substance holds it open lives entirely on the shell, and the shell is forbidden from approaching either horizon. The throat must sit strictly inside the window between the black-hole horizon and the void’s cosmological-like boundary.

Because the seed geometry is not isolated—its mass function carries the void’s density profile inside it—every quantity on the shell inherits that cosmic fingerprint. The surface energy density and tangential pressure, and through them the null, weak, dominant and strong energy-condition combinations, can be written explicitly in terms of the void mass function and density profile. The degree of exoticity demanded at the throat is therefore not a free parameter; it is dictated by how empty, how large and how steeply walled the surrounding void happens to be. That ties together three ingredients usually studied in isolation: the statistical structure of the cosmic web, the horizon structure of compact objects and the classical stability theory of wormholes. It also sharpens a conceptual distinction. Unlike a continuous Morris–Thorne wormhole supported by a fluid filling space, this object’s exotic matter is confined to a junction surface whose admissible radius is boxed in on both sides by horizons born of the environment.

The authors also take the thermodynamics of the shell seriously. A static shell hovering at a fixed radius possesses an associated temperature—an Unruh-type temperature felt by observers stationed at the throat—and the paper derives a first law for the configuration. The striking part is what the first law connects. The shell’s entropy is tied directly to the entropies of the two horizons that bracket it: the black-hole horizon on the inside and the cosmological-like horizon on the outside. The throat’s thermodynamic ledger is therefore not self-contained; it knows about the large-scale void through the outer horizon. This dovetails with a recently developed unified thermodynamic framework for thin-shell wormholes, in which a generalized first and second law relate the shell’s temperature to Hawking-like particle creation. In the void setting, the framework gains an environmental dial: alter the void’s density contrast or size, and the thermodynamic budget of the throat shifts with it.

Stability is where the study earns its keep. The radial motion of the throat is recast as a particle rolling in a one-dimensional effective potential; a static shell is an equilibrium point of that potential, and the sign of its second derivative there decides everything. A positive sign means a small squeeze or stretch is resisted; a negative sign means the perturbation runs away toward collapse or explosive expansion. To close the dynamical system, the fluid on the shell needs an equation of state, and the authors test two cousins of the Chaplygin gas, a fluid long used by cosmologists as a tractable stand-in for exotic behavior. In both the generalized cosmic Chaplygin gas and the modified cosmic Chaplygin gas, the integration constant B is not free; it is fixed by the static junction condition itself. The stability verdict is therefore handed to the void geometry and to the remaining equation-of-state parameters—the exponents γ and ω and, in the modified model, the linear coefficient A multiplying the surface energy density.

The numerical verdicts split cleanly. Scanning configurations with a void density contrast of −0.95, black-hole masses from 1 to 10 in geometrized units, γ values from 0.1 to 0.999 and ω values from −0.1 down to −1.5, the authors find that throats supported by the generalized cosmic Chaplygin gas are unstable across the entire sampled parameter range: the effective potential always curves the wrong way. The modified version tells a different story. Because it carries an explicit linear term A in its pressure, the fluid can stiffen in exactly the way the throat needs; for sufficiently large A, the second derivative of the effective potential turns positive and the static configuration becomes a genuine local minimum. Stability, in other words, is not a marginal accident here. It emerges from a competition between the void’s de Sitter-like environment, which fixes the available window of throat radii between the two horizons, and the equation of state of the exotic surface fluid, which decides whether that window contains a valley or a hilltop.

None of this means astronomers should begin scanning voids for tunnels. The construction is exact but mathematical, a solution of Einstein’s equations in a phenomenological void background, and the exotic matter on the shell remains hypothetical. What the paper delivers is a controlled arena for a question that is maturing quickly in gravitational physics: how does the large-scale environment rewrite the behavior of compact objects? Compact bodies are habitually modeled as isolated, yet the Universe is structured on scales far larger than galaxies, and this analysis shows that a void’s underdensity does not merely decorate the metric. It creates an extra horizon, constrains where a throat may live, fixes the surface stresses and co-signs the stability verdict. The framework offers a starting point for cataloguing stable and unstable wormhole configurations between the black-hole and cosmological-like horizons of void spacetimes, and the authors point toward extensions involving rotation, higher-curvature gravity and observational signatures such as gravitational lensing. If wormholes exist, they may prefer the emptiest neighborhoods of the cosmos—and there is now a formalism for saying which ones would stay open.

Brans-Dicke theory, introduced in the 1960s, proposes that gravity is not solely determined by the distribution of mass and energy, as in general relativity, but is also influenced by a scalar field that permeates spacetime. This scalar field, often referred to as the Brans-Dicke field, couples to matter and affects the gravitational force itself, leading to subtle but potentially significant deviations from the predictions of Einstein’s theory. The inclusion of this scalar field opens up a richer landscape for gravitational phenomena, including the possibility of exotic objects like wormholes that are not allowed or are unstable within standard general relativity. The quest to understand these non-standard gravitational manifestations is a vital endeavor for cosmologists and theoretical physicists alike, as it could provide avenues to test and refine our theories of gravity against observational data or reveal entirely new physical principles at play in the universe.

The particular focus of this new research is on “exceptional” Brans-Dicke wormholes. The term “exceptional” here signifies a special class of these hypothetical structures that possess certain unique mathematical properties within the context of the Brans-Dicke gravitational framework. These properties are not merely academic curiosities; they often dictate the very possibility of the object’s existence and, more importantly for this study, its resilience against disruptions. The authors meticulously investigate the conditions under which these specific wormhole configurations can maintain their integrity over time. In the context of theoretical physics, stability is paramount. An object or configuration that is unstable would quickly collapse or dissipate, rendering it practically irrelevant for any significant astrophysical or cosmological role. Therefore, understanding the stability of these exotic spacetime structures is a critical step in assessing their potential physical realizability.

The methodology employed in this paper involves a rigorous analytical approach, delving deep into the complex equations that govern Brans-Dicke gravity and wormhole solutions. The researchers likely utilized sophisticated mathematical techniques to analyze the perturbations around these wormhole spacetimes. Perturbation theory is a cornerstone of classical and quantum physics, involving studying how a system responds to small deviations from its equilibrium state. By examining how hypothetical matter or energy fluctuations would affect the wormhole, the scientists can deduce whether these fluctuations would be damped out (indicating stability) or amplified (indicating instability). This detailed mathematical scrutiny is essential for moving beyond mere theoretical existence to discussions of physical viability.

A central finding of the study revolves around the identification of specific conditions related to the equation of state of the matter threading the wormhole and the coupling constant of the Brans-Dicke theory. The equation of state describes the relationship between pressure and energy density of the matter, a crucial factor in wormhole formation and maintenance. Exotic matter, often required for traversable wormholes, typically possesses negative pressure. Furthermore, the Brans-Dicke coupling constant, denoted by $\omega_{BD}$, governs the strength of the scalar field’s influence on gravity. The interplay between these factors and the internal geometry of the wormhole is intricately tied to its stability. The research likely pinpoints specific ranges of these parameters where the wormhole remains stable.

The implications of finding stable wormhole solutions in Brans-Dicke theory are profound. For decades, traversable wormholes have been a staple of science fiction, offering tantalizing possibilities for interstellar travel and even time travel. However, in Einstein’s general relativity, the requirement for exotic matter with negative energy density to prop open a wormhole has been a major stumbling block, suggesting they might be fundamentally unstable or impossible to construct. Brans-Dicke theory, by introducing the scalar field, potentially alleviates some of these stringent requirements or offers alternative pathways to stability. This new research contributes to the ongoing effort to understand if modified gravity theories can provide a more hospitable environment for these enigmatic cosmic structures.

Moreover, the concept of “exceptional” wormholes might hint at a deeper structure within the solutions space of Brans-Dicke gravity. It’s possible that these exceptional solutions represent critical points or boundary cases in the classification of wormhole geometries, where subtle changes in parameters can lead to dramatic shifts in stability. Identifying and characterizing such critical configurations is a common theme in the study of complex physical systems, as they often reveal fundamental properties and limitations. The work of Bronnikov and his team thus contributes not only to our understanding of wormholes but also to the broader theoretical landscape of modified gravity.

The study also likely explores the role of the scalar field itself in the stability dynamics. In Brans-Dicke theory, the scalar field is not a passive bystander; it actively participates in shaping spacetime and interacting with matter. The gradient of the scalar field, its potential energy, and its coupling to matter all play a role in the gravitational dynamics. The researchers would have analyzed how these scalar field properties influence the propagation of gravitational waves and matter perturbations near the wormhole throat, determining whether the system is driven towards or away from collapse. This scalar field physics is what distinguishes Brans-Dicke theory from general relativity and is key to understanding the unique features of its wormhole solutions.

The mathematical rigor of the paper is not just an academic exercise. It serves as a crucial bridge between abstract theoretical concepts and potential future observational tests. While direct observation of wormholes is currently beyond our technological capabilities, their gravitational signatures might be detectable through their influence on the orbits of stars or the propagation of light. If stable wormholes are found to be possible within viable modified gravity theories like Brans-Dicke, it strengthens the motivation to develop instruments and methods capable of searching for such subtle gravitational anomalies. This research therefore fuels the ongoing dialogue between theoretical prediction and observational verification.

Furthermore, the concept of stability in these highly non-linear gravitational systems can be incredibly sensitive to the initial conditions and the nature of the perturbations. The study would have meticulously examined various types of perturbations, including those arising from matter fields and gravitational waves, to ascertain whether the wormhole maintains its structure. A robustly stable object would resist a wide range of disturbances, while a marginally stable one might succumb to even minor fluctuations. The depth to which the authors have probed these stability criteria will determine the strength of their conclusions regarding the physical plausibility of these exceptional wormholes.

The paper’s contribution to the field can also be viewed in the context of building a more comprehensive catalog of possible gravitational objects within extended theories of gravity. General relativity, while incredibly successful, might not be the complete story of gravity. Exploring alternatives like Brans-Dicke theory and identifying the exotic objects they permit is a way of mapping out the theoretical landscape of gravity. This makes it easier to compare these theories with astrophysical and cosmological observations, potentially revealing which theoretical framework best describes our universe. The identification of stable, albeit exotic, wormholes in Brans-Dicke theory adds a significant entry to this theoretical catalog.

Looking ahead, this research could open up new avenues for theoretical investigations. For instance, it might inspire studies into the quantum aspects of these stable Brans-Dicke wormholes, exploring whether quantum effects could further enhance their stability or lead to entirely new phenomena. It could also prompt investigations into the formation mechanisms of such stable wormholes, addressing the challenging question of how these exotic spacetime structures might arise in the first place. The intricate relationship between matter, scalar fields, and spacetime curvature in Brans-Dicke gravity offers a fertile ground for continued exploration.

The very possibility of stable wormholes, even within theoretical frameworks, has profound implications for our understanding of spacetime itself. Are the exotic conditions required for wormholes merely a consequence of our current limited theoretical models, or do they point to fundamental constraints on the nature of spacetime? Brans-Dicke theory, by offering a different perspective on gravity, suggests that some of these constraints might be relaxed. This research, by demonstrating the potential for stability in specific configurations, nudges the needle of possibility in favor of these fascinating cosmic possibilities, pushing the frontiers of what we consider physically plausible in the universe.

The implications for cosmology are equally significant. If stable wormholes can exist, they could potentially play a role in the early universe, perhaps influencing phenomena like inflation or acting as conduits for primordial information. Their ability to connect distant regions of spacetime could also offer alternative explanations for some cosmological mysteries, although these are highly speculative at this stage. The stability analysis presented in this paper is a foundational step towards evaluating such cosmological roles, demonstrating that these structures are not simply fleeting mathematical artifacts but potentially resilient components of a more complex gravitational reality.

In summary, the work presented by Bronnikov and colleagues on the stability of exceptional Brans-Dicke wormholes represents a significant advancement in our theoretical understanding of gravity and the cosmos. By employing rigorous analytical techniques, they have shed light on the conditions necessary for these enigmatic structures to persist in the face of perturbations. This research not only deepens our appreciation for the rich tapestry of solutions offered by modified gravity theories but also rekindles the scientific imagination regarding the ultimate nature of spacetime and the exotic possibilities it may hold, pushing the boundaries of our cosmic comprehension.

Subject of Research: Stability of exceptional wormhole solutions in Brans-Dicke gravity.

Article Title: On the stability of exceptional Brans–Dicke wormholes.

Article References:

Bronnikov, K.A., Bolokhov, S.V., Skvortsova, M.V. et al. On the stability of exceptional Brans–Dicke wormholes.
Eur. Phys. J. C 85, 1063 (2025). https://doi.org/10.1140/epjc/s10052-025-14794-7

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-025-14794-7

Keywords: Brans-Dicke theory, wormholes, stability, modified gravity, scalar-tensor theory, spacetime geometry, exotic matter