Physicists Find New Exact Solutions in a Theory Where Empty Space Cannot Exist

A radical reformulation of Einstein’s general relativity has just gained two new exact solutions, and they come with a twist that cuts to the heart of what gravity means. In a paper published in the journal General Relativity and Gravitation, mathematicians Maxime Wavasseur and Olivier Minazzoli present two static, axisymmetric solutions of Entangled Relativity, a theory that forbids genuinely empty spacetime and requires one fewer fundamental constant than Einstein’s framework. The work, published as volume 58, article 113 of the journal, is technical in nature, but its implications touch on one of the oldest philosophical debates in physics: whether space and time can exist independently of the matter they contain.

Entangled Relativity is what physicists call a non-linear reformulation of general relativity. In Einstein’s celebrated 1915 theory, geometry and matter are coupled in a carefully balanced way: the curvature of spacetime, encoded in the Ricci scalar R, is sourced by the energy and momentum of matter, but the two remain conceptually distinct entities. Vacuum solutions, spacetimes with no matter at all, are perfectly legitimate in general relativity. The Schwarzschild solution describing a lone black hole, and the Kerr solution describing a rotating one, are famous examples. Entangled Relativity changes this picture fundamentally by merging matter and geometry into a single Lagrangian, the mathematical object from which the field equations are derived. The result is a theory in which the field equations depend on the on-shell value of the matter Lagrangian itself, and in which a true vacuum, a spacetime with no matter whatsoever, is not merely unusual but ill-defined.

The philosophical motivation reaches back to Ernst Mach, the nineteenth-century physicist whose ideas about the origin of inertia famously influenced the young Einstein. Mach’s principle, in one of its many formulations, holds that the local inertial properties of spacetime are entirely determined by the distribution and motion of matter in the universe. Space without matter, in this view, is not a well-defined concept. Einstein wrestled with this idea throughout the development of general relativity, and his 1918 exchange with Willem de Sitter over an empty-universe solution is a classic episode in the history of the subject. General relativity ultimately allows vacuum solutions, so Mach’s principle was never fully realized in Einstein’s theory. Entangled Relativity, its proponents argue, enforces the Machian property by construction: spacetime is entirely determined by matter and cannot exist without it.

This structural difference creates an immediate technical puzzle for anyone trying to construct exact solutions. The new paper starts from two existing static and axisymmetric solutions of general relativity, one in vacuum and one involving a scalar field. Axisymmetric means the solutions possess symmetry about an axis, like a spinning top at rest, and static means they do not change in time. Such solutions form a venerable family in gravitational physics, tracing back to Hermann Weyl’s 1917 work on static axisymmetric metrics and later refined through the transformations of Zipoy, Voorhees, and others. Wavasseur and Minazzoli first derive corresponding solutions of Entangled Relativity in what is known as the Einstein frame, a mathematical representation in which the equations take a form resembling those of general relativity coupled to a scalar field, a structure familiar from the Einstein-Maxwell-dilaton systems studied in string-inspired gravity.

Here the subtlety emerges. Once these solutions are rewritten in the entangled conformal frame, the frame in which the non-linearity of the theory’s action is manifest, they would correspond to vacuum solutions. And vacuum, in Entangled Relativity, is precisely where the theory breaks down. The theory’s defining non-linear coupling introduces a dependence of the field equations on the matter Lagrangian, and when the trace of the metric field equation is taken, this dependence becomes explicit. Unlike in general relativity, where the Ricci scalar vanishes for electromagnetic fields in certain configurations, in Entangled Relativity one has R not equal to zero in the presence of an electric or magnetic field. The geometry simply refuses to decouple from matter.

The authors’ way around this obstacle is as elegant as it is physically motivated. Rather than discarding the solutions, they embed them in a uniform magnetic or electric field following a construction due to Melvin, and only then take the limit in which the energy-momentum tensor of that field tends to zero. The Melvin universe, a well-known solution in which a magnetic field threads and curves spacetime, provides a physically sensible matter background that regularizes the otherwise ill-defined vacuum. The crucial result of the new paper is that this limiting procedure is perfectly well-defined and finite. Even though a vacuum solution is, from the outset, ill-defined in Entangled Relativity, the limit of the embedded solutions exists and yields sensible spacetimes. This mirrors what has been found for all other solutions of the theory constructed so far, from compact objects to charged and radiating spacetimes.

The technical machinery behind the result is considerable. One of the two solutions, associated with the so-called Penney case, requires a dedicated algorithm to verify that the field equations are satisfied. The verification hinges on the linear independence of exponential functions with distinct arguments: a finite weighted sum of exponentials can vanish identically only if every coefficient vanishes. The authors decompose each quantity to be checked into an algebraic part plus a finite sum of exponential terms, group terms with identical arguments, and confirm that every coefficient in the resulting linearly independent set is identically zero. The computations were carried out with the help of SageManifolds, an open-source mathematical software package, and the authors thank Eric Gourgoulhon for guidance in optimizing their notebooks.

The new work does not stand in isolation. It is the latest entry in a growing program to map out the solution space of Entangled Relativity. Previous results from the same collaboration and its collaborators include analytical external spherical solutions, compact object models, charged black hole and radiating solutions, a Schwarzschild black hole immersed in an electric or magnetic background, and slowly rotating charged black holes. Each of these studies has confronted the same fundamental obstacle, the absence of well-defined vacuum, and each has found that embedding the configuration in an electromagnetic field and taking the appropriate limit resolves the difficulty in a controlled way. The consistency of this pattern across such different physical situations is itself a meaningful result, suggesting that the theory’s non-linearity, far from being a nuisance, is a coherent organizing principle.

Why should anyone care about an alternative to general relativity that has passed no experimental tests yet? The answer lies partly in foundational questions and partly in the theory’s surprising economy. Entangled Relativity possesses one less defining constant than general relativity, a reduction with far-reaching consequences explored in related work by Minazzoli and collaborators, including suggestions that Planck’s quantum of action may vary within the theory and that the theory can be derived from minimal assumptions. The theory belongs to the broader class of f(R, Lm) gravities, in which the gravitational action depends on both curvature and the matter Lagrangian, a framework developed by Harko, Lobo, and others. Such theories naturally exhibit scalar-tensor behavior in some regimes while decoupling from the scalar degree of freedom in others, a feature that has been analyzed in the context of solar system constraints and late-time cosmology.

Exact solutions are the lifeblood of gravitational physics. They are the laboratories in which the counterintuitive predictions of a theory, black holes, gravitational waves, singularities, wormholes, are made concrete and testable. The two new axisymmetric solutions extend the known exact landscape of Entangled Relativity into a class of spacetimes with direct astrophysical relevance, since axisymmetric configurations describe the gravitational fields of flattened, non-spherical bodies. Whether Entangled Relativity ultimately survives confrontation with observation remains an open question, and the authors themselves make no claims on that front. What the new paper establishes is narrower but important: the theory’s most distinctive feature, its refusal to admit empty spacetime, does not prevent the construction of meaningful exact solutions. The limit that replaces vacuum is finite, well-defined, and consistent, and that is precisely the kind of internal coherence an alternative theory of gravity must demonstrate before it can be taken seriously as a rival to Einstein’s.

Subject of Research: Exact static axisymmetric solutions of Entangled Relativity, a non-linear reformulation of general relativity that forbids vacuum spacetimes

Article Title: Two static axisymmetric solutions in Entangled Relativity

Article References: Wavasseur, M., & Minazzoli, O. (2026). Two static axisymmetric solutions in Entangled Relativity. General Relativity and Gravitation, 58(10), Article 113. https://doi.org/10.1007/s10714-026-03615-1

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03615-1

Keywords: Entangled Relativity, general relativity, exact solutions, axisymmetric spacetimes, Mach's principle, non-linear gravity, f(R, Lm) gravity, Einstein-Maxwell-dilaton, vacuum solutions, black holes, scalar field, modified gravity