We figure out the explicit expression for the trace of the field equations associated to generic higher derivative theories of gravity endowed with Lagrangians depending upon the metric and its Riemann tensor, together with arbitrary order covariant derivatives of the Riemann tensor. Then an equality linking the Lagrangian density with the covariant divergence of a vector field is put forward in terms of the trace of the field equations. As a significant application, we particularly concentrate on a broad range of higher derivative theories of gravity with the Lagrangian density constructed from the contraction of the product for metric tensors with the product of the Riemann tensors and the arbitrary order covariant derivatives of the Riemann tensor. By utilizing the trace for the equations of motion, such a type of Lagrangian density is expressed as the covariant divergence of a vector field.

Abstract We present a novel equivalence between scale-dependent gravity and scalar-tensor theories that have only a single scalar field with a canonical kinetic term in the Einstein frame and a conformal coupling to the metric tensor. In particular, we show that the set of well-behaved scale-dependent gravity theories can be fully embedded into scalar-tensor theories in a unique way. Conversely, there are multiple ways to write a scalar-tensor theory as a scale-dependent theory. This equivalence is established both on the level of the actions and on the level of field equations. We find that, in the context of this equivalence, the scale-setting relation k(x) is naturally promoted to a dynamical field, which is made manifest by including a corresponding kinetic term in the scale-dependent action. In addition, we demonstrate that the new equivalence fits well into the framework of existing equivalences involving the aforementioned theories and f(R)-gravity. Finally, we apply the equivalence relations to explicit examples from both scale-dependent gravity and scalar-tensor theories.

Abstract We explore the cosmological behavior of the recently proposed f(R, Σ, T) gravity theory, which generalizes the standard f(R, T) framework by incorporating an additional Ricci-torsion scalar Σ to account for torsion-induced effects within the spacetime geometry. Employing the Efstathiou equation of state parametrization in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background, we derive the modified field equations and reconstruct the Hubble parameter analytically. The model parameters are constrained using the cosmic chronometer CC and Pantheon+SH0ES supernova datasets via a Markov Chain Monte Carlo (MCMC) approach. Our analysis reveals that the f(R, Σ, T) framework can successfully reproduce the universe’s late-time acceleration without the need for an explicit cosmological constant. The resulting effective equation of state evolves within the quintessence regime and approaches the ΛCDM limit at low redshift. Energy conditions, sound speed stability, and thermodynamic consistency are examined, confirming the physical viability of the model. These findings suggest that the inclusion of the Ricci-torsion term Σ offers a promising geometrical mechanism for cosmic acceleration and alleviates current observational tensions in Hubble parameter measurements.

Abstract In this paper, using the ensemble-averaged theory, we define the thermodynamic free energy of Einstein-Gauss-Bonnet (EGB) black holes in anti-de Sitter (AdS) spacetime. This approach derives the gravitational partition function by incorporating non-saddle geometries besides the classical solutions. Unlike the sharp transition points seen in free energy calculated via saddle-point approximation, the ensemble-averaged free energy plotted against temperature shows a smoother behavior, suggesting that black hole phase transitions may be viewed as a small-$G_N$ (Newton’s gravitational constant) limit of the ensemble theory. This is similar to the behavior of black hole solutions in Einstein’s gravity theory in AdS spacetime. We have obtained an expression for the quantum-corrected free energy for EGB-AdS black holes, and in the six-dimensional case, we observe a well-defined local minimum after the transition temperature which was absent in the earlier analysis of the classical free energy landscape. Furthermore, we expand the ensemble-averaged free energy in powers of $G_N$ to identify non-classical contributions. Our findings indicate that the similarities in the thermodynamic behavior between five-dimensional EGB-AdS and Reissner-Nordström-AdS (RN-AdS) black holes, as well as between six-dimensional EGB-AdS and Schwarzschild-AdS black holes, extend beyond the classical regime.

A bstract It is expected that when the string coupling is taken to be sufficiently small, a black hole turns into a bound state of self-gravitating fundamental strings. This state would be described by winding strings wrapping around the Euclidean time circle, known as the Horowitz-Polchinski solution. In this paper, we study such a self-gravitating string configuration in two-dimensional dilaton gravity theories. We first derive an analytic expression of the solution describing a winding string in two-dimensions and investigate in detail the geometry of this solution. Our winding string solution in two dimensions describes the geometry near the surface of the bound state of self-gravitating strings in the large-dimension limit, much like a two-dimensional black hole describes the near horizon geometry of the Schwarzschild black hole in the large-dimension limit. To study quantum effects around self-gravitating strings, we obtain an analytic solution of winding strings in the RST model. In a similar fashion to the Hartle-Hawking vacuum around a black hole, our string solution in the RST model contains the background radiation, whose temperature is the same as that of the winding strings.

Abstract The LAGEOS (LAser GEOdynamics Satellite) and LARES (LAser RElativity Satellite) geodetic satellites have been instrumental also in probing fundamental physics and testing gravitational theories in the weak-field and slow-motion limit of General Relativity (GR). This review paper summarizes key scientific challenges and achievements, focusing on the Precise Orbit Determination (POD) of these passive satellites, the accurate modeling of non-gravitational perturbations, and the subsequent use of orbital residuals to constrain post-Newtonian parameters and alternative theories of gravity. We detail the advanced modeling techniques for spin dynamics, neutral drag, and thermal thrust effects, crucial for achieving millimeter-level precision in orbit. The core of the analysis relies on excluding GR effects from the dynamical models during POD, thus preserving relativistic signatures in the orbital residuals. The paper highlights significant results from the LARASE and SaToR-G projects, including precise measurements of the Schwarzschild and Lense-Thirring precessions. These measurements have yielded stringent constraints on combinations of PPN parameters, non-symmetric gravitation theories, torsional theories, and Yukawa-like long-range interactions, considerably advancing our understanding of gravitational physics beyond GR’s predictions. Future directions include leveraging the mean anomaly as an additional observable and exploring violations of Local Lorentz Invariance to further refine these tests.

Abstract We investigate the impact of dark matter, under the form of a fermion condensate, on the properties of accretion disks, in a spherically-symmetric and static background. We focus on a class of models where dark matter originates from a mass mixing among neutrino fields and compute how the total potential is modified accordingly. We find a Yukawa correction to the Newtonian potential. Hence, adopting the Novikov–Thorne formalism, we compute the corresponding disk-integrated luminosity profiles and, assuming a constant mass accretion rate, constituted solely by baryonic matter, we find non-negligible deviations as compared to the standard Schwarzschild case. Afterwards, we discuss physical consequences of our model with respect to recent literature and we conclude that using our results can be useful to distinguish among candidates of dark matter. Indeed, our findings suggest that incoming high-precision observations of accretion disk spectra may provide a tool to probe dark matter’s nature under the form of particles, extended theories of gravity or condensates.

A bstract The Schwarzschild geometry, describing the gravitational field of a spherical mass in classical vacuum, is one of the most famous vacuum solutions of the Einstein field equations. Classical vacuum is an idealization that does not include quantum vacuum fluctuations of quantum fields, and determining the form of the gravitational field of a spherical mass in quantum vacuum is an important step towards understanding the interplay between gravity and quantum field theory. We formulate and prove general results on the space of static, spherically symmetric and asymptotically flat spacetimes sourced by quantum vacuum fluctuations, obtained under the broad assumptions that the quantum vacuum energy density is negative and unbounded on Killing horizons. In particular, we show the generic replacement of Killing horizons by wormhole throats. We discuss how previous calculations in the literature that have used different prescriptions for the regularized vacuum expectation value of the quantum stress-energy tensor are particular cases of our general results.

As a member of modified theories of gravity, f(R) theories of gravity offer a compelling explanation for the observed phenomenon of the late-time cosmic acceleration, which is a puzzle that general relativity cannot solve. Moreover, it has been proved that the field equations of f(R) theories of gravity are equivalent to the first law of thermodynamics. This equivalence holds true even in the presence of matter-geometry coupling. As more and more f(R) gravitational models are proposed, it is crucial to investigate their physical validity. In this paper, three different f(R) gravitational models with arbitrary coupling between matter and geometry are investigated by the generalized second law of thermodynamics on cosmological scales with astronomical observational data. The results show that all the models considered in this work not only satisfy the generalized second law of thermodynamics but also account for the late-time cosmic acceleration. Moreover, the cosmic matter candidate in the models under consideration is radiation, ordinary matter, vacuum energy, quintessence-like fields or phantom-like fields.

The fundamental nature of time at microscopic scales remains an unsolved problem at the intersection of quantum mechanics and general ‎relativity. This study presents Quantum Chronography, a theoretical framework for analyzing the operational and physical limits of time ‎measurement arising from quantum uncertainty, spacetime curvature, and stochastic metric fluctuations. By integrating the energy–time uncertainty principle with Planck-scale constraints and gravitational backreaction, a lower bound on measurable time intervals is derived. The ‎framework predicts an intrinsic, irreducible temporal uncertainty that grows sublinearly with the measured interval, forming a stochastic ‎lattice of time quanta in regions of significant curvature. Implications for high-precision astronomical timing, including pulsar observations ‎and atomic clock networks, are discussed. Rather than proposing a complete theory of quantum gravity, this work focuses on the physically ‎measurable consequences of quantum and gravitational effects on time. The research results provide a novel operational perspective on the ‎emergent nature of time, bridging concepts from quantum gravity and observational chronometry‎.

A bstract We present a novel robust framework for systematically constructing D -dimensional four-point higher-derivative contact amplitudes. Our modular block (“LEGO”-like) approach builds amplitudes directly from manifestly gauge-invariant kinematic blocks, color-weight factors, and scalar Mandelstam polynomials. Symmetries (Bose/Fermi) are imposed algebraically, acting as filters on combinations of compatible pieces. This framework operates entirely in D dimensions, naturally incorporating evanescent operators crucial for loop-level consistency. Scaling to arbitrary mass dimension is achieved in a highly controlled manner using permutation-invariant scalar polynomials, avoiding combinatorial explosion. A key feature is its manifest compatibility with the double-copy program, allowing the systematic generation of operator towers not only for gauge theories but also for gravity and other theories within the double-copy web.

A bstract We first streamline the construction of the unique six-dimensional conformal gravity action found by Lü, Pang and Pope, that admits Einstein metrics as solutions to the field equations. We then prove that there exists a unique eight-dimensional conformal gravity action that admits Einstein metrics as solutions to the field equations, and explicitly build the corresponding action. Finally, we relate these results to Branson’s Q -curvature and the Fefferman-Graham obstruction tensor, to conjecture that on every even-dimensional space there exists a unique — up to boundary terms — conformally-invariant gravity theory that is extremised by Einstein metrics.

The four-dimensional Einstein–Gauss–Bonnet theory of gravity developed by Glavan and Lin (Phys. Rev. Lett. 124 (2020) 081301) has been amply demonstrated to be applicable to spherically symmetric distributions, notwithstanding its other criticisms. This formulation allows us to study the effects of higher curvature geometry on the evolution of stars in the traditional four-dimensional setting. In particular, our interest in this paper is charged distributions. Despite the fact that the associated Einstein–Maxwell–Gauss–Bonnet field equations require two prescriptions to close the system, the task of locating exact solutions is nontrivial. We exploit a technique relating the electric field intensity to one of the gravitational potentials in order to reduce the order of the governing differential equation, and in this context we construct the isothermal fluid configuration exhibiting an inverse square law fall off of the energy density, and for a choice of integration constants also admits a similar behavior in the pressure. In general, the higher curvature terms negate the isothermal behaviour when the spatial metric potential is constant, unlike the Einstein case. The model constructed is shown to have the necessary qualitative features of realistic distributions as depicted in graphical plots for suitably selected parameter spaces. Notably a surface of vanishing pressure is evident and the model is adiabatically stable and causal. Given the isothermal feature, a central singularity is present as expected. This suggests that our fluid model may be interpreted as a shell of charged fluid in [Formula: see text] EMGB enveloping some other physically viable core.

In this work, we explored the dynamics of black holes within the framework of Kalb-Ramond gravity, emphasizing the effects of spontaneous Lorentz symmetry breaking on their perturbative and thermodynamic properties. Starting with a modified spherically symmetric black hole solution, we derived and analyzed the spin-dependent Regge-Wheeler potentials for scalar, vector, and tensor perturbations, uncovering significant deviations from the Schwarzschild black hole due to the Lorentz symmetry-breaking parameter. Our findings revealed that the parameter strongly influences the stability of the black hole and the propagation of perturbative modes. Additionally, we calculated GFs to investigate the transmission coefficients of black hole radiation and demonstrated their dependence on both the spin of the fields and the symmetry-breaking effects. Thermodynamic analyses showed modifications to the temperature, entropy, and specific heat, which highlighted the critical role of Lorentz symmetry-breaking effects on black hole stability. Furthermore, we discussed the area quantization of the black hole horizon, revealing its direct dependence on the symmetry-breaking parameter and providing insights into the quantum nature of black holes. These results collectively enrich our understanding of black hole physics under modified gravity theories and offer potential observational implications.

A bstract Holography and entropy bounds suggest that the ultraviolet (UV) and infrared (IR) cutoffs of gravitational effective theories are related to one another as a form of UV/IR mixing. Motivated by this, we derive a bound on the allowed scalar field range in theories with cosmic horizons. We show how this bound challenges several inflationary scenarios, such as α -attractors and modular-invariant inflation. Besides, we find a relation between the number of extra spatial dimensions and the tensor-to-scalar ratio.

Abstract We investigate the accumulation of null matter at the stable photon sphere in the Mannheim-Kazanas metric, the analogue to the Schwarzschild solution in Weyl's conformal theory of gravity. In our toy problem in which we consider an infinitely-thin shell, we find that a jump in radial pressure T r r is induced across the shell unless the shell has a radius of either the unstable or stable photon sphere radii. We then find that upon loading the stable photon sphere, its area remains invariant. Furthermore, at a critical threshold loading limit for this zero-width null matter shell, we are able to produce a metric containing an extremal horizon with an AdS 2 ×S 2 geometry completely independent of the cosmological curvature. This hitherto unencountered and therefore unexpected result is a phenomenon unseen in standard nonconformal second-order metrics with nonzero cosmological constants.

A bstract The inclusion of higher derivatives is a necessary condition for a renormalizable or superrenormalizable local theory of quantum gravity. On the other hand, higher derivatives lead to classical instabilities and a loss of unitarity at the quantum level. A standard way to detect such issues is by examining the reflection positivity condition and the existence of a Källén-Lehmann spectral representation for the two-point function. We demonstrate that these requirements for a consistent quantum theory are satisfied in a theory we have recently proposed. This theory is based on a six-derivative scalar field action featuring a pair of complex-mass ghost fields that form a bound state. Our results support the interpretation that physical observables can emerge from ghost dynamics in a consistent and unitary framework.

Abstract We develop a metric-free theory of gravitation generated by geometrical defects. Spacetime geometry is described by a velocity-distortion coframe β a =Ψ a μ dφ μ and a spin-bend-twist connection κ a b =Ψ a μ d(Ψ -1 ) μ b , defined in terms of the motion field φ μ and the intrinsic deformation field Ψ a μ . Their field strengths are the intrinsic torsion Σ a =dβ a +κ a b Λβ b and intrinsic curvature Λ a b =dκ a b +κ a c Λκ c b . The fundamental field equations Σ a =α a and Λ a b =Θ a b balance these geometric quantities against spacetime dislocations α a and disclinations Θ a b —incompatibilities in the motion and intrinsic deformation fields. Geometrical point defects π a b correspond to incompatible frame transformations. The Newtonian limit of the time-space field equations Λ i 0 =Θ i 0 results in the conversion between physical mass-energy ρ and the geometric mass density ρ geom =¬ e i ¬ u Θ i 0 time-space wedge disclination. Gravitation thus emerges directly from geometric incompatibility rather than from curvature sourced by matter. An invariant volume capacity —a 4-form constructed from orientation and deformation determinants—replaces √(- g ) and enables variational principles and integration without a metric. Variation of an intrinsic-frame gravitational Lagrangian capacity produces the corresponding stress and couple-stress capacity currents and their dynamical equations. In this formulation, there is no Newtonian gravitational potential: gravitational accelerations are carried by the time-space components of the connection, which act as the fundamental dynamical variables. The theory reproduces gravitational waves, black hole exteriors, and the Newtonian limit in a metric-free formulation; when needed, a metric-compatible sector restriction and metric reconstruction are used only for comparison with standard GR. It provides a natural foundation for extensions such as f(Λ) theories that may remain valid where metric descriptions fail.

Abstract Over the last seventy years, many Finsler‐type geometric and modified gravity theories (MGTs) have been elaborated. They have been formulated in terms of different classes of Finsler generating functions, metric and nonmetric structures, nonlinear and linear connections, and various sets of postulated fundamental geometric objects with corresponding nonholonomic dynamical or evolution equations. In several approaches, the resulting gravitational and matter field equations were not completely defined geometrically, or were developed only for restricted models. A progress report with historical remarks and a summary of new results on Finsler–Lagrange–Hamilton (FLH) geometric flow and gravity theories is presented. Such theories can be constructed in an axiomatic form on (co)tangent Lorentz bundles as nontrivial modifications of Einstein gravity. They are characterized by nonlinear dispersion relations and may encode nonassociative and noncommutative corrections from string theory, quantum effects, or other MGTs. To generate exact and physically relevant solutions of the FLH–modified Einstein equations, the anholonomic frame and connection deformation method is developed. A proof of the general integrability of such FLH geometric flows and MGTs is provided, and new classes of generic off‐diagonal solutions determined by generating functions and effective sources that, in general, depend on all spacetime and (co)fiber coordinates are analyzed. In general, such off‐diagonal configurations do not exhibit horizon/hypersurface duality or holographic structures and thus lie outside the Bekenstein–Hawking thermodynamic paradigm. Instead, by extending G. Perelman's entropy concept to relativistic FLH geometric flows, new classes of geometric thermodynamic variables that characterize different FLH theories and their associated solution spaces are introduce and computed.

Investigating the gross properties of gravastars in generalised cylindrically symmetric space-time within the framework of Rastall theory of gravity