f ( R ) gravity is a well-known modification of General Relativity, which is used in various models of dark matter and dynamical dark energy. We study static spherically symmetric (SSS) asymptotically flat configurations of the f ( R ) gravity in the Einstein frame for three known scalaron potentials with different asymptotic properties. The main attention is paid to the case of astrophysically relevant configuration mass M and scalaron mass μ greater than several meV , according to existing experimental constraints. This means very large values of the dimensionless (in geometrized units) parameter Mμ , leading to specific properties of the SSS solutions. We consider a sufficiently large size of a scalarization region r 0 ≫ r g , where the space-time metric is essentially different from the Schwarzschild one. It turns out that the scalaron field has universal behavior regardless of Mμ and r 0 and is practically the same for different scalaron potentials. Asymptotic parameters of the metric near the naked singularity at the center of the SSS configuration are obtained analytically for all the models. The test body circular orbits are discussed in view of observational signatures, which can distinguish these configurations from a regular black hole.

We present a high-precision joint gravitational-lensing and kinematic analysis of nine massive galaxy clusters from the CLASH and CLASH-VLT surveys to test chameleon screening gravity and its f(R) sub-class at Mpc scales. We investigate the dependence on the assumed parametrization of the total cluster mass profile by adopting three models, namely Navarro–Frenk–White (NFW), Burkert, and Hernquist. When cuspy models (NFW or Hernquist) are assumed in the general chameleon framework, the combined constraints from the nine clusters are fully consistent with General Relativity (GR), excluding large regions of the modified-gravity parameter space (the coupling constant Q and the background chameleon field ϕ∞), providing one of the tightest bounds on general chameleon models with clusters to date. In contrast, adopting a Burkert profile—disfavored by lensing data—leads to a mild (∼2σ) departure from the GR expectation in joint analysis. When considering the f(R) sub-case, we obtain a bound on the background scalaron field of |fR| ≲ 2−5 × 10−5 (95% C.L.) for NFW and Hernquist models, in agreement with current constraints at cosmological scales, and an apparent deviation from standard gravity of log10|fR|=−4.7±1.2 for the Burkert case. We investigate the impact of systematics in the kinematical analysis, showing that the tension is mitigated when clusters exhibiting clear dynamical disturbance are excluded from the sample. Our results show that galaxy clusters provide competitive tests of screened modified gravity at mega-parsec scales, while highlighting the critical role of accurate mass modeling and dynamical-state assessment. The upcoming generation of wide-field lensing surveys and spectroscopic follow-up programs will enable similar analyses on substantially larger samples, offering the prospect of tightening cluster-based constraints on gravity and the dark sector.

Abstract The relativistic precession of the perihelion provides key evidence for general relativity and remains a powerful example for introducing students to gravitational physics. Existing approaches, however, are often too mathematically involved for many undergraduates. In this article, we present a simpler pedagogical framework that requires minimal coding and treats General Relativity and alternative theories as perturbations to the Newtonian central-force problem. The apsidal angle formalism yields compact analytical expressions for perihelion advance in four contexts: the weak-field limit of General Relativity, Yukawa modifications, $f(R)$ power-law corrections, and Brans--Dicke scalar--tensor theory. We solve the orbital equation numerically with and without these corrections, demonstrating that precession emerges universally from any deviation from the inverse-square law, including the retrograde shift characteristic of certain $f(R)$ models. Quantitative precession rates are then extracted and compared across theories through strategic parameter scaling. This approach establishes classical orbital mechanics as a versatile testing ground for gravitational theories, enabling a direct comparison of their predictions through observable dynamical consequences.

We analyze the Taylor expansion of the metric f(R) gravity in the Jordan frame around the General Relativity limit, expanding in the small deviation (ϕ−ϕ0) with ϕ0=1. By relating the scalar–tensor representation to the original f(R) formulation, we derive constraints on the expansion parameters from the observed value of the present-day ΛCDM (Λ Cold Dark Matter) deceleration parameter and from cosmological bounds on the variation of Newton’s constant. We show that these requirements imply that the scalar degree of freedom must have a mass exceeding the Hubble scale by several orders of magnitude. This result challenges the common assumption that the scalar mode can drive cosmological dynamics with a mass of order of the Hubble constant H0. We provide a dynamical interpretation of this hierarchy by emphasizing that a proper definition of the scalar mass, in a field-theoretical sense, requires an adiabatic separation between background evolution and perturbations, which naturally leads to a super-Hubble mass scale.

A bstract In this paper, we further develop a recently proposed theory of time based on wavefunction collapse in general relativity. It is based on the postulations that quantum states, which violate the momentum and Hamiltonian constraints, represent instances of time, and stochastic fluctuations of the lapse and shift generate the time evolution under which an initial state gradually collapses toward a diffeomorphism-invariant state. Under the wavefunction collapse, the scale factor monotonically increases, thus acting as a clock. The scalar, vector, and tensor gravitons arise as physical excitations, and the arrow of time for their evolution is set by the initial state. In the long-time limit, the tensor gravitons exhibit emergent unitary dynamics. However, the extra modes are strongly damped due to the non-unitary dynamics that suppress the constraint-violating excitations. The vector mode is uniformly suppressed over all length scales, but the decay rate of the scalar is proportional to its wave vector. This makes the latter a viable candidate for dark matter; excitations with large wavelengths survive over long periods, contributing to long-range interactions, while the fast decay of short-wavelength modes renders them undetectable without sufficient temporal resolution. These are demonstrated for the cosmological constant-dominated universe through semi-classical and adiabatic approximations, which are controlled in the limit of large space dimension.

In this work we obtained solutions of field equations for perfect fluid spherically symmetric static spacetimes and found their concircular vector fields (CCVFs) in $f(R,T)$ gravity theory. It came out that special classes of these spacetimes possess either $4$-dimensional or $15$-dimensional CCVFs. In [28] the author obtained CCVFs for the same spacetimes in general relativity and it is argued that such spacetimes possess CCVFs of $4-$, $5-$, $6-$ and $15$ dimensions. Our results revealed that the $f(R,T)$ theory restricted the number of CCVFs for the same spacetime. $f(R,T)$ theory allows such spacetimes to admit either the $4$ basic Killing vector fields as CCVFs or compel the spacetime to be conformally flat and admit $15$ CCVFs. We also calculated the energy density, fluid pressure, trace of the energy-momentum tensor $T$, Ricci scalar $R$ and the function $f(R,T)$. It is observed that energy density and fluid pressure of some solutions are related as $p=-\rho$ which means that such particular metrics behave like dark energy models.

This paper reformulates the SEXA Unified Field Theory within a glyph-governed symmetry framework in which admissible physical states are determined not solely by recursive energy closure, but by invariant compatibility across the SEXA glyph system and high-order finite symmetry structure. Rather than replacing the existing SEXA equation, the present work extends it by introducing a Monster-symmetry-constrained admissibility layer acting on the five-dimensional SEXA exciter manifold and its recursive extensions through the SEXA dimensional stack. The framework is organized through six primary glyph operators: Orr, Na, Ka, Sa, Mu, and Wa, corresponding respectively to radiant energy, flow dynamics, manifold logic, symmetry and stress, mass and memory, and conscious observation. These glyphs function as admissibility gates through which recursive field configurations must pass in order to remain physically meaningful under dimensional embedding, thinning, and collapse. Monster and Baby Monster symmetry are introduced as invariant classifiers of the glyph-admitted exciternion state space, restricting stable configurations to discrete orbit classes. Within this formulation, the SEXA unified energy equation is preserved as the physical anchor of the theory, while the glyph layer specifies the logical and structural conditions under which its terms may be admitted. Interdimensional payload interception is formalized as a glyph-filtered and symmetry-constrained compression of higher-dimensional energy contributions into boundary-accessible field structure. The Σ₆₀ system is introduced as a finite admissibility algebra governing recursive evaluation across the exciternion state space. The framework is explicitly falsifiable. Failure of glyph closure, invariant symmetry compatibility, General Relativity reduction, Quantum Field Theory reduction, or Yukawa-type short-range behavior constitutes immediate rejection.

Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation

In classical Newtonian theory, space and time are treated as separate entities, where time flows independently and space is Euclidean. However, this framework becomes inadequate in describing phenomena involving light and strong gravitational fields. This limitation motivates the development of general relativity, in which space and time are unified into a single geometric structure—spacetime, and gravity is interpreted as its curvature. Within this framework, the principles of geometry and symmetry determine the form of physical laws and their observable consequences. In this work, literature analysis and theoretical derivation methods are employed to study two classical observational tests of general relativity—gravitational redshift and light deflection—from a geometric and symmetry-based perspective, along with comparison to Newtonian theory. The results show that both phenomena arise naturally from the geometric structure of spacetime and can be described within a unified framework. Spacetime symmetries lead to conserved quantities, which provide an effective method for analyzing photon trajectories. The comparison with Newtonian theory highlights that the differences in predictions originate from the underlying geometric structure of spacetime.

'The tensor calculus knows physics better than the physicist': Bachelard on the role of 'Covariant Differentiation' in Relativity theory.

A graph-theoretical approach to the analysis of motion and rest in many-body systems is developed. Point bodies are represented as vertices of a complete bi-colored graph, termed the motion–rest graph (MRG). Two vertices are connected by a rust-colored edge when the corresponding bodies are at rest relative to each other; that is, when their mutual distance remains constant in time, bodies moving relative to each other are connected by a cyan edge. It is shown that the logical structure of the relation “to be at rest relative to each other” determines the combinatorial structure of the graph. For one-dimensional motion in classical mechanics and special relativity, this relation is reflexive, symmetric, and transitive, and therefore defines an equivalence relation. As a result, rust edges form disjoint complete cliques corresponding to rest-clusters, and the MRG becomes a semi-transitive complete bi-colored graph that is completely determined by the partition of the bodies into equivalence classes. It is proven that any such graph on five vertices necessarily contains a monochromatic triangle. For two- and three-dimensional motion, the transitivity of relative rest generally fails because constant mutual distance does not imply an equality of velocities in the presence of rotational degrees of freedom. In this case, the MRG is non-transitive, and the Ramsey threshold becomes the classical value R(3,3) = 6. The approach is extended to mixed sets containing moving bodies and reference points, including the center of mass of the system. Generalizations to general relativity and quantum mechanics are also discussed. In general relativity, transitivity of relative rest is generically lost because global rigid congruences do not generally exist. In quantum mechanics, exact transitivity survives only at the level of idealized delocalized eigenstates, whereas for physically realizable localized states, the notion of mutual rest becomes only approximate. The results demonstrate that the interplay between kinematics, logical properties of relational motion, and Ramsey-type combinatorial constraints gives rise to unavoidable ordered substructures in many-body systems.

Abstract We investigate the influence of boundary terms in gravitational field theories, by considering that in the Einstein–Hilbert action the boundary can be described by a non-metric Weyl-type geometry. The gravitational action and the the field equations, are thus generalized to include new geometrical terms, coming from the non-metric nature of the boundary, and depending on the Weyl vector, and its covariant derivatives. The field equations obtained within this framework generalize the standard Einstein equations by including in their mathematical structure the Weyl vector, and its covariant derivatives. As an applications of the general formalism we investigate the cosmological evolution in a flat FLRW geometry. We obtain the generalized Friedmann equations, which contain extra terms depending on the Weyl vector and its derivatives, arising due to the presence of the Weylian boundary, and which describe an effective, time dependent dark energy. By imposing to the dark energy an equation of state parameter of the Barboza–Alcaniz type, the Friedmann equations can be solved numerically. We compare the predictions of the Weylian boundary gravitational theory with late-time observational data and the predictions of the $$\Lambda $$ Λ CDM paradigm. Our results show that the Weylian boundary cosmological models give a good description of the observational data, and they can reproduce almost exactly the predictions of the $$\Lambda $$ Λ CDM paradigm. Hence, the extension of gravitational theories through the addition of Weylian boundary terms, in which dark energy has a purely geometric origin, emerges as a viable alternative to standard general relativity.

The present paper is aimed at studying the convergence of the Yamabe flow in the case of noncompact solitons. The more specified example of locally conformally flat noncompact solitons is addressed with the aim to newly analyse the qualities of the Ricci scalar. The particular case of noncompact pseudo-Riemannian solitons is studied; moreover, in the instances of Schwarzschild and Generalized-Schwarzschild geometries, rescalings of spherically symmetric weights are performed. For this purpose, new results are achieved as far as the considered structures are concerned. The Myers Theorem is upgraded as the new Myers paradigm of spacetime-dimensional manifolds, where the Einstein Field Equations can now be taken into account. In particular, the Myers Theorems are studied here as far as their new implementation in General Relativity Theory is concerned. As a first important result, the Myers mean curvature is found to coincide with the Ricci scalar in General Relativity Theory, where the 4-position of the observer, from which the 4-velocity 4-vector is calculated from, is taken as that of the observer solidal with the reference frame of the photon. The following results are also of relevance. In more detail, the umbilicity conditions are applied. At a further step, the role of the umbilicity conditions in GR after the Myers Theorems are studied for weighted manifolds and specific new implications of weighted manifolds are developed. The description of the weighted Schwarzschild manifolds and that of the weighted Generalized-Schwarzschild manifolds are newly studied as follows: as a new finding, the Birkhoff Theorem is newly reconciled with the rotational ansatz of the metrised solitons, and the comparison with the previous results about the Brendle non-metrised solitons is accomplished with the outcome stressing the new roles of the new rescalings of the metric tensor with respect to the previous known results of the scaling of the metric tensor of the non-metrised solitons. In the present framework, these procedures allow one to prove the reconciliation of the EFEs with the Yamabe flow. The flow on the tipping lightcones is newly written. The umbilicity condition is studied in General Relativity after the upgrade of the Myers Theorems as far as the sectional curvatures are concerned; as a result, the Calabi–Bernstein description is implemented in General Relativity, as well as the Chen–Yau requirements, and the cases of weighted manifolds are taken into account. More specifically, the equal-time 2-dimensional space surfaces are studied analytically, onto which the weighted General-Relativistic solitons which satisfy the Einstein field equations after the Yamabe flow are projected due to the rotational ansatz. As an accessory introductory result, the class of Wu non-metrised solitons are proven to be discarded in several aspects of the Wu description as the conditions provided after the work of Wu are not compatible with metrisation.

The Kerr solution is the cornerstone of General Relativity (GR) for modeling astrophysical rotating black holes and for testing GR through gravitational-wave observations and black hole imaging. Understanding how the Kerr geometry is modified in alternative theories of gravity is therefore a crucial step toward constraining possible deviations from GR. Despite their importance, exact analytical solutions describing rotating black holes in modified gravity are rare, limiting our ability to explore novel phenomenology and to design robust observational tests of new physics.In this work, we present a new exact rotating black hole solution within a specific scalar–tensor theory belonging to the Horndeski class. The solution is obtained via a disformal transformation acting on a Kerr stealth black hole. Crucially, unlike previous constructions, the disformal transformation of our chosen seed configuration preserves circularity, ensuring that many of the geometrical and physical properties that make the Kerr spacetime so compelling are retained. We refer to the resulting geometry as the Circular Disformal Kerr solution.Remarkably, key features such as the structure of Killing horizons, the ergosphere, and the absence of causality violations closely mirror those of the Kerr metric. The spacetime is algebraically general, corresponding to Petrov type I. This new exact solution therefore provides a rare example of a rotating black hole beyond GR that closely mimics the Kerr geometry, offering a valuable theoretical laboratory to investigate the phenomenology of Kerr-like black holes in modified gravity.

Einstein general relativity (GR) can be formally represented by a Lagrangian in a flat Minkowski space-time where light speed is not a universal constant and vanishes at the horizon of black holes. These appear as virtual singularities where no matter or energy can enter. We revisit the fundamental concepts of GR. Due to the principle of mass-energy equivalence, certain physical quantities, such as rest mass-energy, time, and light speed, must be rescaled according to the gravitational potential. We obtain a Lagrangian for a static gravitational potential (called germinal) different from the Schwarzschild Lagrangian but with the same black hole singularity. We develop a method that transforms a germinal Lagrangian into a fully relativistic Lagrangian. Then, the electrostatic Lagrangian becomes the well-known electromagnetic Lagrangian. The same method shows that the obtained germinal Lagrangian of gravitation generates a fully relativistic Lagrangian where gravitation is represented by a relativistic 4×4 tensor potential related to the Einstein stress-energy tensor. This theory is not equivalent to GR. In the weak gravitation limit, the Lagrangian generated by a stationary source only depends on a four-potential formally identical to that of the Heaviside theory of gravitation but different. As an application, we use this result in a simple model explaining the observed anomalies in the galaxy's stellar velocities without requiring hypothetical dark matter.

In future space-borne gravitational wave (GW) observations, matter around the sources might influence the evolution and GW signals from binary black hole (BBH) inspirals, which can be mistaken as deviations from general relativity (GR). Former research Phys. Rev. D 111 (2025) 104050 proposed a statistic F that characterizes the dispersion of measured parameters to distinguish environmental effect (dynamical friction (DF) from dark matter (DM) spike) and theory of modified gravity effect (varying G). In this work we use the statistic to distinguish other pairs of effects with GW corrections at -4 PN order: DF from DM spike and the extra dimension theory, additionally determine a proper distinguishing threshold via Receiver Operating Characteristic Curve (ROC) curve method to avoid arbitrariness, especially when the two effects to compare have more overlap in the F distribution. Sources of different astronomical models are also considered, and two effects are still distinguishable but less than the example in former work Phys. Rev. D 111 (2025) 104050, so the threshold should be carefully selected. Following these procedures, we finally obtain the statistic thresholds of distinguishment between the three effects with GW corrections at -4 PN order: DF from DM spike, the extra dimension theory, and varying G theory. So each effect corresponds to its own interval of the statistic F, then future observation of F can pick the preferred effect. The method can be used to distinguish other effects among environmental effects and modified theories of gravity effects with the detection of GW events.

In this paper, we examine the behavior of the charged compact star model PSR J0740+6620. To achieve this, we solve the Einstein field equations within an anisotropic framework. We model the star’s structure using a Gaussian density profile. Our analysis focuses on assessing the stability of the star and verifying whether its physical properties are consistent with the conditions required for equilibrium. In particular, we consider the Reissner-Nordstr o ¨ m spacetime configuration as an outer line element for equilibrium conditions. The important aspect of the work is to investigate the acceptable or non-acceptable regions for radial and tangential sound speeds using the contour plot. This helps us to choose the domain of free parameters involved in our work for further analysis. Conclusively, charged compact stars with a Gaussian density profile do exist in general relativity with Reissner-Nordstr o ¨ m exterior geometry.

The subject of the research is the methodological crisis of modern science, characterized by a gap between the verification and validation of knowledge. The problem of replacing a deep philosophical understanding of theoretical models with predominantly empirical verification is analyzed, which leads to a loss of coherence and ontological validity in scientific knowledge. Fundamental theories such as quantum mechanics, general relativity, and the Standard Model of elementary particles are critically examined, which, despite a high degree of experimental verification, remain philosophically unreflected and operate with indeterminate concepts of uncertain content. The role of primary properties of being as the ontological foundation for the validation of knowledge is explored, along with their bi-effectiveness as a tool for the representation of reality and their ontological nature as an indicator of correspondence to the objective world. The mechanism of isomorphism is analyzed as a tool for understanding and integrating knowledge into a cohesive system of the universe. An integrative approach is applied, combining the ontological analysis of primary properties of being with the epistemological analysis of the structure of scientific knowledge, as well as the method of isomorphic comparison of patterns at different levels of system organization. The scientific novelty consists in the development of criteria for the validation of knowledge that go beyond logical consistency and experimental verification. A distinction is proposed between verification (checking the correspondence of theory to experimental data) and validation (checking the congruence of knowledge with primary properties of being and its integration into a cohesive system). Based on the Empirical-Metaphysical General Theory of Systems (EMSGTS), a natural philosophical determinism is formed – a doctrine according to which all the diversity of forms and relations in the universe is conditioned by a finite set of primary properties of being that are bi-effective for reality. Conclusions: validation and understanding must take a central place in the methodology of science, serving as the philosophical foundation of a cohesive system of knowledge; modern science needs to restore the demand for structural causality and the congruence of knowledge with the ontological foundations of being; isomorphism serves not just as an observable regularity, but as a fundamental mechanism for the integration of knowledge into a unified system of the universe.

The future space-based gravitational wave observatories are expected to provide unprecedented opportunities to explore intricate characteristics of black hole binaries, particularly for extreme mass-ratio inspirals (EMRIs), in which a stellar-mass compact object slowly inspirals into a supermassive black hole. These systems are very prominent sources for testing gravity in the strong gravity fields and for probing potential deviations from general relativity, including those arising from the presence of fundamental scalar fields. In this work, we examine the impact of a scalar charge carried by the inspiraling object within the context of EMRIs. We focus on generic orbits that present both eccentricity and inclination to evaluate how these parameters affect the modifications induced by the scalar charge to the gravitational wave signal. Our results demonstrate that the inclusion of orbital inclination, in particular, enhances the detectability of scalar field effects by introducing richer waveform features that deviate from the purely general relativistic case. The interplay among scalar charge, eccentricity and inclination provides a more complete sampling of the black hole spacetime, suggesting that EMRIs with such generic orbits represent compelling systems for stringently constraining or discovering new fundamental fields through future gravitational wave observations.

Modified Gravity Theories (MGTs) are extensions of General Relativity (GR) in its standard formulation. Therefore, within this framework, we will investigate a system composed of a black hole (BH) surrounded by Maxwell-Higgs vortices, forming the BH-vortex system. In the case of linear f(R) gravity it is adopted showing the existence of a three-dimensional ring-like BH-vortex system with quantized magnetic flux. Within this system, one notes the BH at r = 0 and its event horizon at , while the magnetic vortices are at . A remarkable result is the constancy of the Bekenstein-Hawking temperature (T H ), regardless of MGTs and vortex parameters. This invariance of T H suggests that the BH-vortex system reaches thermodynamic stability. Unlike the standard theory of Maxwell-Higgs vortices in flat spacetime, in f(R) gravity, the vortices suffer the influence of the BH's event horizon. This interaction induces perturbations in the magnetic vortex profile, forming cosmological ring-like magnetic structures.