In this work, we study the random metric for the critical long-range percolation on Z d \mathbb {Z}^d . A recent work by Bäumler (2022) implies the subsequential scaling limit, and our main contribution is to prove that the subsequential limit is uniquely characterized by a natural list of axioms. Our proof method is hugely inspired by recent works of Gwynne and Miller (2021b), and Ding and Gwynne (2023a) on the uniqueness of Liouville quantum gravity metrics.

Abstract We develop a strictly geometric reformulation of General Relativity (GR) in which a scalar clock field ϕ defines the spacetime foliation through its timelike gradient nµ = -∇µϕ/ -(∇ϕ) 2 . The field ϕ is not a new dynamical degree of freedom and does not enter the action; its sole role is to generate a covariant, hypersurface-orthogonal slicing that renders the ADM lapse, shift, and extrinsic curvature algebraic functionals of (gµν , ϕ).Starting from the pure Einstein-Hilbert action, we carry out a complete 3 + 1 decomposition adapted to the clock-field foliation and derive the Hamiltonian constraint, momentum constraint, and full constraint algebra. All coincide exactly with their canonical GR counterparts, confirming that the theory propagates only the two tensorial graviton modes and contains no additional scalar or vector excitations. The Einstein equations are recovered identically, the ADM algebra remains first class, and both cosmological dynamics and linear perturbations match those of standard GR.The novelty of the construction is conceptual rather than dynamical: ϕ provides a covariant, relational time variable that fixes the foliation non-perturbatively, offering a clean geometric notion of time for canonical quantization and Wheeler-DeWitt evolution. Because the framework introduces no new fields, couplings, or potentials, it avoids the instabilities and extra degrees of freedom typical of scalar-tensor or modified-gravity theories.This yields a mathematically consistent and fully covariant formulation of GR with an internally defined geometric clock, opening new directions for relational observables, deparametrized quantum gravity, and emergent-time frameworks.

This paper explores whether primordial black hole hydrogen-like atoms (PBH-H protoatoms) could be detectable through their distinctive spectroscopic signatures. Background Building on theoretical work proposing that gravitational binding between primordial black holes and electrons may create exotic quantum systems, we investigate the observational prospects for these hypothetical atomic structures. Methods We develop a comprehensive detection framework combining simulated spectroscopic signatures, sensitivity analyses for current space-based infrared telescopes, and machine-learning classification algorithms to distinguish genuine signals from astrophysical contaminants. Our approach integrates deep spectroscopic surveys targeting dark matter halos, multi-messenger coordination with gravitational wave triggers, and time-domain analysis of spectral evolution. Results Our simulations predict that PBH-H protoatoms would emit characteristic far-infrared transitions spanning one to fifty micrometers, with the dominant n equals two to n equals one line occurring at five point four micrometers, placing it within the detection range of the James Webb Space Telescope Mid-Infrared Instrument. The spectral features exhibit significant broadening from Hawking radiation-induced quantum blur and rapid temporal evolution on timescales of minutes to hours as electrons spiral toward the black hole nuclei. Sensitivity analysis confirms that the James Webb Space Telescope can probe PBH-H densities as low as one thousand per cubic parsec in the Galactic Center, while the Atacama Large Millimeter Array and Very Large Array can access higher-order millimeter and centimeter wavelength transitions. Conclusions Successful detection of PBH-H protoatoms would validate primordial black holes as dark matter constituents and provide unprecedented tests of quantum gravity at atomic scales, representing a transformative advancement in understanding dark matter composition and quantum gravitational phenomena.

Abstract Noncommutative (NC) geometry may open an alternative route to quantum gravity. We study the signatures that quantum structure of spacetime may leave on Dirac quasinormal mode spectrum in the setting defined by a common astrophysical background. For that purpose we examine the influence of spacetime noncommutativity on the Dirac quasinormal modes in modified Reissner–Nordström black hole spacetime. The framework for the latter study is provided by the effective model of NC gravity coupled to fermions introduced in Dimitrijević Ćirić et al. (Eur Phys J C 83:387, 2023). This model describes a classical Dirac field coupled to a modified Reissner–Nordström geometry where the corresponding metric acquires an additional nonvanishing $$r-\varphi $$ r - φ component. As the earlier study shows, this model appears to be equivalent to a model of semiclassical NC gauge theory in which a NC gauge field is coupled to a NC fermion field on the one side and the classical Reissner–Nordström background on the other. We compute the resulting Dirac quasinormal modes and compare them with those of the undeformed Reissner–Nordström spacetime. The results show that spacetime noncommutativity modifies both the oscillation frequencies and damping rates, and induces features in the effective potential and quasinormal mode spectrum reminiscent of a Zeeman-like splitting. Since such geometric modifications are expected to become relevant only near the Planck scale, these effects are more naturally associated with microscopic rather than astrophysical black holes.

This paper presents the Ze → Twistor → Spin Network framework, a unified conceptual pathway from a primitive discrete ontology to the emergence of relativistic spacetime. The framework begins with fundamental events, denoted ΔC_i, each characterized by dual aspects: a temporal component C_i^{temporal} governing participation in sequential causal chains, and a spatial component C_i^{spatial} governing participation in parallel structural configurations. The first transition, Ze → Twistor, encodes these aspects in a complex representation Z_i = C_i^{temporal} + i C_i^{spatial}, drawing on Penrose's twistor theory. The Hermitian norm |Z_i|^2 = (ΔC_i^{spatial})^2 - γ(ΔC_i^{temporal})^2 carries the Minkowski signature intrinsically, emerging from the SU(2,2) invariant structure of twistor space rather than being inserted by hand. This addresses the fundamental question of how a discrete substrate can give rise to a Lorentzian manifold without violating Lorentz invariance. The second transition, Twistor → Spin Network, discretizes the twistor representation into a labeled graph. Nodes correspond to events (antichains representing spatial slices), edges correspond to causal links, and each edge carries a spin label j determined by j(j+1) ∝ |Z_i|^2, connecting directly to loop quantum gravity where spin networks provide an orthonormal basis for kinematical states. From this structure, relativistic spacetime emerges in the continuum limit: proper time along a worldline is given by the sum of spin labels τ = Σ √[j(j+1)] × τ_Planck, velocity emerges from the ratio of accumulated spatial to temporal increments, and the twin paradox resolves combinatorially through different total spin sums along distinct worldlines. The framework synthesizes insights from causal set theory, twistor theory, and loop quantum gravity, demonstrating how these approaches complement rather than compete with one another. It offers resolutions to long-standing puzzles including the origin of the Lorentzian signature, the compatibility of discreteness with Lorentz invariance, and the combinatorial definition of proper time. Open questions regarding dynamics, the quantum measure, and phenomenological predictions are discussed as directions for future investigation.

Abstract The gravity in the context of General Relativity (GR) as dynamical curvature of spacetime remains a principal impediment not only to its unification with the other fundamental interactions, but also to the formulation of quantum gravity. One possible resolution involves considering gravity in the context of Special Relativity (SR). This paper presents a procedure, which correlates the GR metrics of curved spacetime and the SR Gravitational Scalar Generalized Potential (GSGP). The GR time dilation is the key-point for the correlation of the two gravities, which implies the corresponding SR Lagrangian. Previous papers have already demonstrated the procedure and the results in cases of FLRW metric, wormholes with spherical symmetry, and Schwarzschild metric (where not only the gravitational motions (free falls) in the context of SR are exactly the same as those in the context of GR with stationary metric, but also the SR and GR Gravitational Red Shift). This paper analytically derives the general formulae of Equations of Motion in Spacetime endowed with Stationary metric in the context of GR and the ones in the context of SR, proving that they are exactly the same. Finally, it presents the case studies of gravitational motions around Schwarzschild blackhole, Kerr rotating blackhole and Standard Ellis-Bronnikov wormhole. Thus, it is shown that these motions have an equivalent SR viewpoint.

We determine the maximal gauge groups arising in E 8 heterotic string theory on T 2 . Our analysis proceeds along two approaches. First, we start the moduli space of the supersymmetric heterotic string theory on T 2 focusing on points of maximal gauge enhancement. At these special points, the charge lattice can exhibit a Z 2 outer automorphism corresponding to the bulk gauge symmetry. By orbifolding the worldsheet theory by it with the fermion parity, we obtain the maximal gauge group of the E 8 theory. Second, we directly study the toroidal compactification of 10D E 8 heterotic string. Both approaches agree, yielding a classification of 22 maximal gauge groups. For each case, we present the corresponding massless spectrum. In light of the no global symmetry/cobordism conjecture in quantum gravity, our result also offers a classification of nonsupersymmetric branes in 8D supersymmetric heterotic string theories.

Abstract In this paper, we investigate the quantum gravitational corrections to the thermodynamical quantities of Reissner-Nordström black holes within the framework of effective field theory. The effective action originates from integrating out massless particles, including gravitons, at the one-loop level. We perform a complete thermodynamic analysis for both non-extremal and extremal black holes, and are mainly concerned about the shift in the charge-to-mass ratio $q/M$ that plays an important role in analyzing the weak gravity conjuecture. For non-extremal black holes, we identify a relationship between the shift in the charge-to-mass ratio and the thermodynamic stability of the black holes. For extremal black holes, we show that quantum gravity effects naturally lead to the super-extremality $q/M>1$ of charged black holes.

In this study, we investigate electromagnetic and Dirac test field perturbations of a charged regular black hole arising from quantum gravity effects, commonly referred to as the Frolov black hole, a regular (nonsingular) black hole solution. We derive the master wave equations for massless electromagnetic and Dirac perturbations and solve them using the standard Wentzel-Kramers-Brillouin (WKB) method along with Pad´e Averaging. From these solutions, we extract the dominant and overtone quasinormal mode (QNM) frequencies along with the associated grey-body factors, highlighting the deviations introduced by quantum gravity corrections compared to the classical case of Reissner–Nordstr¨om black hole. Furthermore, we analyze the Unruh-Verlinde temperature of this spacetime, providing quantitative estimates of how quantumgravity effects influence both quasinormal ringing and particle emission in nonsingular black hole models.

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 Recent arguments based on the quantum extremal surface formula or the gravitational path integral have given fairly compelling evidence that the Hilbert space of quantum gravity in a closed universe is one-dimensional and real. How can this be consistent with the complexity of our own experiences? In this paper we propose that the experiences of any observer Ob in a closed universe can be approximately described by a quantum mechanical theory with a Hilbert space whose dimension is roughly $$ {e}^{S_{Ob}} $$ e S Ob , where S Ob is the number of degrees of freedom of Ob . Moreover we argue that the errors in this description are exponentially small in S Ob . We give evidence for this proposal by incorporating it into the gravitational path integral and the coding interpretation of holography in simple models and seeing that it works, and we explain how similar effects arise in black hole physics in appropriate circumstances.

Background: The reconciliation of General Relativity with Quantum Mechanics remains the primary challenge in modern theoretical physics. Traditional approaches often assume a fixed background geometry, yet recent developments in string theory and loop quantum gravity suggest that spacetime is not fundamental but emergent. Specifically, the Holographic Principle implies that the information defining the bulk universe is encoded on a lower-dimensional boundary, raising the question of how a singular, classical reality arises from a quantum superposition of geometries. Purpose: This paper proposes a novel model of quantum cosmology where the observed spacetime is defined not as a pre-existing manifold, but as a macroscopic "ensemble average" of all possible spacetime fabrics. We aim to demonstrate that the perception of a unique physical reality is a result of holographic projection rather than intrinsic geometric properties. Methods: We utilize the AdS/CFT correspondence to model the universe as a holographic projection arising from a single, universal quantum state. By applying Feynman’s Path Integral formulation to the "superspace" of all possible metrics, we calculate the sum over histories for these geometric projections. We treat the emergence of classical spacetime as a process of constructive interference among infinite holographic realizations, filtering out unstable geometries through environmental decoherence.

The geometric-differential paradigm of spacetime, while foundational to modern physics, faces intrinsic challenges in quantum gravity and provides a limited framework for understanding state evolution as a computational process. This article proposes a shift in ontological foundations, presenting Ze, a discrete computational framework that implements the dynamics of a fundamental state vector Ψ. We argue that Ze is not merely an algorithm but a model of physical becoming, where space and time are not pre-existing coordinates but emerge as anti-parallel channels for the redistribution of information flow. Within Ze, the state vector is represented by a global configuration of statistical counters, and its evolution is driven by a process of passive learning through the minimization of prediction error. The framework naturally maps key physical entities: the temporal component corresponds to the sequential, irreversible flow of events and counter growth, while the spatial component corresponds to the synchronous, correlational structure of the internal model. This anti-parallelism is formalized in a discrete Ze-Lagrangian, a variational principle balancing structural stabilization against predictive error. We demonstrate how this setup allows for the emergence of relativistic invariants, a statistical and vector-based causality akin to causal sets, and quantum phenomena such as quantization (from discrete accounting) and interference (from superposition of predictive contexts). The theory reframes the observer as a self-norm-preserving system and suggests that Special and General Relativity are limiting descriptions of stable regimes in the Ze dynamics. The work offers a unified, vector-based ontology that derives physics from first principles of information processing.

Comparing Early Modern and Contemporary Emergent Space Ontologies: Kant, Quantum Gravity, and the Spatial Presence Problem

We establish a rigorous correspondence between the Krein space regularization method and the Lagrange multiplier (LM) approach to quantum gravity, extending both to the full Einstein-Hilbert action within the Unified Standard Model with Emergent Gravity-Effective Field Theory (USMEG-EFT) framework. Through explicit calculation of the one-loop effective action using the heat kernel expansion in the LM framework and the modified propagator structure in Krein quantization, we demonstrate that both methods yield identical finite results. The one-loop effective action takes the form [Formula: see text], with coefficients derived from the Seeley-DeWitt heat kernel expansion. The parameter correspondence [Formula: see text] emerges naturally from the regularization structure, with both encoding the finite domain of validity characteristic of an effective field theory. We demonstrate that both methods independently restrict quantum gravitational corrections to one-loop order through structurally isomorphic cancellation mechanisms: opposite-sign propagators in the LM formalism correspond to negative-norm states in Krein space, with higher-loop contributions vanishing identically via binomial summation. We provide detailed analysis of why standard Hilbert space quantization fails due to positive-definiteness constraints, and how the indefinite metric structure of Krein spaces enables systematic divergence cancellation while preserving unitarity in the physical sector. This equivalence provides mutual theoretical support for the USMEG-EFT framework representing the first successful unification of quantum gravity with the Standard Model.

This paper scrutinises contemporary developments in quantum cosmology in dialogue with Qur’anic metaphysics, proposing Divine Storytelling as a conceptual lens through which time, space, and cosmic order may be meaningfully understood. Advances in modern physics - particularly in quantum cosmology, inflationary theory, and approaches to quantum gravity - have increasingly challenged classical assumptions of linear time, absolute space, and material primacy. These scientific developments expose profound questions at the limits of empirical explanation concerning origins, causality, and the intelligibility of the universe. In parallel, the Quran and other scriptural traditions articulate a metaphysical vision in which the cosmos unfolds as a structured and purposeful order, communicated through āyāt (signs) and grounded in Divine knowledge. However, drawing especially on Quranic conceptions of God as the First and the Last, the Outer (visible) and the Inner (hidden) (Quran 57:3) and the creation of universe as per Bible (Genesis creation), this paper adopts a philosophical and hermeneutic approach that resists scientific concordism while allowing for conceptual resonance. Rather than treating revelation as a source of empirical prediction, scriptural metaphysics is approached as offering an interpretive framework in which time, space, and order are contingent dimensions within a deeper narrative unity. By placing quantum cosmology and scriptural metaphysics in constructive dialogue, the study argues that modern physics does not render spiritual cosmologies obsolete but instead reopens the question of meaning at the foundations of reality. Scientific inquiry clarifies how the universe behaves, while Quranic or Biblical and scriptural perspectives illuminate why such an ordered yet mysterious cosmos may be understood as intelligible, purposeful, and narratively coherent.

This paper introduces a novel ontological framework, termed the Ze model, which reframes the foundations of Einstein's relativity. It posits a unified vectorial substance, the state vector Ψ, as the fundamental entity, with its invariant norm ‖Ψ‖² serving as the primary conserved quantity. Space and time are not independent dimensions but emerge as antiparallel projections of Ψ. I demonstrate that Special Relativity (STR)—its invariant interval, time dilation, and the role of the speed of light c—arises as the kinematic limit of the dynamics governing the reallocation of Ψ's magnitude. General Relativity (GR) is recovered as the classical continuum limit, where spacetime curvature is reinterpreted as a smooth gradient in the orientation field of Ψ vectors, effectively unifying matter and geometry into manifestations of a single substrate. The framework exhibits deep conceptual affinities with pre-geometric approaches: it shares the primacy of a deeper space with Twistor Theory and grounds causality in vector directionality, paralleling Causal Set Theory. This synthesis suggests that STR and GR are not fundamental descriptions of an arena but are highly effective theories emergent from a monistic, vector-based reality. The model provides a new pathway for conceptualizing quantum gravity through the proposed quantization of Ψ's orientation.

Abstract The main goal of this work is to investigate how relevant quantum gravity corrections can be, at an effective level, in the geometry describing the exterior of a black hole, and whether such corrections can be tested observationally. For this purpose, we employ Bozza's method to calculate the deflection angle of light in presence of the strong gravitational field generated by an improved Schwarzschild-like black hole whose metric, regular throughout the entire spacetime, was derived using the improved Generalized Uncertainty Principle (GUP). This framework incorporates effective quantum gravity corrections that resolve the physical singularity inside the black hole, quantified by a model parameter | Qc |. In addition, the event horizon, the photon sphere, and the shadow radius receive modifications characterized by a second model parameter Qb . Using observational properties of the supermassive black holes Messier 87* and Sagittarius A * reported by the Event Horizon Telescope, we derive constraints on the parameter | Qb |, namely 0 ≤ | Qb | ≤ 0.3. To the best of our knowledge, these are the first constraints reported in the literature for this improved GUP parameter. Since | Qc | does not play a significant role in the correction of the shadow radius, it was not possible to impose restrictions on its allowed values, however, it is important to consider a non-zero | Qc | in order to avoid a black hole singularity.

A bstract Computing the gravitational effective action provides a direct route to charting the landscape of admissible black hole spacetimes and their alternatives, which we will collectively call “gravitationally localized objects” (GLOBs). In this work, we provide a proof of principle of this idea within the framework of asymptotically safe gravity. Focusing on the Einstein-Weyl truncation, we identify the unique ultraviolet-complete trajectory emanating from the asymptotically safe fixed point and use it to extract the Wilson coefficient of the Weyl-squared term. This allows us to chart the corresponding GLOBs in a “phase diagram”, showing that wormholes dominate a large portion of it, whereas the classical Bachian naked singularities become disfavored. Our results illustrate how quantum gravity can constrain effective field theory and the associated set of allowed spacetimes, yielding a rich landscape of beyond-general-relativity solutions rather than a single alternative to classical black holes.

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.