Classical General Relativity Research Articles

Quantum geometric approach: Nature and characteristics of emerged quantum-conditioned spacetime curvatures on three-sphere

General relativity (GR) in its current classical formulation is fundamentally different from quantum mechanics (QM). We argue that the everlasting battle for exploring and understanding the Universe and then privileging a consistent perception of reality is therefore awaiting a consolidator rather than a conqueror. This should be capable of either unifying these two different benchmarks or at least bringing one closer to another! The latter conservatively describes the consolidating quantum geometric approach, which combines generalization of QM to relativistic energies and gravitational fields and continuous Riemann to discretized Finsler geometry. We found that this type of quantum geometric approach seems to unveil additional quantum-conditioned spacetime curvatures (QCC), sources of gravitation), which obviously could not be disclosed in Einstein’s GR, especially at large scales. This study introduces analytical and numerical analyses of QCC. We conclude that (i) nature and characteristics of QCC are fundamentally distinguishable from the classical GR’s spacetime curvatures, (ii) the QCC are intrinsic, real and essential, i.e. not artifacts that would be removed in a certain coordinate transformation and (iii) the magnitude of QCC are nonnegligible to be underestimated. We also conclude that the spacetime at quantum scales seems to be no longer smooth or continuous so that the proposed quantum geometric approach would be regarded as a novel mathematical framework for the emergence of quantum gravity. Last but not least, a maximal proper force is predicted as a new physical constant, which is responsible for the maximal and gravitational acceleration of a quantum particle that would live in the emerged spacetime curvatures. Thereby, the quantum geometric nature of QCC can be accessed and assessed.

Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity

Classical General Relativity is a dynamical theory of spacetime metrics of Lorentzian signature. In particular the classical metric field is nowhere degenerate in spacetime. In its initial value formulation with respect to a Cauchy surface the induced metric is of Euclidian signature and nowhere degenerate on it. It is only under this assumption of non-degeneracy of the induced metric that one can derive the hypersurface deformation algebra between the initial value constraints which is absolutely transparent from the fact that the inverse of the induced metric is needed to close the algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface deformation algebra in quantum gravity has to address the issue of non-degeneracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop Quantum Gravity (LQG) most emphasis has been laid to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator topologies, dual spaces (habitats) and density weights have been proposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard constructions.

On the feasibility of detecting quantum delocalization effects on relativistic time dilation in optical clocks

Abstract We derive the predicted time dilation of delocalized atomic clocks in an optical lattice setup in the presence of a gravitational field to leading order in quantum relativistic corrections. We investigate exotic quantum states of motion whose relativistic time dilation is outside of the realm of classical general relativity, finding a regime where 24 Mg optical lattice clocks currently in development would comfortably be able to detect the special-relativistic contribution to the quantum effect (if the technical challenge of generating the necessary states can be met and the expected accuracy of such clocks can be attained). We find that the gravitational contribution, on the other hand, is negligible in this setup. We provide a detailed experimental protocol and analyse the effects of noise on our predictions. We also show that the magnitude of our predicted quantum time dilation effect remains just out of detectable reach for the current generation of 87 Sr optical lattice clocks. Our calculations agree with the predicted time dilation of classical general relativity when restricting to Gaussian states.

Quantum geometric perspective on the origin of quantum-conditioned curvatures

Abstract The quantization of the gravitational field, which includes the metric field, has been investigated using various methods such as loop quantum gravity, quantum field theory, and string theory. Nevertheless, an alternative strategy to tackle the challenge of merging the fundamentally different theories of general relativity (GR) and quantum mechanics (QM) is through a quantum geometric approach. This particular approach entails extending QM to relativistic energies and finite gravitational fields, while also expanding the continuous Riemann to a discretized (quantized) Finsler-Hamilton geometry. By embracing this method, it may be feasible to bridge the gap between GR and QM or even achieve their unification. The resulting fundamental tensor appears to blend its original classical and quantum characteristics, effectively integrating quantum-mechanically induced revisions to the affine connections and spacetime curvatures. Our study primarily focuses on investigating the Ricci curvature tensor in the context of the Einstein-Gilbert-Straus metric. By employing both analytical and numerical methods, we have identified quantum-conditioned curvatures (QCC) that act as additional sources of gravitation. These QCC exhibit a fundamental difference from the traditional curvatures described by Einsteinian GR. While the Ricci curvatures are predominantly positive across most regions, the quantized Ricci curvatures display negativity. We conclude that the QCC a) possess an intrinsic, essential, and real character, b) should not be disregarded due to their significant magnitude, and c) are fundamentally different from the curvatures found in classical GR. Moreover, we conclude that the proposed quantum geometric approach may offer an alternative mathematical framework for understanding the emergence of quantum gravity.

Black holes and wormholes beyond classical general relativity

In the paper only Static Spherically Symmetric space–times in four dimension are considered within modified gravity models. The non-singular static metrics, including black holes not admitting a de Sitter core in the center and traversable wormholes, are reconsidered within a class of higher-order F(R), satisfying the constraints F(0)=dFdR(0)=0. Furthermore, making use of the so called effective field theory formulation of gravity, the quantum corrections to Einstein–Hilbert action due to higher-derivative terms related to curvature invariants are investigated. In particular, in the case of Einstein–Hilbert action plus cubic curvature Goroff–Sagnotti contribution, the second order correction in the Goroff–Sagnotti coupling constant is computed. In general, it is shown that the effective metrics, namely Schwarzschild expression plus small quantum corrections are related to black holes, and not to traversable wormholes. In this framework, within the approximation considered, the resolution of singularity for r=0 is not accomplished. The related properties of these solutions are investigated.

Influence of pressure anisotropy on mass-radius relation and stability of millisecond pulsars in f(Q) gravity

In this study we explore the astrophysical implications of pressure anisotropy on the physical characteristics of millisecond pulsars within the framework of f(Q) gravity, in particular f(Q) = - α Q - β,where α and β are constants. Starting off with the field equations for anisotropic matter configurations, we adopt the physically salient Durgapal-Fuloria ansatz together with a well-motivated anisotropic factor for the interior matter distribution. This leads to a nonlinear second order differential equation which is integrated to give the complete gravitational and thermodynamical properties of the stellar object. The resulting model is subjected to rigorous tests to ensure that it qualifies as a physically viable compact object within the f(Q)-gravity framework. We study in detail the impact of anisotropy on the mass, radius and stability of the star. Our analyses indicate that our models are well-behaved, singularity-free and can account for the existence of a wide range of observed pulsars with masses ranging from 2.08 to 2.67 M ⊙, with the upper value being in the so-called mass gap regime observed in gravitational events such as GW190814. A comparison of the so-called Symmetric Teleparallel Equivalent to GR (STEGR) models with classical General Relativity (GR) models reveal that the anisotropy parameter and the sign of β impact on the predicted radii of pulsars. In particular, STEGR models have larger radii than their GR counterparts.

Graviton Physics: A Concise Tutorial on the Quantum Field Theory of Gravitons, Graviton Noise, and Gravitational Decoherence

The detection of gravitational waves in 2015 ushered in a new era of gravitational wave (GW) astronomy capable of probing the strong field dynamics of black holes and neutron stars. It has opened up an exciting new window for laboratory and space tests of Einstein’s theory of classical general relativity (GR). In recent years, two interesting proposals have aimed to reveal the quantum nature of perturbative gravity: (1) theoretical predictions on how graviton noise from the early universe, after the vacuum of the gravitational field was strongly squeezed by inflationary expansion; (2) experimental proposals using the quantum entanglement between two masses, each in a superposition (gravitational cat, or gravcat) state. The first proposal focuses on the stochastic properties of quantum fields (QFs), and the second invokes a key concept of quantum information (QI). An equally basic and interesting idea is to ask whether (and how) gravity might be responsible for a quantum system becoming classical in appearance, known as gravitational decoherence. Decoherence due to gravity is of special interest because gravity is universal, meaning, gravitational interaction is present for all massive objects. This is an important issue in macroscopic quantum phenomena (MQP), underlining many proposals in alternative quantum theories (AQTs). To fully appreciate or conduct research in these exciting developments requires a working knowledge of classical GR, QF theory, and QI, plus some familiarity with stochastic processes (SPs), namely, noise in quantum fields and decohering environments. Traditionally a new researcher may be conversant in one or two of these four subjects: GR, QFT, QI, and SP, depending on his/her background. This tutorial attempts to provide the necessary connective tissues between them, helping an engaged reader from any one of these four subjects to leapfrog to the frontier of these interdisciplinary research topics. In the present version, we shall address the three topics listed in the title, excluding gravitational entanglement, because, despite the high attention some recent experimental proposals have received, its nature and implications in relation to quantum gravity still contain many controversial elements.

Quantum gravity modifications to the accretion onto a Kerr black hole

In the framework of the Asymptotic Safety scenario for quantum gravity, we analyze quantum gravity modifications to the thermal characteristics of a thin accretion disk spiraling around a renormalization group improved (RGI-) Kerr black hole in the low energy regime. We focused on the quantum effects on the location of the innermost stable circular orbit (ISCO), the energy flux from the disk, the disk temperature, the observed redshifted luminosity, and the accretion efficiency. The deviations from the classical general relativity due to quantum effects are described for a free parameter that arises in the improved Kerr metric as a consequence of the fact that the Newton constant turns into a running coupling G(r) depending on the energy scale. We find that, both for rapid and slow rotating black holes with accretion disks in prograde and retrograde circulation, increases in the value of this parameter are accompanied by a decreasing of the ISCO, by a lifting of the peaks of the radiation properties of the disk and by an increase of the accretion mass efficiency, as compared with the predictions of general relativity. Our results confirm previously established findings in Zuluaga and Sánchez (Eur Phys J C 81:840, 2021) where we showed that these quantum gravity effects also occur for an accretion disk around a RGI-Schwarzschild black hole.

Calabi-Yau periods for black hole scattering in classical general relativity

The high-precision description of black hole scattering in classical general relativity using the post-Minkowskian (PM) expansion requires the evaluation of single-scale Feynman integrals at increasing loop orders. Up to 4PM, the scattering angle and the impulse are expressible in terms of polylogarithmic functions and Calabi-Yau (CY) twofold periods. As in QFT, periods of higher dimensional CY n-folds are expected at higher PM order. We find at 5PM in the dissipative leading order self-force sector (5PM-1SF) that the only nonpolylogarithmic functions are the K3 periods encountered before and the ones of a new hypergeometric CY threefold. In the 5PM-2SF sector further CY twofold and threefold periods appear. Griffiths transversality of the CY period motives allows one to transform the differential equations for the master integrals into ε-factorized form and to solve them in terms of a well-controlled function space, as we demonstrate in the 5PM-1SF sector. Published by the American Physical Society 2024

No U(1) ‘electric-magnetic’ duality in Einstein gravity

We revisit the question of whether classical general relativity obeys, beyond the linearised order, an analogue of the global U(1) electric-magnetic duality of Maxwell theory, with the Riemann tensor playing the role analogous to the field strength. Following contradictory claims in the literature, we present a simple gauge-invariant argument that the duality does not hold. The duality condition is the conservation of the helicity charge. Scattering amplitudes of gravitons in general relativity, and of gluons in Yang-Mills theory, violate this selection rule already at tree level. Indeed, the maximally-helicity-violating (MHV) amplitudes are famous for their simplicity. The duality in the linearised theories is, therefore, broken by the interactions. In contrast, the tree-level scattering amplitudes in duality-invariant theories of non-linear electromagnetism are known to obey helicity conservation. While the duality is not a symmetry of the full theory of general relativity, it does hold within a sector of the solution space, including vacuum type D solutions, where the duality is known to rotate between mass and NUT charge.

Gravitational collapse via Wheeler–DeWitt equation

We analyze the Wheeler–DeWitt (WDW) equation in the context of a gravitational collapse. The physics of an expanding/collapsing universe and many details of a collapsing star can classically be described by the Robertson–Walker metric in which the WDW equation takes the form of a times-less Schrödinger equation. We set up the corresponding WDW potential for the collapse and study the solutions of the wave function. The results show that the central singularity appearing in classical general relativity is avoided, the density is quantized in terms of the Planck density and the expectation value of the scale factor exhibits a discrete behavior.

A four parameters quark star in quadratic [formula omitted]action

In this collaborative work, we model charged compact objects within the framework of f(Q) gravity. We assume a quadratic form for the metricity function cast as f(Q)=Q+aQ2 where a can be thought of as a ‘switch’, regaining the classical general relativity limit when it vanishes. In order to close the system of governing equations, we adopt the Krori–Barua ansatz for the metric functions and the MIT bag model equation of state (EOS) determines the evolution of the electric field. The resulting models are singularity-free and describe physically realizable stellar structures such as neutron stars. We demonstrate that the physical attributes of our models are sensitive to four parameters, namely, the EOS, metricity, and charge parameters together with the bag constant. We observe that an increase in the quadratic contribution culminates in a decrease of the density, radial pressure, electric field, and sound speeds while ωr,ωt, anisotropy, and pt increase. The novelty in our work centers on a possible mechanism of increasing the anisotropy of the stellar interior without resorting to exotic fluids or anisotropisation via gravitational decoupling. The robustness of our models can account for observed mass–radius relations of well-known compact stars such as PSR J1614-2230, PSR J1903+6620, Cen X-3 and LMC X-4 and the secondary component of GW190814 event.

Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type Σp×R, p > 1, where Σp is a closed Riemann surface of genus p, in the regime of 2 + 1 dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space (TΣp≈R6p−6) of Σp. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations (PMLPMF), the Thurston boundary of the Teichmüller space.

On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

The relation between KMOC and worldline formalisms for classical gravity

We demonstrate the equivalence between observables in the KMOC and worldline formalisms for classical general relativity, highlighting the relation between the initial conditions in the two frameworks and how the Keldysh-Schwinger in-in formalism is contained in both of them even though the KMOC representation conventionally leads to the evaluation of scattering amplitudes with Feynman propagators. The relationship between the two approaches is illustrated in detail for the momentum kick at second Post-Minkowskian order.

Planck length in classical and quantum Hamiltonian formulations of general relativity

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A

Horizon-scale images of black holes (BHs) and their shadows have opened an unprecedented window onto tests of gravity and fundamental physics in the strong-field regime. We consider a wide range of well-motivated deviations from classical general relativity (GR) BH solutions, and constrain them using the Event Horizon Telescope (EHT) observations of Sagittarius A (Sgr A), connecting the size of the bright ring of emission to that of the underlying BH shadow and exploiting high-precision measurements of Sgr A’s mass-to-distance ratio. The scenarios we consider, and whose fundamental parameters we constrain, include various regular BHs, string-inspired space-times, violations of the no-hair theorem driven by additional fields, alternative theories of gravity, novel fundamental physics frameworks, and BH mimickers including well-motivated wormhole and naked singularity space-times. We demonstrate that the EHT image of Sgr A places particularly stringent constraints on models predicting a shadow size larger than that of a Schwarzschild BH of a given mass, with the resulting limits in some cases surpassing cosmological ones. Our results are among the first tests of fundamental physics from the shadow of Sgr A and, while the latter appears to be in excellent agreement with the predictions of GR, we have shown that a number of well-motivated alternative scenarios, including BH mimickers, are far from being ruled out at present.

Emergent Diffeomorphism Invariance in Toy Models

AbstractConceptual difficulties in semiclassical and quantum gravity arise from diffeomorphism invariance of classical general relativity. With a motivation to shed some light on these difficulties, a class of toy models for which one‐dimensional diffeomorphism invariance, namely time‐reparametrization invariance, emerges at the classical level from energy conservation are studied. An attempt to quantize the models while taking the invariance seriously leads to toy versions of the problem of time in quantum gravity, of the cosmological constant problem, and of the black hole firewall problem. Nevertheless, all these problems are easily resolved by taking into account that the invariance emerges only at the classical level, while the fundamental theory that needs to be quantized is not diffeomorphism invariant.

On the Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

The theoretical prediction of the stochastic property of the quantum cosmological constant and the quantum stochastic nature of event horizons has crucial implications on the physics of space-time and black holes in particular. One of these consequences concerns a new mechanism, which is investigated here, for the stochastic regularization of singular black-hole solutions of classical general relativity. The problem is posed in the context of the theory of covariant quantum gravity (CQG-theory), namely the manifestly covariant, constraint-free and finite graviton-mass quantum Hamiltonian approach developed by Cremaschini and Tessarotto (2015–2022), which permits to cast the theory in a frame-independent setting. It is precisely the trajectory-dependence feature of the theory and the intrinsic stochastic property of quantum gravity which turn out to be crucial properties for reaching quantum regularization of classical singular solutions.

The sub-leading scattering waveform from amplitudes

We compute the next-to-leading order term in the scattering waveform of uncharged black holes in classical general relativity and of half-BPS black holes in mathcal{N} = 8 supergravity. We propose criteria, generalizing explicit calculations at next-to-leading order, for determining the terms in amplitudes that contribute to local observables. For general relativity, we construct the relevant classical integrand through generalized unitarity in two distinct ways, (1) in a heavy-particle effective theory and (2) in general relativity minimally-coupled to scalar fields. With a suitable prescription for the matter propagator in the former, we find agreement between the two methods, thus demonstrating the absence of interference of quantum and classically-singular contributions. The classical mathcal{N} = 8 integrand for massive scalar fields is constructed through dimensional reduction of the known five-point one-loop integrand. Our calculation exhibits novel features compared to conservative calculations and inclusive observables, such as the appearance of master integrals with intersecting matter lines and the appearance of a classical infrared divergence whose absence from classical observables requires a suitable definition of the retarded time.