Einstein-Hilbert Action Research Articles

Relational Lorentzian Asymptotically Safe Quantum Gravity: Showcase Model

In a recent contribution, we identified possible points of contact between the asymptotically safe and canonical approaches to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity, which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantized, and one can construct the generating functional of time-ordered Wightman (i.e., Feynman) or Schwinger distributions, respectively, from the corresponding time-translation unitary group or contraction semigroup, respectively, as a path integral. For the unitary choice, that path integral can be rewritten in terms of the Lorentzian Einstein–Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantization and a reduction term that encodes the reduction of the full phase space to the phase space of observables. This path integral can then be treated with the methods of asymptotically safe quantum gravity in its Lorentzian version. We also exemplified the procedure using a concrete, minimalistic example, namely Einstein–Klein–Gordon theory, with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper, we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature, which has a strong impact on the convergence of “heat” kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation.

Gravastars formation with back-reaction effects from extended general relativity

Using an extended theory of General Relativity that incorporates normalized relativistic velocities, where the boundary terms in the varied Einstein–Hilbert action are considered. Within this context, I investigate the dynamic evolution of a collapsing spherical system characterized by a metric with spatial curvature and variable time-scale, aiming to describe the process of Gravastar formation. To illustrate the system’s dynamics, I analyze a power-law collapsing scenario and derive the equation of state for both cases, with and without geometrical perturbations. In particular, I derive the equation of state, incorporating back-reaction effects related with the boundary terms of the varied Einstein–Hilbert action.

Continuation of Bianchi spacetimes through the big bang

In this paper we present a framework in which the relational description of general relativity can be used to smoothly continue cosmological dynamical systems through the big bang without invoking quantum gravity effects. Cosmological spacetimes contain as a key dynamical variable a notion of scale through the volume factor ν. However, no cosmological observer is ever able to separate their measuring apparatus from the system they are measuring, in that sense every measurement is a relative one, and measurable dynamical variables are, in fact, dimensionless ratios. This is manifest in the identification of a scaling symmetry or “dynamical similarity” in the Einstein-Hilbert action associated with the volume factor. By quotienting out this scaling symmetry, we form a relational system defined on a contact manifold whose dynamical variables are decoupled from scale. When the phase space is reduced to shape space, we show that there exists unique solutions to the equations of motion that pass smoothly through the initial cosmological singularity in flat Friedmann-Lemaître-Robertson-Walker, Bianchi I, and Quiescent Bianchi IX cosmologies. Published by the American Physical Society 2024

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.

Discrete Z4 Symmetry in Quantum Gravity

We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, i^, the emerging tetrads are multiplied by the imaginary unit, i^eμa=−ieμa. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid 3He-B. The order parameter in 3He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The i^-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z4 discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z4 symmetry, in which the i^-operation changes sign of the scalar curvature, i^R=−R, and thus the Einstein–Hilbert action violates the i^-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16πG plays the role of the order parameter, which changes sign under i^-transformation.

Diverging black hole entropy from quantum infrared non-localities

Local higher-derivative corrections to the Einstein-Hilbert action yield sub-leading corrections to the Bekenstein-Hawking area law. Here we show that if the quantum effective action comprises a certain class of infrared non-localities, the entropy of large black holes generally diverges to either positive or negative infinity. In such theories, large spherically symmetric black holes would be either highly chaotic or thermodynamically impossible, respectively. In turn, this puts strong constraints on the Laurent expansion of the form factors in the effective action.

Is dark energy necessary for the sustainability of traversable wormholes?

Abstract Dark energy is usually the principal ingredient needed for traversability of wormholes by providing the negative gravity effect to keep the throat open. But can the same be achieved without dark energy? It turns out that if we couple the trace of energy-momentum with the standard Einstein-Hilbert Lagrangian and utilise a specific equation of state then dark energy may be obviated. The Casimir stress energy is known to result in the violation of the null energy condition (NEC) on the energy momentum tensor. This phenomenon makes such an equation of state (EoS) an ideal candidate to generate traversable wormhole (WH) geometries. The laboratory proven phenomenon provides a natural mechanism to sustain an open WH throat without having to appeal to dark energy. We generate two classes of WH solutions with this in energy-momentum trace-coupling gravity.
 For the specific choice of the Casimir EoS relating the energy-momentum tensor components [ Kar and Sahdev: Phys. Rev D {\bf 52} 2030 (1995)] and different choices for redshift functions, we determine the WH geometry completely. The obtained WH solutions violate the NECs and all qualitative constraints demanded of physically realisable WHs are satisfied. This is demonstrated via graphical plots for a suitably chosen range of values of the coupling parameter. Furthermore, our study involved an investigation into the repulsive effect of gravity, which revealed that its presence leads to a negative deflection angle for photons traveling along null geodesics. Notably, we observed a consistent pattern of negative values for the deflection angle across all values of $r_0$ in the three scenarios considered, thus indicating the clear manifestation of the repulsive gravity effect. All of this is possible without invoking the existence of dark energy.

The cosmological constant emerging from a symmetry invariant

The cosmological constant is normally introduced as an additional term entering the Einstein–Hilbert (EH) action. In this letter, we demonstrate that, instead, it appears naturally from the standard EH action as an invariant term emerging from spacetime symmetries. We then demonstrate that the same constraint emerging from this invariant suppresses the short wavelength modes and it favors the long wavelength ones. In this way, inside the proposed formulation, the observed value for the vacuum energy density is obtained naturally from the zero-point quantum fluctuations.

Warm inflation in a Universe with a Weylian boundary

We investigate the influence of boundary terms in the warm inflationary scenario, by considering that in the Einstein–Hilbert action the boundary can be described in terms of a Weyl-type geometry. The gravitational action, as well as the field equations, are thus extended to include new geometrical terms, coming from the non-metric nature of the boundary, and depending on the Weyl vector, and its covariant derivatives. We investigate the effects of these new boundary terms by considering the warm inflationary scenario of the early evolution of the Universe, in the presence of a scalar field. We obtain the generalized Friedmann equations in the Universe with a Weylian boundary by considering the Friedmann–Lemaitre–Robertson–Walker metric. We consider the simultaneous decay of the scalar field, and of the creation of radiation, by appropriately splitting the general conservation equation through the introduction of the dissipation coefficient, which can depend on both the scalar field, and the Weyl vector. We consider three distinct warm inflationary models, in which the dissipation coefficients are chosen as different functions of the scalar field and of the Weyl vector. The numerical solutions of the cosmological evolution equations show that the radiation is created during the very early phases of expansion, and, after the radiation reaches its maximum value, the transition from an accelerating inflationary phase to a decelerating one takes place. Moreover, it turns out that the Weyl vector, describing the boundary effects on the cosmological evolution, plays a significant role during the process of radiation creation.

Isolated compact star RXJ1856.5 − 3754 in f(R,T) modified gravity in Tolman-Kuchowicz spacetime

Our main motive is to model and study the mass-radius relation of the specific isolated compact star RXJ1856.5−3754 in the background of the late time accelerated Universe and to achieve the equation of state of the core matter under stable equilibrium in f(R,T) modified gravity using Tolman-Kuchowicz metric potentials. Einstein-Hilbert action is taken as f(R,T)=R+2ξT, where R and T are the Ricci scalar and the trace of energy-momentum tensor respectively and ξ as coupling parameter. Modified Tolman-Oppenheimer-Volkoff equations are numerically solved by considering the presence of modified Chaplygin gas inside the star and also solving the corresponding field equations. We have got an explicit scenario of the structural evolution of isolated neutron stars through accelerating spacetime. Interestingly, all the derived outcomes of this investigation have become compatible with physically adopted regimes which reveal the physical viability of this current model.

A Causal Net approach to Relativistic Quantum Mechanics and Gravitation

In this paper we discuss a causal network model to describe quantum mechanics and gravittation, where space–time is built solely from point events and connecting probabilities. A causal net provides a representation which combines the quantum complementary features of both partticles and waves as each vertex on the causal net represents a possible point event or particle observation. Simultaneity on the causal net is defined by hyperplanes of equivalent proper time. The causal net model when constructed in Minkowksi space–time is shown to lead to a formulation for relativistic quantum mechanics including the Dirac equation. In a curved Riemann space–time the causal paths of the causal net are geodesics and in the local Lorentz frame the net probabilities and the Dirac formalism are preserved. The variation of space–time density of events in the causal paths modifies the metric and provides a space–time curvature leading to the Hilbert action associated with general relativity. Consideration of the Schwarzchild metric shows that in classical limit the perturbation in the density of possible events is proportional to the Newtonian gravitational potential.

Seminal electromagnetic fields from preinflation

We investigate the geometric dynamics of the primordial electric and magnetic fields during the early stages of the universe by extending a recently introduced quantum algebra (Bellini et al., 2023). We work on an extended model of gravity that considers the boundary terms from the Einstein–Hilbert action as geometric quantum fluctuations of the spacetime. We propose that the extended Riemann manifold is generated by a new connection δΓˆαβμ. This connection contains geometric information about the fluctuations of gravitational and electromagnetic fields in the vacuum, which could have been crucial during the primordial stages of the universe’s evolution. We revisit a preinflationary cosmological model (Bellini, 2023) with a variable time scale and negative spatial curvature, such that the universe begins with a null initial background energy density. We observed the emergence of large scale magnetic fields starting from small values during the early phases of the universe’s evolution. Subsequently, these fields decrease to reach present day values on the order of δBˆ≃10−12G on cosmological scales (between 1024 and 1026 meters). This significant deviation from inflationary models eliminates the need to impose excessively large initial values on these fields.

The Einstein–Hilbert action for entropically dominant causal sets

In the path integral formulation of causal set quantum gravity, the quantum partition function is a phase-weighted sum over locally finite partially ordered sets, which are viewed as discrete quantum spacetimes. It is known, however, that the number of ‘layered’ sets—a class of causal sets that look nothing like spacetime manifolds—grows superexponentially with the cardinality n, giving an entropic contribution that can potentially dominate that of the action. We show here that in any dimension, the discrete Einstein–Hilbert action for a typical K-layered causal set reduces to the simple link action to leading order in n. Combined with earlier work, this completes the proof that the layered sets, although entropically dominant, are very strongly suppressed in the path sum of causal set quantum gravity whenever the discreteness scale is greater than or equal to a (mildly dimension-dependent) order one multiple of the Planck scale.

Global flows of foliated gravity-matter systems

Asymptotic safety is a promising mechanism for obtaining a consistent and predictive quantum theory for gravity. The ADM formalism allows to introduce a (Euclidean) time-direction in this framework. It equips spacetime with a foliation structure by encoding the gravitational degrees of freedom in a lapse function, shift vector, and a metric measuring distances on the spatial slices. We use the Wetterich equation to study the renormalization group flow of the graviton 2-point function extracted from the spatial metric. The flow is driven by the 3- and 4-point vertices generated by the foliated Einstein-Hilbert action supplemented by minimally coupled scalar and vector fields. We derive bounds on the number of matter fields cast by asymptotic safety. Moreover, we show that the phase diagram obtained in the pure gravity case is qualitatively stable within these bounds. An intriguing feature is the presence of an IR-fixed point for the graviton mass which prevents the squared mass taking negative values. This feature persists for any number of matter fields and, in particular, also in situations where there is no suitable interacting fixed point rendering the theory asymptotically safe. Our work complements earlier studies of the subject by taking contributions from the matter fields into account.

On the variation of the Einstein–Hilbert action in pseudohermitian geometry

Abstract In this paper, we compute the first and second variation of the normalized Einstein–Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater than or equal to five, in three dimensions, we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

Closed time-like curves in [formula omitted] modified gravity theory

In this manuscript, we investigate solutions that allow closed time-like curves (CTCs) in general relativity (GR) within the framework of a modified gravity theory. We consider here an alternative theory recently proposed in gravity theories known as f(R,A)=(R+αA) gravity or Ricci-inverse (RI) gravity, α is the coupling constant. This theory incorporates modifications into the Einstein-Hilbert action through a coupling with an anti-curvature scalar derived from the anti-curvature tensor. We investigate a family of Petrov type-N solutions that are non-vacuum solutions of the field equations, characterized by a negative cosmological constant. The matter-energy content is Vaidya radiation, and its energy density may either remain constant or vary with axial distance. One distinctive feature of the considered solutions is the formation of CTCs, which violate the causality condition. Another observation is that the determinant of the Ricci tensor, Rμν, differs from zero, confirming the existence of an anti-curvature tensor, Aμν, the inverse of the Ricci tensor, defined as Aμν=Rμν−1, and hence, the anti-curvature scalar A=gμνAμν. We demonstrate that the family of cosmological models serves as valid solutions within the framework of Ricci-inverse gravity model, with Vaidya radiation as the matter-energy content whose energy density is modified by the coupling parameter. Thus, our results affirm that the Ricci-inverse gravity theory allows for the existence of CTCs, similar to the case in general relativity.

Internal symmetries in Kaluza-Klein models

The usual approach to Kaluza-Klein considers a spacetime of the form M4 × K and identifies the isometry group of gK0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_K^0 $$\\end{document}, the internal vacuum metric, with the gauge group in four dimensions. In these notes we discuss a variant approach where part of the gauge group does not come from full isometries of gK0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_K^0 $$\\end{document}, but instead comes from weaker internal symmetries that only preserve the Einstein-Hilbert action on K. Then the weaker symmetries are spontaneously broken by the choice of vacuum metric and generate massive gauge bosons within the Kaluza-Klein framework, with no need to introduce ad hoc Higgs fields. Using the language of Riemannian submersions, the classical mass of a gauge boson is calculated in terms of the Lie derivatives of gK0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_K^0 $$\\end{document}. These massive bosons can be arbitrarily light and seem able to evade the standard no-go arguments against chiral fermionic interactions in Kaluza-Klein. As a second main theme, we also question the traditional assumption of a Kaluza-Klein vacuum represented by a product Einstein metric. This should not be true when that metric is unstable. In fact, we argue that the unravelling of the Einstein metric along certain instabilities is a desirable feature of the model, since it generates inflation and allows some metric components to change length scale. In the case of the Lie group K = SU(3), the unravelling of the bi-invariant metric along an unstable perturbation also breaks the isometry group from (SU(3) × SU(3))/ℤ3 down to (SU(3) × SU(2) × U(1))/ℤ6, the gauge group of the Standard Model. We briefly discuss possible ways to stabilize the internal metric after that first symmetry breaking and produce an electroweak symmetry breaking at a different mass scale.

Setting the connection free in the Galilei and Carroll expansions of gravity

We obtain a Palatini-type formulation for the Galilei and Carroll expansions of general relativity, where the connection is promoted to a variable. Known versions of these large and small speed of light expansions are derived from the Einstein-Hilbert action and involve dynamical Newton-Cartan or Carroll geometry, along with additional gauge fields at subleading orders. The corresponding Palatini actions that we obtain in this paper are derived from an appropriate expansion of the Einstein-Palatini action, and the connection variable reduces to the Galilei- or Carroll-adapted connection on shell. In particular, we present the Palatini form for the next-to-leading-order Galilean action and recover the known equations of motion. We also compute the leading-order Palatini-type action for the Carrollian case and show that, while it depends on the connection variable, it reduces on shell to the known action of electric Carroll gravity, which only depends on extrinsic curvature. Published by the American Physical Society 2024

Investigating the equation-of-state, stability and mass–radius relationship of anisotropic and massive neutron stars embedded in f(R,T) modified gravity

In this study, our main focus is to investigate the mass–radius relation and several important properties of massive neutron stars to realize the nature, behavior and evolution of these kinds of compact objects at present time. Also, we want to understand the equation-of-state of the core nuclear matter precisely with their stable equilibrium configuration. We have chosen a few massive binary pulsars and investigated on maximum attainable mass and lowest possible radius of them. We have considered Einstein–Hilbert action as [Formula: see text], where [Formula: see text] is the Ricci scalar and [Formula: see text], the trace of energy–momentum tensor with [Formula: see text] as the coupling parameter and also used modified TOV equations. The interior space-time of the spherical neutron star is matched to the exterior Schwarzschild line element at the surface of the star. The [Formula: see text]–[Formula: see text] curve can predict the maximum achievable mass is about [Formula: see text] with lowest possible radius of around [Formula: see text] km for the massive compact stars under stable equilibrium. We have obtained a clear picture of structural evolution of massive neutron stars through accelerating space-time and can put some constraints on several quantities related to them. It is depicted from our present investigation that all the derived outcomes are compatible with physically adopted regimes which reveal the viability of our current model in the context of [Formula: see text] modified gravity.

Nonlocal Massive Gravity from Einstein Gravity.

We present a top-down construction of a three-dimensional nonlocal theory of massive gravity. This "nonlocal massive gravity" (NLMG) is obtained as the gravitational theory induced by Einstein gravity on a brane inserted in anti-de Sitter space modified by an overall minus sign. The theory involves an infinite series of increasingly complicated higher-derivative corrections to the Einstein-Hilbert action, with the quadratic term coinciding with new massive gravity. We obtain an analytic formula for the quadratic action of NLMG and show that its linearized spectrum consists of an infinite tower of positive-energy massive spin-2 modes. We compute the Newtonian potential and show that the introduction of the infinite series of terms makes it behave as ∼1/r at short distances, as opposed to the logarithmic behavior encountered when the series is truncated at any finite order. We use this and input from brane-world holography to argue that the theory may contain asymptotically flat black hole solutions.