Curved Spacetime Research Articles

Spontaneous Lorentz Violation from Infrared Gravity

In this paper, we investigate a novel implication of the non-negligible spacetime curvature at large distances when its effects are expressed in terms of a suitably modified form of the Heisenberg uncertainty relations. Specifically, we establish a one-to-one correspondence between this modified uncertainty principle and the Standard Model Extension (SME), a string-theoretical effective field theory that accounts for both explicit and spontaneous breaking of Lorentz symmetry. This tight correspondence between string-derived effective field theory and modified quantum mechanics with extended uncertainty relations is validated by comparing the predictions concerning a deformed Hawking temperature derived from the two models. Moreover, starting from the experimental bounds on the gravity sector of the SME, we derive the most stringent constraint achieved so far on the value of the free parameter in the extended Heisenberg uncertainty principle.

Ricci cosmology in light of astronomical data

Recently, a new cosmological framework, dubbed Ricci cosmology, has been proposed. Such a framework has emerged from the study of relativistic dynamics of fluids out of equilibrium in a curved background and is characterised by the presence of deviations from the equilibrium pressure in the energy–momentum tensor which are due to linear terms in the Ricci scalar and the Ricci tensor. The coefficients in front of such terms are called the second order transport coefficients and they parametrise the fluid response to the pressure terms arising from the spacetime curvature. Under the preliminary assumption that the second order transport coefficients are constant, we find the simplest solution of Ricci cosmology in which the presence of pressure terms causes a departure from the perfect fluid redshift scaling for matter components filling the Universe. In order to test the viability of this solution, we make four different ansätze on the transport coefficients, giving rise to four different cases of our model. On the physical ground of the second law of thermodynamics for fluids with non-equilibrium pressure, we find some theoretical bounds (priors) on the parameters of the models. Our main concern is then the check of each of the case against the standard set of cosmological data in order to obtain the observational bounds on the second order transport coefficients. We find those bounds also realising that Ricci cosmology model is compatible with Lambda CDM cosmology for all the ansätze.

Holographic JT gravity with quartic couplings

We construct the most general theory of 2D Einstein-dilaton gravity coupled with U(1) gauge fields that contains all the 2-derivative and the 4-derivative interactions allowed by the diffeomorphism invariance. We renormalise the 2D action and obtain the vacuum solution as well as the black hole solution. The vacuum solution in the UV is dominated by Lifshitz2 with dynamical exponent (z = frac{7}{3} ) while on the other hand, the spacetime curvature diverges as we move towards the deep IR limit. We calculate the holographic stress tensor and the central charge for the boundary theory. Our analysis shows that the central charge goes as the inverse power of the coupling associated to 4-derivative interactions. We also compute the Wald entropy for 2D black holes and interpret its near horizon divergence in terms of the density of states. We compare the Wald entropy with the Cardy formula and obtain the eigen value of Virasoro operator (L0) for our model. Finally, we explore the near horizon structure of 2D black holes and calculate the central charge corresponding to the CFT near horizon. We further show that the near horizon CFT may be recast as a 2D Liouville theory with higher derivative corrections. We study the Weyl invariance of this generalised Liouville theory and identify the Weyl anomaly associated to it. We also comment on the classical vacuum structure of the theory.

Topological deviation of membrane waves as a model of gravitational lens effect

Elastic membranes are often used as didactic demonstration of gravitation from the general relativity perspective. Indeed, trajectories of rolling spheres such as billiard balls influence each other through the deformation their mass print within the membrane tissue as would the space-time curvature of gravity. The analogy is pushed here using membrane waves. Indeed, rules of topology apply to masses and waves. We show through experiments and curved manifold simulations that wave propagation due to topological deviation of a two-dimensional flat fabric membrane, justifies classical analogy with gravitational lenses. This allows revisiting through membrane waves, the famous 1919 Eddington experiment that demonstrated light deviations of stars in the vicinity of the sun. It is our hope that the experiment described here will provide new insight for the testing of geometry effects on wave propagation. Also, inspired by acoustic metamaterials, it is our hope to use a more correctly constructed membrane to observe quantified exchanges of geometric waves with their elementary mesh.

Power-law cosmology in Weyl-type f(Q, T) gravity

Gravity is attributed to the spacetime curvature in classical general relativity (GR). But, other equivalent formulations or representations of GR, such as torsion or non-metricity, have altered the perception. We consider the Weyl-type f(Q, T) gravity, where Q represents the non-metricity and T is the trace of energy momentum tensor, in which the vector field $$\omega _{\mu }$$ determines the non-metricity $$Q_{\mu \nu \alpha }$$ of the spacetime. In this work, we employ the well-motivated $$f(Q, T)= \alpha Q+ \frac{\beta }{6k^{2}} T$$ , where $$\alpha $$ and $$\beta $$ are the model parameters. Furthermore, we assume that the universe is dominated by the pressure-free matter, i.e., the case of dust ( $$p=0$$ ). We obtain the solution of field equations similar to a power-law in Hubble parameter H(z). We investigate the cosmological implications of the model by constraining the model parameter $$\alpha $$ and $$\beta $$ using the recent 57 points Hubble data and 1048 points Pantheon supernovae data. To study various dark energy models, we use statefinder analysis to address the current cosmic acceleration. We also observe the Om diagnostic describing various phases of the universe. Finally, it is seen that the solution which mimics the power-law fits well with the Pantheon data than the Hubble data.

Casimir densities induced by a sphere in the hyperbolic vacuum of de Sitter spacetime

Complete set of modes and the Hadamard function are constructed for a scalar field inside and outside a sphere in (D+1)-dimensional de Sitter spacetime foliated by negative constant curvature spaces. We assume that the field obeys Robin boundary condition on the sphere. The contributions in the Hadamard function induced by the sphere are explicitly separated and the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor are investigated for the hyperbolic vacuum. In the flat spacetime limit the latter is reduced to the conformal vacuum in the Milne universe and is different from the maximally symmetric Bunch-Davies vacuum state. The vacuum energy-momentum tensor has a nonzero off-diagonal component that describes the energy flux in the radial direction. The latter is a purely sphere-induced effect and is absent in the boundary-free geometry. Depending on the constant in Robin boundary condition and also on the radial coordinate, the energy flux can be directed either from the sphere or towards the sphere. At early stages of the cosmological expansion the effects of the spacetime curvature on the sphere-induced VEVs are weak and the leading terms in the corresponding expansions coincide with those for a sphere in the Milne universe. The influence of the gravitational field is essential at late stages of the expansion. Depending on the field mass and the curvature coupling parameter, the decay of the sphere-induced VEVs, as functions of the time coordinate, is monotonic or damping oscillatory. At large distances from the sphere the fall-off of the sphere-induced VEVs, as functions of the geodesic distance, is exponential for both massless and massive fields.

Induced curvature effects in Maxwell’s equations

We have studied in this work extended electrodynamics with a relaxed Lorenz gauge condition. We then compare it with the electrodynamics in curved space–time. Interestingly, the interaction of the electromagnetic field with the space curvature induces electric and magnetic charges and currents. The coupling of the electromagnetic field with the particle charge gives rise to its spin. Two spins are found to emerge in addition to their associated helicities. The electromagnetic field is found to be coupled to curvature through its electric and magnetic spins. The electrodynamics in curved space is found to be analogous to massive electrodynamics. When the electromagnetic field travels in curved space–time, it acquires mass depending on the strength of the space curvature. In curved space, the space in which the electromagnetic field travels got an effective electric conductivity proportional to the temporal variation of the metric determinant. The curved space appears to behave like a conducting medium with variable electric conductivity. Moreover, an additional current perpendicular to the magnetic field is induced whenever the space is curved. Axion and Chern–Simons electrodynamics are found to be connected to massive electrodynamics and standard electrodynamics in curved space–time.

General features of the thermalization of particle detectors and the Unruh effect

We study the thermalization of smeared particle detectors that couple locally to any operator in a quantum field theory in curved spacetimes. We show that if the field state satisfies the Kubo-Martin-Schwinger condition with inverse temperature $\ensuremath{\beta}$ with respect to the detector's local notion of time evolution, reasonable assumptions ensure that the probe thermalizes to the temperature $1/\ensuremath{\beta}$ in the limit of long interaction times. Our method also imposes bounds on the size of the system with respect to its proper acceleration and spacetime curvature in order to accurately probe the Kubo-Martin-Schwinger temperature of the field. We then apply this formalism to a uniformly accelerated detector probing the Minkowski vacuum of any $CPT$ symmetric quantum field theory, and show that the detector thermalizes to the Unruh temperature, independently of the operator it couples to. This exemplifies yet again the robustness of the Unruh effect, even when arbitrary smeared detectors are used to probe general operators in a quantum field theory.

Analytical investigation of pre-inflationary effects in the primordial power spectrum: from General Relativity to hybrid Loop Quantum Cosmology

This work is an analytical study of the main differences between classical inflationary effects of a fast-roll regime and imprints of hybrid Loop Quantum Cosmology (LQC) on the primordial power spectrum. Effective LQC solutions of phenomenological interest typically contain a classical period of kinetic dominance prior to slow-roll inflation, with consequences on the power spectrum that here we try and tell apart from those of phenomena occuring in the vicinity of a bounce in hybrid LQC. For this comparison, we use the same criterion for the choice of a vacuum state of the primordial perturbations, both in the context of relativistic cosmology and on the LQC cosmological background. As a first approximation to the problem, our study ignores the influence of a quasi-de Sitter evolution (rather than de Sitter) in the slow-roll inflationary regime, and of a typically short transition between the kinetically dominated era and the slow-roll period. Our results show that, while the two studied types of effects can lead to a drastic power suppression in the infrared sector of the spectrum, the mode scale at which this happens is much larger for LQC than for classical relativistic cosmology, the corresponding scales being related to the spacetime curvature at the bounce and at the onset of inflation, respectively.

Geometric transformations on a topological black hole and their applications

Topology and geometry can be used to classify interpretation of many astronomical phenomena in our universe. We characterize the spherical topology of black holes in a special case of spherical symmetry. For this purpose, we carefully consider the metrics space of topological black holes B4. By adjusting some of the parameters, we infer the appropriate components and the equatorial geodesics on the line element of the topological black holes. We also construct the minimal retraction ρ2:B4→C23. As a result, the minimal deformation retract has also been inferred to explain the effects on their center of mass in binary system. This would help us to better understand some of possible applications in astrophysics and cosmology, such as the natural phenomenon of a distinctive gravitational wave signature of merging two black holes associated with curved space–time in general relativity. In addition, we discuss the relation between limit folding and limit minimal retraction on the n-dimension. Under the folding map on the connected sum of essential sub-manifolds, the topological black hole can be constructed for different types and characteristics.

Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities

The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein’s equations with an (in this case inevitably smooth) energy–momentum–stress tensor of a ‘matter field’ as the source of spacetime curvature, this identity implies the physical laws of energy and momentum conservation for the ‘matter field’. The present work inquires into whether such a Bianchi identity can still hold in a weak sense for spacetimes with curvature singularities associated with timelike singularities in the ‘matter field’. Sufficient conditions that establish a distributional version of the twice-contracted second Bianchi identity are found. In our main theorem, a large class of spherically symmetric static Lorentzian metrics with timelike one-dimensional singularities is identified, for which this identity holds. As an important first application we show that the well-known Reissner–Weyl–Nordström spacetime of a point charge does not belong to this class, but that Hoffmann’s spacetime of a point charge with negative bare mass in the Born–Infeld electromagnetic vacuum does.

Lorentz symmetry violation and beyond semiclassical theory in the curved space–time of the arbitrarily dimensional Reissner–Nordström black hole

The quantum tunneling radiation of arbitrarily dimensional Reissner–Nordström black hole is corrected by beyond semiclassical theory and Lorentz symmetry violation theory, resulting in new expressions for the tunneling rate of scalar particles and the Hawking temperature at the horizon of the black hole. The physical significance of our results and the thermodynamic evolution characteristics of the black hole are also discussed.

Asymptotically Safe Gravity-Fermion Systems on Curved Backgrounds

We set up a consistent background field formalism for studying the renormalization group (RG) flow of gravity coupled to Nf Dirac fermions on maximally symmetric backgrounds. Based on Wetterich’s equation, we perform a detailed study of the resulting fixed point structure in a projection including the Einstein–Hilbert action, the fermion anomalous dimension, and a specific coupling of the fermion bilinears to the spacetime curvature. The latter constitutes a mass-type term that breaks chiral symmetry explicitly. Our analysis identified two infinite families of interacting RG fixed points, which are viable candidates to provide a high-energy completion through the asymptotic safety mechanism. The fixed points exist for all values of Nf outside of a small window situated at low values Nf and become weakly coupled in the large Nf-limit. Symmetry-wise, they correspond to “quasi-chiral” and “non-chiral” fixed points. The former come with enhanced predictive power, fixing one of the couplings via the asymptotic safety condition. Moreover, the interplay of the fixed points allows for cross-overs from the non-chiral to the chiral fixed point, giving a dynamical mechanism for restoring the symmetry approximately at intermediate scales. Our discussion of chiral symmetry breaking effects provides strong indications that the topology of spacetime plays a crucial role when analyzing whether quantum gravity admits light chiral fermions.

Modification of the Exterior and Interior Solution of Einstein’s G22 Field Equation for a Homogeneous Spherical Massive Bodies whose Fields Differ in Radial Size, Polar Angle, and Time.

In general theory of relativity, Einstein’s field equations relate the geometry of space-time with the distribution of matter within it. These equations were first published by Einstein in the form of a tensor equation which related the local space-time curvature with the local energy and momentum within this space-time. In this article, Einstein’s geometrical field equations interior and exterior to astrophysically real or hypothetical distribution of mass within a spherical geometry were constructed and solved for field whose gravitational potential varies with time, radial distance and polar angle. The exterior solution was obtained using power series. The metric tensors and the solution of the Einstein’s exterior field equations used in this work has only one arbitrary function f(t,r,θ) , and thus put the Einstein’s geometrical theory of gravitation on the same bases with the Newton’s dynamical theory of gravitation. The gravitational scalar potential f(t,r,θ) obtained in this research work to the order of co, c-2 , contains Newton dynamical gravitational scalar potential and post Newtonian additional terms much importance as it can be applied to the study of rotating bodies such as stars. The interior solution was obtained using weak field and slow-motion approximation. The obtained result converges to Newton’s dynamical scalar potential with additional time factor not found in the well-known Newton’s dynamical theory of gravitation which is a profound discovery with the dependency on three arbitrary functions. Our result obeyed the equivalence principle of Physics.

Partial Differential Equations and Quantum States in Curved Spacetimes

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

A wavefunction description for a localized quantum particle in curved spacetimes

We reduce Dirac’s spinor formalism for a spin 1/2 particle to a complex wavefunction description in curved spacetimes. We consider a localized fermionic particle in curved spacetimes and perform an expansion in terms of the acceleration and curvature around the center of mass of the system, generalizing the results of Parker (1980 Phys. Rev. Lett. 44 1559), Parker (1980 Phys. Rev. D 22 1922–1934). Under a non-relativistic approximation, one obtains a quantum description in a Hilbert space of complex wavefunctions defined in the rest space of the system. The wavefunction of the particle then evolves according to a modified Schrödinger equation associated with a symmetric Hamiltonian. When compared to the standard Schrödinger equation for a wavefunction, we obtain corrections in terms of the acceleration of the system’s center of mass and curvature of spacetime along its trajectory. In summary, we provide a formalism for the use of a complex wavefunction to describe a localized quantum particle in curved spacetimes.

Derivation of general relativistic gravitational potential energy using principle of equivalence and gravitational time dilation

Einstein’s theory of general relativity, which has been experimentally proved to be a true theory of gravity does not need gravitational potential energy to predict the trajectory of particles in space. This is because general relativity is a purely geometric theory. Objects move along the geodesics in the curved space–time. The energy–momentum tensor that warps space–time as per Einstein’s field equations takes into account only the energy–momentum of matter and radiation. Thus, gravitational potential energy does not come into the picture in Einstein’s theory of gravity and its role is taken over by the curvature of space–time. However, a general relativistically correct expression for gravitational potential energy is required for energy conservation and some energy-based approaches in physics. Conventionally, the correct form of gravitational potential energy is derived by using the full mathematical formality of general relativity. In this paper, we derive the same general relativistic expression for gravitational potential energy simply by using the principle of equivalence and gravitational time dilation.

Toward the Gravitational Redshift Detection in NGC 4258 and the Estimation of Its Black Hole Mass-to-distance Ratio

Abstract We construct from first principles a general relativistic approach to study Schwarzschild black hole (BH) rotation curves and estimate the mass-to-distance ratio of the active galactic nucleus of NGC 4258 in terms of astrophysical observable quantities. The presented method allows one to clearly distinguish and quantify the general and special relativistic contributions to the total redshift expression. The total relativistic redshift/blueshift comprises three components: the gravitational redshift due to the spacetime curvature generated by the mass of the BH in its vicinity, the kinematic shift, originated by the photons’ local Doppler effect, and the redshift due to a special relativistic boost that describes the motion of a galaxy from a distant observer. We apply our method to the largest data set of highly redshifted water megamaser measurements on the accretion disk of the NGC 4258 active galaxy and use this general relativistic method to estimate its BH mass-to-distance ratio: M/D = (0.5326 ± 0.00022) × 107 M ⊙ Mpc−1.

Towards communication in a curved spacetime geometry

The current race in quantum communication – endeavouring to establish a global quantum network – must account for special and general relativistic effects. The well-studied general relativistic effects include Shapiro time-delay, gravitational lensing, and frame dragging which all are due to how a mass distribution alters geodesics. Here, we report how the curvature of spacetime geometry affects the propagation of information carriers along an arbitrary geodesic. An explicit expression for the distortion onto the carrier wavefunction in terms of the Riemann curvature is obtained. Furthermore, we investigate this distortion for anti de Sitter and Schwarzschild geometries. For instance, the spacetime curvature causes a 0.10 radian phase-shift for communication between Earth and the International Space Station on a monochromatic laser beam and quadrupole astigmatism; can cause a 12.2% cross-talk between structured modes traversing through the solar system. Our finding shows that this gravitational distortion is significant, and it needs to be either pre- or post-corrected at the sender or receiver to retrieve the information.

Spherically Symmetric Exact Vacuum Solutions in Einstein-Aether Theory

We study spherically symmetric spacetimes in Einstein-aether theory in three different coordinate systems, the isotropic, Painlevè-Gullstrand, and Schwarzschild coordinates, in which the aether is always comoving, and present both time-dependent and time-independent exact vacuum solutions. In particular, in the isotropic coordinates we find a class of exact static solutions characterized by a single parameter c14 in closed forms, which satisfies all the current observational constraints of the theory, and reduces to the Schwarzschild vacuum black hole solution in the decoupling limit (c14=0). However, as long as c14≠0, a marginally trapped throat with a finite non-zero radius always exists, and on one side of it the spacetime is asymptotically flat, while on the other side the spacetime becomes singular within a finite proper distance from the throat, although the geometric area is infinitely large at the singularity. Moreover, the singularity is a strong and spacetime curvature singularity, at which both of the Ricci and Kretschmann scalars become infinitely large.