Curved Spacetime Research Articles

Characterizations of a Spacetime of Quasi‐Constant Sectional Curvature and F(R)$\mathcal {F}(\mathcal {R})$‐Gravity

AbstractThe main aim of this article is to investigate a spacetime of quasi‐constant sectional curvature. At first, the existence of such a spacetime is established by several examples. We have shown that a spacetime of quasi‐constant sectional curvature agrees with the present state of the universe and it represents a Robertson Walker spacetime. Moreover, if the spacetime is Ricci semi‐symmetric or Ricci symmetric, then either the spacetime represents a spacetime of constant sectional curvature, or the spacetime represents phantom era. Also, we prove that a Ricci symmetric spacetime of quasi‐constant sectional curvature represents a static spacetime and the spacetime under consideration is of Petrov type I, D or O. Finally, we concentrate on a quasi‐constant sectional curvature spacetime solution in ‐gravity. As a result, various energy conditions are studied and analyzed our obtained outcomes in terms of a ‐gravity model.

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

Weak Gravitational Lensing around Bardeen Black Hole with a String Cloud in the Presence of Plasma

The effect of spacetime curvature on optical properties may provide an opportunity to suggest new tests for gravity theories. In this paper, we investigated gravitational weak lensing around a Bardeen black hole with the string clouds parameter. First, we examined the horizon structure in the presence of string clouds around the gravitational compact object defined by Bardeen spacetime. The effect of gravitational weak lensing in a plasma medium is also discussed. According to the findings, the influence of the string cloud parameter on the circular orbits of a light ray around the black hole is greater than that in the Schwarzschild case, while the influence of the charge is reversed. The deflection angle of light rays in weak lensing is also used to study how much the image is magnified.

Codazzi tensors and their space-times and Cotton gravity

We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdziński. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. We show that the equation of Cotton gravity is Einstein’s equation modified by the presence of a Codazzi tensor, which can be chosen freely and constrains the space-time where the theory is staged. In doing so, the tensor (chosen in forms appropriate for physics) implies the form of the Ricci tensor. The two tensors specify the energy-momentum tensor, which is the source in the equation of Cotton gravity for the metric implied by the Codazzi tensor. For example, we show that the Stephani, Nariai and Bertotti-Robinson space-times are characterized by a “current flow” Codazzi tensor. Because of it, they solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in constant curvature space-times.

ПРИЛИВНЫЕ СИЛЫ ВБЛИЗИ ЧЕРНЫХ ДЫР В ГРАВИТАЦИОННОЙ ТЕОРИИ С НАРУШЕНИЕМ СИММЕТРИИ ЛОРЕНЦА

It is generally accepted that the curvature of space-time is small near the event horizon of a large static black hole, and test particles can fall into it without being destroyed. However, in most cases, at a small distance from the event horizon, the curvature of the black hole can be so huge that the test particle will stretch and even collapse under the influence of tidal forces. Thus, any object that falls into black holes experiences huge tidal forces beyond the horizon. This is a result of the fact that the curvature is very large near black holes and almost zero near the horizon. When measurements are made in a static frame (which also becomes zero), the curvature components remain small. This means that all curvature invariants are small, and any perturbing corrections, such as charge, angular momentum, Ricci scalar, etc., can be neglected. However, in a freely falling frame (the reference frame of a distant asymptotic observer), the curvature components are very large. In our paper, we consider the influence of the perturbing parameter ℓ on the curvature invariants of black holes in the gravitational theory with Lorentz symmetry breaking. The results are compared with the Schwarzschild black hole solution, which can be obtained by substituting ℓ =0.

The Space-Time Properties of Three Static Black Holes

In the curved space-time, the neutral test particle is not affected by any other force except for the influence of the curved space-time. Similar to the free sub in the flat space, the Lagrangian of the test particle only contains the kinetic energy term—the kinetic energy term of the four-dimensional curved space-time. In the case of small space-time curvature, linear approximation can be made. That is, under the weak field approximation, the Lagrangian quantity degenerates into the Lagrangian quantity in the axisymmetric gravitational field in Newtonian mechanics. In this paper, the curved space-time composed of axisymmetric equidistant black holes is taken as a model. We study the geodesic motion of the test particles around three black holes with equal mass and static axisymmetric distribution, including time-like particles and photons. The three extreme Reissner–Nordstrom black holes are balanced by electrostatic and gravitational forces. We first give the geodesic motion equation of particles in Three black holes space-time, give the relativistic effective potential, discuss the possible motion state of particles, and classify their motion trajectories. Then, the particle motion of the special plane (equatorial plane) is studied. The circular orbits of the two types of particles in the symmetric plane are studied, respectively. The circular orbits outside the symmetric plane are also studied, and their stability is also discussed. We will show the influence of the separation distance of the three black holes on the geodesic motion and explore the change of the relativistic effective potential. Then, the relationship between the inherent quantity and the coordinate quantity in space-time is analyzed. Finally, the chaos of the test particle orbit is explored.

Semicooperation under curved strategy spacetime

ABSTRACT Mutually beneficial cooperation is a common part of economic systems as firms in partial cooperation with others can often make a higher sustainable profit. Though cooperative games were popular in 1950s, recent interest in noncooperative games is prevalent despite the fact that cooperative bargaining seems to be more useful in economic and political applications. In this paper we assume that the strategy space and time are inseparable with respect to a contract. Furthermore, it is assumed that each firm’s strategy polygon is a geodesic polygon which changes its shape every point of time with the stubbornness strategy surface of firm’s executive board follow a Gaussian free field. This gives us more flexibility to deal with generalized geodesic cooperative games which is the main contribution of this paper. Under this environment we show that the strategy spacetime is a dynamic curved Liouville-like 2-brane quantum gravity surface under asymmetric information and that traditional Euclidean geometry fails to give a proper feedback Nash equilibrium. Cooperation occurs when two firms’ strategies fall into each other’s influence curvature in this strategy spacetime. Small firms in an economy dominated by large firms are subject to the influence of large firms. We determine an optimal feedback semicooperation of the small firm in this case using a Liouville-Feynman path integral method.

Curvature and dynamical spacetimes: can we peer into the quantum regime?

Stationary compact astrophysical objects such as black holes and neutron stars behave as classical systems from the gravitational point of view. Their (observable) curvature is everywhere ‘small’. Here we investigate whether mergers of such objects, or other strongly dynamical spacetimes such as collapsing configurations, may probe the strong-curvature regime of general relativity. Our results indicate that dynamical black hole spacetimes always result in a modest increase in the Kretschmann scalar, relative to the stationary state. In contrast, we find that the Kretschmann scalar can dynamically increase by orders of magnitude, during the gravitational collapse of scalar fields, and that the (normalized) peak curvature does not correspond to that of the critical solution. Nevertheless, without fine tuning of initial data, this increase lies far below that needed to render quantum-gravity corrections important.

Dynamical emergence of SUq(2) from the regularization of (2+1)D gravity with a cosmological constant

The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in $2+1D$ has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an $SU_q(2)$ quantum group symmetry structure. We hereby tackle the same problem, but start from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$ gravity with non-zero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the $SU_q(2)$ quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of $SU_q(2)$ that are recovered for any Wilson loop evaluated at the $SU(2)$ group element that encodes the discretization of the space-time curvature induced by the cosmological constant. The extension to the spin-network basis, and consequently to any transition amplitude between its generic states, enables us to derive in full generality the recoupling theory of $SU_q(2)$. We provide constructive examples for the scalar product of two loop states and spin-networks encoding trivalent vertices. We further comment on the diffeomorphism symmetry generated by the implementation of the curvature constraint, and finally derive explicitly the partition function amplitude of the Turaev-Viro model.

Vacuum Polarization of a Quantized Scalar Field in the Thermal State on the Short Throat Wormhole Background

Vacuum polarization of a scalar field on the short throat wormhole background is investigated. The scalar field is assumed to be massless, having an arbitrary coupling to the scalar curvature of spacetime. In addition, it is supposed that the field is in a thermal state with an arbitrary temperature.

The LARES 2 satellite, general relativity and fundamental physics

LARES 2, successfully launched on July 13, 2022, is a new generation laser-ranged satellite. LARES is an acronym for LAser RElativity Satellite. The first LARES satellite was successfully launched on February 13, 2012 with the ESA-ASI-AVIO launch vehicle VEGA. LARES 2 was injected with extremely high precision onto a high-altitude orbit at about 5900 km altitude with the new ESA-ASI-AVIO launch vehicle VEGA C. Laser-ranged satellites have many applications, including to test Einstein’s theory of general relativity. The main general relativistic phenomenon that LARES 2 will test with high accuracy is the dragging of inertial frames, or frame-dragging. It will also test other aspects and principles of fundamental physics and general relativity, such as the weak equivalence principle at the foundation of viable gravitational theories. Frame-dragging is the name Einstein himself gave in 1913 to an intriguing phenomenon of general relativity which implies that a current of mass-energy, such as the rotation of a body, will generate spacetime curvature. Frame-dragging has a key role in high energy astrophysics, e.g., in the generation of gravitational waves by the collision of two black holes to form a rotating black hole. Frame-dragging by the rotating Earth was measured to a few percent accuracy by combining the data of the satellites LARES, LAGEOS and LAGEOS 2 (Ciufolini et al. in Eur Phys J C 79:872, 2019). LARES 2, thanks to its extremely high injection precision, is projected to improve the test of frame-dragging by at least an order of magnitude. LARES 2 has also relevant applications in space geodesy and geodynamics, e.g., in the study of the shape of the Earth and in the determination of the International Terrestrial Reference Frame (ITRF) by improving the determination of the Earth center of mass and by contributing to a better determination of its rotation axis.

Normal and quasinormal modes of holographic multiquark star

The quadrupole normal-mode oscillation frequency ${f}_{n}$ of a multiquark star are computed for $n=1--5$. At the transition from low to high density multiquark in the core region, the first two modes jump to larger values, a distinctive signature of the presence of the high-density core. When the star oscillation couples with spacetime, gravitational waves (GW) will be generated and the star will undergo damped oscillation. The quasinormal modes (QNMs) of the oscillation are computed using two methods, direct scan and WKB, for QNMs with small and large imaginary parts respectively. The small imaginary QNMs have frequencies 1.5--2.6 kHz and damping times 0.19--1.7 secs for a multiquark star with mass $M=0.6--2.1\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ (solar mass). The WKB QNMs with large imaginary parts have frequencies 5.98--9.81 kHz and damping times 0.13--0.46 ms for $M\ensuremath{\simeq}0.3--2.1\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$. They are found to be the fluid $f\ensuremath{-}\mathrm{modes}$ and spacetime curvature $w\ensuremath{-}\mathrm{modes}$, respectively.

Doppler signature in electrodynamic retarded potentials

The presence of the term v/c that characterizes the electrodynamic retarded potentials, also known as Liénard–Wiechert potentials, is thought to be reminiscent of a Doppler effect. Here, we show that these potentials are consistent with the Doppler shifted electromagnetic (e.m.) field generated by a charge moving along a generic trajectory. Of course, the retarded potentials derived here are formally the same as those reported in the current literature. Nonetheless, this work sheds a new light on the origin of the electrodynamics retarded fields, offering a direct physical interpretation of the term v/c characterizing the related potentials and making more evident the geometry underlying the radiation field associated to a moving charge. This could inspire new visions for calculating radiation and fields from accelerated charges, including particle physics, Cherenkov radiation and electromagnetic fields in curved space–time.

Proposed method of combining continuum mechanics with Einstein Field Equations

The paper proposes an amendment to the relativistic continuum mechanics which introduces the relationship between density tensors and the curvature of spacetime. The resulting formulation of a symmetric stress–energy tensor for a system with an electromagnetic field leads to the solution of Einstein Field Equations indicating a relationship between the electromagnetic field tensor and the metric tensor. In this EFE solution, the cosmological constant is related to the invariant of the electromagnetic field tensor, and additional pulls appear, dependent on the vacuum energy contained in the system. In flat Minkowski spacetime, the vanishing four-divergence of the proposed stress–energy tensor expresses relativistic Cauchy’s momentum equation, leading to the emergence of force densities which can be developed and parameterized to obtain known interactions. Transformation equations were also obtained between spacetime with fields and forces, and a curved spacetime reproducing the motion resulting from the fields under consideration, which allows for the extension of the solution with new fields.

Optical realization of one-dimensional generalized split-step quantum walks

Quantum walks are more than tools for building quantum algorithms. They have been used effectively to model and simulate quantum dynamics in many complex physical processes. Particularly, a variant of discrete-time quantum walk known as split-step quantum walk is closely related to Dirac cellular automata and topological insulators, whose realizations rely on position-dependent control of evolution operators. Owing to the ease of manipulating multiple degrees of freedom of photons, we provide an optical setup of split-step operators which, in combination with position-dependent coin (PDC) operation, can accomplish a table-top setup of generalized split-step walks. Also, we propose an optical implementation for PDC operation that allows, for instance, realizing electric quantum walks, control localization dynamics, and emulate space-time curvature effects. In addition, we propose a setup to realize any t-step split-step quantum walk involving 2 J-plates, 2 variable waveplates, a half-waveplate, an optical switch, and an optical delay line.

Searches for Shapiro delay in seven binary pulsars using the MeerKAT telescope

ABSTRACT Precision timing of millisecond pulsars (MSPs) in binary systems enables observers to detect the relativistic Shapiro delay induced by space–time curvature. When favourably aligned, this enables constraints to be placed on the component masses and system orientation. Here, we present the results of timing campaigns on seven binary MSPs observed with the 64-antenna MeerKAT radio telescope that show evidence of Shapiro delay: PSRs J0101−6422, J1101−6424, J1125−6014, J1514−4946, J1614−2230, J1732−5049, and J1909−3744. Evidence for Shapiro delay was found in all of the systems, and for three the orientations and data quality enabled strong constraints on their orbital inclinations and component masses. For PSRs J1125−6014, J1614−2230, and J1909−3744, we determined pulsar masses to be $M_{\rm p} = 1.68\pm 0.17$, $1.94\pm 0.03$, and $1.45 \pm 0.03 \, {\rm M_{\odot }}$, and companion masses to be $M_{\rm c} = 0.33\pm 0.02$, $0.495\pm 0.005$, and $0.205 \pm 0.003 \, {\rm M_{\odot }}$, respectively. This provides the first independent confirmation of PSR J1614−2230’s mass, one of the highest known. The Shapiro delays measured for PSRs J0101−6422, J1101−6424, J1514−4946, and J1732−5049 were only weak, and could not provide interesting component mass limits. Despite a large number of MSPs being routinely timed, relatively few have accurate masses via Shapiro delays. We use simulations to show that this is expected, and provide a formula for observers to assess how accurately a pulsar mass can be determined. We also discuss the observed correlation between pulsar companion masses and spin period, and the anticorrelation between recycled pulsar mass and their companion masses.

The Whole Theory of This Universe—A Step Forward to Einstein, Part-3<sup>rd</sup>: “The Universal Theory of Visible and Invisible Universe”

We believe that the universe is of two types: visible and invisible. Nothing is at rest between the invisible and visible universes. All microscopic bodies as well as all macroscopic bodies are in motion along curved paths (i.e. in circles). The universe inside an atom is as vast as the visible universe. An atom consists of millions of particles or particle galaxies which contain central energy pools or central energy cores. Energy pools present in the centre of the invisible universe inside atomic or subatomic particles from which particles and energy are continuously interconverting. In a dense central energy pool, two opposite charges are created due to the swirling motion of microscopic energy droplets. Small microscopic energy droplets may swirl either clockwise or anti-clockwise to produce microscopic tornadoes which are non-superimposable mirror images of each other and gain the property of positive and negative charges. Hence, the electrostatic force is originated between these two opposite charges, which are then changed into a pair of particles, i.e. catitron, which carries a positive charge, and anitron which carries a negative charge. All the other millions of subatomic particles or particle galaxies are produced in the same way. So, the electrostatic force is the basic force, and all other forces originate from this basic electrostatic force of attraction. When charged particles move, they produce an oscillating electric field, and the spinning of these particles produces oscillating magnetic fields. These oscillating electric and magnetic fields are perpendicular to each other and, by their interaction, an oscillating gravitational field is produced which is also perpendicular to both the oscillating electric field and the oscillating magnetic field. The Earth’s axial tilt, which causes the Earth’s precessional motion, is caused by the parallel alignment of the Earth’s magnetic field with the magnetic field of the Sun. Gravity is not a cause of space-time curvature, but gravity causes space-time curvature. Space-time curvature is nothing but a curved path around a heavy object. The Universal Theory of Visible and Invisible Universe—The Whole Theory of This Universe—A Step Forward to Einstein, opens new windows in the challenging fields of science and research, i.e. visible and invisible universe, universe inside an atom, what is the stuff of the entire universe? What will happen at the end of this whole universe?

The Significance of Generalized Gauge Transformation across Fundamental Interactions

The author of this paper has put forward a unified program of gauge field from the mathematical and physical picture of the principal associated bundles: thinking that our universe may have more fundamental interactions than the four fundamental interactions, and these basic interaction gauge fields are only the projection components to the base manifold, that is our universe, from a unified gauge potential or connection of the principal associated bundle manifold on the base manifold. These components can satisfy the transformation of gauge potential, and can even be transformed from one basic interaction gauge potential to another basic interaction gauge potential, and can be summarized into a unified equation, that is, the generalized gauge Equation (GGE), but the gauge potential or connection on the principal bundle is invariant, corresponding to the invariance of gauge transformation [1]. In this paper, we will continue to discuss this aspect concretely, and specifically construct a spatiotemporal model with the frame bundle as the principal bundle, and the tensor bundle as the associated bundle, so that the four fundamental interactions, especially the electromagnetic interaction and the gravitational interaction, can be reflected in the bottom manifold, that is, the regional distributions in our universe. Furthermore, this paper studies the existence of gauge transformation across basic interactions by establishing a model of gauge transformation of basic interaction field; it is found that the unified expression formula is GGE and the expression relation on the curvature of space-time. Therefore, the author discusses the feasibility of the generalized gauge transformation across the basic electromagnetic interaction and the basic gravitational interaction, and on this basis, specifically determines a method or way to find the generalized gauge transformation, so as to try to realize the last step of the “unification” of the four fundamental interactions in physics, that is, the “unification” of electromagnetism and gravity.

Gravity Is Not Attraction; It’s a Push (Space-Time Expansion Theory)

The space-time expansion has a new perspective on the universe phenomena. In this article, the key features of the Space-Time Expansion Theory are summarized and discussed, with three postulates incorporating different insights into the behavior of space-time expansion, gravity, space-time curvature, and time itself. Gravity is not an attraction; it is a push. Inertia, free fall, the principles of the theory of relativity and some other phenomena support the author’s assertions. The expansion of space-time is universal, occurs everywhere, and produces gravity instead of counteracting it. Being immersed in the same space-time acceleration, we only perceive the disturbance caused by the matter, which is the massive objects push. This study aims to offer scientists an alternative investigation. “Gravitation is not attraction; it’s a push.” All in accordance with hydrodynamic gravitation and expanding Universe.

The bumblebee field excitations in a cosmological braneworld

We investigate the effects of the spacetime curvature and extra dimensions on the excitations of the Lorentz violating bumblebee field B μ . By assuming the bumblebee field living in a five-dimensional AdS 5 bulk, we found an exponential suppression of the bumblebee self-interacting constant λ and of the bumblebee vacuum expectation value (VEV) b M along the fifth extra dimension. The fluctuations of the bumblebee field upon the VEV can be decomposed into transverse and longitudinal modes with respect to b M . For a spacelike b M along the extra dimension and assuming a thin FRW 3-brane embedded in the AdS 5, the transverse mode is localized on the brane. The bulk geometry leads to a propagating and unbound longitudinal mode along the extra dimension. On the brane, the cosmological expansion leads to the decay of the longitudinal mode in time, as Δt ≈ H −1.