Einstein Equations Research Articles

Hadrons, superconductor vortices, and cosmological constant

We explore the roles of the trace anomaly in several hadron properties. We derive the scale invariant expression for the pressure from the gravitational form factors (GFF) of QCD which results in consistent results for the mass and rest energy from the GFF and those from the trace and the Hamiltonian of the energy-momentum tensor (EMT) operators. It is shown that the energy-equilibrium correspondence of hadrons infers an equation of state where the trace anomaly matrix element, emerging from the glue condensate in the vacuum, gives a negative constant pressure that leads to confinement, much like the confinement mechanism for the vortices in type II superconductors where the negative constant pressure is due to the cost of depleting the superconducting condensate. We also note that both the trace anomaly in the QCD energy-momentum tensor and the cosmological constant in Einstein's equation are associated with the metric term which contributes to both energy and pressure. Their difference in terms of the role the pressure plays is discussed. Finally, we note that a lattice calculation of the trace anomaly distribution in the pion has addressed a question about the trace anomaly contribution to the pion mass and suggests that there might be a connection between the conformal symmetry breaking and chiral symmetry breaking in this case.

One-Dimensional Subspaces of the SL(n,mathbb {R}) Chiral Equations

In this work we find solutions of the (n+2)-dimensional Einstein Field Equations (EFE) with n commuting Killing vectors in vacuum. In the presence of n Killing vectors, the EFE can be separated into blocks of equations. The main part can be summarized in the chiral equation (alpha g_{, bar{z}} g^{-1})_{, z} + (alpha g_{, z} g^{-1})_{, bar{z}} = 0 with gin SL(n,mathbb {R}). The other block reduces to the differential equation , (ln f alpha ^{1-1/n})_{, z} = 1/2 , alpha ,{textrm{tr}}( g_{, z} g^{-1})^2 and its complex conjugate. We use the ansatz g = g(xi ) , where xi satisfies a generalized Laplace equation, so the chiral equation reduces to a matrix equation that can be solved using algebraic methods, turning the problem of obtaining exact solutions for these complicated differential equations into an algebraic problem. The different EFE solutions can be chosen with desired physical properties in a simple way.

Determination of Protein Amount in Nanosized Synthetic Liposomes by Surface Effect Raman Spectroscopy (SERS)

Accurate characterization of synthetic liposomes is essential since they give information about the vesicu-lar structures in bodily fluids such as extracellular vesicles. The characterization tasks are generally the determination of the sizes of the liposomes and the profiling of the liposomes' content. Optical tweezers and Surface Enhanced Raman Spectroscopy (SERS) were used to profile the nanosized liposomes. The size distribution of the trapped liposomes (140 nm on average) was found by using Einstein's Brownian motion equation, consistent with the size distribution obtained from dynamic light scattering measure-ments. Besides, Gramicidin-encapsulated liposomes were measured using SERS, and statistically signifi-cant differentiation was found in Raman intensities between liposome populations with altering concentra-tions of proteins. This study uniquely measured size distributions of nano-sized liposomes with conven-tional optical tweezers (without plasmonics) and determined the chemical differences between empty and protein encapsulated liposomes with high accuracy using Raman spectroscopy

On curvature related geometric properties of Hayward black hole spacetime

This paper is devoted to the study of curvature properties of Hayward black hole (briefly, HBH) spacetime, which is a solution of Einstein field equations (briefly, EFE) having non-vanishing cosmological constant. We have proved that the HBH spacetime is an Einstein manifold of level 2, 2-quasi Einstein, generalized quasi-Einstein and Roter type manifold. Also, it is shown that the nature of the HBH spacetime is pseudosymmetric and it obeys several types of pseudosymmetries, such as, pseudosymmetry due to concircular, conformal and conharmonic curvature (i.e., F⋅F=LQ(g,F) for F=W,C,K with a smooth scalar function L), and it also possesses the relation R⋅R−LQ(g,C)=Q(S,R). It is engrossing to mention that the nature of energy momentum tensor of the HBH spacetime is pseudosymmetric. On the basis of curvature related properties, we have made a comparison among Reissner–Nordström spacetime, interior black hole spacetime and HBH spacetime. It is noteworthy to mention that the gravitational force of the point-like global monopole spacetime is much stronger than that of HBH spacetime. Also, it is shown that the HBH spacetime admits an almost η-Ricci soliton as well as an almost η-Ricci-Yamabe soliton. Finally, an elegant comparative study is delineated between the HBH spacetime and the point-like global monopole spacetime with respect to different kinds of symmetry, such as, motion, curvature collineation, curvature inheritance etc.

What is a real black hole?

Abstract Four dimensional Einstein equation with non-vanishing energy momentum tensor is solved to give a solution where the event horizon of a celestial body is outside of the body, which corresponds to a black hole by definition. There is no singularity at the center of the object unlike the case of Schwarzschild solution. We do not have the gravitational collapse. We do not have the gravitational collapse thus avoiding the Penrose-Hawking singularity theorems. The main feature of our black hole solution is that it has the horizon inside the black hole in addition to the usual outside horizon. This inner horizon singularity is not removable by a coordinate transformation because the determinant of metric tensor is singular at this point but there is no singularity neither in the density nor in the pressure indicating there is no gravitational collapse.

On the dynamics of a test particle in the field of the de Broglie gravitational waves

The de Broglie gravitational waves are a solution of the linearized Einstein equations with properties that are absent in the standard gravity waves because they are also longitudinal waves and obey a dispersion relation that leads to an effective mass. Furthermore, they represent a classical realization of a form of dynamics, proposed for quantum particles by de Broglie in 1927. In this paper, we discuss the dynamics of a massive particle in presence of the de Broglie gravity wave. We will compute the analytical expression for the linear and angular momentum of the corpuscle. As an application, we will consider the case in which these oscillations of spacetime are associated to an electron travelling with a velocity equal to [Formula: see text] of the light speed and we will estimate the order of magnitude of the quantities involved. We will show that a nearby positron oscillates and radiates an energy of 1[Formula: see text]eV in [Formula: see text][Formula: see text]s, an effect that is, in principle, measurable.

Relativistic configurations of Tolman Stellar structures in Gauss–Bonnet gravity

The purpose of this study is to discuss the charged stellar structure in the background of the [Formula: see text] theory of gravity. The Einstein–Maxwell equations defined in the appearance of charge by utilizing the Tolman–Kuchowicz spacetime are described by spherically symmetric spacetime. To analyze the charged stellar structure, we use the matching conditions of spherically symmetric spacetime with the Reissner–Nordström exterior metric and develop some expressions for examining the energy density, radial pressure and tangential pressure, respectively. We address the energy conditions to verify our model’s viability for the SAX J1808.4-3658 star. Some other physical features, such as energy density, pressure components, anisotropic parameters, equation of state parameters, stability analysis, adiabatic index, Tolman–Oppenheimer–Volkoff equation, mass-radius relationship, compactness and redshift, have been investigated. We observe that our results are physically viable, free from geometrical and physical singularities, which provide circumstantial evidence for the existence of viable and stable compact star.

Gravity of gluonic fluctuations and the value of the cosmological constant

We analyze the classical linear gravitational effect of idealized pion-like dynamical systems, consisting of light quarks connected by attractive gluonic material with a stress–energy in one or more dimensions. In one orbit of a system of total mass M, quarks of mass expand apart initially with , slow due to the gluonic attraction, reach a maximum size , then recollapse. We solve the linearized Einstein equations and derive the effect on freely falling bodies for two systems: a gluonic bubble model where uniform gluonic stress–energy fills a spherical volume bounded by a 2D surface comprising the quarks’ rest mass, and a gluonic string model where a thin string connects two pointlike quarks. The bubble model is shown to produce a secular mean outward residual velocity of test particles that lie within its orbit. It is shown that the mean gravitational repulsion of bubble-like virtual-pion vacuum fluctuations agrees with the measured value of the cosmological constant, for a bubble with a radius equal to about twice the pion de Broglie length. These results support the view that the gravity of standard QCD vacuum fluctuations is the main source of cosmic acceleration.

A new solution for a generalized cosmological wormhole

We find a new exact solution to Einstein field equations that represents a cosmological wormhole embedded in a flat Friedmann–Lemaître–Robertson–Walker universe. The new metric is a generalization of a previous cosmological wormhole solution found by Kim. We explicitly show that the flaring out condition is satisfied at the throat at all cosmic times; in addition, the null energy condition is violated at the throat regardless of the background cosmological model; thus, the spacetime geometry presented here describes a wormhole coupled to the cosmic dynamics that exists at all cosmic times and whose throat remains open in any cosmological model.

Timelike Ricci bounds for low regularity spacetimes by optimal transport

We prove that a globally hyperbolic smooth spacetime endowed with a [Formula: see text]-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn–Minkowski, Bonnet–Myers, and Bishop–Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even [Formula: see text], in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.

Einstein field equations extended to fractal manifolds: A fractal perspective

This paper provides a framework for understanding and analyzing non-differentiable fractal manifolds. By introducing specialized mathematical concepts and equations, such as the Metric Tensor, Curvature Tensors, Analogue Arc Length, and Inner Product, it enables the study of complex patterns that exhibit self-similarity across different scales and dimensions. The Analogue Geodesic and Einstein Field Equations, among others, offer practical applications in physics, highlighting the relevance and potential of fractal geometry.

On the Fourier Analysis of the Einstein–Klein–Gordon System: Growth and Decay of the Fourier Constants

We consider the (1+3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+3)$$\\end{document}-dimensional Einstein equations with negative cosmological constant coupled to a spherically symmetric, massless scalar field and study perturbations around the anti-de Sitter spacetime. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator as well as on some oscillatory integrals.

Cosmological models with variable parameters

The present study deals with the investigation of the Friedmann-Lemaitre-Robertson-Walker models (often FLRW-models) with time varying G and ∧ in the framework of General theory of Relativity. The Einstein field equations have been solved by considering the time-varying deceleration parameter q(t) and Scale factor , where m and β are arbitrary constants. We have analysed the value of m, which will generate a transition for universe from early decelerating phase to recent acceleration phase. The physical and graphic behaviour have also been planned to study in this communication.

Double field theory, twistors, and integrability in 4-manifolds

The search for a geometrical understanding of dualities in string theory, in particular T-duality, has led to the development of modern T-duality covariant frameworks such as Double Field Theory, whose mathematical structure can be understood in terms of generalized geometry and, more recently, para-Hermitian geometry. In this work we apply techniques associated to this doubled geometry to four-dimensional manifolds, and we show that they are particularly well-suited to the analysis of integrability in special spacetimes, especially in connection with Penrose's twistor theory and its applications to general relativity. This shows a close relationship between some of the geometrical structures in the para-Hermitian approach to double field theory and those in algebraically special solutions to the Einstein equations. Particular results include the classification of four-dimensional, possibly complex-valued, (para-)Hermitian structures in different signatures, the Lie and Courant algebroid structures of special spacetimes, and the analysis of deformations of (para-)complex structures. We also discuss a notion of “weighted algebroids” in relation to a natural gauge freedom in the framework. Finally, we analyze the connection with two- and three-dimensional (real and complex) twistor spaces, and how the former can be understood in terms of the latter, in particular in terms of twistor families.

On some new black hole, wormhole and naked singularity solutions in the free Dirac–Born–Infeld theory

In this paper, we present some new static and spherically symmetric solutions of the Einstein equation in which the matter sector is accounted for by a free Dirac–Born–Infeld field. Our novel spacetimes can describe either a black hole, a wormhole, or a naked singularity depending on certain boundary conditions. By tracking the dynamical gravitational collapse, we enlighten the importance of the isotropy of the pressure for having an horizon as a result, as required by the Cosmic Censorship Conjecture. Our new spacetime solutions, the amount of exotic matter, its “complexity”, and the equation of state along the tangential direction are analytical and written in closed forms. We identify a taming of the breaking of the null energy condition, customary for wormhole spacetimes in General Relativity, along both the radial and tangential direction. We assess the astrophysical applicability and perform a comparative analysis between our solutions and other literature ones, by identifying an ISO-like density profile of the matter field, which provides a flattening of the rotation curves, by discussing the motion of test particles, and the shadow properties. In our model, those effects are interpreted as a manifestation of a topological defect, and since they can observationally mimic the signatures of other spacetimes, a study of the perturbations is performed within the quasi-normal modes formalism. Having identified the Reissner–Nordström-like quasi-resonance, our paper is intended also to provide some insights on which combinations of background and perturbation properties should be observed, for claiming the nature of astrophysical compact objects.

Compact stars with dark matter induced anisotropy in complexity-free background and effect of dark matter on GW echoes

ABSTRACT In this work, we consider the vanishing complexity factor scenario which has opened up a whole new way of generating solutions to the Einstein field equations for the spherically symmetric structure of celestial bodies. By using this very rare condition on the system of two metric potentials, viz. gtt and grr, we make reduce it to a uni-metric potential system satisfying all physical conditions. Along with this, we further have considered that the space–time is deformed by dark matter (DM) content in DM haloes resulting into perturbations in the gtt and grr metric potentials. This DM deformation is mathematically done by the complete geometric decoupling method where the decoupling parameter β decides the amount of DM content. In connection to the claimed post-merger object in the GW170817 event we have argued that if these compact stars were in galactic DM haloes with the assumption that the radius remains the same, the compactness factor can grow within the range 1/3 to 4/9 and therefore can generate gravitational waves (GW) echoes. Additionally, we have presented effect of β on the generation of GW echoes in accordance with the observational constraints related to the compact stars GW190814, PSR J0740+6620, PSR J1614−2230, Cen X-3, and LMC X-4.

Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordström Spacetime

We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, |Q|=M,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ |Q|=M, $$\\end{document} Reissner–Nordström spacetime, as a solution to the Einstein–Maxwell equations. Our work uses and extends the framework [28, 32] of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}. In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}, the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component α̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\underline{\\alpha }} $$\\end{document} not decaying asymptotically along the event horizon H+,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+, $$\\end{document} a result previously unknown in the literature.

Casimir Wormholes driven by non-commutative geometries in higher dimensional Einstein Gauss–Bonnet gravity

This manuscript investigates the construction of a new class of wormhole solutions motivated by non-commutative geometry in fusion with Casimir wormholes in Einstein Gauss–Bonnet (EGB) gravity . It is not likely to ensure that the violation at small scales is significant enough to produce a macroscopic wormhole by depending merely on the Casimir effect. In our strategy, we solely utilised modifications to the energy–momentum tensor (EMT) in the non-commutative background, coupled with the impact of Casimir phenomena. On the other hand, the Einstein tensor remains unchanged in the Einstein field equations. In this regard, we have assumed four cases of non-commutative effects combined with the Casimir wormholes. We have investigated the non-commutative effects using Gaussian and Lorentzian distributions. Also, we incorporated the Casimir and Generalised Uncertainty Principle (GUP) corrected Casimir wormholes to study the basic features of wormhole solutions. The dynamics of a theory that relates the Gauss–Bonnet (GB) parameters (μ) and non-commutative parameter (β) have been addressed graphically, with a particular focus on analysing null energy conditions (NEC) through contour plots to identify the presence of exotic matter. Furthermore, embedding diagrams have been used to investigate the geometry of newly produced asymptotically flat wormholes. Additionally, the active gravitational mass has been calculated and plotted up to the radius r in the area around the wormhole throat. We have discussed equilibrium conditions to analyse the stability of obtained wormhole solutions. Finally, to understand the complexity of wormhole solutions, we have examined the complexity factor graphically.

Gravitational decoupling of anisotropic stars in the Brans–Dicke theory

Anisotropic spherically symmetric solutions within the framework of the Brans–Dicke theory are uncovered through a unique gravitational decoupling approach involving a minimal geometric transformation. This transformation effectively divides the Einstein field equations into two separate systems, resulting in the alteration of the radial metric component. The first system encompasses the influence of the seed source, derived from the metric functions of the isotropic Tolman IV solution. Meanwhile, the anisotropic source is subjected to two specific constraints in order to address the second system. By employing matching conditions to determine the unknown constants at the boundary of the stellar object, a comprehensive examination of the internal structure of stellar systems ensues. This investigation delves into the impact of the decoupling parameter, the Brans–Dicke parameters, and a scalar field on the structural characteristics of anisotropic spherically symmetric spacetimes, all while considering the strong energy conditions.

Some new relativistic charged models with anisotropic pressure

In this paper, we found new classes of solutions to the Einstein-Maxwell field equations with matter anisotropic distribution incorporating a particular form of electric field intensity within the framework of general relativity. We use a metric potential or ansatz that depends on an adjustable parametern in order to get the new solutions. We generated new models of compact stars with n=1 and n=2. Graphical analysis allows us to conclude that the new models satisfy all the physical characteristics for astrophysical objects and can be very useful in the study and description of compact structures. We obtained models consistent with the pulsars PSR J1311-3430 and PSR J0952–0607.