Einstein Equations Research Articles

Note on holographic torus stress tensor correlators in AdS3 gravity

In the AdS3/CFT2 framework, the Euclidean BTZ black hole corresponds to the dominant high-temperature phase of its dual field theory. We initially employ perturbative methods to solve the Einstein equations as boundary value problems, providing correlators for the energy-momentum tensor operator at low points. Utilizing operator equations established in our previous work, we further compute arbitrary high-point correlators for the energy-momentum tensor operator in the high-temperature phase and recursive relations for these high-point functions. Concurrently, we employ the Chern-Simons formalism to derive consistent results. Further, using the cut-off AdS/TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}-deformed CFT duality, we calculate the energy-momentum tensor correlators, contributing to the comprehensive understanding of the system’s dynamics. Finally, stress tensor correlators enable us to ascertain the corresponding KdV operator correlators at low-temperature.

Minimal Einstein-Aether theory

We show that there is a phenomenologically and theoretically consistent limit of the generic Einstein-Aether theory in which the Einstein-Aether field equations reduce to Einstein field equations with a perfect fluid distribution sourced by the aether field. This limit is obtained by taking three of the coupling constants of the theory to be zero but keeping the expansion coupling constant to be nonzero. We then consider the further reduction of this limited version of Einstein-Aether theory by taking the expansion of the aether field to be constant (possibly zero), and thereby we introduce the Minimal Einstein-Aether theory that supports the Einstein metrics as solutions with a reduced cosmological constant. The square of the expansion of the unit-timelike aether field shifts the bare cosmological constant and thus provides, via local Lorentz symmetry breaking inherent in the Einstein-Aether theories, a novel mechanism for reconciling the observed, small cosmological constant (or dark energy) with the large theoretical prediction coming from quantum field theories. The crucial point here is that minimal Einstein-Aether theory does not modify the well-tested aspects of General Relativity such as solar system tests and black hole physics including gravitational waves.

Quantum Field Theory of Black Hole Perturbations with Backreaction: I General Framework

In a seminal work, Hawking showed that natural states for free quantum matter fields on classical spacetimes that solve the spherically symmetric vacuum Einstein equations are KMS states of non-vanishing temperature. Although Hawking’s calculation does not include the backreaction of matter on geometry, it is more than plausible that the corresponding Hawking radiation leads to black hole evaporation which is, in principle, observable. Obviously, an improvement of Hawking’s calculation including backreaction is a problem of quantum gravity. Since no commonly accepted quantum field theory of general relativity is available yet, it has been difficult to reliably derive the backreaction effect. An obvious approach is to use the black hole perturbation theory of a Schwarzschild black hole of fixed mass and to quantize those perturbations. However, it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations. In recent work, we proposed a new approach to this problem that applies when the physical situation has an approximate symmetry, such as homogeneity (cosmology), spherical symmetry (Schwarzschild), or axial symmetry (Kerr). The idea, which is surprisingly feasible, is to first construct the non-perturbative physical (reduced) Hamiltonian of the reduced phase space of fully gauge invariant observables and only then apply perturbation theory directly in terms of observables. The task to construct observables is then disentangled from perturbation theory, thus allowing to unambiguously develop perturbation theory to arbitrary orders. In this first paper of the series we outline and showcase this approach for spherical symmetry and second order in the perturbations for Einstein–Klein–Gordon–Maxwell theory. Details and generalizations to other matter and symmetry and higher orders will appear in subsequent companion papers.

Many body gravity and the galaxy rotation curves

A novel theory was proposed earlier to model systems with thermal gradients, based on the postulate that the spatial and temporal variation in temperature can be recast as a variation in the metric. Combining the variation in the metric due to the thermal variations and gravity, leads to the concept of thermal gravity in a 5-D space-time-temperature setting. When the 5-D Einstein field equations are projected on to a 4-D space, they result in additional terms in the field equations. This may lead to unique phenomena such as the spontaneous symmetry breaking of scalar particles in the presence of a strong gravitational field. This theory, originally conceived in a quantum mechanical framework, is now adapted to explain the galaxy rotation curves. A galaxy is not in a state of thermal equilibrium. A parameter called the “degree of thermalization” is introduced to model partially thermalized systems. The generalization of thermal gravity to partially thermalized systems, leads to the theory of many-body gravity. The theory of many-body gravity is now shown to be able to explain the rotation curves of the Milky Way and the M31 (Andromeda) galaxies, to a fair extent. The radial acceleration relation (RAR) for 63 galaxies, with their galactic masses spanning three orders of magnitude, has been replicated. Finally, the wide binary star (WBS) system is touched upon.

Equivalence of magnetic flux and energy

Magnetic flux and energy equivalence is the principle that everything that has magnetic flux has an equivalent amount of energy, and vice versa. The main methods used: transformation of natural units, algebra, analogy. The equivalence of magnetic flux and energy, despite its widespread use in describing the principles of physics, has not yet been formulated. This paper presents the formula for this equivalence. This is done based on known measurement systems, parameters and principles of physics. Five examples (Stoney units, Planck units, standard gravitational parameter, Lorentz force, and Ampere force) of the algebraic representation of this principle show its universality. Five examples should justify the universality of the new principle and its use in physics and astrophysics. Using the equivalence of magnetic flux and energy, new physical units of mass and magnetic flux are proposed, each of which is suitable for measuring both mass and magnetic flux. Natural values of electric current and voltage, linear density of electric capacitance and inductance, linear density of magnetic flux, linear density of electric charge, as well as natural units of electric voltage and current, electrical resistance, magnetic flux, electrical capacitance and inductance are also described. The article presents the Einstein field equation (additional magnetic stress–energy–momentum tensor) and a standard astrophysical parameter for refining the orbits of celestial bodies.

Holographic reconstruction of flat spacetime

The flat/CFT dictionary between the bulk gravitational theory and boundary conformal field theory is systematically developed in this paper. Asymptotically flat spacetime is built up by asymptotically AdS hyperboloid slices in terms of Fefferman Graham coordinates together with soft modes propagating between different slices near the null boundary. Then we construct the flat holography dictionary based on studying the Einstein equation at zero and first order and it turns out that these correspond to the description of hard and soft sector for the field theory from the boundary point of view. The explicit expression for energy-stress tensor is also determined by performing holographic renormalisation on the Einstein Hilbert action. By studying the anomalies of the energy-stress tensor, we obtain the leading and subleading contribution to the central charge. Einstein equations in the bulk are related to the Ward identities of the boundary theory and we find that the boundary CFT energy-stress tensor is not conserved due to the existence of radiative soft modes which will generate the energy flow through the null boundary.

Linearized gravity and soft graviton theorem in de Sitter spacetime

We study the linearized gravity theory in the Newman-Unti gauge in the near horizon region of the de Sitter spacetime. The linearized Einstein equation involves the cosmological constant. The near horizon symmetry consists of near horizon supertranslation and near horizon superrotation. We compute the near horizon supertranslation charge and find the proper near horizon falloff conditions that uncover a soft graviton theorem from the Ward identity of the near horizon supertranslation. Published by the American Physical Society 2024

Most general isotropic charged fluid solution for Buchdahl model in ℱ(𝒬) gravity

In this work, we investigated a most general isotropic charged fluid solution for the Buchdahl model via a two-step method in ℱ(𝒬)-gravity framework for the first time. In this context, a linear function of the form ℱ(𝒬) = ζ 1 𝒬 + ζ 2 and a particular transformation is used to solve the Einstein-Maxwell Equations (EMEs) employing the Buchdahl ansatz: e Υ(r) = μ(1+λ r 2)/μ+λ r 2, where ζ 1, ζ 2, λ and μ are constant parameters. The Schwarzschild de Sitter (AdS) exterior solution is joined to the interior solution at the boundary to determine the constant parameters. It should be emphasized that, for a given transformation, the Buchdahl ansatz only offers a mathematically feasible solution in the context of electric charge, where pressure and density are maximum at the center and decrease monotonically towards the boundary when 0 < μ < 1. We taken into account the compact star EX01785-248 with M = (1.3±0.2)M ⊙; Radius = 12.02+0.55 -0.55 km for graphical analysis.The physical acceptability of the model in the context of ℱ(𝒬) gravity has been evaluated by looking at the necessary physical properties, including energy conditions, causality condition, hydrostatic equilibrium, pressure-density ratio, etc. Additionally, we predicted the maximum mass limit of different compact objects for various parameter values along with the mass-radius relation. The maximum masses range (1.927 - 2.321) M ⊙ are obtained for our solution. It can be observed that when the coupling parameter ζ 1 for ℱ(𝒬 gravity is smaller, then our solution yields massive stars. The present investigation provides novel insights and realistic implications regarding the formation of compact astrophysical objects.

Topological dressing method for the Einstein–Maxwell equations

Topological dressing method for the Einstein–Maxwell equations

How the non-metricity of the connection arises naturally in the classical theory of gravity

Spacetime geometry is described by two–a priori independent–geometric structures: the symmetric connection Γ and the metric tensor g. Metricity condition of Γ (i.e. ∇g = 0) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of Γ. Traditionally, instead of the (first order) Palatini principle, we use in this case the (second order) Hilbert principle, assuming metricity condition a priori. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein’s theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions.

Analytical solutions to Einstein field equations for spherically symmetric anisotropic matter: a comparative study using Tolman VII metric potential

In this paper, we present analytical solutions to the Einstein field equations for spherically symmetric anisotropic matter distributions using the well-established Tolman VII metric potential, chosen for its strong physical and mathematical foundations. Our solutions are derived using three distinct approaches: the vanishing complexity factor condition (VCC), the embedding class I condition (ECC), and the conformally flat condition (CFC). We conduct a comprehensive comparative analysis of these three approaches. By ensuring a smooth match between the interior spacetime metric and the exterior Schwarzschild metric, and applying the condition of vanishing radial pressure at the boundary, we determine the model parameters. We graphically assess the model’s stability by examining conditions such as causality, the adiabatic index, equations of state, and the generalized Tolman–Oppenheimer–Volkov (TOV) equation, considering the forces acting within the system. Additionally, the effects of anisotropy on the stars’ physical characteristics are investigated through graphical representations.

The role of dimensions in gravitating relativistic shear-free fluids

We study the dynamics of relativistic shear-free gravitating fluids in higher dimensions for both neutral and charged matter. We reduce the Einstein–Maxwell equations to a single second order nonlinear partial differential equation which contains two arbitrary functions. This generalizes the condition of pressure isotropy to higher dimensions; the new condition is functionally different from four dimensions. Our result in higher dimensions reduces to known results in four dimensions. The presence of higher dimensions affects the dynamics of relativistic fluids in general relativity. The dynamical behaviour of the gravitating fluid in higher dimensions is qualitatively different to the four dimensional case. Higher dimensions affect astrophysical and cosmological processes in gravitating shear-free fluids.

Framework for charged compact objects admitting conformal motion in higher dimension.

We propose a new framework for spherical charged compact objects admitting conformal motion in five-dimensional spacetime. The outer spacetime is considered as Reissner-Nordström to obtain matching conditions. The behavior of model characteristics like stress, pressure, and surface tension for the specific density profile is investigated by using Einstein's Maxwell field equations in a five-dimensional framework. For the proposed solution, all physical parameters behave very well even for variations in electric charge parameters. The existence of charged compact stars is also predicted by this study.

Modified Einstein equations from the 1-loop effective action of the IKKT model

Abstract We derive the equations of motion that arise from the one-loop effective action for the geometry of 3+1 dimensional quantum branes in the IKKT matrix model. These equations are cast into the form of generalized Einstein equations, with extra contributions from dilaton and axionic fields, as well as a novel anharmonicity tensor Cµν capturing the classical Yang-Mills-type action. The resulting gravity theory approximately reduces to general relativity in some regime, but differs significantly at cosmic scales, leading to an asymptotically flat Friedmann–Lemaître–Robertson–Walker cosmological evolution governed by the classical action.&amp;#xD;

Stability analysis of the cosmological dynamics of O(D, D)-complete stringy gravity

The massless fields in the universal NS-NS sector of string theory form O(D, D) multiplets of Double Field Theory, which is a theory that provides a T-duality covariant formulation of supergravity, leading to a stringy modification of General Relativity. In this framework, it is possible to write down the extensions of the Einstein field equations and the Friedmann equations in such a way that the coupling of gravitational and matter sectors is dictated by the O(D, D) symmetry universally. In this paper, we obtain the autonomous form of the O(D, D)-complete Friedmann equations, find the critical points and perform their stability analysis. We also include the phase portraits of the system. Cosmologically interesting cases of scalar field, radiation, and matter are separately considered and compared with the Chameleon models in a similar setting. Accelerating phases and the conditions for their existence are also given for such cases.

The de Broglie-Einstein-Rosen gravitational wave

de Broglie gravitational waves are solutions of the linearized Einstein's field equations in vacuum, with intriguing properties. They are axially symmetric and have an effective mass, which is responsible for longitudinal effects that are absent in standard gravity waves. Moreover, they represent a classical realization of a form of dynamics proposed for quantum particles by de Broglie one hundred years ago. In this paper we will show that this perturbation field can be obtained, apart from a proportionality constant, in the weak field limit of a particular Einstein-Rosen field, which we call the de Broglie-Einstein-Rosen wave. Some properties of this exact solution are also discussed.

On the same origin of quantum physics and general relativity from Riemannian geometry and Planck scale formalism

It has been a long time to reconcile quantum physics and general relativity. To date, no globally accepted theory has been proposed to explain all physical observations. In this work, we reformulated the Riemannian geometry in terms of curvature and energy tensors using the Planck scale formalism. The proposed equation can be transformed into Dirac equations in electrodynamic and chromodynamic fields with a reduction in the background curvature. We redefined the mass and charge of leptons in terms of the interactions between the energy of the field and the curvature of the spacetime. The obtained equation is covariant in space–time and invariant with respect to any Planck scale. Therefore, the constants of the universe can be reduced to only two quantities: Planck length and Planck time. We proved that the Einstein field equation from general relativity is actually a relativistic quantum mechanical equation. We further modeled the universe using the equation with Einstein's lambda formalism and found that the universe dynamics could be considered as harmonic oscillators entangled with lambda curvature. This equation can be used to describe the energy transfer between two entangled spacetimes between the same universe and between any two universes (ER=EPR). The singularity of black holes can be avoided at the Planck scale, because space and time are no longer entangled. This equation predicts that information of light from the entangled universe can be transferred to our universe. The gravitational wave background was predicted, and its spectrum was close to that of the observation.

From the Janis–Newman–Winicour Naked Singularities to the Einstein–Maxwell Phantom Wormholes

The Janis–Newman–Winicour spacetime corresponds to a static spherically symmetric solution of Einstein equations with the energy momentum tensor of a massless quintessence field. It is understood that the spacetime describes a naked singularity. The solution has two parameters, b and s. To our knowledge, the exact physical meaning of the two parameters is still unclear. In this paper, starting from the Janis–Newman–Winicour naked singularity solution, we first obtain a wormhole solution by a complex transformation. Then, letting the parameter s approach infinity, we obtain the well-known exponential wormhole solution. After that, we embed both the Janis–Newman–Winicour naked singularity and its wormhole counterpart in the background of a de Sitter or anti-de Sitter universe with the energy momentum tensor of massive quintessence and massive phantom fields, respectively. To our surprise, the resulting quintessence potential is actually the dilaton potential found by one of us. It indicates that, by modulating the parameters in the charged dilaton black hole solutions, we can obtain the Janis–Newman–Winicour solution. Furthermore, a charged wormhole solution is obtained by performing a complex transformation on the charged dilaton black hole solutions in the background of a de Sitter or anti-de Sitter universe. We eventually find that s is actually related to the coupling constant of the dilaton field to the Maxwell field and b is related to a negative mass for the dilaton black holes. A negative black hole mass is physically forbidden. Therefore, we conclude that the Janis–Newman–Winicour naked singularity solution is not physically allowed.

Shell Universe: Reducing Cosmological Tensions with the Relativistic Ni Solutions

Recent discoveries of massive galaxies existing in the early universe, as well as apparent anomalies in Ωm and H0 at high redshift, have raised sharp new concerns for the ΛCDM model of cosmology. Here, we address these problems by using new solutions for the Einstein field equations of relativistic compact objects originally found by Ni. Applied to the universe, the new solutions imply that the universe’s mass is relatively concentrated in a thick outer shell. The interior space would not have a flat, Minkowski metric, but rather a repulsive gravitational field centered on the origin. This field would induce a gravitational redshift in light waves moving inward from the cosmic shell and a corresponding blueshift in waves approaching the shell. Assuming the Milky Way lies near the origin, within the KBC Void, this redshift would make H0 appear to diminish at high redshifts and could thus relieve the Hubble tension. The Ni redshift could also reduce or eliminate the requirement for dark energy in the ΛCDM model. The relative dimness of distant objects would instead arise because the Ni redshift makes them appear closer to us than they really are. To account for the CMB temperature–redshift relation and for the absence of a systematic blueshift in stars closer to the origin than the Milky Way, it is proposed that the Ni redshift and blueshift involve exchanges of photon energy with a photonic spacetime. These exchanges in turn form the basis for a cosmic CMB cycle, which gives rise to gravity and an Einsteinian cosmological constant, Λ. Black holes are suggested to have analogous Ni structures and gravity/Λ cycles.

Revisit Birkhoff’s Theorem: The Post-Newtonian Metric of a Self-Gravitating and Collapsing Thin Spherical Shell

We calculate the metric of a self-gravitating and collapsing infinitely thin spherical shell in the weak-field and slow-motion limits, and we demonstrate that Birkhoff’s theorem is not consistent with the theory of the post-Newtonian approximation. More importantly, it is illustrated that performing a coordinate transformation in solving Einstein field equations may change the matter energy-momentum tensor, making the resultant solution not correspond to the original problem.