Einstein Equations Research Articles

On boundary conditions for linearised Einstein’s equations

We investigate the properties of a fairly large class of boundary conditions for the linearised Einstein equations in the Riemannian setting, ones which generalise the linearised counterpart of boundary conditions proposed by Anderson. Through the prism of the quest to quantise gravitational waves in curved spacetimes, we study their properties from the point of view of ellipticity, gauge invariance, and the existence of a spectral gap.

Spherically symmetric electrovac spacetimes in conformal Killing gravity

Recently Harada proposed a third-order gravitational theory for the derivatives of the metric. Mantica and Molinari showed that Harada’s theory may be recast into the form of Einstein’s field equations (EFEs) with an additional source term which is a second-order conformal Killing tensor. Accordingly they named the theory conformal Killing gravity. However, they overlooked the fact that all solutions of the new theory (except those satisfying EFEs) admit a non-trivial second-order Killing tensor. Harada derived an analog of the Schwarzschild solution. Recently Tarciso et al obtained a generalization of Harada’s vacuum solution analogous to the Reissner–Nordström solution. However, like Harada they assumed a restricted form for a static spherically symmetric metric. In this study the most general spherically symmetric static vacuum solution of Harada’s theory and its generalization with a Maxwell electromagnetic field as source were obtained. The validity of Birkhoff’s theorem for static spherically symmetric electrovac fields in conformal Killing gravity is investigated.

Visualisation of counter-rotating dust disks using ray tracing methods

Physical aspects of stationary axisymmetric vacuum spacetimes given by exact solutions of the Einstein equations are discussed via ray tracing. A detailed study of the spacetime generated by a disk of counter-rotating dust is presented. The spacetime is given in explicit form in terms of hyperelliptic theta functions. The numerical approach to ray tracing is set up for general stationary axisymmetric spacetimes and tested at the well-studied example of the Kerr solution. Similar features as in the case of a rotating black hole, are explored in the case of a dust disk. The effect of the central redshift varying between a Newtonian disk and the ultrarelativistic disk, where the exterior of the disk can be interpreted as the extreme Kerr solution, and the transition from a single component disk to a static disk is explored. Frame dragging, as well as photon spheres, are discussed.

Newtonian limit of Einstein equation with a cosmological constant* *Supported by the National Natural Science Foundation of China (12275350), and the Natural Science Foundation of Hunan Province, China (2023JJ30179)

Traditionally, the cosmological constant has been viewed as dark energy that mimics matter with negative energy. Given that matter with negative energy provides a repulsive force, which fundamentally differs from typical gravitational forces, it has been believed that the cosmological constant effectively contributes a repulsive force. However, it is important to note that the concept of gravitational force is valid only within the framework of Newtonian dynamics. In this study, we demonstrate that the traditional understanding of the gravitational force contributed by the cosmological constant is not entirely correct. Our approach involves investigating the Newtonian limit of the Einstein equation with a cosmological constant. The subtleties involved in this analysis are discussed in detail. Interestingly, we find that the effect of the cosmological constant on Newtonian gravity is an attractive force rather than a repulsive one for ordinary matter. As expected, this corrective force is negligibly small. However, our findings may offer a way to distinguish between dark energy and the cosmological constant, as one contributes a repulsive force while the other contributes an attractive force.

Modified gravity/entropic gravity correspondence due to graviton mass

Some time ago, it has been suggested that gravitons can acquire mass in the process of spontaneous symmetry breaking of diffeomorphisms through the condensation of scalar fields [Chamseddine and Mukhanov, JHEP, 2010]. Taking this possibility into account, in the present paper, first we show how the graviton mass intricately reshapes the gravitational potential akin to a Yukawa-like potential at large distances. Notably, this long-range force modifies the Newton’s law in large distances and might explain the phenomena of dark matter. The most important finding in the present paper is the derivation of a modified Newtons law of gravity by modifying the Verlinde’s entropic force relation due to the graviton contribution. The graviton contribution to the entropy basically measures the correlation of graviton and matter fields which then reproduces the Bekenstein–Hawking entropy at the horizon. This result shows the dual description of gravity: in the language of quantum information and entropy the gravity can be viewed as an entropic force, however in terms of particles and fields, it can be viewed as a long range force. Further we have recovered the corrected Einstein field equations as well as the ΛCDM where dark matter emerges as an apparent effect.

Stress and Geometry for Isotropic Singularities.

We develop the mathematics needed to treat the interaction of geometry and stress at any isotropic spacetime singularity. This enables us to handle the Einstein equations at the initial singularity and characterize allowed general relativistic stress-energy tensors. Their leading behaviors are dictated by an initial hypersurface conformal embedding. We also show that an isotropic big bang determines a canonical nonsingular metric on and about the initial hypersurface as well as a cosmological time. This assigns a volume and energy to the initial point singularity.

Cosmos MIND and matter: Is mind in spacetime?

We attempt in this article to formulate a conceptual and testable framework weaving Cosmos, Mind and Matter into a whole. We build on three recent discoveries, each requiring more evidence: i. The particles of the Standard Model, SU(3) x SU(2) x U(1), are formally capable of collective autocatalysis. This leads us to ask what roles such autocatalysis may have played in Cosmogenesis, and in trying to answer, Why our Laws? Why our Constants? A capacity of the particles of SU(3) x SU(2) x U(1) for collective autocatalysis may be open to experimental test, stunning if confirmed. ii. Reasonable evidence now suggests that matter can expand spacetime. The first issue is to establish this claim at or beyond 5 sigma if that can be done. If true, this process may elucidate Dark Matter, Dark Energy and Inflation and require alteration of Einstein's Field Equations. Cosmology would be transformed. iii. Evidence at 6.49 Sigma suggests that mind can alter the outcome of the two-slit experiment. If widely and independently verified, the foundations of quantum mechanics must be altered. Mind plays a role in the universe. That role may include Cosmic Mind. Our considerations concern1. Ontologically Real Potentia and the Unmanifest; 2. Nonlocality as Fundamental; 3. Res potentia, Res extensa, and Actualization; 4. Mind and Qualia, Mind is not in Spacetime; 5. Quantum Vacuum = Potentia not in Spacetime = Mind not in Spacetime; 6. Mind can Actualize Potentia; 7. The emergence of the classical world; 8. Co-evolution of evermore complex matter; 9. Why “My Mind”?; 10. Each embodied mind is coupled bilaterally to the Quantum Vacuum that is Cosmic Mind; 11. Responsible Free Will.

Inflationary initial conditions for the cosmological gravitational wave background

The initial conditions on the anisotropies of the stochastic gravitational-wave background of cosmological origin (CGWB) largely depend on the mechanism that generates the gravitational waves. Since the CGWB is expected to be non-thermal, the computation of the initial conditions could be more challenging w.r.t. the Cosmic Microwave Background (CMB), whose interactions with other particles in the early Universe lead to a blackbody spectrum. In this paper, we show that the initial conditions for the cosmological background generated by quantum fluctuations of the metric during inflation deviate from adiabaticity. These primordial gravitational waves are indeed generated by quantum fluctuations of two independent degrees of freedom (the two polarization states of the gravitons). Furthermore, the CGWB plays a negligible role in the Einstein's equations, because its energy density is subdominant w.r.t. ordinary matter. Therefore, the only possible way to compute the initial conditions for inflationary gravitons is to perturb the energy-momentum tensor of the gravitational field defined in term of the small-scale tensor perturbation of the metric. This new and self-consistent approach shows that a large, non-adiabatic initial condition is present even during the single-field inflation. Such a contribution enhances the total angular power spectrum of the CGWB compared to the standard adiabatic case, increasing also the sensitivity of the anisotropies to the presence of relativistic and decoupled particles in the early Universe. In this work we have also proved that our findings are quite general and apply to both single-field inflation and other scenarios in which the CGWB is generated by the quantum fluctuations of the metric, like the curvaton.

Deriving Schwarzschild metric from vacuum zero-point energy

In this paper, we study the correspondence between the three-dimensional and one-dimensional physical quantity in the spherically symmetric gravitational fields and we obtain a conjecture that the total energy enclosed by the two-dimensional sphere is equal to twice the change in vacuum zero-point energy of a one-dimensional circle that is the boundary of a hemispherical sphere, we can naturally obtain Schwarzschild metric without relying on Einstein's gravitational field equation under this conjecture, the result leads us to conclude that gravity originates from the change of zero-point energy in the one-dimensional vacuum.

Apparent dark matter inspired by the Einstein equation of state

The purpose of this article is twofold. First, by means of Padmanabhan's proposal on the emergence nature of gravity, we recover the ΛCDM model and the effect of the dark matter in the context of cosmology. Toward this goal, we use the key idea of Padmanabhan that states cosmic space emerges as the cosmic time progresses and links the emergence of space to the difference between the number of degrees of freedom on the boundary and in the bulk. Interestingly enough, we show that the effect of the cold dark matter in the cosmological setup can be understood by assuming an interaction between the numbers of degrees of freedom in the bulk. In the second part, we follow Jacobson's argument and obtain the modified Einstein field equations with additional dark matter component emerging due to the interaction term between dark energy and baryonic matter related by , where α is a coupling constant. Finally, a correspondence with the Yukawa cosmology is pointed out, and the role of massive gravitons as a possibility in explaining the nature of the dark sector as well as the theoretical origin of the Modified Newtonian Dynamics (MOND) are addressed. We speculate that the interaction coupling α fundamentally measures the entanglement between the gravitons and matter fields and there exists a fundamental limitation in measuring the gravitons wavelength.

An exact stationary axisymmetric vacuum solution within a metric-affine bumblebee gravity

Within the framework of the spontaneous Lorentz symmetry breaking (LSB), we consider a metric-affine generalization of the gravitational sector of the Standard Model Extension (SME), including the Lorentz-violating (LV) coefficients u and sμν . In this model, we derive the modified Einstein field equations in order to obtain a new axisymmetric vacuum spinning solution for a particular bumblebee's profile. Such a solution has the remarkable property of incorporating the effects of LSB through the LV dimensionless parameter X = ξb 2, with ξ is the nonminimal coupling constant, and b 2 = bμbμ , with bμ is the vacuum expectation value of the bumblebee field; as the LSB is turned off, X = 0, we recover the well-established result, the Kerr solution, as expected. Afterwards, we calculate the geodesics, the radial acceleration and thermodynamic quantities for this new metric. We also estimate an upper bound for X by using astrophysical data of the advance of Mercury's perihelion.

Time Functions on Lorentzian Length Spaces

AbstractIn general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by K-causality, show that a modified notion of Geroch’s volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

Application of the Pythagorean Theorem as a Quick Solution in Solving Relativistic Momentum Problems

The Pythagorean theorem is a fundamental concept in mathematics that is often used to solve various geometric problems. In physics studies, especially in relativistic momentum analysis, the application of this theorem can be a fast and effective solution. This article explores the application of the Pythagorean theorem in the context of relativistic momentum, where the speed and energy of objects approach the speed of light. By exploiting the quadratic relationship between the components of the relativistic momentum and energy vectors, the Pythagorean theorem allows for simpler and more efficient calculations. This research identifies the relationship between the Pythagorean theorem equation and the relativistic momentum equation through several steps. The first step involves calculating the Pythagorean theorem equation. The second step is to calculate Einstein's special relativity equations. The final step is to analyze the relationship between the two equations. In this case, the relativistic and Pythagorean momentum equations have similarities and one of the formulas can be used.

Late time acceleration in Bianchi type-V dark energy cosmological models with linear deceleration parameter

We present a Bianchi type-V cosmological model with deceleration parameter as a simple linear function of the Hubble parameter interacting with a perfect fluid in the general theory of relativity. For Bianchi type-V space-time, the available solutions of Einstein field equations have been procure under the assumption of linear deceleration parameter, q = A + BH, where H is the Hubble parameter. Also,during the evolution of the Bianchi-V universe, dark energy(DE) with a variable EoS parameters has been studied in detail and w slowly varies around w = –0.33 for a 10 GYr. lapse, which involves an accelerated expansion, through an expansion not compatible with current. In this article, we discuss the physical and geometrical facets of the cosmological parameter of the model. Several elements of the model universe are explained by the jerk parameter and energy conditions, including the recent cosmological expansion and singularity theories. We also deliberated state-finder parameters, which forecast that the Universe created from Einstein’s static era rallies towards ΛCDM model (r = 1, s = 0).

Universal Basic Mass Density inside Dark Matter Halos

Abstract In this work we propose a universal basic mass density and a universal basic metric inside dark matter halos in the framework of Einstein equations providing an analytical ground for learning about dark matter. In a previous work the authors have proposed a simplified model for galaxies: a Schwarzschild black hole (that contains all the baryonic matter of the galaxy) immersed inside a dark matter halo. The solution solved the Einstein equations perturbatively and successfully gave the flat rotation curve and the baryonic Tully-Fisher relation, the two signatures of spiral galaxies. In this work we take the black hole mass (the baryonic mass) of the solution to be zero in order to focus our study on the dark matter halo exclusively. Among the results (1) is the prediction of a universal basic mass density of dark matter, ρ = a_0/2πGr, where a_0 ≈ 2.8×10^{-11} m/s^2 is a universal constant whose value was deduced from observations, (2) we show that the mass and velocity curve of the dark halo agree excellently with observational data at intermediate distances where the baryonic matter contribution is negligible, (3) we show that the constant a0is the origin of the constant surface density of dark matter and the origin of the scale-radius/scale-density scaling relation of the Navarro-Frenk-White profile (4) we show how the inclusion of baryonic matter in our model solves the core-cusp problem.

Gravitational waves effects in a Lorentz–violating scenario

This paper focuses on how the production and polarization of gravitational waves are affected by spontaneous Lorentz symmetry breaking, which is driven by a self–interacting vector field. Specifically, we examine the impact of a smooth quadratic potential and a non–minimal coupling, discussing the constraints and causality features of the linearized Einstein equation. To analyze the polarization states of a plane wave, we consider a fixed vacuum expectation value (VEV) of the vector field. Remarkably, we verify that a space–like background vector field modifies the polarization plane and introduces a longitudinal degree of freedom. In order to investigate the Lorentz violation effect on the quadrupole formula, we use the modified Green function. Finally, we show that the space–like component of the background field leads to a third–order time derivative of the quadrupole moment, and the bounds for the Lorentz–breaking coefficients are estimated as well.

Energetic processes and thermodynamic analysis of the spinning Kiselev black hole with cloud of strings

Here we study the impact of the string clouds, the quintessence dark energy and other spacetime parameters on the center-of-mass energy, Penrose-process and thermodynamic properties of a stationary black hole solution of the Einstein field equations. We observe that the three distinct horizons i.e the Cauchy horizon, the event horizon and the cosmological horizon exist when the spin parameter a of the black hole is less than its extremal value aE. Interestingly we see that the infinite redshift surface of the black hole is not much sensitive to the quintessence parameter γ. It is sensitive to the parameter a and it increases as the parameter a increases. Further, we notice that an increase in the string cloud parameters b, quintessence parameter γ and a make the radius of the ergosphere of the black hole larger. We also notice that a rise in the parameter a makes the innermost stable circular orbits (ISCO) smaller and for an increase in the parameter b and γ the ISCO becomes larger than that for the Kerr spacetime. For the center-of-mass energy of the spinning black hole with quintessence and cloud of strings we observe that it can be arbitrarily high only for the extremal case of the black hole for a specific range of values of the angular momentum of the colliding particles. We notice that the Penrose-process is more efficient for smaller values of the string cloud parameter b and the parameter γ has no significant impact on this process. Moreover we see that an increase in the parameters b, γ and a cause the Helmholtz free energy to increase and opposite is true for the temperature of this type of rotating black hole. We learn that the area and entropy of the black hole are also depend on the parameters b, γ and a, and their effect make the entropy smaller. Interestingly we find a range of the horizon radius for the thermodynamic stability of the black hole. To get more information about the thermodynamic quantities of the rotating black hole with quintessence and cloud of strings, we plot them for different ranges of the spacetime parameters.

Tetrad formalism in the solution of spherically symmetric spacetime in general relativity

Abstract Spherically symmetric solutions in general relativity are the most fundamental solutions to the Einstein field equation. The first exact solution of the Einstein field equation is the spherically symmetric solution given by the Schwarzschild metric, as easily found in any standard textbook on general relativity. The FLRW (Friedmann-Lemaitre-Robertson-Walkers) metric is another spherically symmetric solution of Einstein’s equation describing the standard model in Cosmology. The standard approach to solving Einstein’s equations is by considering the metric. However, we can also adopt a tetrad-based method or tetrad formalism. We review these two solutions by the tetrad formalism as an alternative approach. In addition, we give some more cases, including the cosmological constant and the Taub-NUT metric.

Complete solution of the Einstein field equations for a spherical shell of truly incompressible liquid

Complete solution of the Einstein field equations for a spherical shell of truly incompressible liquid

The Einstein-harmonic equations and constant scalar curvature Kähler metrics

Let (M,J) be a compact complex surface. In his paper [9], LeBrun showed that J-invariant solutions of the Einstein-Maxwell equations correspond to conformally Kähler constant scalar curvature metrics whose Ricci tensors are J-invariant. In the present paper, we prove that constant scalar curvature Kähler manifolds of even complex dimension give solutions of Einstein equations with matter fields which we call the Einstein-harmonic equations.