Field Equations Of General Relativity Research Articles

Extending gravitational potentials from the surface boundaries of compact objects

The solutions of the Einstein field equations of general relativity (GR) are prone to rendering systems which are physically non-viable as shown by Delgaty and Lake (1998). This does not detract from the success of general relativity but rather suggests the possibility for implementing new solution methods while being cautious that the resulting systems are indeed physically viable. This emphasizes the importance of applying appropriate criteria in closing the system of non-linear, coupled differential equations of GR. In the case of static systems describing massive compact bodies, an intrinsic property is the vanishing of the radial pressure at the surface boundary of the object which is embedded in an empty Schwarzschild exterior spacetime. In attempting to complete the gravitational description, we first assume a standard, physically plausible metric potential such as that of Finch and Skea. We then attempt to solve for the other metric component by imposing an ansatz which assists in extrapolating from the vanishing pressure boundary condition. Our model is guided by the physicality of the sound speed profiles and associated stability conditions.

An approximate application of quantum gravity to the rotation problem

Arbitrary initial conditions allow solutions of Einstein’s field equations for General Relativity to have arbitrarily large relative rotation of matter and inertial frames. The ‘Rotation Problem’ is to explain why the measured relative rotation rate is so small. As it turns out, nearly any reasonable theory of quantum gravity can solve the rotation problem by phase interference. Even as early as about a quarter of a second after the initial simgularity, quantum cosmology would limit the cosmologies that contribute significantly to a path integral calculation to have relative rms rotation rates less than about 10−51 radians per year. Those calculations are based on using 50 e-foldings during inflation. For 55 or 60 e-foldings, the cosmologies contributing significantly to the path integral would have even smaller relative rotation rates. In addition, although inflation dominates the calculation, even if there had been no inflation, the cosmologies contributing significantly to the path integral would have relative rotation rates less than about 10−32 radians per year at about a quarter of a second after the initial singularity. These calculations are insensitive to the details of the theory of quantum gravity because the main factor depends only on the size of the visible Universe, the Planck time, the free-space speed of light, the Hubble parameter, and the number of e-foldings during inflation. These calculations use the Einstein–Hilbert action in quantum gravity, including large-scale relative rotation of inertial frames and the matter distribution, in which each ‘path’ is a cosmology with a different rms relative rotation rate. The calculations include inflation for 50, 55, and 60 e-foldings, and for values of the dependence of relative rotation rate on cosmological scale factor a as a −m for various values of m. The calculation shows that the action is an extremum at zero rms relative rotation rate.

Finch–Skea dark energy stars with vanishing complexity factor

We make use of the complexity factor formalism for static and self-gravitating objects, recently proposed by L. Herrera (2018), for setting up a relativistic, anisotropic dark energy stellar model satisfying the vanishing complexity condition. The condition serves as an aid in closing the system of field equations for General Relativity. The popular Finch–Skea ansatz is used for the metric component, grr, with the temporal counterpart determined by imposing the vanishing complexity condition. We assess the physical acceptability of our dark energy stellar model according to the standard requirements for static, spherically symmetric compact matter, and we study the associated matter quantities with respect to the dark energy coupling parameter, α. The effect of α on our stellar model has been studied in detail. It is established that our model is regular and stable, and the radial pressure vanishes at the surface boundary as required by selecting a suitable range for α. We find that the coupling parameter influences the baryonic energy density and pressures; however, the pressure anisotropy remains largely unaffected. This is a novel feature of our dark-energy star model, obtained via the vanishing complexity condition. We further demonstrate the robustness of our model in predicting the observed radii of well-known compact objects such as 4U 1820-30, Vela X-1, LMC X-4, PSR J1614-2230, and PSR J1903+0327 by varying a free parameter arising in the Finck–Skea ansatz. Our findings show that our solution predicts masses beyond the standard general relativity limit of 2.0 M⊙ for neutron stars. The prediction of compact objects with masses in the range of 2.5 M⊙ within our solution-generating framework augers well for the observation of gravitational waves arising in binary mergers.

Classical and quantum gravity from the axioms of relativistic quantum mechanics

It is common practice to describe elementary particles by irreducible unitary representations of the Poincaré group. In the same way, multi-particle systems can be described by irreducible unitary representations of the Poincaré group. Representations of the Poincaré group are characterised by fixed eigenvalues of two Casimir operators corresponding to a fixed mass and a fixed angular momentum. In multi-particle systems (of massive spinless particles), fixing these eigenvalues leads to correlations between the particles. In the quasi-classical approximation of large quantum numbers, these correlations take on the structure of a gravitational interaction described by the field equations of conformal gravity. A theoretical value of the corresponding gravitational constant is calculated. It agrees with the empirical value used in the field equations of general relativity.

On the mathematical structure of Einstein field equations and the existence of dark fields

In this work, we examine the possible existence of dark fields by showing that an energy‐momentum tensor associated with a dark field can be introduced into Einstein field equations of general relativity, in which the trace of the energy‐momentum of the dark field is identified with the cosmological constant. The introduction of dark fields into Einstein field equations is made possible by establishing field equations for the Ricci curvature tensor, which have similar mathematical structure to Maxwell field equations of electromagnetism. We also establish a system of field equations for the Riemann curvature tensor and investigate the possibility to represent physical systems consisting of dark fields and observable fields as spaces of constant scalar curvature, which are maximally symmetric spaces that admit the maximal number of Killing vectors. As an illustration, we show that if a dark field is considered as a dark fluid, then the pressure associated with the dark field can take negative values if the cosmological constant is assigned with positive values.

Constrained deformations of positive scalar curvature metrics, II

AbstractWe prove that various spaces of constrained positive scalar curvature metrics on compact three‐manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean‐convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.

Entropic force for quantum particles

Entropic force has been drawing the attention of theoretical physicists following E Verlinde’s work in 2011 to derive Newton’s second law and Einstein’s field equations of general relativity. In this paper, we extend the idea of entropic force to the distribution of quantum particles. Starting from the definition of Shannon entropy for continuous variables, here we have derived quantum osmotic pressure as well as the consequent entropic forces for bosonic and fermionic particles. The entropic force is computed explicitly for a pair of bosons and fermions. The low temperature limit of this result show that the entropic force for bosons is similar to Hooke’s law of elasticity revealing the importance of this idea in the formation of a Bose–Einstein condensate. For fermions, the low temperature limit boils down to the well known Neumann’s radial force and also reveals the Pauli’s exclusion principle. The classical limit of the entropic force between quantum particles is then discussed. As a further example, the entropic force for quantum particles in noncommutative space is also computed. The result reveals a violation of the Pauli exclusion principle for fermions in noncommutative space.

Unimodular gravity traversable wormholes

Wormholes are outstanding solutions of Einstein’s General Relativity. They were worked out in the late 1980s by Morris and Thorne, who have figured out a recipe that wormholes must obey in order to be traversable, that is, safely crossed by travelers. A remarkable feature is that General Relativity Theory wormholes must be filled by exotic matter, which Morris and Thorne define as matter satisfying $$-p_r>\rho$$ , in which $$p_r$$ is the radial pressure and $$\rho$$ is the energy density of the wormhole. In the present article, we introduce, for the first time in the literature, traversable wormhole solutions of Einstein’s Unimodular Gravity Theory. Unimodular Gravity was proposed by Einstein himself as the theory for which the field equations are the traceless portion of General Relativity field equations. Later, Weinberg has shown that this approach elegantly yields the solution of the infamous cosmological constant problem. The wormhole solutions here presented satisfy the metric conditions of “traversability” and remarkably evade the exotic matter condition, so we can affirm that Unimodular Gravity wormholes can be filled by ordinary matter.

On Rastall gravity formulation as a f(R,mathcal {L}_m) and a f(R, T) theory

Rastall introduced a stress-energy tensor whose divergence is proportional to the gradient of the Ricci scalar. This proposal leads to a change in the form of the field equations of General Relativity, but it preserves the number of degrees of freedom. Rastall’s field equations can be either interpreted as GR with a redefined SET, or it can imply different physical consequences inside the matter sector. We investigate limits under which the Rastall field equations can be directly derived from an action, in particular from two f(R)-gravity extensions: f(R,mathcal L_m) and f(R, T). We show that there are similarities between these theories, but the Rastall SET cannot be fully recovered from them, apart from certain particular cases here discussed. It is remarkable that a simple, covariant and invertible redefinition of the SET, as the one proposed by Rastall, is hard to be directly implemented in the action.

Simulation of Closed Timelike Curves in a Darwinian Approach to Quantum Mechanics

Closed timelike curves (CTCs) are non-intuitive theoretical solutions of general relativity field equations. The main paradox associated with the physical existence of CTCs, the so-called grandfather paradox, can be satisfactorily solved by a quantum model named Deutsch-CTC. An outstanding theoretical result that has been demonstrated in the Deutsch-CTC model is the computational equivalence of a classical and a quantum computer in the presence of a CTC. In this article, in order to explore the possible implications for the foundations of quantum mechanics of that equivalence, a fundamental particle is modelled as a classical-like system supplemented with an information space in which a randomizer and a classical Turing machine are stored. The particle could then generate quantum behavior in real time in case it was controlled by a classical algorithm coding the rules of quantum mechanics and, in addition, a logical circuit simulating a CTC was present on its information space. The conditions that, through the action of evolution under natural selection, might produce a population of such particles with both elements on their information spaces from initial sheer random behavior are analyzed.

Space probe of the gravitational constant G

The empirical Gravitational constant G is of fundamental interest in metrology, Newtonian physics, and the field equations in General Relativity, particle physics, geophysics, and astrophysics. The precise value of G has yet to be known with the desired level of accuracy. In the 233 years since Cavendish, hundreds of experiments sought more accurate results but produced inconsistent results. Although they have had much more advanced supporting technology, past experiments have mostly stayed the same from the Cavendish approach. This paper shows that a different method is feasible to enhance the accuracy of G. A sphere made of metal with a large uniform mass density having a channel through the center with a bead oscillating back and forth in a simple-harmonic fashion. Monitoring the positions of the bead experimentally and comparing the predictions of the channel model will allow refining the value of G. The study develops a gravitational force function for the bead in a channeled sphere to provide more accuracy in the prediction of the bead's motion. This study determines that the best place to perform future gravitation constant experiments is in space far away from interfering effects. A near-perfect place is likely to be in outer space at the Sun-Earth Lagrange point L1, located about 1.5 million kilometers from the Earth on a line toward the Sun where other probes reside. It is estimated that three to six significant figures may be added to the current value of G.

The Speed of Light Is Not Constant in Basic Big Bang Theory

Starting from the basic assumptions and equations of Big Bang theory, we present a simple mathematical proof that this theory implies a varying (decreasing) speed of light, contrary to what is generally accepted. We consider General Relativity, the first Friedmann equation and the Friedmann-Lema?tre- Robertson-Walker (FLRW) metric for a Comoving Observer. It is shown explicitly that the Horizon and Flatness Problems are solved, taking away an important argument for the need of Cosmic Inflation. A decrease of 2.1 cm/s per year of the present-day speed of light is predicted. This is consistent with the observed acceleration of the expansion of the Universe, as determined from high-redshift supernova data. The calculation does not use any quantum processes, and no adjustable parameters or fine tuning are introduced. It is argued that more precise laboratory measurements of the present-day speed of light (and its evolution) should be carried out. Also it is argued that the combination of the FLRW metric and Einstein’s field equations of General Relativity is inconsistent, because the FLRW metric implies a variable speed of light, and Einstein’s field equations use a constant speed of light. If we accept standard Big Bang theory (and thus the combination of General Relativity and the FLRW metric), a variable speed of light must be allowed in the Friedmann equation, and therefore also, more generally, in Einstein’s field equations of General Relativity. The explicit form of this time dependence will then be determined by the specific problem.

Cosmic topology. Part I. Limits on orientable Euclidean manifolds from circle searches

The Einstein field equations of general relativity constrain the local curvature at every point in spacetime, but say nothing about the global topology of the Universe. Cosmic microwave background anisotropies have proven to be the most powerful probe of non-trivial topology since, within ΛCDM, these anisotropies have well-characterized statistical properties, the signal is principally from a thin spherical shell centered on the observer (the last scattering surface), and space-based observations nearly cover the full sky. The most generic signature of cosmic topology in the microwave background is pairs of circles with matching temperature and polarization patterns. No such circle pairs have been seen above noise in the WMAP or Planck temperature data, implying that the shortest non-contractible loop around the Universe through our location is longer than 98.5% of the comoving diameter of the last scattering surface. We translate this generic constraint into limits on the parameters that characterize manifolds with each of the nine possible non-trivial orientable Euclidean topologies, and provide a code which computes these constraints. In all but the simplest cases, the shortest non-contractible loop in the space can avoid us, and be shorter than the diameter of the last scattering surface by a factor ranging from 2 to at least 6. This result implies that a broader range of manifolds is observationally allowed than widely appreciated. Probing these manifolds will require more subtle statistical signatures than matched circles, such as off-diagonal correlations of harmonic coefficients.

Defect Wormhole: A Traversable Wormhole Without Exotic Matter

We present a traversable-wormhole solution of the gravitational field equation of General Relativity without need of exotic matter (exotic matter can, for example, have negative energy density and vanishing isotropic pressure). Instead of exotic matter, the solution relies on a 3-dimensional "spacetime defect" characterized by a locally vanishing metric determinant.

Representation of Physical Fields as Einstein Manifold

In this work we investigate the possibility to represent physical fields as Einstein manifold. Based on the Einstein field equations in general relativity, we establish a general formulation for determining the metric tensor of the Einstein manifold that represents a physical field in terms of the energy-momentum tensor that characterises the physical field. As illustrations, we first apply the general formulation to represent the perfect fluid as Einstein manifold. However, from the established relation between the metric tensor and the energy-momentum tensor, we show that if the trace of the energy-momentum tensor associated with a physical field is equal to zero then the corresponding physical field cannot be represented as an Einstein manifold. This situation applies to the electromagnetic field since the trace of the energy-momentum of the electromagnetic field vanishes. Nevertheless, we show that a system that consists of the electromagnetic field and non-interacting charged particles can be represented as an Einstein manifold since the trace of the corresponding energy-momentum of the system no longer vanishes. As a further investigation, we show that it is also possible to represent physical fields as maximally symmetric spaces of constant scalar curvature.

Spontaneous nonlinear scalarization of Kerr black holes

As it became well known in the past years, Einstein-scalar-Gauss-Bonnet (EsGB) theories evade no-hair theorems and allow for scalarized compact objects including black holes (BH). The coupling function that defines the theory is the main character in the process and nature of scalarization. With the right choice, the theory becomes an extension of general relativity (GR) in the sense any solution to the GR field equations remains a solution in the EsGB theory, but it can destabilize if a certain threshold value of the spacetime curvature is exceeded. Thus BHs can spontaneously scalarized. The most studied driving mechanism to this phenomenon is a tachyonic instability due to an effective negative squared mass for the scalar field. However, even when the coupling is chosen such that this mass is zero, higher order terms with respect to the scalar field can lead to what is coined nonlinear scalarization. In this paper we investigate how Kerr BHs spontaneously scalarize by evolving the scalar field on a fixed background via solving the nonlinear Klein-Gordon equation. We consider two different coupling functions with higher order terms, one that yields a non-zero effective mass and another that does not. We sweep through the Kerr parameter space in its mass and spin and obtain the scalar charge by the end of the evolution when the field settles in an equilibrium stationary state. When there is no tachyonic instability present, there is no probe limit in which the BH scalarizes with zero charge, i.e. there is a gap between bald and hairy BHs and they only connect when the mass goes to zero together with the charge.

From maximum force to the field equations of general relativity and implications

There are at least two ways to deduce Einstein’s field equations from the principle of maximum force [Formula: see text] or from the equivalent principle of maximum power [Formula: see text]. Tests in gravitational wave astronomy, cosmology, and numerical gravitation confirm the two principles. Apparent paradoxes about the limits can all be resolved. Several related bounds arise. The limits illuminate the beauty, consistency and simplicity of general relativity from an unusual perspective.

A new class of charged spherically symmetric and static superdense star configurations

In this paper, we have obtained a new family of interior solutions of the Einstein–Maxwell field equations in general relativity for static, spherically symmetric distribution of charged fluid. The electric density is assumed to be [Formula: see text] where [Formula: see text] is a positive constant. The solution is well behaved for a wide range of [Formula: see text] hence, it is suitable for modeling a superdense star. The surface density is assumed to be [Formula: see text], for the validation of the solution set present, a comparison has also been made with the stars EXO 1785-248, SMC X-1, Her X-1, 4U 1538-52. For these particular stars, the maximum value of surface redshift is obtained to be 0.3251 for the star EXO 1785-248 and the minimum value is obtained as 0.2027 for the star Her X-1. We subject our solution to rigorous physical viability tests. In the absence of charge, we obtain the regular and well-behaved sixth model of Durgapal.

On the Field Equations of General Relativity

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

Exact Static Plane Symmetric Soliton-Like Solutions to the Nonlinear Interacting Electromagnetic and Scalar Field Equations in General Relativity

This research work is related to soliton solutions considered as models that can describe the complex configuration of elementary particles from the study of the interactions of their fields. It is interested in the interaction of fields between two different elementary particles by expressing their physical properties. For that, we have obtained, exact static plane symmetric soliton-like solutions to the nonlinear equations of interacting electromagnetic and scalar fields taking into account the own gravitational field of elementary particles using the calibrated invariance function P(I). It has been proved that all solutions of the Einstein, nonlinear electromagnetic and scalar field equations are regular with the localized energy density. Moreover, the total charge of particles is finite and the total energy of the interaction fields is bounded. It have been emphasized the importance to the own gravitational field of elementary particles and the role of the nonlinearity of fields in the determination of these solutions. In flat space-time, soliton-like solutions exist but the total energy of the interaction fields is equal to zero. We have also shown that in the linear case, soliton-like solutions are absent.