General Relativistic Equations Research Articles

Re-evaluating the cosmological redshift: Insights into inhomogeneities and irreversible processes

Understanding the expansion of the Universe remains a profound challenge in fundamental physics. The complexity of solving general relativity equations in the presence of intricate, inhomogeneous flows has compelled cosmological models to rely on perturbation theory in a homogeneous Friedmann-Lemaître-Robertson-Walker background. This approach accounts for a redshift of light encompassing contributions from both the cosmological background expansion along the photon's trajectory and Doppler effects at emission due to peculiar motions. However, this computation of the redshift is not covariant, as it hinges on specific coordinate choices that may distort physical interpretations of the relativity of motion. In this study we show that peculiar motions, when tracing the dynamics along time-like geodesics, must contribute to the redshift of light through a local volume expansion factor, in addition to the background expansion. By employing a covariant approach to redshift calculation, we address the central question of whether the cosmological principle alone guarantees that the averaged local volume expansion factor matches the background expansion. We establish that this holds true only in scenarios characterised by a reversible evolution of the Universe, where inhomogeneous expansion and compression modes compensate for one another. In the presence of irreversible processes, such as the dissipation of large-scale compression modes through matter virialisation and associated entropy production, the averaged expansion factor becomes dominated by expansion in voids that can no longer be compensated for by compression in virialised structures. Furthermore, for a universe in which a substantial portion of its mass has undergone virialisation, adhering to the background evolution on average leads to significant violations of the second law of thermodynamics. Our approach shows that entropy production due to irreversible processes during the formation of structures plays the same role as an effective, time-dependent cosmological constant (i.e. dynamical dark energy) without the need to invoke new unknown physics. Our findings underscore the imperative need to re-evaluate the influence of inhomogeneities and irreversible processes on cosmological models, shedding new light on the intricate dynamics of our Universe.

Neutron stars in f(R,Lm,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R,L_m,T)$$\\end{document} gravity

This study explores the behavior of compact stars within the framework of f(R,Lm,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R,L_m,T)$$\\end{document} gravity, focusing on the functional form f(R,Lm,T)=R+αTLm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R,L_m,T) = R + \\alpha TL_m$$\\end{document}. The modified Tolman–Oppenheimer–Volkoff (TOV) equations are derived and numerically solved for several values of the free parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} by considering both quark and hadronic matter—described by realistic equations of state (EoSs). Furthermore, the stellar structure equations are adapted for two different choices of the matter Lagrangian density (namely, Lm=p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_m= p$$\\end{document} and Lm=-ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_m= -\\rho $$\\end{document}), laying the groundwork for our numerical analysis. As expected, we recover the traditional TOV equations in General Relativity (GR) when α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document}. Remarkably, we found that the two choices for Lm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_m$$\\end{document} have appreciably different effects on the mass-radius diagrams. Results showcase the impact of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on compact star properties, while final remarks summarize key findings and discuss implications, including compatibility with observational data from NGC 6397’s neutron star. Overall, this research enhances comprehension of f(R,Lm,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R,L_m,T)$$\\end{document} gravity’s effects on compact star internal structures, offering insights for future investigations.

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.

Cosmological implications of the Weyl geometric gravity theory

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Attractors in [formula omitted]-gravity

We investigate the asymptotic behavior of the cosmological field equations in Symmetric Teleparallel General Relativity, where a nonlinear function of the boundary term is introduced instead of the cosmological constant to describe the acceleration phase of the universe. Our analysis reveals constraints on the free parameters necessary for the existence of an attractor that accurately represents acceleration. However, we also identify asymptotic solutions depicting Big Rip and Big Crunch singularities. To avoid these solutions, we must impose constraints on the phase-space, requiring specific initial conditions.

Accretion discs onto supermassive compact objects: A portal to dark matter physics in active galaxies

Context. The study of the physics of the accretion discs that develop around supermassive black hole (BH) candidates provides essential theoretical tools to test their nature. Aims. Here, we study the accretion flow and associated emission using generalised α-discs accreting onto horizonless dark compact objects in order to make comparisons with the traditional BH scenario. The BH alternative proposed here consists in a dense and highly degenerate core made of fermionic dark matter (DM) and surrounded by a more diluted DM halo. This dense core–diluted halo DM configuration is a solution of Einstein’s equation of general relativity (GR) in spherical symmetry, which naturally arises once the quantum nature of the DM fermions is duly accounted for. Methods. The methodology followed in this work consists in first generalising the theory of α-discs to work in the presence of regular and horizonless compact objects, and then applying it to the case of core–halo DM profiles typical of active-like galaxies. Results. The fact that the compactness of the dense and transparent DM core scales with particle mass allows the following key findings of this work: (i) There is always a given core compacity – corresponding particle mass – that produces a luminosity spectrum that is almost indistinguishable from that of a Schwarzschild BH of the same mass as the DM core. (ii) The disc can enter deep inside the non-rotating DM core, allowing accretion-powered efficiencies of as high as 28%, which is comparable to that of a highly rotating Kerr BH. Conclusions. These results, together with the existence of a critical DM core mass of collapse into a supermassive BH, open new avenues of research for two seemingly unrelated topics: AGN phenomenology and dark matter physics.

Gluing Small Black Holes into Initial Data Sets

We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document} spatial dimensions. We glue an ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-rescaling of an asymptotically flat data set (γ^,k^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{\\gamma }},{\\hat{k}})$$\\end{document} into the neighborhood of a point p∈X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}\\in X$$\\end{document} inside of another initial data set (X,γ,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\\gamma ,k)$$\\end{document}, under a local genericity condition (non-existence of KIDs) near p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document}. As the scaling parameter ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} tends to 0, the rescalings xϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{x}{\\epsilon }$$\\end{document} of normal coordinates x on X around p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document} become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document} on the other hand, the glued initial data converge back to (γ,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma ,k)$$\\end{document}. The initial data we construct enjoy polyhomogeneous regularity jointly in ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets (X,γ,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\\gamma ,k)$$\\end{document} and appropriate boosted Kerr initial data sets (γ^,k^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{\\gamma }},{\\hat{k}})$$\\end{document} produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} black hole. The proof combines a variant of the gluing method introduced by Corvino and Schoen with geometric singular analysis techniques originating in Melrose’s work. On a technical level, we present a fully geometric microlocal treatment of the solvability theory for the linearized constraints map.

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.

Variable brane tension and dark energy

In this letter, we show that in a particular braneworld scenario with variable brane tension, we obtain matter acting as dark energy while the gravitational constant G promoted to a scalar field on the brane plays the role of matter (both in the sense that they have an “effective” equation of state equivalent to that of dark energy and matter, respectively). This result is interpreted from the Friedmann equation obtained from our model that exactly matches the standard Friedmann equation of general relativity with a cosmological constant Λ in terms of the aforementioned quantities. The universe is assumed to consist of only matter and dark energy in this model which is a good approximation for our universe.

COMPARISON BETWEEN TWO INSPIRAL MODELS

Gravitational waves are related to black holes in the context of two-body motion. However, the two-body problems are not held in general relativity. Consequently, analytical models have been developed to describe the behavior of binary systems before they merge. During this process, gravitational waves emit as the separation between the components of the binary system decreases. These models can be categorized into two parts: models for the inspiral phase, which rely on the post-Newtonian approximation, and models for the merger phase, which are based on numerical solutions to the equations of general relativity. We implemented two distinct models for the inspiral phase to compare their agreement. We found a complete agreement in all aspects except for the timing of the detection of gravitational waves, even though both models were conducted under the same conditions.

A New Gravitational Radiation from General Relativity

Actual cosmological and galactic data is showing the possible existence of a non-visible gravitational mass. This mass has been called dark matter, and researchers are looking for an explanation of its physical nature. A new solution to Einstein’s general relativity equations is introduced here. This new solution assumes that gravity has density and can produce pressure as any other known field does. Our hypothesis does not create any mathematical contradiction with anything that is known and points to the conclusion that dark matter is a non-linear manifestation of the gravitational field.

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.

Deriving the cosmological constant from the Euler–Lagrange equation of second-order differentiable gravitational field Lagrangian

The principles underlying the variational approach prove to be invaluable tools in articulating physical phenomena, particularly when dealing with conserved quantities. The derivation of gravitational field equations from the Euler–Lagrange equation, a result of the first-order derivative of Lagrange field density, is a well-established concept. However, within the context of this paper, the field equations stem from the second-order differentiable field Lagrangian in the Euler–Lagrange equation. The Euler–Lagrange equation of the first-order differentiable field Lagrangian yields the gravitational field equations incorporating matter fields and the cosmological constant. On the other hand, the Euler–Lagrange equation of the second-order differentiable field Lagrangian gives rise to the equation governing the cosmological constant. The procedure of determining the cosmological constant through the Euler–Lagrange equation adheres to fundamental mathematical theory encompassing higher-order Euler–Lagrange equations. This theory implies that utilizing higher-order Euler–Lagrange equations may result in instability due to negative energy. The cosmological constant is postulated to be intertwined with negative vacuum energy density. The value assigned to the gravitational constant, as ascertained through the employment of the general relativistic equation, is not only significantly small and nonzero but also demonstrates remarkable compatibility with the value inferred from observational deductions.

Combined Screw and Wedge Dislocations

Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear parallel sources. It describes elastic media with parallel combined wedge and screw dislocations.

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.

Towards Multifractality through an Ernst-Type Potential in Complex Systems Dynamics.

Some possible correspondences between the Scale Relativity Theory and the Space-Time Theory can be established. Since both the multifractal Schrödinger equation from the Scale Relativity Theory and the General Relativity equations for a gravitational field with axial symmetry accept the same SL(2R)-type invariance, an Ernst-type potential (from General Relativity) and also a multi-fractal tensor (from Scale Relativity) are highlighted in the description of complex systems dynamics. In this way, a non-differentiable description of complex systems dynamics can become functional, even in the case of standard theories (General Relativity and Quantum Mechanics).

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.

Global high-order numerical schemes for the time evolution of the general relativistic radiation magneto-hydrodynamics equations

Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the grey approximation, which reduces Boltzmann seven-dimensional equation to a system of equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem. Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit Runge-Kutta commonly employed for the magneto-hydrodynamics and Einstein equations. Finally, we analyze their performance in several numerical tests.