Schwarzschild Interior Research Articles

Compact anisotropic stellar distribution with linear barotropic equation of state in energy–momentum trace coupling modification of general relativity

There are no known exact isotropic or anisotropic stellar models with an equation of state in energy–momentum trace-coupling (EMTC) modifications of general relativity — the simplest linear case of f(R,T) gravity. The difficulty lies in the intricate entanglement of the density and pressure functions that generate intractable governing equations when an equation of state is introduced. For example there is no interior Schwarzschild incompressible star analogue in the well studied f(R,T) theory. In Einstein’s theory it is straightforward to find anisotropic stellar models with a linear equation of state since the system is under-determined and there remains one more choice to nominate any of the variables. This is also true in EMTC theories however, the master field equation is more formidable. If interpreted as a linear second order equation, no viable solutions emerge by prescribing one of the gravitational potentials or by suppressing some terms in the spirit of Tolman (1939). Rewriting as a nonlinear first order equation and speculating on a power-law form of the temporally directed gravitational potential results in success in finding a physically viable compact star distribution. Specifically the model has monotonic decrease of both pressure and density and a surface of vanishing pressure exists. Several stability tests are imposed and the model performs according to expectations. The singularity at the stellar center may be cured by insertion of a regular core enveloped by our fluid model in a multi-layered star.

A generalized static spherically symmetric anisotropic compact stellar model

We report here a new exact solution of the Einstein field equations that describes a static spherically symmetric, anisotropic compact stellar fluid sphere. A generalized metric form corresponding to the grr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_{rr}$$\\end{document} component and a specific anisotropic profile generates the model. The new class of generalized solutions reduces to Finch-Skea, Vaidya Tikekar and Schwarzschild interior solutions for specific cho-ices of the model parameters. The model parameters have been determined from the continuity of the interior and exterior metrics at the surface of the star and setting the radial pressure at the boundary to zero. Recent observed measurements from some pulsars as sources have been utilized to validate our model.

The Post-Quasi-Static Approximation: An Analytical Approach to Gravitational Collapse

A seminumerical approach proposed many years ago for describing gravitational collapse in the post-quasi-static approximation is modified in order to avoid the numerical integration of the basic differential equations the approach is based upon. For doing that we have to impose some restrictions on the fluid distribution. More specifically, we shall assume the vanishing complexity factor condition, which allows for analytical integration of the pertinent differential equations and leads to physically interesting models. Instead, we show that neither the homologous nor the quasi-homologous evolution are acceptable since they lead to geodesic fluids, which are unsuitable for being described in the post-quasi-static approximation. Also, we prove that, within this approximation, adiabatic evolution also leads to geodesic fluids, and therefore, we shall consider exclusively dissipative systems. Besides the vanishing complexity factor condition, additional information is required for a full description of models. We shall propose different strategies for obtaining such an information, which are based on observables quantities (e.g., luminosity and redshift), and/or heuristic mathematical ansatz. To illustrate the method, we present two models. One model is inspired in the well-known Schwarzschild interior solution, and another one is inspired in Tolman VI solution.

Diffeomorphism Covariance and the Quantum Schwarzschild Interior

We introduce a notion of residual diffeomorphism covariance in quantum Kantowski–Sachs (KS) describing the interior of a Schwarzschild black hole. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, as well as parity invariance, preservation of the Bohr Hilbert space of the Loop Quantum KS and a correct (naïve) classical limit. We further explore the imposition of minimality for the number of terms and compare the solution with those of other Hamiltonian constraints proposed for the Loop Quantum KS in the literature. In particular, we discuss a lapse that was recently commonly chosen due to the resulting decoupling of the evolution of the two degrees of freedom and the exact solubility of the model. We show that such a choice of lapse can indeed be quantized as an operator that is densely defined on the Bohr Hilbert space and that any such operator must include an infinite number of shift operators.

Analogy of the interior Schwarzschild metric from transformation optics.

In this paper, we make an analogy of the interior Schwarzschild metric from transformation optics (we call the method transformation cosmology). It is shown that a simple refractive index profile is sufficient to capture the behavior of the metric to bend light. There is a critical value of the ratio of the radius of the massive star to the Schwarzschild radius, which is exactly related to the condition of collapsing into a black hole. We demonstrate the light bending effect for three cases from numerical simulations as well. Especially, we find that a point source at the photon sphere will form an image inside the star approximately, and the equivalent lens is like Maxwell's fish-eye lens. This work will help us to explore the phenomena of massive stars with laboratory optical tools.

New exact models of ideal gas in 5D EGB using curvature coordinates

Exploiting the use of curvature coordinates (also called Schwarzschild coordinates), new classes of exact solutions are discovered. By prescribing the timelike potential two new solutions are found one of which displays all the necessary qualitative features demanded of closed compact astrophysical objects. Since the analysis reduces to a single first order nonlinear differential equation, the single integration constant is obtained in terms of the mass and radius through matching of the interior and exterior metrics. Invoking the second matching condition results in constraining the value of the Gauss–Bonnet coupling constant in terms of the mass and radius of the star. Stability, causality and energy conditions are satisfied for a suitable choice of parameter space. In attempting to specify the radially oriented potential it was only possible to find a defective spacetime and to regain the interior Schwarzschild metric for an incompressible fluid sphere.

Consistent gauge-fixing conditions in polymerized gravitational systems

For classical gravitational systems the lapse function and the shift vector are usually determined by imposing appropriate gauge fixing conditions and then demanding their preservation with respect to the dynamics generated by a canonical Hamiltonian. Effective descriptions encoding quantum geometric effects motivated by loop quantum gravity for symmetry reduced models are often captured by polymerization of connection (or related) variables in gauge fixing conditions as well as constraints. Usually, one chooses the same form of polymerization in both cases. A pertinent question is if the dynamical stability of the effective gauge fixing conditions under the effective dynamics generated by the polymerized canonical Hamiltonian is provided by the lapse function and the shift vector obtained from the polymerization of their classical counterparts. If this is the case, then we say that gauge fixing and polymerization commute. In this manuscript we investigate these issues and obtain consistency conditions for the commutativity of gauge fixing and polymerization. Our analysis shows that such a commutativity occurs in rather special situations and reveals pitfalls in making seemingly well motivated choices which turn out to be inconsistent with the effective dynamics. We illustrate these findings via examples of symmetry reduced models in the loop quantization of the Schwarzschild interior and Lema\^{\i}tre-Tolman-Bondi (LTB) spacetimes and report the non-commutativity of gauge fixing and polymerization, and inherent limitations of some choices made in the literature with a consistent effective dynamics.

Compact stars in the Einstein dark energy model

We investigate the properties of high density compact objects in a vector type theory, inspired by Einstein's 1919 theory of elementary particles, in which Einstein assumed that elementary particles are held together by gravitational as well as electromagnetic type forces. From a modern perspective, Einstein's theory can be interpreted as a vector type model, with the gravitational action constructed as a linear combination of the Ricci scalar, of the trace of the matter energy-momentum tensor, and of a massive self-interacting vector type field. To obtain the properties of stellar models we consider the field equations for a static, spherically symmetric system, and we investigate numerically their solutions for different equations of state of quark and neutron matter, by assuming that the self-interaction potential of the vector field either vanishes or is quadratic in the vector field potential. We consider quark stars described by the MIT bag model equation of state and in the Color Flavor Locked phase, as well as compact stars consisting of a Bose-Einstein Condensate of neutron matter, with neutrons forming Cooper pairs. Constant density stars, representing a generalization of the Interior Schwarzschild solution of general relativity, are also analyzed. Also, we consider the Douchin-Haensel (SLy) equation of state. The numerical solutions are explicitly obtained in both standard general relativity, and the Einstein dark energy model and an in depth comparison between the astrophysical predictions of these two theories are performed. As a general conclusion of our study, we find that for all the considered equations of state a much larger variety of stellar structures can be obtained in the Einstein dark energy model, including classes of stars that are more massive than their general relativistic counterparts.

Loop quantum deparametrized Schwarzschild interior and discrete black hole mass

We present the detailed analyses of a model of loop quantum Schwarzschild interior coupled to a massless scalar field and extend the results in our previous rapid communication arXiv:2006.08313 to more general schemes. It is shown that the spectrum of the black hole mass is discrete and does not contain zero. This indicates the existence of a black hole remnant after Hawking evaporation due to loop quantum gravity effects. Besides to show the existence of a stable black hole remnant in the vacuum case, the quantum dynamics for the non-vacuum case is also solved and compared with the effective one.

Junction conditions for generalized hybrid metric-Palatini gravity with applications

The generalized hybrid metric-Palatini gravity is a theory of gravitation that has an action composed of a Lagrangian given by $f(R,\mathcal{R})$, where $f$ is a function of the metric Ricci scalar $R$ and a new Ricci scalar $\mathcal{R}$ formed from a Palatini connection, plus a matter Lagrangian. This theory can be rewritten by trading the new geometric degrees of freedom that appear in $f(R,\mathcal{R})$ into two scalar fields, $\ensuremath{\varphi}$ and $\ensuremath{\psi}$, yielding a dynamically equivalent scalar-tensor theory. Given a spacetime theory, the next important step is to find solutions within it. To construct appropriate solutions it is often necessary to know the junction conditions between two spacetime regions at a separation hypersurface $\mathrm{\ensuremath{\Sigma}}$, with each spacetime region being an independent solution of the theory. The junction conditions for the generalized hybrid metric-Palatini gravity are found here, both in the geometric representation and in the scalar-tensor representation, and in addition, for each representation, the junction conditions for a matching with a thin-shell of matter and for a smooth matching at the separation hypersurface are worked out. These junction conditions are then applied to three configurations, namely, a star, a quasistar with a black hole, and a wormhole. The star is made of a Minkowski interior, a thin shell at the interface with all the matter energy conditions being satisfied, and a Schwarzschild exterior with mass $M$, and unlike general relativity where the matching can be performed at any radius ${r}_{\mathrm{\ensuremath{\Sigma}}}$, for this theory the matching can only be performed at a specific value of the shell radius, namely ${r}_{\mathrm{\ensuremath{\Sigma}}}=\frac{9M}{4}$, that corresponds to the general relativistic Buchdahl radius. The quasistar with a black hole is made of an interior Schwarzschild black hole surrounded by a thick shell that matches smoothly to a mass $M$ Schwarzschild exterior at the light ring radius ${r}_{\mathrm{\ensuremath{\Sigma}}e}=3M$, and with the matter energy conditions being satisfied for the whole spacetime. The wormhole is made of some interior with matter that contains the throat, a thin shell at the interface, and a Schwarzschild-AdS exterior with mass $M$ and negative cosmological constant $\mathrm{\ensuremath{\Lambda}}$, with the matter null energy condition being obeyed everywhere within the wormhole.

New solution generating algorithm for isotropic static Einstein-Gauss-Bonnet metrics

The static isotropic gravitational field equation, governing the geometry and dynamics of stellar structure, is considered in Einstein–Gauss–Bonnet (EGB) gravity. This is a nonlinear Abelian differential equation which generalizes the simpler general relativistic pressure isotropy condition. A gravitational potential decomposition is postulated in order to generate new exact solutions from known solutions. The conditions for a successful integration are examined. Remarkably we generate a new exact solution to the Abelian equation from the well known Schwarzschild interior seed metric. The metric potentials are given in terms of elementary functions. A physical analysis of the model is performed in five and six spacetime dimensions. It is shown that the six-dimensional case is physically more reasonable and is consistent with the conditions restricting the physics of realistic stars.

Spin-1/2 Particles under the Influence of a Uniform Magnetic Field in the Interior Schwarzschild Solution

The relativistic wave equation for spin-1/2 particles in the interior Schwarzschild solution in the presence of a uniform magnetic field is obtained. The fully relativistic regime is considered, and the energy levels occupied by the particles are derived as functions of the magnetic field, the radius of the massive sphere and the total mass of the latter. As no assumption is made on the relative strengths of the particles’ interaction with the gravitational and magnetic fields, the relevance of our results to the physics of the interior of neutron stars, where both the gravitational and the magnetic fields are very intense, is discussed.

Wave packet treatment of neutrino flavor oscillations in various spacetimes

We study the effect of gravity on neutrino flavor oscillations when each mass eigenstate is described by a wave packet instead of a plane wave. Two different approaches for implementing the wave packet formalism in the study of neutrino flavor oscillations in curved spacetime are examined. We work with a general static and spherically symmetric spacetime before applying our results to a few spacetime metrics of interest. We first focus on general relativity by examining the effect of the exterior and interior Schwarzschild solutions, as well as the de Sitter-Schwarzschild metric, and then we examine selected metrics from modified gravity models.

(b,v)-type variables for black to white hole transitions in effective loop quantum gravity

Quantum gravity effects in effective models of loop quantum gravity, such as loop quantum cosmology, are encoded in the choice of so-called polymerisation schemes. Physical viability of the models, such as an onset of quantum effects at curvature scales near the Planck curvature, severely restricts the possible choices. An alternative point of view on the choice of polymerisation scheme is to choose adapted variables so that the scheme is the simplest possible one, known as μ0-scheme in loop quantum cosmology. There, physically viable models with μ0-scheme polymerise the Hubble rate b that is directly related to the Ricci scalar and the matter energy density on-shell. Consequently, the onset of quantum effects depends precisely on those parameters. In this letter, we construct similar variables for black to white hole transitions modelled using the description of the Schwarzschild interior as a Kantowski-Sachs cosmology. The resulting model uses the μ0-scheme and features sensible physics for a broad range of initial conditions (= choices of black and white hole masses) and favours symmetric transitions upon invoking additional qualitative arguments. The resulting Hamiltonian is very simple and at most quadratic in its arguments, allowing for a straightforward quantisation.

Dark energy stars with a phantom field

Dark energy is the constituent with an enormous abundance in the present Universe, responsible for the Universe's accelerated expansion. Therefore, it is plausible that dark energy may interact within any compact astrophysical object. The author in [S. S. Yazadjiev, Phys. Rev. D 83, 127501 (2011)] constructs an exact star solution consisting of ordinary matter and a phantom field from a constant density star (CDS) known as the Schwarzschild interior solution. The star denotes a dark energy star (DES). The author claims that the phantom field represents dark energy within the star. So far, the role of the phantom field as dark energy in a DES is not systematically studied yet. Related to this issue, we analyze the energy condition of a DES. We expect that DESs shall violate the strong energy condition (SEC) for a particular condition. We discover that a SEC is fully violated only when the compactness reaches the Buchdahl limit. Furthermore, we also investigate the causal conditions and stabilities due to the convective motion and gravitational cracking. We also find that those conditions are violated. These results indicate that DES is not physically stable. However, we may consider DES as an ultracompact object of which we can calculate the gravitational wave echo time and echo frequency and compare them to those of CDS. We find that the contribution of the phantom field delays the gravitational wave echoes. The effective potential of the perturbed DES is also studied. The potential also enjoys a potential well like CDS but with a deeper well. We also investigate the possibility that DES could form a gravastar when $C=1$. It is found that a gravastar produced from DES possesses no singularity with a dS-like phase as the interior. These results could open more opportunities for the observational study of dark energy in the near future, mostly from the compact astrophysical objects.

Investigating new forms of gravity-matter couplings in the gravitational field equations

This paper proposes a toy model where, in the Einstein equations, the right-hand side is modified by the addition of a term proportional to the symmetrized partial contraction of the Ricci tensor with the energy-momentum tensor, while the left-hand side remains equal to the Einstein tensor. Bearing in mind the existence of a natural length scale given by the Planck length, dimensional analysis shows that such a term yields a correction linear in $\ensuremath{\hbar}$ to the classical term that is instead just proportional to the energy-momentum tensor. One then obtains an effective energy-momentum tensor that consists of three contributions: pure energy part, mechanical stress, and thermal part. The pure energy part has the appropriate property for dealing with the dark sector of modern relativistic cosmology. Such a theory coincides with general relativity in vacuum, and the resulting field equations are here solved for a Dunn and Tupper metric, for departures from an interior Schwarzschild solution as well as for a Friedmann-Lemaitre-Robertson-Walker universe.

All conformally flat Einstein–Gauss–Bonnet static metrics

It is known that the standard Schwarzschild interior metric is conformally flat and generates a constant density sphere in any spacetime dimension in Einstein and Einstein–Gauss–Bonnet (EGB) gravity. This motivates the questions: in EGB does the conformal flatness criterion yield the Schwarzschild metric? Does the assumption of constant density generate the Schwarzschild interior spacetime? The answer to both questions turn out in the negative in general. In the case of the constant density sphere, a generalised Schwarzschild metric emerges. When we invoke the conformal flatness condition the Schwarschild interior solution is obtained as one solution and another metric which does not yield a constant density hypersphere in EGB theory is found. For the latter solution one of the gravitational metrics is obtained explicitly while the other is determined up to quadratures in 5 and 6 dimensions. The physical properties of these new solutions are studied with the use of numerical methods and a parameter space is located for which both models display pleasing physical behaviour.

Embedding class one solutions of anisotropic fluid spheres in modified $$f({\mathcal {G}})$$ gravity

The main purpose of current analysis is to study three different compact spheres LMC X-4, Cen X-3, and EXO 1785-248 in modified $$f({\mathcal {G}})$$ gravity. In this context, the well-known Karmarkar condition is used to choose the suitable metric coefficients. The Schwarzschild interior solution is used to determine the matching conditions, and all the unknown coefficients in metric potentials are calculated through matching conditions for different values of the model parameter K. The necessary properties of stellar objects are discussed in this study, and a detailed graphical analysis is provided to check the physical viability. The validity of obtained anisotropic solutions confirms that the results are physically well fitted with observed data and free from any singularity.

An electromagnetic extension of the Schwarzschild interior solution and the corresponding Buchdahl limit

We wish to construct a model for charged star as a generalization of the uniform density Schwarzschild interior solution. We employ the Vaidya and Tikekar ansatz (Astrophys Astron 3:325, 1982) for one of the metric potentials and electric field is chosen in such a way that when it is switched off the metric reduces to the Schwarzschild. This relates charge distribution to the Vaidya–Tikekar parameter, k, indicating deviation from sphericity of three dimensional space when embedded into four dimensional Euclidean space. The model is examined against all the physical conditions required for a relativistic charged fluid sphere as an interior to a charged star. We also obtain and discuss charged analogue of the Buchdahl compactness bound.

Lemaître Transformations of the Interior Schwarzschild Metric

A Lemaître transformation is set up for the free fall in the interior of a stellar object using the frame of the interior Schwarzschild solution. The metric is calculated in comoving coordinates and field strengths are derived for this metric.