Schwarzschild Line Element Research Articles

Study of gravastar admitting Tolman IV spacetime in Rastall theory

In this paper, we present a novel solution representing the gravastar (or gravitational vacuum star) model within the framework of non-conservative Rastall gravity. As a viable alternative to black holes, the geometry of a gravastar comprises three distinct regions: the inner sector, the intermediate layer, and the exterior domain. Utilizing the temporal component of Tolman IV spacetime, we derive the singularity-free radial metric potentials for both the inner and intermediate regions. Further, by aligning the thin shell with the external Schwarzschild line element, we establish boundary conditions in order to determine unknown constants that appear due to the chosen ansatz and performing some integrations. Subsequently, we investigate various properties for the gravastar’s shell, including the equation of state parameter, proper length, energy, entropy and gravitational redshift for four different values of the Rastall parameter. It is concluded that the gravastar structure exists and provides a feasible substitute of black holes in the framework of Rastall theory.

Particle and light motion in Lyra–Schwarzschild spacetime

Abstract In this paper we recall the static and spherically symmetric solution derived within LyST gravity. LyST stands for “Lyra Scalar-Tensor” and is a scalar-tensor proposal for the gravitational interaction modifying general relativity by adopting Lyra manifold as the spacetime substrate. Lyra manifold is characterized both by the metric tensor gµν(x) and the scale φ(x); the latter being an integral part of the definition of reference frame. In a previous work, we have launched the formal geometrical basis of LyST, we have proposed an associated action integral for the gravitational interaction, we have derived the field equations generalizing Einstein field equation, and we have solved these equation to obtain the generalized version of Schwarzschild line element in Lyra manifold. This lead us to Lyra-Schwarzschild spacetime, which is further explored in the present paper. Herein, the trajectories of test particles is thoroughly studied. This is done for both massive particles and massless particles. The geodesic equations are built and solved. The effective potential associated to the motion of these particles around the source is determined and characterized. The possible trajectories include gravitational capture, scattering, bounded orbits, stable and unstable circular orbits, near-horizon motion. The innermost circular orbit allowed for massive particles is determined and compared to the one predicted by GR. The last photon orbit is also calculated for Lyra-Schwarzschild metric. The equation for the periastron shift in the context of Lyra-Schwarzschild solution is constructed; its application to observational data could be useful to constrain the parameters typical of LyST, telling it apart from GR.

Investigating the equation-of-state, stability and mass–radius relationship of anisotropic and massive neutron stars embedded in f(R,T) modified gravity

In this study, our main focus is to investigate the mass–radius relation and several important properties of massive neutron stars to realize the nature, behavior and evolution of these kinds of compact objects at present time. Also, we want to understand the equation-of-state of the core nuclear matter precisely with their stable equilibrium configuration. We have chosen a few massive binary pulsars and investigated on maximum attainable mass and lowest possible radius of them. We have considered Einstein–Hilbert action as [Formula: see text], where [Formula: see text] is the Ricci scalar and [Formula: see text], the trace of energy–momentum tensor with [Formula: see text] as the coupling parameter and also used modified TOV equations. The interior space-time of the spherical neutron star is matched to the exterior Schwarzschild line element at the surface of the star. The [Formula: see text]–[Formula: see text] curve can predict the maximum achievable mass is about [Formula: see text] with lowest possible radius of around [Formula: see text] km for the massive compact stars under stable equilibrium. We have obtained a clear picture of structural evolution of massive neutron stars through accelerating space-time and can put some constraints on several quantities related to them. It is depicted from our present investigation that all the derived outcomes are compatible with physically adopted regimes which reveal the viability of our current model in the context of [Formula: see text] modified gravity.

The space–time line element for static ellipsoidal objects

In this paper, we solved the Einstein’s field equation and obtained a line element for static, ellipsoidal objects characterized by the linear eccentricity ( $$\eta $$ ) instead of quadrupole parameter (q). This line element recovers the Schwarzschild line element when $$\eta $$ is zero. In addition to that it also reduces to the Schwarzschild line element, if we neglect terms of the order of $$r^{-2}$$ or higher which are present within the expressions for metric elements for large distances. Furthermore, as the ellipsoidal character of the derived line element is maintained by the linear eccentricity ( $$\eta $$ ), which is an easily measurable parameter, this line element could be more suitable for various analytical as well as observational studies.

Modelling a new class of anisotropic compact stars satisfying the Karmakar’s condition

The present paper provides a new class of interior solutions which satisfies the Karmakar condition (K.R. Karmarkar, Proc. Indian Acad. Sci. A 27, 56 (1948)). The model is generated by assuming a reasonable choice for the metric potential $ g_{rr}$ and the model parameters are obtained accordingly since, in the case of embedding class-I, the metric co-efficients are dependent on each other. We match our interior solution to the exterior Schwarzschild line element in the presence of thin shell. The proposed model does not suffer from central singularity. The model is discussed for a neutron star of mass $ 1.4 M_{\odot}$ and radius 10km. The potential stability condition of the model is also discussed.

Strange star admitting Chaplygin equation of state in Finch–Skea spacetime

In the present paper we propose a new model of an anisotropic strange star which admits the Chaplygin equation of state. The exterior spacetime is described by a Schwarzschild line element. The model is developed by assuming the Finch–Skea ansatz (Finch and Skea in Class. Quantum Gravity 6:467, 1989). We obtain the model parameters in closed form. Our model is free from a central singularity. Choosing some particular values for the parameter we show that our model corroborates the observational data of the strange star PSR J1614-2230 (Gangopadhyay et al. in Mon. Not. R. Astron. Soc. 431:3216, 2013).

A Comment on Background Independence in Quantum Theory

AbstractIn this communication we take up the significance and purpose of selecting the proper coordinate system from the flat space‐time of non‐relativistic theories to the quantum theoretic formulation of general relativity. The universal background problem is straight forwardly framed as a momentum‐energy portrait in nexus with its space‐time conjugates. The description is based on operator matrix algebra, where the related analogue of the secular equation yields a Klein‐Gordon type equation and the associated Minkowski eigentime element. The energy‐momentum and their conjugate partners are represented by spaces that have (+,−) signatures. The general theory implicates both non‐zero‐ and zero rest‐mass entities, and it is proved that the conjugate relationship between energy and time provide a simple derivation of the Schwarzschild line element for the case of a gravitational field outside a spherical non‐rotational uncharged mass. This result, indicating the appearance of a black hole as a true singularity in the energy‐time formulation, and obtained as a direct consequence of their conjugate relationship, manifests background independence in concert with Einstein’s equivalence principle. Inducing a reformulation of the Lorentz Transformation respecting the indefinite Minkowski metric, displays an interesting relation between complex dilations and indefinite metric spaces, validating the complex symmetric ansatz.

A geometric representation for visualizing relativistic length and time measurement

The simplest solution of Einstein's field equations is Schwarzschild solution. This solution is not able to describe for any nonspherical shaped objects. Some stars and galaxies are in ellipsoidal. Consequently, the gravitational field around these objects should be different in compare with spherical form. This paper is considering a new line element so that we are able to construct not only spherical objects, but also we are able to explain an ellipsoidal object too. This new line element is more accurate and complete than Schwarzschild line element. In this research we see that Schwarzschild line element and its solution is only a part of whole work, which we have done. For more consideration we applied this metric to an arbitrary object in the next step. Moreover, we used this line element for solution of a planetary orbit of an ellipsoid planet by using Einstein’s field equations. These equations used for the exterior solution of an ellipsoidal celestial object.

Charged gravastars admitting conformal motion

We propose a new model of a gravastar admitting conformal motion. While retaining the framework of the Mazur–Mottola model, the gravastar is assumed to be internally charged, with an exterior defined by a Reissner–Nordström instead of a Schwarzschild line element. The solutions, obtained by exploiting an assumed conformal Killing vector, involve (i) the interior region, (ii) the shell, and (iii) the exterior region of the sphere. Of these three cases the first one is of primary interest since the total gravitational mass here turns out to be an electromagnetic mass under some specific conditions. This suggests that the interior de Sitter vacuum of a charged gravastar is essentially an electromagnetic mass model that must generate gravitational mass which provides a stable configuration by balancing the repulsive pressure arising from charge with its attractive gravity to avert a singularity. Therefore the present model, like the Mazur–Mottola model, results in the construction of a compact astrophysical object, as an alternative to a black hole. We have also analyzed various other aspects such as the stress energy tensor in the thin shell and the entropy of the system.

Behavior of elliptical objects in general theory of relativity

The simplest solution to Einstein's field equations is the Schwarzschild solution. This solution is not able to describe any non-spherical shaped objects. Some stars and galaxies are ellipsoidal. Consequently, the gravitational field around these objects should be different in comparison with the spherical form. This paper is considering a new line element so that we are able to construct not only spherical objects but also we are able to explain an ellipsoidal object too. This new line element is more accurate and complete than the Schwarzschild line element. In this research, we see that the Schwarzschild line element and its solution is only a part of the whole work, which we have done. For more consideration, we applied this metric to an arbitrary object in the next step. Moreover, we used this line element for the solution of a planetary orbit of an ellipsoid planet by using Einstein’s field equations. These equations used for the exterior solution of an ellipsoidal celestial object.

The solution of complete Schwarzschild's planetary equation with the method of successive approximations

The Einstein's solution of the planetary equation of motion from Schwarzschild's line element is well known. In this paper, we solve the complete Schwarzschild's planetary equation with the method of successive approximation for the corresponding precession and compare the result with that from the line element. Keywords: precession angle, successive approximation, line element, planetary equation, perturbation term Nigerian Journal of Physics Vol. 17, 2005: 7-10

Bending of Light and Gravitational Signals in Certain On-Brane and Bulk Geometries

In this article, we first investigate the bending of light rays in 4D line elements representing spherically symmetric, static on–brane geometries. The amount of bending in these four dimensional, strong and weak field solutions is derived. Signatures, which appear in the bending formulae, due to the presence of extra dimensions are discussed. Subsequently, as a separate exercise, we calculate the bending of null geodesic trajectories in a bulk five dimensional spacetime with a Schwarzschild line element on the 3-brane section. We interpret the deviation of null trajectories as that of gravitational signals or five dimensional ‘photons,’ which are, unlike light rays confined to four dimensions, allowed to propagate along the fifth (extra) dimension. Features of the presence of extra dimensions in the effective potentials and the bending formula are analysed in this context.

The Rate of Entropy Change: Gravitational Effects

The covariant form of entropy change (or balance) is expressed in the metric of the Schwarzschild line element. In this metric the rate of entropy change depends only on the time component of the metric tensor; it is then different at different points in space and appears to be slower in the presence than in the absence of a gravitational field. The linear equations originated by the internal entropy sources depend on both the time and radial components of the metric tensor and the transport processes behave anisotropically, being slower along the radial coordinate than in the directions of the others. The extent of rate reduction is expressed by the kinetic (transport) quantities which assume different values at different points in space.

THE SOLITON STARS EVOLUTION

The evolution of a soliton star filled with fermions is studied in the framework of general relativity. Such a system could be described by the surface tension σ0, the bag constant B, and the fermion number density ρ. One of the parameters mentioned above could prevail in the system and thus affect the spacetime inside the bubble. Whether it is described by Friedman or de Sitter metric depends on the prevailing parameter. In broad outline, the whole spacetime is divided by the surface of the soliton into the false vacuum region inside the soliton and the true vacuum region outside, the latter being described by the Schwarzschild line element. The aim of this paper is to study the equations of motion of the domain wall in two cases. In both of them, the Schwarzschild metric is outside the soliton. The de Sitter metric describes the interior in the first case, and in the second case, it is replaced by the Friedman metric. Analyzing obtained equations, one can draw conclusions concerning the further evolution of the soliton star.

On a Stationary Cosmology in the Sense of Einstein's Theory of Gravitation

A world is to be considered stationary in the sense of general relativity if the coefficients of its metric are independent of time in a coordinate system in which the masses are at rest on average. The remark on the system of coordinates is important because time itself is no invariant notion but is taken only in the sense of proper time. Our definition is unique, in the form given above. On the other hand it is also possible to have points where no matter is present. At such points we may place a test body of infinitesimally small mass and analyse whether it remains at rest in our coordinate system. A necessary and sufficient condition for this is that the time lines of our coordinate system are geodesics. Therefore the static solution given by de Sitter is not an example of a stationary world. The Schwarzschild line element which, from a cosmological point of view, is a world with a single central body can also not be considered a stationary solution. Indeed, there are no stationary solutions which are also spherically symmetric for the original field equations. The only such solution for the cosmological equations is Einstein's cylinder world. It is, to my knowledge, the only stationary world known so far. In that case the average matter density and the total mass of the world has to have a well defined value given by the cosmological constant which doubtless would be purely coincidental and is thus not a satisfactory assumption. In the following we shall discuss a new solution which is in accord with the original field equations without the need of an a priori relation between mass and cosmological constant. However, we shall find that its mass cannot be less than the mass of the cylinder world.

Approximate black holes for numerical relativity.

Spherically symmetric solutions in Brans-Dicke theory of relativity with zero coupling constant, $\omega=0$, are derived in the Schwarzschild line-element. The solutions are obtained from a cubic transition equation with one small parameter. The exterior space-time of one family of solutions is arbitrarily close to the exterior Schwarzschild space-time. These nontopological solitons have some similarity with soliton stars, and are proposed as candidates for {\em approximate black holes} for the use in numerical relativity, in particular for treatment of horizon boundary conditions.

A hypercontinuous hypersmooth Schwarzschild line element transformation

In this paper, a new derivation for one of the black hole line elements is given since the basic derivation for this line element is flawed mathematically. This derivation postulates a transformation procedure that utilizes a transformation function that is modeled by an ideal nonstandard physical world transformation process that yields a connection between an exterior Schwarzschild line element and distinctly different interior line element. The transformation is an ideal transformation in that in the natural world the transformation is conceived of as occurring at an unknown moment in the evolution of a gravitationally collapsing spherical body with radius greater than but near to the Schwarzsclfild radius. An ideal transformation models this transformation in a manner independent of the objects standard radius. It yields predicted behavior based upon a Newtonian gravitational field prior to the transformation, predicted behavior after the transformation for a field internal to the Schwarzschild surface and predicted behavior with respect to field alteration processes during the transformation.

The equivalence principle in the Schwarzschild geometry

Written in its standard form the Schwarzschild line element obscures the equivalence principle which is hidden in its geometrical structure. Through a coordinate transformation we re-express the line element in a form composed of terms of two different types: (1) terms that correspond to a line element in a uniformly accelerated reference frame in flat space–time, and (2) terms that correspond to curvature and are related to geodesic deviation. The dissection of the Schwarzschild line element into two parts, one corresponding to acceleration and the other to curvature, provides a good specific example of Einstein’s equivalence principle.

Einstein's evolution equations as a system of balance laws.

The evolution system in the space plus time (3+1) decomposition of Einstein's field equations is explicitly written as a system of balance laws. This is achieved by demanding that the time coordinate be harmonic (harmonic synchronization) and the space coordinate lines be normal to the constant-time hypersurfaces. No symmetry nor special form of the metric has been assumed, so that the equations may be used as a part of a three-dimensional numerical code for general relativity. The particular case of spherical symmetry is also considered and a numerical test of this case is provided by using a nonstandard form of the Schwarzschild line element.

Analytical derivation of the embedding of the Schwarzschild solution in a flat space-time

The embedding of the Schwarzschild line element in a flat space-time of six dimensions is univocally worked out on the basis of i) the static nature of the Schwarzschild field, ii) the Minkowski structure of the flat space-time and iii) the requirement that the embedding itself be analytic on the whole interval 0<r<∞. We arrive then at Fronsdal's embedding, which is therefore the sole possible complete embedding of the Schwarzschild solution in the six-dimensional Minkowski space-time.