Higher dimensional black hole solutions with scalar hair/dilaton field and Toroidal horizons and the interior solution

In this paper we investigate the black hole solutions with toroidal horizons in scalar hair/dilaton gravity. First, we obtain the field equations in n-dimensions, then we propose some different models(Ansatz) and find the exact solutions for these type of ansatzs. These solutions are not asymptotically (anti-)de Sitter or flat, except in one special case. We also show that the BTZ and BTZ-like solutions will emerge from some of these solutions as a special case. We also show that when the event horizon radius gets bigger and bigger, the temperature will be the same in various dimensions. The only difference is noticeable near the origin(this statement is clear in diagrams). For these solutions, we obtained a new version of the Smarr formula as well. Also, we show that the presence of the scalar field makes the black holes to be more stable near the origin except for the BTZ case. We can say in general that the presence of scalar field is an important factor in black hole’s stability investigations. In the critical behavior analysis we find that there is no evidence to show the existence of P-V criticality. We present here a class of interior solutions corresponding to the solution in scalar hair gravity exterior. The solution which is obtained in linear case is regular and well-behaved at the stellar interior.

Minimally deformed anisotropic stars by gravitational decoupling in Einstein–Gauss–Bonnet gravity

In this article, we develop a theoretical framework to study compact stars in Einstein gravity with the Gauss–Bonnet (GB) combination of quadratic curvature terms. We mainly analyzed the dependence of the physical properties of these compact stars on the Gauss–Bonnet coupling strength. This work is motivated by the relations that appear in the framework of the minimal geometric deformation approach to gravitational decoupling (MGD-decoupling), we establish an exact anisotropic version of the interior solution in Einstein–Gauss–Bonnet gravity. In fact, we specify a particular form for gravitational potentials in the MGD approach that helps us to determine the decoupling sector completely and ensure regularity in interior space-time. The interior solutions have been (smoothly) joined with the Boulware–Deser exterior solution for 5D space-time. In particular, two different solutions have been reported which comply with the physically acceptable criteria: one is the mimic constraint for the pressure and the other approach is the mimic constraint for density. We present our solution both analytically and graphically in detail.

Modification of the Exterior and Interior Solution of Einstein’s G22 Field Equation for a Homogeneous Spherical Massive Bodies whose Fields Differ in Radial Size, Polar Angle, and Time.

In general theory of relativity, Einstein’s field equations relate the geometry of space-time with the distribution of matter within it. These equations were first published by Einstein in the form of a tensor equation which related the local space-time curvature with the local energy and momentum within this space-time. In this article, Einstein’s geometrical field equations interior and exterior to astrophysically real or hypothetical distribution of mass within a spherical geometry were constructed and solved for field whose gravitational potential varies with time, radial distance and polar angle. The exterior solution was obtained using power series. The metric tensors and the solution of the Einstein’s exterior field equations used in this work has only one arbitrary function f(t,r,θ) , and thus put the Einstein’s geometrical theory of gravitation on the same bases with the Newton’s dynamical theory of gravitation. The gravitational scalar potential f(t,r,θ) obtained in this research work to the order of co, c-2 , contains Newton dynamical gravitational scalar potential and post Newtonian additional terms much importance as it can be applied to the study of rotating bodies such as stars. The interior solution was obtained using weak field and slow-motion approximation. The obtained result converges to Newton’s dynamical scalar potential with additional time factor not found in the well-known Newton’s dynamical theory of gravitation which is a profound discovery with the dependency on three arbitrary functions. Our result obeyed the equivalence principle of Physics.

Anisotropic compact stars in f(R) gravity

We derive a new interior solution for stellar compact objects in $$f\mathcal {(R)}$$ f ( R ) gravity assuming a differential relation to constrain the Ricci curvature scalar. To this aim, we consider specific forms for the radial component of the metric and the first derivative of $$f\mathcal {(R)}$$ f ( R ) . After, the time component of the metric potential and the form of $$f({\mathcal {R}})$$ f ( R ) function are derived. From these results, it is possible to obtain the radial and tangential components of pressure and the density. The resulting interior solution represents a physically motivated anisotropic neutron star model. It is possible to match it with a boundary exterior solution. From this matching, the components of metric potentials can be rewritten in terms of a compactness parameter C which has to be $$C=2GM/Rc^2<<0.5$$ C = 2 G M / R c 2 < < 0.5 for physical consistency. Other physical conditions for real stellar objects are taken into account according to the solution. We show that the model accurately bypasses conditions like the finiteness of radial and tangential pressures, and energy density at the center of the star, the positivity of these components through the stellar structure, and the negativity of the gradients. These conditions are satisfied if the energy-conditions hold. Moreover, we study the stability of the model by showing that Tolman–Oppenheimer–Volkoff equation is at hydrostatic equilibrium. The solution is matched with observational data of millisecond pulsars with a withe dwarf companion and pulsars presenting thermonuclear bursts.

The structure of black holes

Contrary to the general belief that a black hole has only an event horizon and a singularity it is shown that it has a structure. The structure of a black hole is worked out using Schwarzschild interior solution. It is shown that there is an attraction towards the centre of the distribution when the radius r of the sphere is greater than 9m /4 where m= GM/c2 , M being the mass of the distribution of matter. When the radius is between 9m /4 and 2m the attraction is replaced by a repulsion first at the centre and then in a core gradually increasing in radius up to 2m. The matter is repelled towards the surface ending up with a spherical shell of radius 2m and of infinite density, forming a black hole.

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.

On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density ρ and pressure field p(r) located in a ball r≤r0. We find a 1-parameter family of time-independent and radially symmetric solutions ga,ρa,pa:−2m<a<a1 satisfying the boundary conditions g=gS and p=0 on r=r0, where gS is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at r=0) and containing (for a=0) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement r0>9κM/(4c2) identifies the “physical” (i.e., such that pa(r)≥0 and pa(r) is bounded in 0≤r≤r0) solutions {pa:a∈U0} for some neighbourhood U0⊂(−2m,+∞) of a=0. For every star model {ga:a0<a<a1}, we compute the volume V(a) of the region r≤r0 in terms of abelian integrals of the first, second, and third kind in Legendre form.

Nonsingular black holes from charged dust collapse: A concrete mechanism to evade interior singularities in general relativity

In this paper, we examine the gravitational collapse of a nonrelativistic charged perfect fluid interacting with a dark energy component. Given a simple factor for the energy transfer, we obtain a nonsingular interior solution which naturally matches the Reissner–Nordström–de Sitter exterior geometry. We also show that the interacting parameter is proportional to the overall charge of the final black hole thus formed. For the case of quasi-extremal configurations, we propose a statistical model for the entropy of the collapsed matter. This entropy extends Bekenstein’s geometrical entropy by an additive constant proportional to the area of the extremal black hole.

Collapse in f(R) gravity and the method of R matching

Collapsing solutions in f(R) gravity are restricted due to junction conditions that demand continuity of the Ricci scalar and its normal derivative across the time-like collapsing hypersurface. These are obtained via the method of R-matching, which is ubiquitous in f(R) collapse scenarios. In this paper, we study spherically symmetric collapse with the modification term $$\alpha R^2$$ α R 2 , and use R-matching to exemplify a class of new solutions. After discussing some mathematical preliminaries by which we obtain an algebraic relation between the shear and the anisotropy in these theories, we consider two metric ansatzes. In the first, the collapsing metric is considered to be a separable function of the co-moving radius and time, and the collapse is shear-free, and in the second, a non-separable interior solution is considered, that represents gravitational collapse with non-zero shear viscosity. We arrive at novel solutions that indicate the formation of black holes or locally naked singularities, while obeying all the necessary energy conditions. The separable case allows for a simple analytic expression of the energy-momentum tensor, that indicates the positivity of the pressures throughout collapse, and is further used to study the heat flux evolution of the collapsing matter, whose analytic solutions are presented under certain approximations. These clearly highlight the role of modified gravity in the examples that we consider.