Solutions Of General Relativity Research Articles

Generating 4-dimensional wormholes with Yang–Mills Casimir sources

This work presents a new static and spherically symmetric traversable wormhole solution in General Relativity, which is supported by the quantum vacuum fluctuations associated with the Casimir effect of the Yang–Mills field confined between perfect chromometallic mirrors in (3+1) dimensions, recently fitted using first-principle numerical simulations. Initially, we employ a perturbative approach for x=mr≪1, where m represents the Casimir mass and r is the radial coordinate. This approach has proven to be a reasonable approximation when compared with the exact case in this regime. To find well-behaved redshift functions, we impose constraints on the free parameters. As expected, this solution recovers the electromagnetic-like Casimir solution for m=0. Analyzing the traversability conditions, we graphically find that all are satisfied for 0≤m≤0.17. On the other hand, all the energy conditions are violated, as usual in this context due to the quantum origin of the source. Stability from Tolman–Oppenheimer–Volkov (TOV) equation is guaranteed for all r and from the speed of sound for 0.16≤m≤0.18. Therefore, for 0.16≤m≤0.17, we will have a stable solution that satisfies all traversability conditions.

On black bounce space-times in non-linear electrodynamics

One of the main issues in gravitation is the presence of singularities in the most common space-time solutions of General Relativity, as the case of black holes. A way of constructing regular solutions that remove spacelike singularities consists in implement a bounce on such space-time, leading to what is usually known as black bounce space-times. Such space-times are known to describe regular black holes or traversable wormholes. However, one of the main issues lies on reconstructing the appropriate source that leads to such a solution. In this paper, a reconstruction method is implemented to show that such types of metrics can be well accommodated in non-linear electrodynamics with the presence of a scalar field. Some of the most important black bounces solutions are reconstructed in this framework, both in 3 + 1 as in 2 + 1 dimensions. For the first time in the literature, these solutions have an electrically charged source of matter from non-linear electrodynamics. Specific features are indicated that distinguish electric sources from magnetic ones, previously found for the same space-times.

Higher-spin self-dual General Relativity: 6d and 4d pictures, covariant vs. lightcone

We study the higher-spin extension of self-dual General Relativity (GR) with cosmological constant, proposed by Krasnov, Skvortsov and Tran. We show that this theory is actually a gauge-fixing of a 6d diffeomorphism-invariant Abelian theory, living on (4d spacetime)×(2d spinor space) modulo a finite group. On the other hand, we point out that the theory respects the 4d geometry of a self-dual GR solution, with no backreaction from the higher-spin fields. We also present a lightcone ansatz that reduces the covariant fields to one scalar field for each helicity. The field equations governing these scalars have only cubic vertices. We compare our lightcone ansatz to Metsaev’s lightcone formalism. We conclude with a new perspective on the lightcone formalism in (A)dS spacetime: not merely a complication of its Minkowski-space cousin, it has a built-in Lorentz covariance, and is closely related to Vasiliev’s concept of unfolding.

Quadratic perturbations of the Schwarzschild black hole: the algebraically special sector

We investigate quadratic algebraically special perturbations (ASPs) of the Schwarzschild black hole. Their dynamics are derived from the expansion up to second order in perturbation of the most general algebraically special twisting vacuum solution of general relativity. Following this strategy, we present analytical expressions for the axial-axial, polar-polar and polar-axial source terms entering in the dynamical equations. We show that these complicated inhomogeneous equations can be solved analytically and we present explicit expressions for the profiles of the quadratic ASPs. As expected, they exhibit exponential growth both at the past and future horizons even in the non-linear regime. We further use this result to analyze the quadratic zero modes and their interpretation in terms of quadratic corrections to mass and spin of the Schwarzschild black hole. The present work provides a direct extension beyond the linear regime of the original work by Couch and Newman.

Self-similar collapse in Painlevé–Gullstrand coordinates

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space: (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve–Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.

Tetrad formalism in the solution of spherically symmetric spacetime in general relativity

Abstract Spherically symmetric solutions in general relativity are the most fundamental solutions to the Einstein field equation. The first exact solution of the Einstein field equation is the spherically symmetric solution given by the Schwarzschild metric, as easily found in any standard textbook on general relativity. The FLRW (Friedmann-Lemaitre-Robertson-Walkers) metric is another spherically symmetric solution of Einstein’s equation describing the standard model in Cosmology. The standard approach to solving Einstein’s equations is by considering the metric. However, we can also adopt a tetrad-based method or tetrad formalism. We review these two solutions by the tetrad formalism as an alternative approach. In addition, we give some more cases, including the cosmological constant and the Taub-NUT metric.

Testing disformal non-circular deformation of Kerr black holes with LISA

There is strong observational evidence that almost every large galaxy has a supermassive black hole at its center. It is of fundamental importance to know whether such black holes are described by the standard Kerr solution in General Relativity (GR) or by another black hole solution. An interesting alternative is the so-called disformal Kerr black holes which exist within the framework of degenerate higher-order scalar-tensor (DHOST) theories of gravity. The departure from the standard Kerr black hole spacetime is parametrized by a parameter D, called disformal parameter. In the present work, we discuss the capability of LISA to detect the disformal parameter. For this purpose, we study Extreme Mass Ratio Inspirals (EMRI's) around disformal Kerr black holes within the framework of the quadrupole hybrid formalism. Even when the disformal parameter is very small, its effect on the globally accumulated phase of the gravitational waveform of an EMRI can be significant due to the large number of cycles in the LISA band made by the small compact object. We show that LISA will in principle be able to detect and measure extremely small values of the disformal parameter which in turn, can be seen as an assessment of LISA's ability to detect very small deviations from the Kerr geometry.

Classical observables using exponentiated spin factors: electromagnetic scattering

In [1], the authors argued that the Newman-Janis algorithm on the space of classical solutions in general relativity and electromagnetism could be used in the space of scattering amplitudes to map an amplitude with external scalar states to an amplitude associated to the scattering of “infinite spin particles”. The minimal coupling of these particles to the gravitational or Maxwell field is equivalent to the classical coupling of the Kerr blackhole with linearized gravity or the so-called Kerr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{\ extrm{Kerr}} $$\\end{document} charged state with the electromagnetic field. The action of the Newman-Janis mapping on scattering amplitudes was then used to compute the linear impulse at first post-Minkowskian (1PM) order, via the Kosower, Maybee, O’Connell (KMOC) formalism. In this paper, we continue with the idea of using the Newman-Janis mapping on the space of scalar QED amplitudes to compute classical observables such as the radiative gauge field and the angular impulse. We show that for tree-level amplitudes, the Newman-Janis action can be reinterpreted as a dressing of the photon propagator. This turns out to be an efficient way to compute these classical observables. Along the way, we highlight a subtlety that arises in proving the conservation of angular momentum for scalar −Kerr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\sqrt{\ extrm{Kerr}} $$\\end{document} scattering.

Nonlinear gravitational waves in Horndeski gravity: scalar pulse and memories

We present and analyze a new non-perturbative radiative solution of Horndeski gravity. This exact solution is constructed by a disformal mapping of a seed solution of the shift-symmetric Einstein-Scalar system belonging to the Robinson-Trautman geometry describing the gravitational radiation emitted by a time-dependent scalar monopole. After analyzing in detail the properties of the seed, we show that while the general relativity solution allows for shear-free parallel transported null frames, the disformed solution can only admit parallel transported null frames with a non-vanishing shear. This result shows that, at the nonlinear level, the scalar-tensor mixing descending from the higher-order terms in Horndeski dynamics can generate shear out of a pure scalar monopole. We further confirm this analysis by identifying the spin-0 and spin-2 polarizations in the disformed solution using the Penrose limit of our radiative solution. Finally, we compute the geodesic motion and the memory effects experienced by two null test particles with vanishing initial relative velocity after the passage of the pulse. This exact radiative solution offers a simple framework to witness nonlinear consequences of the scalar-tensor mixing in higher-order scalar-tensor theories.

Constant velocity physical warp drive solution

Abstract Warp drives are exotic solutions of general relativity that offer novel means of transportation. In this study, we present a solution for a constant-velocity subluminal warp drive that satisfies all of the energy conditions. The solution involves combining a stable matter shell with a shift vector distribution that closely matches well-known warp drive solutions such as the Alcubierre metric. We generate the spacetime metric numerically, evaluate the energy conditions, and confirm that the shift vector distribution cannot be reduced to a coordinate transformation. This study demonstrates that classic warp drive spacetimes can be made to satisfy the energy conditions by adding a regular matter shell with a positive ADM mass.

Exact Solution for Rotating Black Holes in Parity-Violating Gravity

Abstract It has recently been pointed out that one can construct invertible conformal transformations with a parity-violating conformal factor, which can be employed to generate a novel class of parity-violating ghost-free metric theories from general relativity. We obtain exact solutions for rotating black holes in such theories by performing the conformal transformation on the Kerr solution in general relativity, which we dub conformal Kerr solutions. We explore the geodesic motion of a test particle in the conformal Kerr spacetime. While null geodesics remain the same as those in the Kerr spacetime, timelike geodesics exhibit interesting differences due to an effective external force caused by the parity-violating conformal factor.

Testing beyond-Kerr spacetimes with GWTC-3

The Kerr spacetime is a fundamental solution of general relativity (GR), describing the gravitational field around a rotating, uncharged black hole (BH). Kerr spacetime has been crucial in modern astrophysics and it serves as a foundation for the study of gravitational waves (GWs). Possible deviations in Kerr geometry may indicate deviations from GR predictions. In this work, we consider the Johannsen–Psaltis metric, which is a beyond-Kerr metric characterized by a single free parameter, and then we probe this theory framework using several GWs observations from the third Gravitational-wave Transient Catalog (GWTC-3). We find that, for most of the events analyzed, there are no significant deviations from the null hypothesis, i.e. the Kerr metric. Our main findings demonstrate alignment and certain enhancements when compared to previous estimates documented in the literature.

Spherically symmetric vacuum solutions in one-parameter new general relativity and their phenomenology

Spherically symmetric vacuum solutions in one-parameter new general relativity and their phenomenology

The weak gravity conjecture, overcharged shells and gravitational traps

The Weak Gravity Conjecture (WGC) predicts that in quantum gravity there should exist overcharged states, that is states with charge larger than their mass. Extending this to large masses and charges, we are expecting similar overcharged classical solutions. This has been demonstrated in higher-derivative extensions of General Relativity. In this paper we investigate the existence of overcharged solutions in General Relativity. We study the dynamics of a thin shell of mass m and charge Q under the action of its own gravitational and U(1) fields. We show that shells with surface energy σ and pressure P obeying P=wσ with 0⩽w⩽1 are necessarily undercharged m⩾|Q| and always collapse to form Reissner–Nordström black holes. Nevertheless, if −1⩽w<0 , we find that overcharged m⩽|Q| shells exist, which however, are inevitably stabilized at finite radial distance. Therefore they never form naked singularities in accordance with cosmic censorship and the conjectured relation between cosmic censorship and the WGC. An intriguing consequence of the existence of such overcharged shells is that gravitational modes may form bound states due to the peculiar form of the Regge–Wheeler–Zerilli potential. This might lead to gravitational traps close the surface of near-overcharged shells.

The equivalence principle for a plane gravitational wave in torsion-based and non-metricity-based teleparallel equivalents of general relativity

We study the energy–momentum characteristics of the plane “+”-polarized gravitational wave solution of general relativity in the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR) using the previously constructed Noether currents. The current components describe energy–momentum locally measured by an observer if the displacement vector ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} is equal to the observer’s 4-velocity. To determine the non-dynamical connection in these theories, we use the unified “turning off” gravity principle. For a constructive analysis of the values of Noether currents and superpotentials in TEGR and STEGR, we use the concept of “gauges”. The gauge changing can affect the Noether current values. We study under what conditions the Noether current for the freely falling observer is zero. When they are established, the zero result can be interpreted as a correspondence to the equivalence principle, and it is a novelty for gravitational waves in the TEGR and STEGR. We highlight two important cases with positive and zero energy, which reproduce the results of previous works with a different approach for determining gravitational energy–momentum in the TEGR, and give their interpretation.

Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and F(R,T,Q,Tm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(R,T,Q,T_{m})$$\\end{document} gravity

We extend the anholonomic frame and connection deformation method, AFCDM, for constructing exact and parametric solutions in general relativity, GR, to geometric flow models and modified gravity theories, MGTs, with nontrivial torsion and nonmetricity fields. Following abstract geometric or variational methods, we can derive corresponding systems of nonmetric gravitational and matter field equations which consist of very sophisticate systems of coupled nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and applying the AFCDM, we prove that such systems of PDEs can be decoupled and integrated in general forms for generic off-diagonal metric structures and generalized affine connections. We generate new classes of quasi-stationary solutions (which do not depend on time like coordinates) and study the physical properties of some physically important examples. Such exact or parametric solutions are determined by nonmetric solitonic distributions and/or ellipsoidal deformations of wormhole hole configurations. It is not possible to describe the thermodynamic properties of such solutions in the framework of the Bekenstein–Hawking paradigm because such metrics do not involve, in general, certain horizons, duality, or holographic configurations. Nevertheless, we can always elaborate on associated Grigori Perelman thermodynamic models elaborated for nonmetric geometric flows. In explicit form, applying the AFCDM, we construct and study the physical implications of new classes of traversable wormhole solutions describing solitonic deformation and dissipation of non-Riemannian geometric objects. Such models with nontrivial gravitational off-diagonal vacuum are important for elaborating models of dark energy and dark matter involving wormhole configurations and solitonic-type structure formation.

Static spherically symmetric solutions in new general relativity

We give a pedagogical introduction to static spherically symmetric solutions in models of New GR, both explaining the basics and showing how all such vacuum solutions can be obtained in elementary functions. In doing so, we coherently introduce the full landscape of these modified teleparallel spacetimes, and find a few special cases. The equations of motion are turned into a very nice and compact form by using the Levi-Civita divergence of the torsion-conjugate; and generalised Bianchi identities are briefly discussed. Another important point we make is that a convenient choice of the radial variable might be instrumental for success of similar studies in other modified gravity models.

Global phase space analysis for a class of single scalar field bouncing solutions in general relativity

We carry out a compact phase space analysis of a non-canonical scalar field theory whose Lagrangian is of the form F(X)-V(ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X)-V(\\phi )$$\\end{document} within general relativity. In particular, we focus on a kinetic term of the form F(X)=βXm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X)=\\beta X^m$$\\end{document} (m≠1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\ e 1/2$$\\end{document}) with power-law potential V0ϕn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0 \\phi ^n$$\\end{document} and exponential potential V0e-λϕ/MPl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0 e^{-\\lambda \\phi /M_{Pl}}$$\\end{document} of the scalar field. The Cuscuton case m=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=1/2$$\\end{document} where the scalar field is non-dynamical is left out of consideration. The main aim of this work is to investigate the genericity of nonsingular bounce in these models and to investigate the cosmic future of the bouncing cosmologies when they are generic. A global dynamical system formulation that is particularly suitable for investigating nonsingular bouncing cosmologies is used to carry out the analysis. We show that when F(X)=βXm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X)=\\beta X^m$$\\end{document} (β<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta <0$$\\end{document}), nonsingular bounce is generic for a power law potential V(ϕ)=V0ϕn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\phi ) = V_0 \\phi ^n$$\\end{document} only within the parameter range 12<m<1,n<2mm-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ \\frac{1}{2}<m<1,\\,n<\\frac{2\\,m}{m-1}\\right\\} $$\\end{document} and for an exponential potential V(ϕ)=V0e-λϕ/MPl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\phi ) = V_0 e^{-\\lambda \\phi /M_{Pl}}$$\\end{document} only within the parameter range 12<m≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ \\frac{1}{2}<m\\le 1\\right\\} $$\\end{document}. Except in these cases, nonsingular bounce in these models is not generic due to the non-existence of global past or future attractors. Our analysis serves to show the importance of a global phase space analysis to address important questions about nonsingular bouncing solutions, an idea that may and must be adopted for such solutions even in other theories.

Gravitational Light Bending in Weyl Gravity and Schwarzschild–de Sitter Spacetime

The topic of gravitational lensing in the Mannheim–Kazanas solution of Weyl conformal gravity and the Schwarzschild–de Sitter solution in general relativity has featured in numerous publications. These two solutions represent a spherical massive object (lens) embedded in a cosmological background. In both cases, the interest lies in the possible effect of the background non-asymptotically flat spacetime on the geometry of the local light curves, particularly the observed deflection angle of light near the massive object. The main discussion involves possible contributions to the bending angle formula from the cosmological constant Λ in the Schwarzschild–de Sitter solution and the linear term γr in the Mannheim–Kazanas metric. These effects from the background geometry, and whether they are significant enough to be important for gravitational lensing, seem to depend on the methodology used to calculate the bending angle. In this paper, we review these techniques and comment on some of the obtained results, particularly those cases that contain unphysical terms in the bending angle formula.

Extension of the Schwarzschild black hole solution in f(R) gravitational theory and its physical properties

The successes of f(R) gravitational theory as a logical extension of Einstein’s theory of general relativity (GR) encourage us to delve deep into this theory and continue our study to attempt to derive an extension of the Schwarzschild black hole (BH) solution. In this study, in order to solve the output nonlinear differential equation, we closed the form of the system by assuming the derivative of f(R) with respect to the scalar curvature R to have the form F(r)=df(R(r))dR(r)=1-αr4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(r)=\\frac{\ extrm{d}f(R(r))}{\ extrm{d}R(r)}=1- \\frac{\\alpha }{r^4}$$\\end{document}, where 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} is a dimensional constant. Our study shows that 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}, we obtain the Schwarzschild BH solution of GR assuming some constraints on the constant of integration, and if these constraints are bounded, we obtain the anti-de Sitter (AdS)/de Sitter (dS) spacetime. For the general case, i.e., 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}$$\\alpha \ e 0$$\\end{document}, we obtain a BH solution that tends asymptotically to AdS/dS spacetime. Moreover, we derive the timelike and null particle geodesics of the BH solution studied in this article. The equation of motion and effective potential of test particles are calculated to study the stability of radial orbits (trajectories). The energy and angular momentum are calculated to study the circular motion and stability of circular orbits. We also derive the stability condition using the geodesic deviation. Moreover, we discuss the physics of the output BH solutions through calculation of the thermodynamic quantities including entropy, the Hawking temperature, and Gibbs free energy. Finally, we check the validity of the first law of thermodynamics applied to the BH of this study. Although we can derive a Schwarzschild black hole solution in the lower order of f(R), specifically when f(R)=R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R)=R$$\\end{document}, where the gravitational mass is generated from the source of gravity, we demonstrate that in the higher orders of f(R), when f(R)≠R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R)\ e R$$\\end{document}, the source of gravity is attributed primarily to higher-order corrections, and the source of gravity that was originally derived from the Schwarzschild black hole has ceased to be dominant.