In this study, we explore the realization of spherically symmetric solutions in the context of modified teleparallel gravity, particularly in <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='script'>T</mi><mo stretchy='false'>)</mo></math> theory, where <math altimg='si2.svg'><mi mathvariant='script'>T</mi></math> denotes the torsion scalar. Traditional mimetic gravity and its extension using two scalar fields fail to reproduce general spherically symmetric spacetimes within <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='script'>T</mi><mo stretchy='false'>)</mo></math> gravity due to the requirement of a constant torsion scalar or a linear form of <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='script'>T</mi><mo stretchy='false'>)</mo></math>, which restricts the theory to the teleparallel equivalent of general relativity (TEGR). To address this limitation, we propose a four-scalar field model that extends the two-scalar framework, allowing the construction of arbitrary spherically symmetric spacetimes. We show that this model eliminates ghost degrees of freedom through suitable constraints enforced by Lagrange multipliers. As a test of our procedure, we consider a specific form of a spherically symmetric spacetime and derive the associated four scalars, analyzing their behavior. Within the framework of a quadratic correction term, <math altimg='si3.svg'><mi mathvariant='script'>T</mi><mo linebreak='goodbreak' linebreakstyle='after'>+</mo><mfrac><mrow><mi>&#945;</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi mathvariant='script'>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math>, our method demonstrates the ability to reconstruct physically relevant solutions with non-ghost scalar fields. These results underscore the broader applicability of scalar field models in constructing viable geometries within modified gravity theories, extending beyond the scope of standard formulations.

Egypt

South Africa

G.G.L. Nashed

Saudi Arabia

A. Eid

Four-scalar model and spherically symmetric solution in <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='script'>T</mi><mo stretchy='false'>)</mo></math> theory

We introduce novel black hole (BH) solutions, charge/non-charge, within the framework of <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>)</mo></math> gravity, a theory that does not inherently include a cosmological constant, using equal/different metric ansatzs. Remarkably, these solutions exhibit asymptotically Anti-de Sitter (AdS) or de Sitter (dS) behavior, depending on their parameter values. Unlike the BTZ solutions of General Relativity, which feature a causal singularity and constant scalar invariants, our solutions display strong spacetime singularities, as shown by their scalar invariants. We construct <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>)</mo></math> functions that behave as polynomial functions, emphasizing the unique nature of these solutions. We demonstrate the stability of these solutions in two ways: first, by showing that their heat capacity is positive, which ensures thermodynamic stability; and second, by proving that the second derivative of <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>)</mo></math> is positive, meeting the Ostrogradski criterion for dynamical stability. Furthermore, the solutions satisfy the first law of thermodynamics, confirming their consistency with fundamental thermodynamic principles.

General form of <italic>f</italic>( <italic>R</italic>) and (2&#8239;+&#8239;1)-dimensional charge/non-charge black hole solutions

In this study, we introduce a toroidal solution for charged anti-de Sitter black holes in <italic>N</italic> dimensions within the framework of the quadratic form of <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='double-struck'>Q</mi><mo stretchy='false'>)</mo></math> gravity, employing the coincident gauge condition <cross-ref refid='br0010' id='crf0070'>[1]</cross-ref>. We assume <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='double-struck'>Q</mi><mo stretchy='false'>)</mo></math> to take the form <math altimg='si2.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='double-struck'>Q</mi><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mi mathvariant='double-struck'>Q</mi><mo linebreak='goodbreak' linebreakstyle='after'>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>&#945;</mi><msup><mrow><mi mathvariant='double-struck'>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo linebreak='goodbreak' linebreakstyle='after'>&#8722;</mo><mn>2</mn><mi mathvariant='normal'>&#923;</mi></math>, where <math altimg='si3.svg'><mi>N</mi><mo>&#8805;</mo><mn>4</mn></math>. These black hole solutions are characterized by flat or cylindrical horizons. A notable feature of these solutions is the presence of both electric monopole and quadrupole components in the potential field. These monopole and quadrupole components are inseparable and exhibit interconnected momenta, distinguishing them from the known charged solutions in the linear case of non-metricity theory. Furthermore, we demonstrate that the curvature singularities of these solutions are less severe than those in charged general relativity solutions. Finally, we calculate thermodynamic parameters, including entropy, Hawking temperature, and Gibbs free energy. These calculations confirm the stability of our model.

Special <italic>N</italic>-dimensional charged anti-de-Sitter black holes in <math altimg='si1.svg'><mi>f</mi><mo stretchy='false'>(</mo><mi mathvariant='double-struck'>Q</mi><mo stretchy='false'>)</mo></math> gravitational theory

We revisit slow rotating black hole (BH) solutions in Chern-Simons modified gravity (CSMG) by considering perturbative solution about Schwarzschild BH. In particular, the case when nondynamical CSMG with noncanonical CS scalar is considered. We provide a new solution different from the previously obtained one [1] which we refer to as KMT model. The present solution accounts for frame dragging effect which includes not only radial dependence as in the KMT. Nevertheless, it reduces to KMT as a particular case. We show that the tidal gravitational force (Kretschmann invariant) associated to the present solution contains a term of order <math><mn>1</mn><mo>/</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msup></math> additional to Schwarzschild but absent from directional divergence, unlike KMT model which diverges along the axis of symmetry. We derive the corresponding circular velocity of a massive test particle in which the KMT velocity is recovered in addition to an extra term ∝r. We investigate possible constraints on KMT and the present solutions from the observed rotation curve of UGC11455 galaxy as an example. We show that perturbation solutions cannot physically explain the flattening of galactic rotation curves.

Waleed El Hanafy

Mahmoud Hashim

Revisiting flat rotation curves in Chern-Simons modified gravity

This research aims to incorporate an axionic field within the context of dynamical Chern-Simons (CS) gravity and investigate the behavior of a slowly rotating black hole (BH) within this theoretical framework. To the best of our knowledge, this is the first instance where such a problem has been tackled within the context of (CS) gravitational theory coupled with the axionic field. The Lagrangian of this theory includes two CS scalar field functions, denoted as <math><mi>ω</mi><mo>(</mo><mi>ϑ</mi><mo>)</mo></math> and <math><mi>U</mi><mo>(</mo><mi>ϑ</mi><mo>)</mo></math>, where ϑ represents the CS scalar field. Our investigation involves calculating the equation related to the CS scalar field and demonstrating that deriving a solution necessitates imposing constraints on U and ω, specifically in the form of <math><msup><mrow><mi>U</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>ω</mi></math>, where <math><msup><mrow><mi>U</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≡</mo><mfrac><mrow><mi>d</mi><mi>U</mi><mo>(</mo><mi>ϑ</mi><mo>)</mo></mrow><mrow><mi>d</mi><mi>ϑ</mi></mrow></mfrac></math>. By utilizing the solution for the CS scalar field in other field equations, we reveal that the enigmatic function U can be expressed explicitly as <math><mi>U</mi><mo>=</mo><msup><mrow><mi>ϑ</mi></mrow><mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></math>, where p represents an integer with unrestricted values. In the instance where <math><mi>p</mi><mo>=</mo><mn>0</mn></math>, the CS Axion Einstein gravity theory [1] aligns with the study carried out in [2]. Furthermore, our analysis establishes that for positive values of p, the theory is capable of generating a significantly potent CS BH compared to the model derived in [2]. Moreover, our study delves into the geodesics of this BH, deriving its conserved quantities, orbital period, as well as its geodesic precession angle.

A new slow-rotating black hole solution in Chern-Simons axion of Einstein gravity

The slow rotation of Kerr-NUT spacetime is explored by taking into account the linear form of rotation and NUT parameters in the dynamical Chern-Simon gravity theory, which can be formulated from a scalar field describing the background. We show that in the absence of the potential scalar field, the metric potential does not respect the effect of the NUT parameter, although the scalar field is affected by the rotation and NUT parameters. As a result, unlike the slowly rotating black hole solution presented in [1], the cross term of the metric potential, which has both rotation and NUT parameters, has no contribution up to the leading order of the first order perturbation.

Japan

Kazuharu Bamba

Slow Kerr-NUT black hole solution in dynamical Chern-Simons modified gravity

Is Rastall's theory of gravity equivalent to Einstein's general relativity theory? This question has sparked a significant debate, prompting researchers to delve into the topic. To investigate further, we apply Rastall's theory's field equation to a spacetime characterized by spherical symmetry. This leads us to encounter a system of non-linear differential equations that is overdetermined. To address this, we make assumptions about the specific form of the metric potential's temporal component, denoted as <math><msub><mrow><mi>g</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub></math>. Additionally, we impose constraints to eliminate the anisotropic condition, resulting in a vanishing effect. These steps allow us to determine the form of <math><msub><mrow><mi>g</mi></mrow><mrow><mi>r</mi><mi>r</mi></mrow></msub></math> and ultimately achieve an isotropic spacetime. Furthermore, our investigation focuses on the potential of obtaining a set of parameters that align with the observed behavior of pulsars. To achieve this, we employ junction conditions to match the interior spacetime with the exterior Schwarzschild configuration, thereby constraining the model's relevant constants. Subsequently, we employ the pulsar <math><mi>S</mi><mi>A</mi><mi>X</mi><mi>J</mi><mn>1748.9</mn><mo>−</mo><mn>2021</mn></math>, characterized by a measured mass of <math><mi>M</mi><mo>=</mo><mn>1.81</mn><mo>±</mo><mn>0.31</mn><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mo>⊙</mo></mrow></msub></math> and a radius of <math><mi>R</mi><mo>=</mo><mn>11.7</mn><mo>±</mo><mn>1.7</mn></math> km, to numerically explore the physical properties of the model. Stability is assessed using the Tolman-Oppenheimer-Volkoff equation and the adiabatic index. Our findings suggest that Rastall's parameter, a key distinction of Rastall's theory from Einstein's general relativity, can play a crucial role in forming a realistic, compact object consistent with observational data. Furthermore, we verify the model's validity by comparing it with various observed masses and radii of different pulsars, ensuring a satisfactory fit between the model proposed in this study and the observed data.

Isotropic compact stellar model in Rastall's gravitational theory

Recently, the mimetic gravitational theory has gained much attention in the frame of cosmology as well as in the domain of astrophysics. In this study, we show that in the frame of mimetic gravitation theory we are not able to derive an isotropic model. Therefore, we turn our attention to mimetic gravitational theory coupled with the Lagrangian multiplier. The field equations of a static isotropic gravitational system that controls the geometry and dynamics of star structure are studied in the frame of mimetic theory coupled with a Lagrangian multiplier using a non-linear equation of state. An energy density is assumed from where all the other unknowns are fixed and a new isotropic model is derived. The physical analysis of this model is studied from different viewpoints and consistent results compatible with a realistic isotropic star are investigated analytically and graphically. Finally, we show that the model under consideration is stable using the adiabatic index procedure.

The key role of Lagrangian multiplier in mimetic gravitational theory in the frame of isotropic compact star

We show that the mimetic theory with the constraint <math><msup><mrow><mi>g</mi></mrow><mrow><mi>ρ</mi><mi>σ</mi></mrow></msup><msub><mrow><mo>∂</mo></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ϕ</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>σ</mi></mrow></msub><mi>ϕ</mi><mo>=</mo><mn>1</mn></math> cannot realize the black hole geometry with the horizon(s). To overcome such issue, we may change the mimetic constraint a little bit by <math><mi>ω</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>ρ</mi><mi>σ</mi></mrow></msup><msub><mrow><mo>∂</mo></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ϕ</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>σ</mi></mrow></msub><mi>ϕ</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, where <math><mi>ω</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></math> is a function of the scalar field ϕ. As an example, we consider <math><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></math>-dimensional mimetic gravity with the mimetic potential and construct black hole (BH) solutions by using this modified constraint. We study three different classes: In the first class, we assume the Lagrange multiplier and mimetic potential are vanishing and obtain a BH solution that fully matches the BH of GR despite the non-triviality of the mimetic field which ensures the study presented in Nashed et al. (2019) [42]. In the second class, we obtain a BH having constant mimetic potential and a non-trivial form of the Lagrange multiplier. In the third class, we obtain a new BH solution with non-vanishing values of the mimetic field, the Lagrange multiplier, and the mimetic potential. In any case, the solutions correspond to the space-time with only one horizon but we show that the formalism for the constraint works.

Shin'ichi Nojiri

Consistency between black hole and mimetic gravity – Case of (2 + 1)-dimensional gravity

In this study, we address the issue of a spherically symmetrical interior solution to the quadratic form of <math><mi>f</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>=</mo><mi>T</mi><mo>+</mo><mi>ϵ</mi><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math> gravitational theory using a physical tetrad that provides vanishing components of the off-components of the field equation, in contrast to what exists in the current literature. To be able to formulate the resulting differential equation in a closed form, we employ the Krori-Barua (KB) ansatz. Using the KB spacetime form, we derive the analytic form of the energy-density, radial, and tangential pressures and the anisotropic form. All of these quantities are affected by the dimensional parameter ϵ, which causes them to have a noted difference from those given in the frame of Einstein general relativity. The derived model of this study exhibits a non-trivial form of torsion scalar, and it also contains three constants that we drew from the matching of the boundary condition with a line element that also features a non-trivial form of torsion scalar. Having established the physical conditions that are needed for any real stellar, we check our model and show in detail that it bypasses all of these. Finally, we analyze the model's stability utilizing the Tolman-Oppenheimer-Volkoff equation and adiabatic index and show that our model satisfies these.