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# Exact Einstein-Gauss-Bonnet spacetime in six dimensions

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## Abstract

A new exact solution to the field equations in the Einstein-Gauss-Bonnet modified theory of gravity, for a six-dimensional spherically symmetric static distribution of a perfect fluid source is presented. The pressure isotropy equation is integrated after a form for the temporal potential proportional to the radius is postulated to close the system of equations. For a specific choice of the coupling parameter it is demonstrated that the matching of the interior and exterior spacetimes is explicitly achievable. The general model has been tested to be physically acceptable in this framework using criteria extrapolated from the standard four dimensional theory and after locating a suitable parameter space through fine-tuning. A vanishing pressure hypersurface signifying a boundary exists and the speed of sound is subluminal throughout the interior of the matter distribution. Furthermore, all energy conditions are satisfied. Finally, the Chandrasekhar adiabatic stability bound is satisfied.

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... For a study of the physical features in stellar models it is necessary to find exact solutions to the EGB field equations. Particular classes of exact solutions in static metrics have been found mainly in five and six spacetime dimensions [30][31][32][33][34][35] for neutral matter distributions with isotropic pressure. Other interesting models have been studied by [36][37][38][39]. ...
... Hence the dimension N has a dramatic effect on the gravitational behaviour of the model for N ≥ 6. Note that the condition of pressure isotropy for N = 5 was also presented by Hansraj et al. [31], and by Hansraj and Mkhize [33] when N = 6. Our result (11) holds for all dimensions N. Our intention is to find exact solutions for all N. ...
... This choice was also made by Hansraj and Mkhize [33] for the particular spacetime dimension N = 6. The integral in (16) can be evaluated and we obtain ...
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We generate the Einstein–Gauss–Bonnet field equations in higher dimensions for a spherically symmetric static spacetime. The matter distribution is a neutral fluid with isotropic pressure. The condition of isotropic pressure, an Abel differential equation of the second kind, is transformed to a first order nonlinear canonical differential equation. This provides a mechanism to generate exact solutions systematically in higher dimensions. Our solution generating algorithm is a different approach from those considered earlier. We show that a specific choice of one potential leads to a new solution for the second potential for all spacetime dimensions. Several other families of exact solutions to the condition of pressure isotropy are found for all spacetime dimensions. Earlier results are regained from our treatments. The difference with general relativity is highlighted in our study.
... Tangphati et al. [25] have also found out that an anisotropic quark star in the context of EGB gravity leads to considerable change both in the structure of the star and the mass-radius relation. Hansraj and Mkhize [26] recently obtained exact solutions in EGB gravity in a six dimensional fluid sphere and used it to construct stellar models, using barotropic fluid in a higher dimensional Krori-Barua metric. However, for a given equation of state (EoS) the metric solutions can be obtained from the gravitational field equations. ...
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We present a new class of relativistic anisotropic stellar models with spherically symmetric matter distribution in Einstein Gauss–Bonnet (EGB) gravity. A higher dimensional Finch–Skea geometry in the theory is taken up here to construct stellar models in hydrostatic equilibrium. The Gauss–Bonnet term is playing an important role in accommodating neutron stars. We study the physical features namely, the energy density, the radial and tangential pressures and the suitability of the models. It is found that the equation of state of such stars are non-linear which is determined for a given mass and radius of known stars. The stability of the stellar models are also explored for a wide range of values of the model parameters.
... A static charged anisotropic fluid sphere described by Krori-Barua metric is also studied [29] considering a coupled Einstein-Maxwell-Gauss-Bonnet field equations with a linear equation of state (EoS) different from MIT Bag model. Recently, Hansraj and Mkhize [30] obtained exact solutions for constructing stellar models in the Einstein-Gauss-Bonnet gravity in a six-dimensional spacetime. Considering Finch-Skea geometry, a new class of interior solutions of compact objects in five dimensional Einstein Gauss-Bonnet (EGB) gravity is obtained [31] with a linear equation of state (EoS) which permits compact objects. ...
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We obtain a class of new anisotropic relativistic solution in Einstein Gauss-Bonnet (EGB) gravity with Finch-Skea metric in hydrostatic equilibrium. The relativistic solutions are employed to construct anisotropic stellar models for strange star with MIT Bag equation of state $p_{r}= \frac{1}{3} \left( \rho - 4 B_{g}\right)$, where $B_{g}$ is the Bag constants. Considering the mass and radius of a known star PSR J0348+0432 we construct stellar models in the framework of higher dimensions. We also predict the mass and radius of stars for different model parameters. The Gauss-Bonnet coupling term ($\alpha$) plays an important role in determining the density, pressure, anisotropy profiles and other features. The stability of the stellar models are probed analyzing the different energy conditions, variation of sound speed and adiabatic stability conditions inside the star. The central density and pressure of a star in EGB gravity are found to have higher values compared to that one obtains in Einstein gravity ($\alpha =0$). We also explore the effect of extra dimensions for the physical features of a compact object. For this we consider $D=5$ and $D=6$ to obtain a realistic stellar model and found that in the formal case both positive and negative values of $\alpha$ are allowed. But in the later case, only $\alpha <0$ permits compact object in the Finch-Skea metric. We determine the best fit values of the model parameters for a number of observed stars for acceptable stellar models.
... A static charged anisotropic fluid sphere described by Krori-Barua metric is also studied [28] considering a coupled Einstein-Maxwell-Gauss-Bonnet field equations with a linear equation of state (EoS) different from MIT Bag model. Recently, Hansraj and Mkhize [29] obtained exact solutions for constructing stellar models in the Einstein-Gauss-Bonnet gravity in a six-dimensional spacetime. Considering Finch-Skea geometry, a new class of interior solutions of compact objects in five dimensional Einstein Gauss-Bonnet (EGB) gravity is obtained [30] with a linear equation of state (EoS) which permits compact objects. ...
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... It is seriously difficult to locate exact solutions for perfect fluid matter in EGB because the extra curvature terms make the governing differential equations intractable. An additional solution for constant potentials in six dimensional EGB spacetimes was found in [27] and recently Hansraj and Mkhize generated a physically viable six dimensional model with variable potentials and density [28]. A greater number of the extra curvature terms survive in 6D as opposed to 5D making the differential equations even more difficult to work with. ...
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Preface; List of tables; Notation; 1. Introduction; Part I. General Methods: 2. Differential geometry without a metric; 3. Some topics in Riemannian geometry; 4. The Petrov classification; 5. Classification of the Ricci tensor and the energy-movement tensor; 6. Vector fields; 7. The Newman-Penrose and related formalisms; 8. Continuous groups of transformations; isometry and homothety groups; 9. Invariants and the characterization of geometrics; 10. Generation techniques; Part II. Solutions with Groups of Motions: 11. Classification of solutions with isometries or homotheties; 12. Homogeneous space-times; 13. Hypersurface-homogeneous space-times; 14. Spatially-homogeneous perfect fluid cosmologies; 15. Groups G3 on non-null orbits V2. Spherical and plane symmetry; 16. Spherically-symmetric perfect fluid solutions; 17. Groups G2 and G1 on non-null orbits; 18. Stationary gravitational fields; 19. Stationary axisymmetric fields: basic concepts and field equations; 20. Stationary axisymmetiric vacuum solutions; 21. Non-empty stationary axisymmetric solutions; 22. Groups G2I on spacelike orbits: cylindrical symmetry; 23. Inhomogeneous perfect fluid solutions with symmetry; 24. Groups on null orbits. Plane waves; 25. Collision of plane waves; Part III. Algebraically Special Solutions: 26. The various classes of algebraically special solutions. Some algebraically general solutions; 27. The line element for metrics with κ=σ=0=R11=R14=R44, Θ+iω≠0; 28. Robinson-Trautman solutions; 29. Twisting vacuum solutions; 30. Twisting Einstein-Maxwell and pure radiation fields; 31. Non-diverging solutions (Kundt's class); 32. Kerr-Schild metrics; 33. Algebraically special perfect fluid solutions; Part IV. Special Methods: 34. Applications of generation techniques to general relativity; 35. Special vector and tensor fields; 36. Solutions with special subspaces; 37. Local isometric embedding of four-dimensional Riemannian manifolds; Part V. Tables: 38. The interconnections between the main classification schemes; References; Index.
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We explore static spherically symmetric stars in the Gauss-Bonnet gravity without cosmological constant, and present an exact internal solution which attaches to the exterior vacuum solution outside stars. It turns out that the presence of the Gauss-Bonnet term with a positive coupling constant completely changes thermal and gravitational energies, and the upper bound of red shift of spectral lines from the surface of stars. Unlike in general relativity, the upper bound of red shift is dependent on the density of stars in our case. Moreover, we have proven that two theorems for judging the stability of equilibrium of stars in general relativity can be hold in Gauss-Bonnet gravity.
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Expansion of supersymmetric string theory suggests that the leading quadratic curvature correction to the Einstein action is the Gauss-Bonnet invariant. It is shown that this model has both flat and anti-de Sitter space as solutions, but that the cosmological branch is unstable, because the graviton becomes a ghost there: the theory solves its own cosmological problem. The general static spherically symmetric solution is exhibited; it is asymptotically Schwarzschild. The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the string-generated sign) or a naked singularity. The effects of higher-curvature corrections, and an explicit cosmological term on stability, are discussed.
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Conventional wisdom states that Newton's force law implies only four non-compact dimensions. We demonstrate that this is not necessarily true in the presence of a non-factorizable background geometry. The specific example we study is a single 3-brane embedded in five dimensions. We show that even without a gap in the Kaluza-Klein spectrum, four-dimensional Newtonian and general relativistic gravity is reproduced to more than adequate precision.
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Models with a scalar field coupled to the Gauss-Bonnet Lagrangian appear naturally from Kaluza-Klein compactifications of pure higher-dimensional gravity. We study linear, cosmological perturbations in the limits of weak coupling and slow-roll, and derive simple expressions for the main observable sub-horizon quantities: the anisotropic stress factor, the time-dependent gravitational constant, and the matter perturbation growth factor. Using present observational data, and assuming slow-roll for the dark energy field, we find that the fraction of energy density associated with the coupled Gauss-Bonnet term cannot exceed 15%. The bound should be treated with caution, as there are significant uncertainies in the data used to obtain it. Even so, it indicates that the future prospects for constraining the coupled Gauss-Bonnet term with cosmological observations are encouraging.
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In spacetimes of dimension greater than four it is natural to consider higher order (in R) corrections to the Einstein equations. In this letter generalized Israel junction conditions for a membrane in such a theory are derived. This is achieved by generalising the Gibbons-Hawking boundary term. The junction conditions are applied to simple brane world models, and are compared to the many contradictory results in the literature. Comment: 4 pages
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We ask the following question: Of the exact solutions to Einstein's equations extant in the literature, how many could represent the field associated with an isolated static spherically symmetric perfect fluid source? The candidate solutions were subjected to the following elementary tests: i) isotropy of the pressure, ii) regularity at the origin, iii) positive definiteness of the energy density and pressure at the origin, iv) vanishing of the pressure at some finite radius, v) monotonic decrease of the energy density and pressure with increasing radius, and vi) subluminal sound speed. A total of 127 candidate solutions were found. Only 16 of these passed all the tests. Of these 16, only 9 have a sound speed which monotonically decreases with radius. The analysis was facilitated by use of the computer algebra system GRTensorII.