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![Plot of the root γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} against β:=α/R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta :=\alpha /R^2$$\end{document}. The top dashed line indicates the horizon, whereas the lower dashed line indicates the general relativistic bound. The point shows the location of vanishing 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}, agreeing with the five dimensional general relativistic result. Positive β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} gives a less stricter bound than in general relativity, and approaches the horizon as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} increases, whereas negative β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} gives a stricter bound and approaches 0 as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} decreases](publication/280243011/figure/fig2/AS:961488680394762@1606248173129/Plot-of-the-root-gdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym.gif)
Plot of the root γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} against β:=α/R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta :=\alpha /R^2$$\end{document}. The top dashed line indicates the horizon, whereas the lower dashed line indicates the general relativistic bound. The point shows the location of vanishing 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}, agreeing with the five dimensional general relativistic result. Positive β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} gives a less stricter bound than in general relativity, and approaches the horizon as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} increases, whereas negative β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} gives a stricter bound and approaches 0 as β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} decreases
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![Plot of the root γ\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/280243011/figure/fig2/AS:961488680394762@1606248173129/Plot-of-the-root-gdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
The Buchdahl limit for static spherically symmetric isotropic stars is
generalised to the case of five dimensional Gauss-Bonnet gravity. Our result
depends on the sign of the Gauss-Bonnet coupling constant $\alpha$. When
$\alpha>0$, we find, unlike in general relativity, that the bound is dependent
on the stellar structure, in particular the centra...
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Citations
... Buchdahl star shares almost all the property of the black hole [26][27][28]. The compactness limit of Buchdahl star has been considered in extensive literature like the inclusion of Λ [29][30][31][32], different conditions than Buchdahl's [33,34], brane-world gravity [35,36], modified gravity theories including Lovelock gravity and higher dimensions [37][38][39][40][41][42]. One can obtain the limit on the maximum mass of the Buchdahl-star by appealing to the dominant energy condition where the sound velocity is subliminal [43,44]. ...
We find static fluid solutions of Einstein and pure Lovelock equations with , , which could be possible models for the interior of a Buchdahl-like star. Buchdahl star is a limiting stellar configuration without a horizon whose formation does not need any exotic matter.
... Moreover, the mass function m(r) in 5D Einstein-Gauss-Bonnet gravity is given as [73][74][75] ...
In astronomy, the study of compact stellar remnants — white dwarfs, neutron stars, black holes — has attracted much attention for addressing fundamental principles of physics under extreme conditions in the core of compact objects. In a recent argument, Maurya et al. [Eur. Phys. J. C 77, 45 (2017)] have proposed an exact solution depending on a specific spacetime geometry. Here, we construct equilibrium configurations of compact stars for the same spacetime that make it interesting for modeling high density physical astronomical objects. All calculations are carried out within the framework of the five-dimensional Einstein–Gauss–Bonnet gravity. Our main interest is to explore the dependence of the physical properties of these compact stars depending on the Gauss–Bonnet coupling constant. The interior solutions have been matched to an exterior Boulware–Deser solution for 5D spacetime. Our finding ensures that all energy conditions hold, and the speed of sound remains causal, everywhere inside the star. Moreover, we study the dynamical stability of stellar structure by taking into account the modified field equations using the theory of adiabatic radial oscillations developed by Chandrasekhar. Based on the observational data for radii and masses coming from different astronomical sources, we show that our model is compatible and physically relevant.
... Here "prime" denotes the differentiation with respect to r, only. Moreover, the gravitational mass function m(r) in 5D-EGB gravity can be calculated by the following formula (see ref. [59] for more details): ...
In this paper, we investigated a new anisotropic solution for the strange star model in the context of 5 D Einstein-Gauss-Bonnet (EGB) gravity. For this purpose, we used a linear equation of state (EOS), in particular p r = β ρ + γ , (where β and γ are constants) together with a well-behaved ansatz for gravitational potential, corresponding to a radial component of spacetime. In this way, we found the other gravitational potential as well as main thermodynamical variables, such as pressures (both radial and tangential) with energy density. The constant parameters of the anisotropic solution were obtained by matching a well-known Boulware-Deser solution at the boundary. The physical viability of the strange star model was also tested in order to describe the realistic models. Moreover, we studied the hydrostatic equilibrium of the stellar system by using a modified TOV equation and the dynamical stability through the critical value of the radial adiabatic index. The mass-radius relationship was also established for determining the compactness and surface redshift of the model, which increases with the Gauss-Bonnet coupling constant α but does not cross the Buchdahal limit.
... However all these discussions were mostly in the context of Einstein gravity in four spacetime dimensions (for similar scenario in five dimensional Einstein-Gauss-Bonnet gravity, see [13,14]). Since it is natural to expect that the spacetime will inherit additional spacelike dimensions at a higher energy scale, it seems worthwhile to understand the fate of stellar structure in pure Lovelock theories. ...
We study the bound on the compactness of a stellar object in pure Lovelock theories of arbitrary order in arbitrary spacetime dimensions, involving electromagnetic field. The bound we derive for a generic pure Lovelock theory, reproduces the known results in four dimensional Einstein gravity. Both the case of a charged shell and that of a charge sphere demonstrates that for a given spacetime dimension, stars in general relativity are more compact than the stars in pure Lovelock theories. In addition, as the strength of the Maxwell field increases, the stellar structures become more compact, i.e., the radius of the star decreases. In the context of four dimensional Einstein-Gauss-Bonnet gravity as well, an increase in the strength of the Gauss-Bonnet coupling (behaving as an effective electric charge), increases the compactness of the star. Implications are discussed.
... However all these discussions were mostly in the context of Einstein gravity in four spacetime dimensions (for similar scenario in five dimensional Einstein-Gauss-Bonnet gravity, see [13,14]). Since it is natural to expect that the spacetime will inherit additional spacelike dimensions at a higher energy scale, it seems worthwhile to understand the fate of stellar structure in pure Lovelock theories. ...
We study the bound on the compactness of a stellar object in pure Lovelock theories of arbitrary order in arbitrary spacetime dimensions, involving electromagnetic field. The bound we derive for a generic pure Lovelock theory, reproduces the known results in four dimensional Einstein gravity. Both the case of a charged shell and that of a charge sphere demonstrates that for a given spacetime dimension, stars in general relativity are more compact than the stars in pure Lovelock theories. In addition, as the strength of the Maxwell field increases, the stellar structures become more compact, i.e., the radius of the star decreases. In the context of four dimensional Einstein-Gauss-Bonnet gravity as well, an increase in the strength of the Gauss-Bonnet coupling (behaving as an effective electric charge), increases the compactness of the star. Implications are discussed.
... There have also been various alternative derivations involving various situations like inclusion of Λ [5][6][7][8], different conditions than Buchdahl's [4,9], brane-world gravity [10,11], modified gravity theories including Lovelock gravity and higher dimensions [12][13][14][15][16]. The limit on maximum mass has been obtained by appealing to the dominant energy condition and sound velocity being subliminal. 1 The Buchdahl limit defines an overriding state which is obtained under very general conditions while more compact distributions are allowed under specific circumstances and conditions. ...
The main aim of this paper is essentially to point out that the Buchdahl compactness limit of a static object is given by gravitational field energy being less than or equal to half of its non-gravitational matter energy. It is thus entirely determined without any reference to interior distribution by the unique exterior solutions, the Schwarzschild for neutral and the Reissner-Nordström for charged object. In terms of surface potential, it reads as Φ(R) = (M−Q²/2R)/R ⩽ 4/9 which translates to surface red-shift being less than or equal to 3. It also prescribes an upper bound on charge an object could have, Q²/M² ⩽ 9/8 > 1.
... In these works, it is obvious to see that the values of the mass-radius ratio bounds are significantly affected by the dimensions of spacetimes. In addition, the cases of modified gravity have also attracted many attentions of researchers, and the massradius ratio bounds were derived in Gauss-Bonnet gravity [16] and dRGT Massive Gravity theory [17]. ...
In this paper, the mass-radius ratio bound for horizonless compact object in higher dimensions is derived. Instead of considering the various matter conditions of the compact objects following Andreasson's approach, we focus on the conditions for the existence of dynamical compact objects. The radius of the outermost circular null geodesic is derived in higher dimensions, which is the lower bound on the minimally allowed radius of dynamically stable horizonless charged compact object. Subsequently, the upper bound on the mass-radius ratio is obtained. Our results are strongly dependent on the dimensions. What's more significant is that the developed bound is proven always to be stronger than the result following Andreasson's approach in higher dimensions.
... This raises an intriguing question, how is the above limit modified if one considers a theory of gravity different from general relativity or if one introduces some additional matter fields. Several such ideas have been explored quiet extensively in recent times, which include -(a) inclusion of cosmological constant [4][5][6][7], (b) effects due to presence of extra dimensions [8][9][10][11], (c) dependence of Buchdahl's limit on higher curvature gravity models [12][13][14][15][16][17][18][19][20][21], such as, Einstein-Gauss-Bonnet gravity, f (R) gravity [22][23][24][25][26][27], pure Lovelock theories [28][29][30][31][32] etc. In some of these cases one had to impose some additional physically motivated assumptions, e.g., imposition of dominant energy condition, sub-luminal propagation of sound etc. ...
... where in the last line we have used one of the Einstein's equations, namely Eq. (13). It is evident that the above equation when divided by a factor of two coincides with Eq. (14) and hence the third Einstein's equation is redundant. Even then there is one ambiguity present in the system, which is worth mentioning. ...
We have derived the Buchdahl's limit for a relativistic star in presence of the Kalb-Ramond field in four as well as in higher dimensions. It turns out that the Buchdahl's limit gets severely affected by the inclusion of the Kalb-Ramond field. In particular, the Kalb-Ramond field in four spacetime dimensions enables one to pack extra mass in any compact stellar structure of a given radius. On the other hand, a completely opposite picture emerges if the Kalb-Ramond field exists in higher dimensions, where the mass content of a compact star is smaller compared to that in general relativity. Implications are discussed.
... This raises an intriguing question, how is the above limit modified if one considers a theory of gravity different from general relativity or if one introduces some additional matter fields. Several such ideas have been explored quiet extensively in recent times, which include -(a) inclusion of cosmological constant [4][5][6][7], (b) effects due to presence of extra dimensions [8][9][10][11], (c) effect of scalar tensor theories [12][13][14][15][16][17][18][19] and (d) dependence of Buchdahl's limit on higher curvature gravity models [14,[20][21][22][23][24][25][26][27][28], such as, Einstein-Gauss-Bonnet gravity, f (R) gravity [29][30][31][32][33][34], pure Lovelock theories [35][36][37][38][39] etc. In some of these cases one had to impose some additional physically motivated assumptions, e.g., imposition of dominant energy condition, sub-luminal propagation of sound etc. ...
... At this stage one can use the two identities introduced in Eq. (22) and Eq. (23) respectively, and then simplifying the resulting expression further, we obtain the following result, ...
We have derived the Buchdahl's limit for a relativistic star in presence of the Kalb-Ramond field in four as well as in higher dimensions. It turns out that the Buchdahl's limit gets severely affected by the inclusion of the Kalb-Ramond field. In particular, the Kalb-Ramond field in four spacetime dimensions enables one to pack extra mass in any compact stellar structure of a given radius. On the other hand, a completely opposite picture emerges if the Kalb-Ramond field exists in higher dimensions, where the mass content of a compact star is smaller compared to that in general relativity. Implications are discussed.
... More importantly, it could be successively pierced downwards by the assumption of dominant energy condition [13] and further requiring sound velocity being subluminal [14]. Recently it has been studied [15] for Einstein-Gauss-Bonnet gravity in five dimensions and it is shown that the limit for positive GB coupling depends upon stellar structure, i.e., on the central density. ...
... In this general situation, it is not possible to integrate Eq. (Eq. (15)) and obtain e ν in a closed form. In the absence of exact solution we will consider the technique used by [30,31]. ...
We obtain the Buchdahl compactness limit for a pure Lovelock static fluid star and verify that the limit following from the uniform density Schwarzschild's interior solution, which is universal irrespective of the gravitational theory (Einstein or Lovelock), is true in general. In terms of surface potential , it means at the surface of the star , where d, N respectively indicate spacetime dimensions and Lovelock order. For a given N, is maximum for d=2N+2 while it is always 4/9, Buchdahl's limit, for d=3N+1. It is also remarkable that for N=1 Einstein gravity, or for pure Lovelock in d=3N+1, Buchdahl's limit is equivalent to the criteria that gravitational field energy exterior to the star is less than half its gravitational mass, having no reference to the interior at all.
