Compact star in Tolman–Kuchowicz spacetime in the background of Einstein–Gauss–Bonnet gravity
Abstract
The present work is devoted to the study of anisotropic compact matter distributions within the framework of five-dimensional Einstein–Gauss–Bonnet gravity. To solve the field equations, we have considered that the inner geometry is described by Tolman–Kuchowicz spacetime. The Gauss–Bonnet Lagrangian \(\mathcal {L}_{GB}\) is coupled to the Einstein–Hilbert action through a coupling constant, namely \(\alpha \). When this coupling tends to zero general relativity results are recovered. We analyze the effect of this parameter on the principal salient features of the model, such as energy density, radial and tangential pressure and anisotropy factor. These effects are contrasted with the corresponding general relativity results. Besides, we have checked the incidence on an important mechanism: equilibrium by means of a generalized Tolman–Oppenheimer–Volkoff equation and stability through relativistic adiabatic index and Abreu’s criterion. Additionally, the behavior of the subliminal sound speeds of the pressure waves in the principal directions of the configuration and the conduct of the energy-momentum tensor throughout the star are analyzed employing the causality condition and energy conditions, respectively. All these subjects are illuminated by means of physical, mathematical and graphical surveys. The M–I and the M–R graphs imply that the stiffness of the equation of state increases with \(\alpha \); however, it is less stiff than GR.
1 Introduction
Nowadays it is of great interest to obtain models that describe compact structures, that is, massive objects with a small size, which configurations due to their high density, such as white dwarfs, neutron stars or more exotic stars such as those formed by quarks, constitute a real laboratory to investigate the regime of strongly coupled gravitational fields. For a long time the development of these models in order to describe and understand the behavior of the aforementioned objects was under the framework of the general relativity theory (GR). With great observational and experimental support [1] GR describes very well the gravitational interaction and its consequences in a four-dimensional spacetime. However, two questions arise: Is it possible to study gravity in less than four dimensions? Is it possible to study gravity in more than four dimensions? If so, what benefits and consequences would such studies bring about? In the first case, for a two-dimensional spacetime the Einstein tensor is zero. This is just the consequence of the Einstein–Hilbert Lagrangian being the two-dimensional Euler characteristic \(\chi \). This is a topological invariant in two dimensions, and therefore we cannot obtain equations of motion for our fields from it. In three dimensions we already have an Einstein tensor not identically zero, but we run into another problem. Now the number of independent components of the Riemann tensor is six: the same as the number of independent components of the Ricci tensor. So, Ricci-plane solutions, i.e., those with \(R_{\mu \nu }=0\), are solutions with vanishing Riemann tensor, not giving place to solutions of gravitational waves for example. We then end in the usual three spatial dimensions plus a temporal one. The previous discussion may already be enough to hope that the study in larger dimensions can bear fruit. Perhaps the dynamics resulting from the action of Einstein–Hilbert in four dimensions hides effects that in larger dimensions could become manifest. In fact, several theories have been favored in part to study a larger number of dimensions, such as Kaluza–Klein theory (adding an extra dimension to unify gravity with electromagnetism) or string theory (reaching a total of 11 dimensions in order to unify all the known interactions). Thus, in larger numbers of dimensions there is no reason to exclude quadratic, cubic terms, etc., of scalars formed from the Riemann tensor and its contractions. In this direction Lanczos [2] was the first to extend the GR including covariant high-order derivatives terms of the metric tensor, in order to study the scale invariance under \(g_{\mu \nu }\rightarrow \lambda g_{\mu \nu }\) transformation, \(\lambda \) being a constant parameter. Nevertheless, the quadratic term combination found by Lanczos in four dimensions did not contribute to the dynamics of the theory. This was because Lanczos was dealing with the four-dimensional Euler characteristic \(\chi \), which is a topological invariant in four dimensions just as the Einstein–Hilbert action is in two dimensions.
Lovelock [3], later generalized the Einstein–Hilbert action including terms of higher order, with the first-order term corresponding to the Einstein–Hilbert action and the second-order one to the Gauss–Bonnet (GB) Lagrangian. In a n-dimensional spacetime (with \(n\ge 5\)) the GB Lagrangian leads to second-order equations of motion, as is required. In the spirit of searching for compact structures, Einstein–Gauss–Bonnet (EGB) theory is promising. In the context of black holes, Boulware and Deser [4] generalized the higher-dimensional solutions in Einstein theory due to Tangherlini [5], obtained the exterior vacuum spacetime, i.e., the equivalent Schwarzschild solition in EGB theory. Moreover, the study by Ghosh and Deshkar [6] of Vaidya radiating black holes in EGB gravity revealed that the location of the horizons is changed from the standard four-dimensional gravity. In the cosmological and modified gravity theories context EGB gravity has received much attention [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Recently, Bamba et al. [18] have investigated the energy conditions in the cosmological scenario employing FLRW spacetime. On the other hand, regarding stellar interiors much interesting work available in the literature has been devoted to the study of the existence of collapsed structures [19, 20, 21, 22]. Besides, Wright [23] has studied the maximum mass–radius ratio (Buchdahl’s limit [24]) in five-dimensional EGB gravity.
The study of a compact object driven by an anisotropic matter distribution has a long history. Since the pioneering work by Bowers and Liang [25] many researchers have been studying the properties and consequences of this type of structures [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. These well-known works explore diverse properties such as: mechanisms of stability and hydrodynamic equilibrium, the behavior of the material content through energy conditions, causality conditions, maximum limit of the mass–radius ratio, maximum value of the superficial redshift, etc. A recent work on the role played by the anisotropy on the properties mentioned above is [42] (see also the references therein).
Following this line, in this paper we construct a well behaved anisotropic fluid sphere in the five-dimensional EGB scenario, by using Tolman–Kuchowicz [43, 44] spacetime. This metric has been used by other authors in the study of anisotropic charged/uncharged interior solutions [45, 46]. So, the plan of this paper is as follows: in Sect. 2 we discuss Einstein–Gauss–Bonnet gravity in a five-dimensional spacetime.
In Sect. 3 we study the field equations and the mathematical solutions of EGB gravity within Tolman–Kuchowicz spacetime, obtaining the main salient features that characterize the model such as the energy-matter density \(\rho \), the radial pressure \(p_{r}\) and the tangential pressure \(p_{t}\) and the anisotropy factor \(\Delta \). In Sect. 4 we analyze the physical and mathematical behavior of the thermodynamic variables. In Sect. 5 we obtain the complete set of constant parameters, joining the inner geometry with exterior spacetime in a smooth way. Several physical properties are studied in Sects. 6, 7, 8 and 9, such as the causality condition, stability, equilibrium under different forces and energy conditions. Finally, in Sect. 11 we provide some remarks of the model obtained.
2 Field equations
3 Solution of the field equations
4 Physical analysis
5 Exterior spacetime and matching conditions
6 Causality condition
7 Stability mechanisms: relativistic adiabatic index and Abreu’s criterion
8 Energy conditions
The matter content that makes up astrophysical bodies can be composed of a large number of material fields. Although the components that constitute the matter distribution are known, it could be very complex to describe the concrete form of the energy-momentum tensor. Indeed, one has some ideas on the behavior of the matter under extreme conditions of density and pressure.
Moreover, from the physical point of view NEC means that an observer traversing a null curve will measure the ambient (ordinary) energy density to be positive. WEC implies that the energy density measured by an observer crossing a timelike curve is never negative. SEC purports that the trace of the tidal tensor measured by the corresponding observers is always non-negative [60]. Furthermore, violations of energy conditions have sometimes been presented as only being produced by unphysical stress energy tensors. Usually SEC as used as a fundamental guide will be extremely idealistic. Nevertheless, SEC is violated in many cases, e.g. in minimally coupled scalar field and curvature-coupled scalar field theories. It may or may not imply the violation of the more basic energy conditions, i.e., NEC and WEC.
9 Generalized Tolman–Oppenheimer–Volkoff equation
In the case of isotropic fluid spheres (\({p}_{r}={p}_{t}\)) and regarding \(\alpha \rightarrow 0\) (GR limit), this equation drives the equilibrium of relativistic compact structures described by isotropic matter distribution. Regarding the presence of anisotropies and the EGB framework, this equation still determines the balance of the system. As was pointed out before, the present model is determined by three forces. To guarantee the equilibrium of the proposed stellar structure, we have shown in Fig. 11 that the balance of the forces is reached at all the values of \({\alpha }\) and GR also. Consequently, Fig. 11 indicates that, in the situation of \(10\le \alpha \le 50\), the resulting impact of the hydrodynamic force (\({F_h}\)) and the anisotropic force (\({F_a}\)) compensates for the internal attraction due to the gravitational force (\({F_g}\)). Furthermore, it is worth mentioning that in the GR case the \(F_{h}\), \(F_{a}\) and \(F_{g}\) forces are greater than the corresponding EGB forces.
10 Rigid rotation, moment of inertia and comparison with M–R graph
11 Concluding remarks
Values of all the parameters corresponding to different values of \(\alpha \) and the corresponding \((M_{max},R)\)
\(\alpha \) | b | a | B | c | \(\rho _c\) | \(p_c\) | \(\rho _b\) | \(M_{max}\) | R |
---|---|---|---|---|---|---|---|---|---|
\(\times 10^{-8}\) (km\(^{-4})\) | \(\times 10^{-3}\) (km\(^{-2})\) | \(\times 10^{-3}\) (km\(^{-2})\) | \(\times 10^{14}\) (gm cm\(^{-3})\) | \(\times 10^{33}\) (dyne cm\(^{-2})\) | \(\times 10^{14}\) (gm cm\(^{-3})\) | \((M_\odot )\) | (km) | ||
0 | 15 | 688655 | 713032 | 0.95768 | 4.43142 | 7.04898 | 4.09734 | – | – |
10 | 4 | 33681 | 33405 | 0.97947 | 2.19653 | 1.80466 | 2.11426 | 2.999 | 12.77 |
20 | 4 | 334567 | 32364 | 0.979853 | 2.21053 | 1.84999 | 2.13184 | 3.027 | 12.784 |
30 | 4 | 332382 | 3047 | 0.980493 | 2.22415 | 1.88714 | 2.14398 | 3.062 | 12.80 |
40 | 4 | 330251 | 31387 | 0.980278 | 2.23742 | 1.92088 | 2.15641 | 3.093 | 12.813 |
50 | 4 | 328172 | 29605 | 0.980881 | 2.25035 | 1.95076 | 2.16905 | 3.12 | 12.83 |
Moreover, the remaining thermodynamic variables that characterize the solution, i.e., the radial pressure \(p_{r}\) and the tangential pressure \(p_{t}\), are well behaved at all points within the configuration (3). Besides, the tangential pressure \(p_{t}\) coincides with the radial pressure \(p_{r}\) at the center and then is always greater than \(p_{r}\) everywhere. Actually, it is a very important fact, because it induces a positive anisotropy factor \(\Delta \) inside the star (Fig. 4). A positive \(\Delta \) brings with it important consequences for the structure. For example, it allows the construction of more compact objects (greater amount of mass contained in a smaller size) and introduces a force (repulsive in nature) that helps sustain the hydrostatic balance by counteracting the gravitational compression. The latter not only prevents the system from being subject to a gravitational re-collapse (as would be the case of \(\Delta <0\), which would introduce an attractive force, contributing to the gravitational gradient to collapse the object, which can take it even below its Schwarzschild’s radius to form a black hole), but it improves the stability of the system as well.
On the other hand, as the material content is confined within the region given by \(\Sigma =r=R\), to find all the constant parameters that describe the solution \(\{a,b,B,C\}\) we have made the junction between the internal geometry and the outer spacetime, the Schwarzschild equivalent solution (free of material content, i.e., the vacuum solution) in EGB. This was performed by applying the first and second fundamental forms.
- 1.
The causality condition (Fig. 7) for the stability of the anisotropic matter distribution as a profile of the difference in squared of subliminal sound speed of the pressure waves, \(| v^2_{t} - v^2_{r}|\) with respect to the radial coordinate r satisfies the inequality \(-1< v^2_{t} - v^2_{r}<0\) which manifests itself in Fig. 9 (lower panel).
- 2.
In Fig. 8 we have displayed the behavior of the adiabatic index \(\Gamma \) with respect to the infinitesimal radial adiabatic perturbation which confirms that when \(\Gamma > 4/3\) our stellar structure is stable in all interior points of the stellar object with spherical symmetry.
- 3.
As regards examination of the energy conditions in order to test the physical validity of the obtained solution, in Fig. 10 we have indicated the behavior of all energy conditions with respect to the radial coordinate r for the stellar system, which shows that our compact stellar structure is well suited for the system in the context of the EGB gravity at various choose values of \(\alpha \), also considering GR theory.
- 4.
We have shown in Fig. 11 that the equilibrium of the forces is reached for all the values of \(\alpha \) (including GR), which confirms that our stellar model is stable with respect to the equilibrium of forces.
- 5.
The stiffness of the corresponding EoS increases with increasing coupling constant \(\alpha \), however, it is less stiff w.r.t. the GR limit. The maximum mass corresponding to \(\alpha =10\)–50 is given in Table 1. As \(\alpha \) increases to 10–50, the moment of inertia also increases. This makes the EoS stiffer and therefore the system can support higher masses (Figs. 12 and 13).
Notes
Acknowledgements
P.B is thankful to IUCAA, Govt of India, for providing a visiting associateship, F. Tello-Ortiz thanks the financial support by the CONICYT PFCHA/DOCTORADO-NACIONAL/2019-21190856, Grant Fondecyt No. 1161192, Chile and project ANT-1855 at the Universidad de Antofagasta, Chile.
References
- 1.C.M. Will, Living Rev. Rel. 9, 3 (2005)CrossRefGoogle Scholar
- 2.C. Lanczos, Ann. Math. 39, 842 (1938)MathSciNetCrossRefGoogle Scholar
- 3.D. Lovelock, J. Math. Phys. 498, (1971)Google Scholar
- 4.D.G. Boulware, S. Deser, Phys. Rev. Lett. 55, 2656 (1985)ADSCrossRefGoogle Scholar
- 5.F.R. Tangherlini, I1 Nuovo Cimento 27, 636 (1963)MathSciNetCrossRefGoogle Scholar
- 6.S.G. Ghosh, D.W. Deshkar, Phys. Rev. D 77, 047504 (2008)ADSMathSciNetCrossRefGoogle Scholar
- 7.S. Nojiri, S.D. Odintsov, Phys. Lett. B 631, 1 (2005)ADSMathSciNetCrossRefGoogle Scholar
- 8.G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov, S. Zerbini, Phys. Rev. D 75, 086002 (2007)ADSMathSciNetCrossRefGoogle Scholar
- 9.S. Nojiri, S.D. Odintsov, S. Ogushi, Int. J. Mod. Phys. A 17, 4809 (2002)ADSCrossRefGoogle Scholar
- 10.B.M. Leith, I.P. Neupane, J. Cosmol. Astropart. Phys. 0705, 019 (2007)ADSCrossRefGoogle Scholar
- 11.A. De Felice, S. Tsujikawa, Phys. Lett. B 675, 1 (2009)ADSCrossRefGoogle Scholar
- 12.S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011)ADSMathSciNetCrossRefGoogle Scholar
- 13.S. Nojiri, S.D. Odintsov, P.V. Tretyakov, Prog. Theor. Phys. Suppl. 172, 81 (2008)ADSCrossRefGoogle Scholar
- 14.K. Bamba, S.D. Odintsov, L. Sebastiani, S. Zerbini, Eur. Phys. J. C 67, 295 (2010)ADSCrossRefGoogle Scholar
- 15.A. De Felice, S. Tsujikawa, Phys. Rev. D 80, 063516 (2009)ADSCrossRefGoogle Scholar
- 16.J.H. Kung, Phys. Rev. D 52, 6922 (1995)ADSCrossRefGoogle Scholar
- 17.J.H. Kung, Phys. Rev. D 53, 3017 (1996)ADSCrossRefGoogle Scholar
- 18.K. Bamba, M. Ilyas, M.Z. Bhatti, Z. Yousaf, Gen. Relativ. Gravit. 49, 112 (2017)ADSCrossRefGoogle Scholar
- 19.P. Bhar, M. Govender, R. Sharma, Eur. Phys. J. C 77, 109 (2017)ADSCrossRefGoogle Scholar
- 20.S. Hansraj, B. Chilambwe, S.D. Maharaj, Eur. Phys. J. C 75, 277 (2015)ADSCrossRefGoogle Scholar
- 21.S.D. Maharaj, B. Chilambwe, S. Hansraj, Phys. Rev. D 91, 084049 (2015)ADSMathSciNetCrossRefGoogle Scholar
- 22.S. Hansraj, Eur. Phys. J. C 77, 557 (2017)ADSCrossRefGoogle Scholar
- 23.M. Wright, Gen. Relativ. Gravit. 48, 93 (2016)ADSCrossRefGoogle Scholar
- 24.H.A. Buchdahl, Phys. Rev. D 116, 1027 (1959)ADSCrossRefGoogle Scholar
- 25.R.L. Bowers, E.P.T. Liang, Astrophys. J. 188, 657 (1974)ADSCrossRefGoogle Scholar
- 26.M. Cosenza, L. Herrera, M. Esculpi, L. Witten, J. Math. Phys. 22, 118 (1981)ADSMathSciNetCrossRefGoogle Scholar
- 27.M. Cosenza, L. Herrera, M. Esculpi, L. Witten, Phys. Rev. D 25, 2527 (1982)ADSMathSciNetCrossRefGoogle Scholar
- 28.L. Herrera, J. Ponce de León, J. Math. Phys. 26, 2302 (1985)ADSMathSciNetCrossRefGoogle Scholar
- 29.J. Ponce de León, Gen. Relativ. Gravit. 19, 797 (1987)ADSCrossRefGoogle Scholar
- 30.J. Ponce de León, J. Math. Phys. 28, 1114 (1987)ADSMathSciNetCrossRefGoogle Scholar
- 31.R. Chan, S. Kichenassamy, G. Le Denmat, N.O. Santos, Mon. Not. R. Astron. Soc. 239, 91 (1989)ADSCrossRefGoogle Scholar
- 32.R. Chan, L. Herrera, N.O. Santos, Class. Quantum Grav. 9, 133 (1992)ADSCrossRefGoogle Scholar
- 33.R. Chan, L. Herrera, N.O. Santos, Mon. Not. R. Astron. Soc. 265, 533 (1993)ADSCrossRefGoogle Scholar
- 34.L. Herrera, Phys. Lett. A 165, 206 (1992)ADSCrossRefGoogle Scholar
- 35.A. Di Prisco, E. Fuenmayor, L. Herrera, V. Varela, Phys. Lett. A 195, 23 (1994)ADSMathSciNetCrossRefGoogle Scholar
- 36.L. Herrera, N.O. Santos, Phys. Rep. 286, 53 (1997)ADSMathSciNetCrossRefGoogle Scholar
- 37.A. Di Prisco, L. Herrera, V. Varela, Gen. Relativ. Gravit. 29, 1239 (1997)ADSCrossRefGoogle Scholar
- 38.L. Herrera, A.D. Prisco, J. Ospino, E. Fuenmayor, J. Math. Phys. 42, 2129 (2001)ADSMathSciNetCrossRefGoogle Scholar
- 39.L. Herrera, J. Ospino, A.D. Prisco, Phys. Rev. D 77, 027502 (2008)ADSMathSciNetCrossRefGoogle Scholar
- 40.M.K. Gokhroo, A.L. Mehra, Gen. Relativ. Gravit. 26, 75 (1994)ADSCrossRefGoogle Scholar
- 41.B.V. Ivanov, Phys. Rev. D 65, 104011 (2002)ADSCrossRefGoogle Scholar
- 42.S.K. Maurya, A. Banerjee, S. Hansraj, Phys. Rev. D 97, 044022 (2018)ADSMathSciNetCrossRefGoogle Scholar
- 43.R.C. Tolman, Phys. Rev. 55, 364 (1939)ADSCrossRefGoogle Scholar
- 44.B. Kuchowicz, Acta Phys. Pol. 33, 541 (1968)Google Scholar
- 45.M.K. Jasim, D. Deb, S. Ray et al., Eur. Phys. J. C 78, 603 (2018)ADSCrossRefGoogle Scholar
- 46.S.K. Maurya, F. Tello-Ortiz, Eur. Phys. J. C 79, 33 (2019)ADSCrossRefGoogle Scholar
- 47.H. Maeda, M. Nozawa, Phys. Rev. D 77, 064031 (2008)ADSMathSciNetCrossRefGoogle Scholar
- 48.W. Israel, Nuovo Cim. B 44, 1 (1966)ADSCrossRefGoogle Scholar
- 49.G. Darmois, Mémorial des Sciences Mathematiques (Gauthier-Villars, Paris, 1927). Fasc. 25, (1927)Google Scholar
- 50.S.C. Davis, Phys. Rev. D 67, 024030 (2003)ADSMathSciNetCrossRefGoogle Scholar
- 51.G.W. Gibbons, S.W. Hawking, Phys. Rev. D 15, 27 (1977)CrossRefGoogle Scholar
- 52.R.C. Myers, Phys. Rev. D 36, 392 (1987)ADSMathSciNetCrossRefGoogle Scholar
- 53.C. Charmousis, J.F. Dufaux, Class. Quant. Grav. 19, 4671 (2002)ADSCrossRefGoogle Scholar
- 54.H. Bondi, Mon. Not. R. Astron. Soc. 281, 39 (1964)Google Scholar
- 55.H. Heintzmann, W. Hillebrandt, Astron. Astrophys. 38, 51 (1975)ADSGoogle Scholar
- 56.H. Abreu, H. Hernández, L.A. Núñez, Calss. Quant. Gravit. 24, 4631 (2007)ADSCrossRefGoogle Scholar
- 57.B.K. Harrison, K.S. Thorne, M. Wakano, J.A. Wheeler, Gravitational Theory and Gravitational Collapse (University of Chicago Press, Chicago, 1965)Google Scholar
- 58.Ya B. Zeldovich, I.D. Novikov, Relativistic Astrophysics Stars and Relativity, vol. 1 (University of Chicago Press, Chicago, 1971)Google Scholar
- 59.J. Ponce de León, Phys. Rev. D 37, 309 (1988)ADSMathSciNetCrossRefGoogle Scholar
- 60.E. Curiel, Einstein Stud. 13, 43 (2017)CrossRefGoogle Scholar
- 61.J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939)ADSCrossRefGoogle Scholar
- 62.M. Bejger, P. Haensel, A & A 396, 917 (2002)ADSCrossRefGoogle Scholar
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