Black Holes with Anisotropic Fluid in Lyra Scalar-Tensor Theory

Süleyman Demirel University Journal of Natural and Applied Sciences Volume 22, Issue 1, 38-44, 2018 Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi Cilt 22, Sayı 1, 38-44, 2018 DOI: 10.19113/sdufbed.38844 Black Holes with Anisotropic Fluid in Lyra Scalar-Tensor Theory Melis ULU DOĞRU*1, Murat DEMİRTAŞ2 1Çanakkale Onsekiz Mart University, Faculty of Arts and Science, Physics Department, 17100, Çanakkale 2Çanakkale Onsekiz Mart University, Graduate School of Natural and Applied Sciences, Physics Program, 17100, Çanakkale (Alınış / Received: 08.06.2017, Kabul / Accepted: 07.11.2017, Online Yayınlanma / Published Online: 05.02.2018) Keywords Lyra scalar-tensor theory, Black hole, Anisotropic fluid Abstract: In this paper, we investigate distribution of anisotropic fluid which is a resource of black holes in regard to Lyra scalar-tensor theory. As part of the theory, we obtain field equations of spherically symmetric space-time with anisotropic fluid. By using field equations, we suggest distribution of anisotropic fluid, responsible for space-time geometries such as Schwarzschild, ReissnerNordström, Minkowski type, de Sitter type, Anti-de Sitter type, BTZ and charged BTZ black holes. Finally, we discuss obtained pressures and density of the fluid for different values of arbitrary constants, geometrically and physically. Lyra Skaler-Tensör Teoride Anizotropik Akışkanlı Karadelikler Anahtar Kelimeler Lyra Skaler-tensör teori, Karadelik, Anizotropik akışkan Özet: Bu çalışmada, Lyra skaler-tensör teorisine göre karadeliklere kaynak olacak anizotropik akışkan dağılımı araştırılmıştır. Bu teori kapsamında, anizotropik akışkanlı küresel simetrik uzay-zaman için alan denklemleri elde edilmiştir. Alan denklemlerinin çözümleri kullanılarak, Schwarzschild, Reissner-Nordström, Minkowski tipi, de Sitter tipi, Anti-de Sitter tipi, BTZ ve yüklü BTZ karadelikleri gibi uzay-zaman geometrilerinden sorumlu olabilecek anizotropik akışkan dağılımı önerilmiştir. Sonuçta, anizotropik akışkan için elde edilmiş olan basınç ve yoğunluklar farklı keyfi sabit seçimleri için fiziksel ve geometrik açıdan tartışılmıştır. 1. Introduction [6,7,8]. Contrary to this, it is seen that alternative ways such as Brans-Dicke Theory, teleparallel gravity, f(R) gravity and other modified theories have been approved in the recent years. It can be considered that Lyra scalar-tensor theory is one of the available alternative ways. The theory has all the makings of Newtonian and general relativity limits [9]. Various space-time models have been paid attention to the theory. Particularly, spherically symmetric space-times and black holes have been gone about with perfect fluid or vacuum cases: Sen and Dunn suggested a serial solution for spherically symmetric models in Lyra scalar-tensor theory [5]. The solution has structure like as Schwarzschild black hole. Afterwards, Rahaman et.al. [10] proposed a different solution of field equations for the spacetime. They showed that the solution, called as Lyra black hole has two singular points. Bhamra [11] proved that static cosmological models are nonrealistic by using the spherically symmetric spacetime together with perfect fluid in Lyra scalar-tensor theory. Reddy and Venkateswarlu [12] investigated conformal spherical symmetric cosmologies. They There are wide range of efforts to combine electromagnetical and gravitational fields within the scope of unification theories. For this purpose, Weyl suggested a new geometry, in which Lagrangian density directly depends on metric tensor and a gauge field [1]. Weyl geometry differs from Riemannian geometry owing to the gauge field [1]. Structure of the vector length can be deformed under the coordinate transformations in the Weyl geometry [2]. The deformation poses a problem in respect to availability of the geometry [3]. Lyra resolved the problem by choosing metric potentials which depend on coordinates and gauge field. The choice allows that the vector length remains to be invariant under the parallel transformations [3]. So, new scalartensor theory of gravitation was constructed by using the Lyra geometry [4,5]. It is known that the general relativity loses its validity because of cosmological constant problem and behaviour of universe in early and/or late stages *Corresponding author: 38 M. Ulu Doğru, M. Demirtaş / Black Holes with Anisotropic Fluid in Lyra Scalar-Tensor Theory where 𝐹(𝑟) is a function which depends on radial coordinate. The relationship between line element and metric tensor is 𝑑𝑠 2 = 𝑔𝑖𝑗 (𝑥 𝑛 )𝑑𝑥 𝑖 𝑑𝑥 𝑗 . So, components of metric tensor can be easly obtained as follows: showed that displacement vector in Lyra scalartensor theory has same role spin density in EinsteinCartan theory. Rahaman et.al. [13] obtained higher dimensional spherically symmetric model without any restriction or singularity. Global monopoles were considered in non-static spherically symmetric spacetimes [14]. It is showed that the monopole has an attractive force on test particle [15].  𝑔𝑖𝑗 = Even though there are above-mentioned studies, behaviour of the anisotropic fluid and black holes, hasn’t been investigated in the context of Lyra scalartensor theory. In this study, it is aimed to define a cosmic matter formed as anisotropic fluid, resource of Schwarzschild, Reissner-Nordström, Minkowski type, de Sitter type, Anti-de Sitter type, BTZ and charged BTZ black holes, in regard to Lyra scalartensor theory. 1 0 2 F (r ) 0 0 0 0  r2 0 0 0 0 0  r 2 sin 2 ( ) 0 2 0 F (r ) (4) It is widely-known that Ricci tensor depends on metric tensor such as 𝑅𝑖𝑗 = 𝜕Γ 𝑛𝑖𝑗 ∂𝑥 𝑛 − 𝜕Γ 𝑛𝑖𝑛 ∂𝑥 𝑗 𝑛 + Γ𝑖𝑗𝑘 Γ𝑘𝑛 − Γ𝑖𝑛𝑘 Γ𝑘𝑗𝑛 (5) where connection coefficient of space-time is This paper has contents as follows: In Section.2, field equations of Lyra scalar-tensor theory is repeated. The equations and their solutions for spherically symmetric space-time and anisotropic fluid, are obtained. By using the solutions, various black hole models, full of anisotropic fluid are suggested in Section.3. Finally, obtained results are discussed in Section.4. 1 𝜕𝑔 𝜈𝛽 2 ∂𝑥 𝜇 𝛼 Γ𝜇𝜈 = 𝑔𝛼𝛽 + 𝜕𝑔 𝜇𝛽 ∂𝑥 𝜈 − 𝜕𝑔 𝜇𝜈 ∂𝑥 𝛽 . By using the connection coefficients together with Equations 4 and 5, required components of Ricci tensor are calculated as follows: 𝐹 𝐹2 2 𝐹𝑟 𝐹 𝐹 𝑟 𝐹 𝑅11 = 𝑟 ,𝑟 + 𝑟2 + (6) 𝑅33 2. Material and Method 𝑅22 = 2𝑟𝐹𝐹𝑟 + 𝐹 2 − 1 = , (7) By using the principle of least action, field equation of Lyra scalar-tensor theory can be written as 𝑅44 = −𝐹 3 𝐹𝑟,𝑟 − 𝐹 2 𝐹𝑟2 − 𝐹 3 𝐹𝑟 . (8) 1 3 3 2 2 4 𝑒𝑓𝑓 𝑅𝑖𝑗 − 𝑔𝑖𝑗 𝑅 = −𝑇𝑖𝑗 − 𝜑𝑖 𝜑𝑗 + 𝑔𝑖𝑗 𝜑𝑘 𝜑𝑘 = −𝑇𝑖𝑗 2 𝑟 where 𝐹𝑟 and 𝐹𝑟,𝑟 symbolize first and second derivatives of 𝐹(𝑟) function regarding to radial coordinate, respectively. Ricci scalar curvature is defined as constructions of Ricci tensor and metric tensor (𝑅 = 𝑅𝑖𝑗 𝑔𝑖𝑗 ). From Equations 4, 5, 6, 7 and 8, one can obtain Ricci scalar curvature: (1) where 𝑅𝑖𝑗 , 𝑅 and 𝑔𝑖𝑗 symbolize Ricci curvature tensor, (...truncated)