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Extended General Relativity for a Curved Universe

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Extended General Relativity for a Curved Universe Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 January 2021 doi:10.20944/preprints202010.0320.v4 Extended General Relativity for a Curved Universe Mohammed. B. Al-Fadhli1* 1College of Science, University of Lincoln, Lincoln, LN6 7TS, UK. Abstract: The recent Planck Legacy release revealed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB). Notably, this amplitude is higher than that estimated by the lambda cold dark matter model (ΛCDM), which endorses the positive curvature of the early Universe with a confidence level greater than 99%. Although General Relativity (GR) performs accurately in the local/present Universe where spacetime is almost flat, its lost boundary term, incompatibility with quantum mechanics and the necessity of dark matter and dark energy might indicate its incompleteness. By utilising the Einstein–Hilbert action, this study presents extended field equations considering the pre-existing/background curvature and the boundary contribution. The extended field equations consist of Einstein field equations with a conformal transformation feature in addition to the boundary term, which could remove singularities from the theory and facilitate its quantisation. The extended equations have been utilised to derive the evolution of the Universe with reference to the scale factor of the early Universe and its radius of curvature. Keywords: Astrometry, General Relativity, Boundary Term. 1. INTRODUCTION Considerable efforts have been devoted to Recent evidence by Planck Legacy recent release modifying gravity in form of Conformal Gravity, (PL18) indicated the presence of an enhanced lensing Loop Quantum Gravity, MOND, ADS-CFT, String amplitude in the cosmic microwave background theory, 퐹(푅) Gravity, etc. (Mannheim, 1997, 2001; (CMB), which is notably higher than that estimated Randall and Sundrum, 1999; Garriga and Tanaka, by the lambda cold dark matter model (ΛCDM). This 1999; Germani and Sopuerta, 2002; Nojiri and endorses the positive curvature of the early Universe Odintsov, 2006, 2010; Amendola et al., 2007; Appleby with a confidence level greater than 99% (Aghanim et and Battye, 2007; Bekenstein, 2007; Nelson, 2010). al., 2020; Di Valentino, Melchiorri and Silk, 2020). Motivation to modify General Relativity (GR) Besides, the gravitational lensing by substructures of aimed to elucidate possible existence or the form of several galaxy clusters is an order of magnitude more dark matter/energy, achieve a better description of than the ΛCDM estimation (Meneghetti et al., 2020; observation data, verify theoretical restrictions in the Umetsu, 2020). This evidence endorses a spatially strong curvature regime such as within back holes as curved Universe in spite of the spacetime flatness of well as to formulate Quantum Gravity (Appleby and the local/present Universe. Further, a closed Universe Battye, 2007; Nelson, 2010). To achieve an efficient can provide an agreement with the observed CMB action for quantum corrections, several theories have anisotropy (Efstathiou, 2003). been formulated on the modification of Lagrangian Accordingly, in this study, the background/pre- gravitational fields. Such modifications appear to be existing curvature has been incorporated into the inevitable, which included higher-order curvature Einstein–Hilbert action. However, to comply with the terms as well as non-minimally coupled scalar fields energy conservation law and the compatibility of the (Vilkovisky, 1992; Capozziello and Lambiase, 2000; action, a new modulus of deformation/curvature of Garattini, 2013). However, these modifications have spacetime continuum is utilised, which is formulated to be consistent with the energy conservation law. on the theory of elasticity (Landau, 1986). The paper is organised as follows. Section 2 discusses the origination of the spacetime continuum modulus of curvature and mathematical derivations of the extended field equations. Sections 3 discusses the derivations of referenced Friedmann equations and boundary contributions. Section 4 summarises *E-mail: [email protected] the outcomes and conclusions. Section 5 suggests the future development of this work. 1 © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 January 2021 doi:10.20944/preprints202010.0320.v4 2. Extended Field Equations By extracting one boundary part from Eq. (5) as 휇휈 4 ∫ 푔 훿푅휇휈√−푔푑 푥 (6) To consider pre-existing/background curvatures signified by the scalar curvature ℛ, a new modulus The variation in the Ricci curvature tensor 훿푅푢푣 can of deformation/curvature of the spacetime continuum be written in terms of the covariant derivative of the 퐸 =(stress/strain) in (푁/푚2) is defined, which can be 퐷 difference between two Levi-Civita connections, the expressed using Einstein field equations as Palatini identity: 푇푣 − 푇훿푣/2 c4 휇 휇 휌 휌 퐸퐷 = 푣 = ℛ (1) 훿푅휇휈 = ∇휌(훿Γ 휇휈) − ∇휈(훿Γ 휇휌) (7) 푅휇/ℛ 8πG where the stress is signified by the stress-energy The Ricci curvature tensor variation with respect to 휇 푔휇휈 tensor 푇휈 of trace 푇, while the strain is signified by the inverse metric tensor can be obtained utilising 휇 the Ricci curvature tensor 푅휈 as the change in the the metric compatibility of the covariant derivative, 푣 휇휈 curvature divided by pre-existing curvature ℛ. 훿푢 is ∇휌푔 = 0 (S. M. Carroll, 2003) as the Kronecker delta (Straumann, 2013). According to 휇휈 휇휈 휌 휇휈 휌 푔 훿푅휇휈 = ∇휌(푔 훿Γ ) − ∇휈(푔 훿Γ ) (8) the theory of elasticity, the modulus times the volume 휇휈 휇휌 equals internal energy of reversible systems (Landau, Thus, the boundary part as a total derivative for any 1986). Thus, 퐸퐷 could represent the internal energy tensor density can be transformed according to the density of the space, i.e., the vacuum energy density. Stokes' theorem with renaming dummy indices as 퐸퐷 is proportional to the fourth-order of the speed of 휎휈 휇 휎휇 휇 휇 light, resembling Quantum field theory (QFT) energy ∭ ∇휇(푔 훿Γ 휈휎 − 푔 훿Γ 휇휎)√−푔 푑푉 ≡ ∭ ∇휇퐴 √−푔 푑푉 푉 푉 cut-off predictions of vacuum energy density (Rugh (9) 휇 3 and Zinkernagel, 2000). According to conservation of = ∯ 퐴 . 푛̂푢√−푔 푑푆 = ∮ 퐾휖√|q| 푑 푥 energy law, the Einstein–Hilbert action is extended to 푆 휕푉 where 퐾 = 퐾 푞휇휈 is the trace of extrinsic curvature as 퐸퐷 푅 푢푣 4 4 푆 = ∫[푇 + ℒ푀 ] √−푔 푑 푥 = ∫ [ + ℒ푀 ] √−푔 푑 푥 (2) 2 ℛ the second fundamental form, 푞 is the determinant of the induced metric on the manifold boundary, and where 푇 is the kinetic term, ℒ푀 is the Lagrangian 휖 푛̂ matter density denoting the potential term, 푅 is the equals 1 when the normal 푢 on the boundary is a spacelike entity and equals -1 when the normal is a Ricci scalar curvature and 푔 is the determinant of the metric 푔푢푣. According to the principle of least action, timelike entity (Dyer and Hinterbichler, 2009). Thus, Eq. (2) should hold for any variation in the metric as the action in Eq. (2) can be rewritten in parts as −1 퐸 훿(푅ℛ −푔) 훿(ℒ −푔) 퐷 4 3 퐸퐷 √ 푀√ 휇휈 4 푆 + 푆 + 푆 = ∫ (푅 −푔 푑 푥 + 퐾휖√|푞| 푑 푥) 훿푆 = ∫ [ ( ) + ] 훿푔 푑 푥 (3) 푐 푏 푚 푐√ 2 훿푔휇휈 훿푔휇휈 2ℛ (10) 4 The differentiation yields + ∫ ℒ푀√−푔 푑 푥 훿푅 −푔 훿ℛ −푔푅 훿 −푔푅 퐸퐷 √ √ √ where 푆푐, 푆푏, and 푆푚 denote the contributions to the 훿푆 = ∫ [ ( 휇휈 − 2 휇휈 + 휇휈 ) 2 ℛ훿푔 ℛ 훿푔 ℛ훿푔 action from the curvature without the boundary, the (4) 훿 −푔ℒ 훿ℒ −푔 √ 푀 푀√ 휇휈 4 boundary alone and the matter fields respectively. + 휇휈 + 휇휈 ] 훿푔 푑 푥 훿푔 훿푔 To solve the parted action, firstly, the action By performing the differentiation of the determinant without boundary parts is reduced to the following 휇휈 according to the Jacobi's formula as 훿푔 = 푔푔 훿푔휇휈 , 훿푆푐 + 훿푆푚 = 휇휈 휇휈 훿√−푔 = −훿푔/2√−푔 while 훿푔 푔휇휈 = −훿푔휇휈푔 (S. M. 휇휈 휇휈 퐸퐷 푅휇휈훿푔 ℛ휇휈훿푔 푔휇휈 Carroll, 2003) in addition to the differentiation of the ( 휇휈 − 2 휇휈 푅 − 푅) 2 ℛ훿푔 ℛ 훿푔 2ℛ 휇휈 (11) scalar curvature 푅 which equals 푅휇휈푔 as ∫ −푔훿푔휇휈푑4푥 √ 푔휇휈 훿ℒ푀 휇휈 푢푣 푅 훿푔 + 푔 훿푅 − ℒ푀 + 휇휈 휇휈 [ 2 훿푔휇휈 ] ℛ훿푔휇휈 퐸 휇휈 휇휈 퐷 ℛ휇휈훿푔 + 푔 훿ℛ휇휈 The stress energy-momentum tensor is proportional − 푅 2 ℛ2훿푔휇휈 to the Lagrangian term in the action by the definition 훿푆 = ∫ −푔훿푔휇휈푑4푥 (5) 푔휇휈 √ (Straumann, 2013) as − 푅 ( 2ℛ ) 2훿ℒ푀 푇푢푣: = 푔휇휈ℒ푀 − 푔 훿ℒ 훿푔휇휈 (12) − 휇휈 ℒ + 푀 2 푀 훿푔휇휈 [ ] Additionally, the energy-momentum tensor includes The lambda/pressure is not considered which might pressure terms according to the equation of state be incorporated implicitly into stress-energy tensors. (Vikman, 2005). 2 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 January 2021 doi:10.20944/preprints202010.0320.v4 By substituting Eqs. (1,12) in Eq. (11) and using the By combining action parts in Eqs. (13, 19) according principle of least action yield to Eq. (10), the new extended field equations are ℛ휇휈 1 8휋퐺 1 푅 − ℛ 1 8휋퐺 푅 − 푅 − 푅푔 = 푇 (13) 푅 − 푅푔̂ + (퐾 − 퐾푞̂ ) = 푇 (20) 휇휈 ℛ 2 휇휈 푐4 휇휈 휇휈 2 휇휈 ℛ 휇휈 2 휇휈 푐4 휇휈 휇휈 here ℛ휇휈/ℛ = ℛ휇휈/ℛ휇휈푔̅ = 푔휇휈̅ corresponds to Weyl’s The boundary tensor/term is only significant at high- conformal transformation of the metric (Kozameh et energy limits such as with black holes (Dyer and al., 1985; Straub, 2006). Eq. (13) can be rewritten as Hinterbichler, 2009) and the early Universe, and can 1 8휋퐺 remove the singularities from the theory. 푅 − 푅푔̂ = 푇 (14) 휇휈 2 휇휈 푐4 휇휈 3. Universe Evolution Model where 푔̂휇휈 = 푔휇휈 + 2푔휇휈̅ is the conformal transformation of the metric tensor due to the fact that Einstein The Friedmann–Lemaîtree–Robertson–Walker spaces are a subclass of conformal spaces (Kozameh et (FLRW) metric model is the standard cosmological al., 1985).
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