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Evolutionary Physical Constants in Einstein's Field Equations and the Redshift

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Evolutionary Physical Constants in Einstein's Field Equations and the Redshift Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 April 2019 doi:10.20944/preprints201904.0119.v1 Evolutionary physical constants in Einstein’s field equations and the redshift Rajendra P. Gupta Macronix Research Corporation, 9 Veery Lane, Ottawa, ON Canada K1J 8X4; [email protected] Abstract We have shown that the use of evolutionary gravitational constants G, the speed of light c, and cosmological constant Λ in the Einstein’s field equations leads to a very simple model that fits the supernovae Ia data with a single parameter as well as the standard ΛCDM model with two parameters, and has the predictive capability superior to the latter. We have shown that at the current time G and c both increase as dG/dt = 5.4GH0 and dc/dt = 1.8cH0 with H0 as the Hubble constant, and Λ decreases as dΛ/dt = -1.2ΛH0. Negative finding on the variation of G with time derived from the analysis of lunar laser ranging data in several studies has been attributed to the fact that these studies did not consider the variation of c simultaneously with G. Keywords: cosmology: theory, cosmological parameters – galaxies: distances and redshifts – variable physical constants 1 INTRODUCTION final section, Sec. 6, we will discuss our findings and present conclusions. Varying physical constant theories have been in existence The essence of this work is that the new model implicitly since time immemorial, especially since Dirac (1937, 1938) includes the cosmological constant (that is varying) and it can suggested such variation based on his large number fit the supernovae Ia data with a single parameter almost as hypothesis. The most comprehensive review of the subject well as the ΛCDM model with two parameter, and it has the was done by Uzan (2003) followed by his more recent review predictive capability superior to the latter. In addition, we not (Uzan 2011). Chiba (2011) has provided an update of the only establish that the physical constants vary, we provide observational and experimental status of the constancy of exactly how much they vary. physical constants. We therefore will not attempt to cover the subject’s current status except to mention a few of direct 2 EVOLUTIONARY CONSTANTS MODEL relevance to this work. Magueijo (2000, 2003) is a strong proponent of the variable speed of light theories. We will develop our model in the general relativistic domain Accordingly, the model developed here will be considered starting from the Robertson-Walker metric with the usual generally Lorentz invariant which is true for any theory based coordinates 푥휇 (푐푡, 푟, 휃, 휙): on varying constants. We will be using the null results of Hofmann and Muller’s (2018) attempts to determine the 푑푟2 푑푠2 = 푐2푑푡2 − 푎(푡)2[ + 푟2(푑휃2 + sin2 휃푑휙2)].(1) variation of gravitational constant 퐺 by analysing several 1−푘푟2 decades of LLR data to establish that when the speed of light variation is included, one finds that 퐺 and 푐 vary exactly in Here 푎(푡) is the scale factor and 푘 determines the spatial the proportion of 3 to 1. geometry of the universe: 푘 = −1 (closed), 0 (flat), +1 (open). We will follow Maharaj and Naidoo’s (1993) approach to The Einstein’s field equations may be written in terms of the 휇휈 휇휈 include variable gravitational constant 퐺 and cosmological Einstein tensor 퐺 , metric tensor 푔 , and energy-momentum 휇휈 constatnt Λ in the Einstein’s field equations, and generalize it tensor 푇 , as: to include the variable speed of light 푐, to obtain their 8휋퐺 solution for the Robert-Walker metric in Sec. 2. We will show 퐺휇휈 + Λ푔휇휈 = − 푇휇휈. (2) 푐4 that Λ̇ /Λ = −1.2퐻 and 퐻̇ /퐻 = −0.6퐻 where 퐻 is the Hubble parameter. In Sec. 3 we will establish that 퐺̇/퐺 = 5.4퐻 and When solved for the Robertson-Walker metric, we get the 푐̇/푐 = 1.8퐻. We will derive the expression for distance following non-trivial equations for the flat universe (푘 = 0) of modulus 휇 vs. redshift 푧 in Sec. 4. Sec. 5 is devoted to state interest to us here, with 푝 as the pressure and 휀 as the energy the methodology for fitting the supernovae Ia data (Scolnic et density: al. 2018) and apply the same to the new model and compare it with the fits obtained with the standard ΛCDM model. In the 1 © 2019 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 April 2019 doi:10.20944/preprints201904.0119.v1 푎̈ 1 푎̇ 2 4휋퐺 1 푎̈ 푎̇ 3+3푤−푘 푎̈ 푎 −1−3푤+푘 + ( ) = − 푝 + Λ (3) = ( ) (1 − ) ; and −푞 ≡ 2 = . (12) 푎 2 푎 푐2 2 푎̇ 푎 2 푎̇ 2 푎̇ 2 8휋퐺 1 = 휀 + Λ (4) 푎2 3푐2 3 Here 푞 is deceleration parameter that does not depend on time, i.e. 푞0 = 푞. As we know the radiation energy density is If we do not regard 퐺, 푐 and Λ to be constant and define negligible at present, so we need be concerned with the matter 퐾 ≡ 퐺/푐2, we may easily derive the continuity equation by only solutions, i.e. with 푤 = 0. taking time derivative of Eq. (4) and substituting in Eq. (3) The deceleration parameter 푞0 has been analytically determined on the premise that expansion of the universe and 3푎̇ 퐾̇ Λ̇ the tired light phenomena are jointly responsible for the 휀̇ + (휀 + 푝) + 휀 + = 0. (5) 푎 퐾 8휋퐾 observed redshift, especially in the limit of very low redshift (Gupta 2018). One could see it as if the tired light effect is It reduces to the standard continuity equation when 퐾 and Λ are superimposed on the Einstein de Sitter’s matter only universe held constant. And since the Einstein’s field equations require rather than the cosmological constant. By equating the that the covariant derivative of the energy-momentum tensor expressions for the proper distance of the source of the redshift 푇휇휈 be zero, we can interpret Equation (5) as comprising of two for the two, one gets 푞0 = −0.4. Then from Eq. (12) we get continuity equations (Maharaj & Naidoo 1993), viz 푘 = 1.8, and also 푙 = −1.2, and 푚 = −0.6. We thus have from Eq. (8) 퐾̇ /퐾 = 1.8퐻, Λ̇ /Λ = −1.2퐻 and 퐻̇ /퐻 = −0.6퐻. 3푎̇ 휀̇ + (휀 + 푝) = 0, and (6) 푎 3 VARYING G AND c FORMULATION 8휋휀퐾̇ + Λ̇ = 0. (7) Having determined the value of 푘 = 1.8, and since the Hubble parameter is defined as 퐻 = 푎̇/푎, we may write from Eqs. (8) This separation simplifies the solution of the field equations and (9) (Eqs. 3 and 4). Eq. (6) yields the standard solution for the −3(1+푤) energy density 휀 = 휀0푎 . Here 푤 is the equation of state 퐾̇ 퐾 = 퐾 푎1.8, and = 1.8퐻. (13) parameter defined as 푝 ≡ 푤휀 with 푤 = 0 for matter, 1/3 for 0 퐾 radiation and −1 for Λ. It has been explicitly discussed by Magueijo in several of his We may also write explicitly papers (e.g. Magueijo 2000) that this approach is not generally 퐾̇ 퐺̇ 2푐̇ Lorentz invariant albeit relativistic. Moreover, one should = − = 1.8퐻. (14) ideally use Einstein-Hilbert action to obtain correct Einstein’s 퐾 퐺 푐 equations with variable 퐺 and 푐. Our approach therefore may be Taking 퐻 at the current time as 퐻 ≃ 70 km s−1 Mpc-1 (2.27 considered quasi-phenomenological. 0 퐾̇ × 10−18 s−1) we get = 4.09 × 10−18 s−1 = 1.29 × Since the universe expansion is determined by 퐻(푡) ≡ 푎̇/푎, it 퐾 is natural to assume the time dependence of any time dependent 10−10 yr−1. parameter to be proportional to 푎̇/푎 – the so called Machian The findings from the Lunar Laser Ranging (LLR) data scenario (Magueijo 2003). Let us therefore write analysis provides the limits on the variation of 퐺̇/퐺 (7.1 ± 7.6 × 10−14) (Hofmann & Müller 2018), which is considered to be 퐾̇ 푎̇ Λ̇ 푎̇ 퐻̇ 푎̇ = 푘 ( ), = 푙 ( ) and = 푚 ( ), i.e. (8) about three orders of magnitude lower than expected. However, 퐾 푎 Λ 푎 퐻 푎 the LLR data analysis is based on the assumption that the speed of light is constant and non-evolutionary. If this constraint is 퐾 = 퐾 푎푘, Λ = Λ 푎푙 and 퐻 = 퐻 푎푚. (9) 0 0 0 dropped then the finding would be very different. As is well known (Merkowitz 2010), a time variation of 퐺 Here 푘 (not the same as in Eq. (1)) 푙 and 푚 are the should show up as an anomalous evolution of the orbital period proportionality constants, and subscript zero indicates the 푃 of astronomical bodies expressed by Kepler’s 3rd law: parameter value at current epoch (푡 = 푡0). With this substitution in Eq. (4) we may write 2 3 2 4휋 푟 푃 = , (15) 퐺푀 푎̇ 2 8휋 1 = 퐻2푎2푚 = (퐾 푎푘)휀 푎−3(1+푤) + Λ 푎푙. (10) 푎2 0 3 0 0 3 0 where 푟 is semi-major axis of the orbit, 퐺 is the gravitational Comparing the exponents of a of all the terms, we may write constant and 푀 is the mass of the bodies involved in the orbital motion considered. If we take time derivative of Eq. (15), 2푚 = 푘 − 3 − 3푤 = 푙, and with 푤 = 0 for matter, we have 2 2푚 = 푘 − 3 = 푙. Thus, if we know 푘, we know 푙 and 푚. divide by 푃 and rearrange, we get We can now have a closed analytical solution of Eq.
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