Energy Conditions with Einstein’s Cosmological Constant Issam Mohanna Department of Physics and Astronomy, FCUP, University of Porto,Rua do Campo Alegre 1021/ 1055, 4169-007 Porto,Portugal March 23, 2020 Abstract A mathematical and physical study of the dynamic mechanism that governs the expansion of the universe via Einstein’s field equations and the Landau-Raychaudhuri equation is shown with the effective contribution of the cosmological constant,which has observationally and theo- retically been calculated and found to be positively constant and small since the Planck epoch. The unabandoned energy conditions, namely the NEC, WEC, SEC, and DEC, which the stress- energy tensor should satisfy, are applied in case of a perfect fluid. A comparison between the QFT vacuum ,the cosmological-constant vacuum, and the thermodynamic vacuum is presented leading to the same equation of state. Keywords:Vacuum energy;cosmological constant;dark energy;Lovelock’s theorem;expansion of the uni- verse;quantum field theory;thermodynamics 1 Introduction Since the Big-Bang beginning, space has been homogeneously and isotropically expanding with time proportionally to the scale factor of the universe and started to accelerate at redshift z ≃ 0.4[27] even before the equality of matter and dark energy at redshift z ≃ 0.3[28].Einstein’s matrical tensor field equation is a kinematic and dynamical equation that describes the expansion of the universe by relating matter and energy to the geometry of spacetime. One of the two essential structural aspects of Einstein’s equa- tion is that the stress-energy tensor of matter and energy is restricted by four energy conditions in order to be of physical sense. The other as- pect,which is basically mathematical,is that the cosmological constant Λ is essentially on the left hand side of Einstein’s field equation as part of the geometry of spacetime;it was physically in- terpreted by Einstein as a mathematical term that allowed the universe to be static, but later astronomical observations showed that a positive value of Λ was needed to explain the accelerating universe. 2 Cosmological Line Elements for a Spherically Symmetric Universe The most general metric that is locally invari- ant under Lorentz transformations and describes a homogeneous,isotropic,spherically symmet- ric,and expanding universe is the Friedmann- Lemaˆ ıtre-Robertson-Walker metric, whose squared line element is ds 2 = dt 2 −a 2 (t)  dr 2 1 − kr 2 + r 2 (dθ 2 + sin 2 θ)dφ 2  , (1) where k = 1 R 2 0 (2) is the Gaussian curvature for a static universe of radius R 0 and a(t) is the time-dependent scale 1