Particle Pair Production in Cosmological General Relativity

Abstract

The Cosmological General Relativity (CGR) of Carmeli, a 5-dimensional (5-D) theory of time, space and velocity, predicts the existence of an acceleration a 0=c/τ due to the expansion of the universe, where c is the speed of light in vacuum, τ=1/h is the Hubble-Carmeli time constant, where h is the Hubble constant at zero distance and no gravity. The Carmeli force on a particle of mass m is F c =ma 0, a fifth force in nature. In CGR, the effective mass density ρ eff =ρ−ρ c , where ρ is the matter density and ρ c is the critical mass density which we identify with the vacuum mass density ρ vac =−ρ c . The fields resulting from the weak field solution of the Einstein field equations in 5-D CGR and the Carmeli force are used to hypothesize the production of a pair of particles. The mass of each particle is found to be m=τc 3/4G, where G is Newton’s constant. The vacuum mass density derived from the physics is ρ vac =−ρ c =−3/8πGτ 2. We make a connection between the cosmological constant of the Friedmann-Robertson-Walker model and the vacuum mass density of CGR by the relation Λ=−8πGρ vac =3/τ 2. Each black hole particle defines its own volume of space enclosed by the event horizon, forming a sub-universe. The cosmic microwave background (CMB) black body radiation at the temperature T o =2.72548 K which fills that volume is found to have a relationship to the ionization energy of the Hydrogen atom. Define the radiation energy ϵ γ =(1−g)mc 2/N γ , where (1−g) is the fraction of the initial energy mc 2 which converts to photons, g is a function of the baryon density parameter Ω b and N γ is the total number of photons in the CMB radiation field. We make the connection with the ionization energy of the first quantum level of the Hydrogen atom by the hypothesis 1 $$ \epsilon_{\gamma} = \frac{ ( 1 - g ) m c^2 }{ N_{\gamma} } = \frac{\alpha^2 \mu c^2}{2}, $$ where α is the fine-structure constant and μ=m p f/(1+f), where f=m e /m p with m e the electron mass and m p the proton mass. We give a model for g≈Ω b (1+f)m p /m n , where m n is the neutron mass. Then ratio η of the number of baryons N b to photons N γ is given by 2 $$ \eta = \frac{N_b}{N_{\gamma}} \approx \frac{ \alpha^2 \varOmega_b f m_p / m_n}{ 2 ( 1 + f ) [ 1 - \varOmega_b ( 1 + f ) m_p / m_n ] } $$ with a value of η≈6.708×10−10. The Bekenstein-Hawking black hole entropy S is given by S=(kc 3 A)/(4ħG), where k is Boltzmann’s constant, ħ is Planck’s constant over 2π and A is the area of the event horizon. For our black hole sub-universe of mass m the entropy is given by 3 $$ S = \frac{\pi k \tau^2 c^5 }{ \hbar G }, $$ which can be put into the form relating to the vacuum mass density 4 $$ \rho_{vac} = \frac{\rho_P }{ ( S / k ) }, $$ where the cosmological Planck mass density $\rho_{P} = -\mathcal{M}_{P} / L^{3}_{P}$ . The cosmological Planck mass $\mathcal{M}_{P} = \sqrt{ \sqrt{ 3 / 8 } \hbar c / G}$ and length $L_{P} = \hbar / \mathcal{M}_{P} c$ . The value of (S/k)≈1.980×10122.