Flat-Spacetime Covariant Gravity, its Quantization and Solar Application

General relativity (GR) is a covariant theory of gravity on curved spacetime (general covariance) which, however, denies a covariant Hamiltonian. Most alternative works of gravity on flat spacetime present non-covariant Lagrangians which treat time t as a parameter entirely distinct from the spatial coordinates. A covariant formulation would require that space and time be considered as entirely similar coordinates in world space and the parameter p involved in the Lagrangian functional L(xα, dxα/dp, p) and the action I = ∫ L(xα, dxα/dp, p)dp should be an invariant parameter rather than t. He (2005c) finds that spacetime must be flat with Minkowski metric ηαβ and suggests a theory of gravity which is the same as GR except that spacetime is flat. That is, the theory is general covariant with respect to all curvilinear coordinate transformation (including non-curvilinear Lorentz transformation). The metric tensor is called effective metric and measures the gravitational “medium“ which is generated by the corresponding mass distribution. The “medium“ curves the motion of (test) particles (i. e., extremizing effective distance s̄) in the similar way the dielectric medium curves the propagation of light waves (extremizing refractive index n). Following the same principle, the present paper considers the metric form to be a Lagrangian defined on flat spacetime. Therefore, the Lagrangian is general covariant, and space and time are considered as entirely similar coordinates in world space. In the case of the gravitational field of single point mass, the common procedure, ~ P → − ih̆∇ leads to the quantization of the corresponding Hamiltonian, where P i are the canonical momentum to xi and h̆ is the quantization constant of macrophysics whose counterpart is the Planck constant h̄. The corresponding wave differential equation has exact solution. For the Schwarzschild radius rg = GM/c 2 of solar mass, the first order approximation of the solution is the well-known wave function Ψnl of Hydrogen atom. Nottale, Schumacher, and Gay (1997) gave an excellent fit of the wave function to the distributions of planetary distances and planetary masses. Because the abovesaid Lagrangian is homogeneous of the generalized velocity components, dxα/dp, the wave function is independent of the quantization constant h̆ and we are not troubled with its determination. The observable quantities depend only on the quantum numbers (e. g., n, l) and the gravitational strength rg, and a consistent description of solar quantization is given. keywords: Relativity – Gravitational Theory -Quantization : Hydrogen Atom