In Newtonian mechanics, space and time are separate but in General Relativity is unified. It is considered that the space in the weak-field approximation is quasi-static and it arises from a perfect field whose particles have very small velocity in comparison to light velocity in this coordinate system and the metric is a gravitational potential tensor of rank two which implies the field of empty space. If each point of an area in N-dimensional space there existed a corresponding definite tensor, where the components of the tensor are the function of space and space acts as the strong or weak gravitational field.
Abstract We study the gravitational potential generated by static, spherically symmetric matter distributions in a quadratic f ( R ) gravity model. In the weak-field regime, the linearized field equations lead to a fourth-order modified Poisson equation whose solutions contain Newtonian and Yukawa-type contributions. Imposing regularity at the origin and asymptotic flatness uniquely fixes the integration constants, yielding potentials fully determined by the mass density. Analytical expressions are derived for several classical profiles, including Plummer, Hernquist, and Navarro-Frenk-White (NFW), as well as for new analytic density models introduced in this work. The dependence on the quadratic gravity parameter α is analyzed, and the Newtonian limit of General Relativity is consistently recovered as α → ∞ . As an application, circular velocity curves are computed and compared with the observed rotation curve of NGC 3198. A chi-squared analysis shows that the linearized quadratic f ( R ) model provides improved fits relative to the Newtonian case in the inner and intermediate galactic regions r ≲ 30 kpc, while predicting a decline at larger radii due to Yukawa suppression.
We propose a theoretical framework in which spacetime curvature and gravitation emerge from a homogeneous compression field endowed with locally shared internal nuclei. In this model, diminution is interpreted as an internal fourth-dimensional direction associated with each fragment of matter. The macroscopic spacetime metric arises from a conformal projection of this internal compression dynamics. By introducing a minimal action coupling the compression field to geometry and matter, we demonstrate that Einstein’s field equations naturally emerge. In the weak-field regime, the Newtonian limit is recovered, identifying the gravitational potential with the spatial profile of the compression field. This work establishes a conceptual bridge between microscopic internal structure and macroscopic gravitational dynamics, providing a geometric interpretation of time as emergent from local diminution.
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