Abstract
We show that results of a simple dynamical gedanken experiment interpreted according to standard Newton's gravitational theory, may reveal that three-dimensional space is curved. The experiment may be used to reconstruct the curved geometry of space, i.e. its non-Euclidean metric. The perihelion of Mercury advance and the light bending calculated from the Poisson equation and the equation of motion in the curved geometry have the correct (observed) values. Independently, we also show that Newtonian gravity theory may be enhanced to incorporate the curvature of three dimensional space by adding an extra equation which links the Ricci scalar with the density of matter. Like in Einstein's general relativity, matter is the source of curvature. In the spherically symmetric (vacuum) case, the metric of space 3gik that follows from this extra equation agrees, to the expected accuracy, with the metric measured by the Newtonian gedanken experiment mentioned above.
Highlights
Newton’s theory of gravity was formulated in a flat, Euclidean 3-D space but its basic laws, i.e. the Poisson equation and the equation of motion,3 gik ∇i ∇k Φ = −4π Gρ, (1) Fi = mai, (2)make perfect sense in a 3-D space with an arbitrary geometry 3gik
The Newtonian physicist may form a table of pairs [r, R], which follow from his measurements of [aG, aC ] at many different orbits
Given by (40), we may calculate the effect of light bending assuming that light travels along geodesic lines in space
Summary
Newton’s theory of gravity was formulated in a flat, Euclidean 3-D space but its basic laws, i.e. the Poisson equation and the equation of motion,. Abramowicz has recently suggested in [3] that a Newtonian physicist could experimentally determine the metric 3gik of the real physical 3-D space and calculate, according to (1) and (2), the perihelion of Mercury advance and the light bending effects. It is connected to the mass M expressed in the standard units by M = GM/c2 and has the dimension of length Another point discussed in this paper is based on the following two remarks:. (ii) They could discover that the curvature of space depends on the distance from the gravity center This would suggest to them, again within the framework of Newton’s theory, that gravity and curvature are not independent, but instead they are somehow linked. Several authors have discussed the idea of Newton’s gravity and dynamics in a curved 3-D space, for example most recently Naresh Dadhich in “Einstein is Newton with space curved” [1] or previously, in the context of the optical geometry, [2]
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