Einstein's Geometrization vs. Holonomic Cancellation of Gravity via Spatial Coordinate-rescale and Nonholonomic Cancellation via Spacetime Boost

Particle's acceleration in static homogeneous gravitational field is cancelled by any reference frame of the same accelerating direction and the same accelerating rate. The frame is commonly called the freely-falling one. The present paper shows that the acceleration is also cancelled by a spatial curvilinear coordinate system. The coordinate system is simply a spatial square-root coordinate rescale in the field direction, no relative motion being involved. This suggests a new equivalence principle. Spacetime is flat which has inertial frame of Minkowski metric η ij. Gravity is a tensor g αβ on the spacetime, which is called effective metric. The effective metric emerges from the coordinate transformation. The gravitational field of an isolated point mass requires a nonholonomic spacetime boost transformation. This generalization of Newtonian gravity shares the properties of Lorentz transformation , which should help quantize gravity. The corresponding effective metric is different from that of Schwarzschild. To first order, its prediction on the deflection of light and the precession of the perihelia of planetary orbits is the same as the one of general relativity (GR). Its further implication is left for future work. 1 Introduction (i) Minkowski metric description of vanishing gravity. The present paper deals with grav-itational interaction only, no other interaction being involved. Newton's first law of motion that a particle experiencing no net force (i. e., vanishing gravitational field) must move in straight direction with a constant (or zero) velocity with respect to inertial frame τ ξηζ, can be proved geometrically by introducing Minkowski metric η αβ to the frame, ds 2 = d˜τ 2 − dξ 2 − dη 2 − dζ 2 = −η αβ dξ α dξ β (1) where ξ 0 = cτ = ˜ τ , ξ 1 = ξ, ξ 2 = η, ξ 3 = ζ, c is light speed, and η 00 = −1, η 11 = η 22 = η 33 = 1, η αβ = 0(α = β). The metric is the basis of special relativity. I call the distance s along the curves of spacetime by real distance because I will introduce a new term, effective distance ¯ s. The real distance is generally called proper distance which can be negative because the matrix η αβ is indefinite. The indefinite quadratic form (1) is the generalization of Pythagoras theorem to Minkowski spactime. It is straightforward to show that the first Newton law of motion (vanishing …