arXiv:math-ph/0603039v1 13 Mar 2006 Einstein metric formalism without Schwarzschild singularities I. Bulyzhenkov-Widicker 1 University of Ottawa, Canada and Institute of Spectroscopy RAS, Russia Abstract. The intrinsic metric symmetries of pseudo-Riemannian space-time universally main- tain strict spatial flatness in the invariant GR interval. The non-linear time element of this interval always depends on the particle velocity or spatial displacement and differs from the proper-time differential of a local observer. The interval is a first integral of motion in a strong central field with Weber-type gravitational potentials, which relieve the metric from the Schwarzschild singularity. These conceptual poten- tials keep the Gauss gravitational flux and the universal Newton force for strong fields. The observed planetary perihelion precession, the radar echo delay, and gravitational light bending can be explained quantitatively without departure from Euclidean spa- tial geometry. Shiff’s frame-drag estimations, but diminished geodetic precessions of the Gravity Probe B gyroscopes are predicted from the standard GR formalism with the metric symmetries. MSC: Primary 83C20; Secondary 83C10, 83C75. Keywords: Metric symmetries, pseudo-Riemannian geometry, flatness, singularities, GR tests, Gravity Probe B, Weber potentials 1 Introduction In 1913 Einstein and Grossman published their metric formalism for a test particle in a gravitational field and in 1915 the Einstein equation for sources accomplished the basic tensor approach to warped space-time with matter[1]. This metric theory of gravity, known as General Relativity (GR), can operate fluently with curved spatial displacement dl N =  γ N ij dx i dx j of the point mass m N by accepting the Schwarzschild or Droste metric solution [2] without specific restrictions on the space metric tensor γ N ij ≡ g N oi g N oj (g N oo ) −1 −g N ij . All GR solutions are related to the space-time interval, ds 2 N ≡ g N µν dx µ dx ν = dτ 2 N − dl 2 N , where the particle time element dτ N ≡ [g N oo (dx o + g −1 Noo g N oi dx i ) 2 ] 1/2 depends on the local pseudo-Riemannian metric tensor g N µν and, consequently, on local gravitational fields. Hereinafter i =1, 2, 3, µ =0, 1, 2, 3, and the vacuum light velocity c = 1. We intend to analyze time and space elements within the GR four-interval ds ≡  g N µν dx µ dx ν and to prove that the time element dτ N of the moving mass m N depends in gravitational fields not only on the world time t (with the in- terval dt ≡ √ δ oo dx o dx o = dx o > 0) and space coordinates x i , but also on the physical velocity dl N /dτ N ≡ v and, ultimately, on the element of spatial 1 email: Igor.Bulyzhenkov@science.uottawa.ca 1