Eur. Phys. J. Plus (2011) 126: 32
https://doi.org/10.1140/epjp/i2011-11032-x

Regular Article

Global embedding via coordinate basis vectors

Physics Department, New Mexico State University, 88003, Las Cruces, NM, USA

Received: 28 November 2010
Accepted: 14 February 2011
Published online: 30 March 2011

Abstract

Following the work reported in an earlier paper (G.H. Goedecke, J. Math. Phys. 15, 789 (1974)), the coordinate basis vector approach to tensor calculus is exploited fully to obtain additional and stronger results, including: i) any curved Riemannian space must be a subspace of a larger flat host space; ii) a free test particle moves uniformly on a geodesic in the host space, but experiences gravity and other pseudoforces as viewed in four-dimensional (4D) spacetime; iii) the 4D Riemann-Christoffel curvature tensor is identically equal to a geometrical tensor associated with the complementary subspace of the host space; iv) Einstein’s field equations are automatically geometrized, with the stress-energy tensor expressed in terms of the contracted complementary tensor; v) there cannot be a conventional cosmological term in these field equations, except as an approximation; vi) other geometrical field equations involving the complementary subspace metric fields exist and should be physical equations; vii) the long-range two-body time-independent central forces due to all metric fields that have localized sources are inverse square or generalized Yukawa forces. Some of the relationships between this theory and Kaluza-Klein (KK) and string theories are noted; for example, some of the results derived herein must be postulated in KK theories.