Normal coordinates based on curved tangent space

Riemann normal coordinates (RNC) at a regular event p0 of a spacetime manifold M are constructed by imposing (i) gab|p0=ηab, and (ii) Γabc|p0=0. There is, however, a third, independent, assumption in the definition of RNC which essentially fixes the density of geodesics emanating from p0 to its value in flat spacetime, viz.: (iii) the tangent space Tp0(M) is flat. We relax (iii) and obtain the normal coordinates, along with the metric gab, when Tp0(M) is a maximally symmetric manifold M˜Λ with curvature length |Λ|−1/2. In general, the "rest" frame defined by these coordinates is noninertial with an additional acceleration a=−(Λ/3)x depending on the curvature of tangent space. Our geometric setup provides a convenient probe of local physics in a universe with a cosmological constant Λ, now embedded into the local structure of spacetime as a fundamental constant associated with a curved tangent space. We discuss classical and quantum implications of the same.