Normal coordinates based on curved tangent space

Abstract

Riemann normal coordinates (RNC) at a regular event $p_0$ of a spacetime manifold $\mathcal{M}$ are constructed by imposing: (i) $g_{\text{ab}}{|}_{p_0}={\eta }_{ab}$, and (ii) ${\Gamma }_{\text{bc}}^{\text{a}}{|}_{p_0}=0$. There is, however, a third, $independent$, assumption in the definition of RNC which essentially fixes the $density$ $of$ $geodesics$ emanating from $p_0$ to its value in flat spacetime, viz.: (iii) the tangent space ${\mathcal{T}}_{p_0}(\mathcal{M})$ is $flat$. We relax (iii) and obtain the normal coordinates, along with the metric $g_{\text{ab}}$, when ${\mathcal{T}}_{p_0}(\mathcal{M})$ is a maximally symmetric manifold ${\tilde{\mathcal{M}}}_{\Lambda }$ with curvature length $|\Lambda {|}^{-1/2}$. In general, the "rest" frame defined by these coordinates is non-inertial with an additional acceleration $\mathbf{a}=-(\Lambda /3)\mathbf{x}$ depending on the curvature of tangent space. Our geometric set-up provides a convenient probe of local physics in a universe with a cosmological constant $\Lambda$, 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.

arXiv comment13 pages, 4 figures, comments added and typos fixed, matches version accepted in Phys. Rev. D

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