Small scale structure of spacetime: van Vleck determinant and equi-geodesic surfaces

Abstract

It has recently been argued that if spacetime $\mathcal{M}$ possesses non-trivial structure at small scales, an appropriate semi-classical description of it should be based on non-local bi-tensors instead of local tensors such as the metric $g_{ab}$. Two most relevant bi-tensors in this context are Synge's World function $\Omega (p,p_0)$ and the van Vleck determinant (VVD) $\Delta (p,p_0)$, as they encode the metric properties of spacetime and (de)focussing behaviour of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian ${}_{p_0}{\square }_p$. We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces ${\Sigma }_{G,p_0}$ defined by $\Omega (p,p_0)=constant$ in a geodesically convex neighbourhood of an event $p_0$, and highlight some elementary identities relating the VVD with geometry of these surfaces. As an aside, we also comment on the contribution of ${\Sigma }_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln \Delta$. We then study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off ${\ell }_0$ using $\Omega$ and $\Delta$, based on some recently developed ideas. We derive a 2nd rank bi-tensor $q_{ab}$ which naturally yields geodesic intervals bounded from below, and present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of ${\Sigma }_{G,p_0}$, (b) structure of the non local d'Alembartian associated with $q_{ab}$, and (c) properties of the VVD. In particular, we show that the Ricci bi-scalar of $q_{ab}$ is completely determined by ${\Sigma }_{G,p_0}$, the tidal tensor, and first two derivatives of the van Vleck determinant, and has a non-trivial "classical" limit given by (constant) $R_{ab}{q}^a{q}^b$ (see text).

v1: 12 pages, 1 figure; abstract formatted (see PDF file for the complete abstract). v2: minor typos corrected, references added

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