Small scale structure of spacetime: The van Vleck determinant and equigeodesic surfaces

It has recently been argued that if spacetime $\mathcal{M}$ possesses nontrivial structure at small scales, an appropriate semiclassical description of it should be based on nonlocal bitensors instead of local tensors such as the metric $g_{ab}(p)$. Two most relevant bitensors in this context are Synge’s world function $\mathrm{\Omega }(p,p_0)$ and the van Vleck determinant (VVD) $\mathrm{\Delta }(p,p_0)$, as they encode the metric properties of spacetime and (de)focusing behavior 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 equigeodesic surfaces ${\mathrm{\Sigma }}_{G,p_0}\equiv \{p\in \mathcal{M}|\mathrm{\Omega }(p,p_0)=\text{constant}\}$ in a geodesically convex neighborhood of an event $p_0$ and highlight some elementary identities relating the VVD with geometry of ${\mathrm{\Sigma }}_{G,p_0}$. As an aside, we also comment on the contribution of ${\mathrm{\Sigma }}_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln \mathrm{\Delta }$. We then proceed to study the small scale structure of spacetime in presence of a Lorentz invariant short distance cutoff ${\ell }_0$ using $\mathrm{\Omega }(p,p_0)$ and $\mathrm{\Delta }(p,p_0)$, based on some recently developed ideas. We derive a second rank bitensor $q_{ab}(p,p_0;{\ell }_0)=q_{ab}[g_{ab},\mathrm{\Omega },\mathrm{\Delta }]$ which naturally yields geodesic intervals bounded from below and reduces to $g_{ab}$ for $\mathrm{\Omega }\gg {\ell }_0^2/2$. We present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of equigeodesic surfaces ${\mathrm{\Sigma }}_{G,p_0}$ of $g_{ab}$, (b) structure of the nonlocal d’Alembartian $\tilde{{{}_{p_0}\square }_p}$ associated with $q_{ab}$, and (c) properties of VVD. In particular, we prove the following: (i) The Ricci biscalar $\widetilde{\mathbf{Ric}}(p,p_0)$ of $q_{ab}$ is completely determined by ${\mathrm{\Sigma }}_{G,p_0}$, the tidal tensor and first two derivatives of $\mathrm{\Delta }(p,p_0)$, and has a nontrivial classical limit (see text for details): $\underset{{\ell }_0\to 0}{\lim }\underset{\mathrm{\Omega }\to 0^{\pm }}{\lim }\widetilde{\mathbf{Ric}}(p,p_0)=\pm D{R}_{ab}{q}^a{q}^b$ (ii) The GHY term in EH action evaluated on equigeodesic surfaces straddling the causal boundaries of an event $p_0$ acquires a nontrivial structure. These results strongly suggest that the mere existence of a Lorentz invariant minimal length ${\ell }_0$ can leave unsuppressed residues independent of ${\ell }_0$ and (surprisingly) independent of many precise details of quantum gravity. For example, the coincidence limit of $\widetilde{\mathbf{Ric}}(p,p_0)$ is finite as long as the modification of distances ${\mathcal{S}}_{{\ell }_0}:2\mathrm{\Omega }\mapsto 2\tilde{\mathrm{\Omega }}$ satisfies (i) ${\mathcal{S}}_{{\ell }_0}(0)={\ell }_0^2$ (the condition of minimal length), (ii) ${\mathcal{S}}_0(x)=x$, and (iii) $[|{\mathcal{S}}_{{\ell }_0}|/{\mathcal{S}}_{{\ell }_0}^{\prime 2}](0)<\infty$. In particular, the function ${\mathcal{S}}_{{\ell }_0}(x)$, which should eventually come from a complete framework of quantum gravity, need not admit a perturbative expansion in ${\ell }_0$. Finally, we elaborate on certain technical and conceptual aspects of our results in the context of entropy of spacetime and classical description of gravitational dynamics based on Noether charge of diffeomorphism invariance instead of the EH lagrangian.

Figure 1

The geodesic structure of spacetime. (a) Equi-geodesic surfaces ${\mathrm{\Sigma }}_{G,p_0}$ attached to an event $p_0$ in an arbitrary curved spacetime; (b) ${\mathrm{\Sigma }}_{G,p_0}$ in Minkowski spacetime.