Dawood Kothawala

If there exists a lower bound l0 to spacetime intervals which is Lorentz-invariant, then the effective description of spacetime that incorporates such a lower bound must necessarily be nonlocal. Such a nonlocal description can be derived using standard tools of differential geometry, but using as basic variables certain bi-tensors instead of the conventional metric tensor gab(x). This allows one to construct a qmetric qab(x; y), using the Synge's world function ?(x,y) and the van Vleck determinant ?(x,y), that incorporates the lower bound on spacetime intervals. The same nonanalytic structure of the reconstructed spacetime which renders a perturbative expansion in l0 meaningless, will then also generically leave a non-trivial "relic" in the limit l0 ? 0. We present specific results derived from qab(x; y) where such a relic term manifests, and discuss several implications of the same. Specifically, we will discuss how these results: (i) suggest a description of gravitational dynamics different from the conventional one based on the Einstein-Hilbert Lagrangian, (ii) imply a dimensional reduction to 2 at small scales and (iii) can be significant for the idea that the cosmological constant itself might be related to some nonlocal vestige of the small-scale structure of spacetime. We will conclude by discussing the ramifications of these ideas in the context of quantum gravity.

All our observations that characterise space and time are expressed in terms of non-local, bi-tensorial objects such as geodesic intervals between events and two-point (Green) functions. In this contribution, I highlight the importance of characterising spacetime geome-try in terms of such non-local objects, focusing particularly on two important bi-tensors that play a particular fundamental role - Synge's World function and the van Vleck determinant. I will first discuss how these bi-tensors help capture information about spacetime geometry, and then describe their role in characterising quantum spacetime endowed with a lower bound, say ℓ 0, on spacetime intervals. Incorporating such a length scale in a Lorentz covariant manner necessitates a description of spacetime geometry in terms of above bi-tensors, and naturally replaces the conventional description based on the metric tensor gab (x) with a description in terms of a non-local bi-tensor qab (x, y). The non-analytic structure of qab (x, y) which renders a perturbative expansion in ℓ 0 meaningless, also generically leaves a non-trivial "relic"in the limit ℓ 0 → 0. I present some results where such a relic term is manifest; specifically, I will discuss how this: (i) suggests a description of gravitational dynamics different from the one based on Einstein-Hilbert lagrangian, (ii) implies dimensional reduction to 2 at small scales, (iii) connects with the notion of cosmological constant itself being a non-local vestige of the small scale structure of spacetime, (iv) helps address the issues of spacetime singularities. I will conclude by discussing the ramifications of these ideas for quantum gravity.