Dawood Kothawala

We consider metrics related to each other by functionals of a scalar field (Formula presented) and it’s gradient (Formula presented), and give transformations of some key geometric quantities associated with such metrics. Our analysis provides useful and elegant geometric insights into the roles of conformal and non-conformal metric deformations in terms of intrinsic and extrinsic geometry of (Formula presented)-foliations. As a special case, we compare conformal and disformal transforms to highlight some non-trivial scaling differences. We also study the geometry of equi-geodesic surfaces formed by points (Formula presented) at constant geodesic distance (Formula presented) from a fixed point (Formula presented), and apply our results to a specific disformal geometry based on (Formula presented) which was recently shown to arise in the context of spacetime with a minimal length.

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.