Small scale structure of spacetime in presence of a minimal length

Abstract

I present some recent results which yields some specific consequences of having a minimal spacetime length L0 while maintaining Lorentz covariance. In particular: (i) the resultant description of geometry must be in terms of a non-local bi-tensor qab(P,p, L20) instead of the conventional metric tensor gab(p), (ii) qab and gab are related disformally through the Synge world function, (iii) the resultant non-local d'Alembartian yields Green's functions with a natural covariant short distance cut-off, (iv) the Ricci bi-scalar R(P,p, L20) corresponding to qab yields a local scalar S(P, L20) = R(P, P, L20) in the coincidence such thatwhere ta are arbitrary normalized vectors representing gauge degrees of freedom at each spacetime event; this limit turns out to be independent of the precise details of the short distance modification, (v) the Gibbons-Hawking-York surface term for qab yields a residual entropy for each spacetimeevent