  The theory of fractional calculus is attracting a lot of attention from
mathematicians as well as physicists. The fractional generalisation of the
well-known ordinary calculus is being used extensively in many fields,
particularly in understanding stochastic process and fractal dynamics. In this
paper, we apply the techniques of fractional calculus to study some specific
modifications of the geometry of submanifolds. Our generalisation is applied to
extend the Israel formalism which is used to glue together two spacetimes
across a timelike, spacelike or a null hypersurface. In this context, we show
that the fractional extrapolation leads to some striking new results. More
precisely we demonstrate that, in contrast to the original Israel formalism,
where many spacetimes can only be joined together through an intermediate thin
hypersurface of matter satisfying some non- standard energy conditions, the
fractional generalisation allows these spacetimes to be smoothly sewed together
without any such requirements on the stress tensor of the matter fields. We
discuss the ramifications of these results for spacetime structure and the
possible implications for gravitational physics.
