To clarify, I know very little about physics and don't pretend to have any insight whatsoever into relativity beyond what has entered the popular imagination; my knowledge is more or less at the level only of someone who has taken vector calculus and beginning differential geometry.
Nevertheless, it has struck me that the warping of space by mass (see picture) is "like" curvature $\kappa$ and that lightspeed $c$ is "like" velocity $v$ in the formula for the normal component of acceleration $\kappa v^2$; in other words that there is a very general similarity in form between the expressions $m c^2$ and $\kappa v^2$.
Also, it seems to me that if one imagines electrons orbiting an atom (with a very high curvature $\kappa$ for the orbit since $\kappa = 1/R$ for an approximate circle), that the correspondingly high normal component to acceleration would equate to a lot of energy, for example if the electrons suddenly were released from their orbits. Is there any correct intuition here at all? Again please remember if answering that this is purely speculative - but the site is here to ask, so I'm asking. Please don't be too hard on me if this is preposterously silly for anyone that really understands relativity!