What are the most precise definitions of "locality" and "causality"?

Causality means that a physical process happening at space time event $x$ is the physical cause of another physical process happening at space-time event $y$

Relativist locality (in special relativity) says that physical quantities at space-time event $x$ cannot be the cause of other physical quantities at space-time event $y$, if the events $x$ and $y$ are separated by a space-like interval : $(\Delta s)^2(x,y)=(x-y)^2<0$ (in a metrics $g = Diag(1,-1,-1,-1)$

In Quantum Field Theory, this is expressed as : $[\Phi(x), \Phi'(y)]_{(x-y)^2<0} = 0$, where $\Phi, \Phi'$ are any physical measurable quantities (hermitian operators).

What are the most precise ways known to justify that gravity is
non-local? How does one reconcile this with the fact that Einstein's
equations local? (any differential equation is local by definition!)

The problem, is that, for any space-time point $x$, we may choose frames, which are (locally) inertial frames. So, locally (at $x$) - and only locally -, the effects of gravity are being cancelled, just by a change of frame. This is the consequence of the invariance by diffeomorphism of the Einstein equations. So, inevitably, there is a pseudo "non-local" appearance character of gravity. But the expression "non-local" is not a good choice, because that does not mean at all that you can send an instantaneous information, or locally violate special relativity. This just comes for the freedom to choose frames at some space-time point $x$. Note that one of the consequences is that you cannot find a total stress-energy tensor which is both covariant and conserved.

This post imported from StackExchange Physics at 2014-03-07 13:47 (UCT), posted by SE-user Trimok