Locality and relativity - a paradox?

The equations of nature are supposed to exhibit locality in the sense that the action depends on fields and their derivatives. i.e. comparing the values of fields at local points.

But two points on a light cone have 4D distance of zero and hence they can also be said to be local. Indeed two points nearly on a light cone will have a 4D distance of nearly zero.

So these points are also 'local'.

But worse, in general relativity, the shape of the light cones depend on the equations of motion so we don't even know which points are 'local' to another point without first solving the equations of GR.

So how can we preserve the concept that equations must be 'local' if we don't know which points are local until after solving the equations? Is this a paradox?

• "We don't know which points are local until after solving the equations." This assumes that points exist prior to the spacetime resulting from solving the GR equations, hence the paradox. But there are no points to begin with if there is no spacetime. The solution creates the means to reason about points, it does not work the other way round: spacetime is not created from points. – Stéphane Rollandin Mar 29 at 9:32
• @Rollandin True, but take something like lattice QCD, you divide spacetime into points and then calculate. And then take a limit as the distance between points goes to zero. The lattice of points is over 4D space-time. Even GR is written as an equation involving derivatives in space and time. Derivatives need to have a concept of locality. – zooby Mar 30 at 0:57