Einstein equivalence principle I know that there are two different formulations of Einstein's equivalence principle: the weak version says that the inertial and gravitational mass are equal. On the other hand  the strong version states that (from Carroll's book):

"In small enough regions of space time, laws of physics reduce to special relativity: it is impossible to detect the existence of a gravitational field by means of local experiments."

I don't really understand what that means in general. Specifically what do you mean by "local experiments"? Also why is the second formulation "stronger" that the first?
 A: Imagine placing a sheet of paper on a basketball. These two objects are of comparable size. Consider the gravitational effect of the basketball on each part the piece of paper by dividing the piece of paper into many small squares for example. The ball's gravitational field points radially inward and hence different squares on the paper will be subject to different gravitational forces, depending on the distance of each square from the ball's center. Now imagine the basketball to be the earth and the paper is a lab where you conduct experiments. The place inside the lab where you conduct the experiment now matters because different parts of the lab will feel different gravitational forces. But if your lab is very small compared to the size of the earth, the gravitational field is basically uniform and points in the same direction everywhere. That's why it says "locally" i.e. in a very small region compared to the size of the gravity source. I'm new to GR too so I hope this answer isn't misleading or, worse, wrong.
A: The second tells you something about radioactive decay (for example). The first does not.
Local experiment means an experiment whose whole apparatus is restricted to some small region of spacetime. The region has to be small enough such that all the significant distance scales in the experiment are small compared to the smallest radius of Gaussian curvature of the region of spacetime in question. (This includes time intervals also, via the speed of light).
