Could someone please help me understand the Einstein hole argument (as outlined by Norton, see below). In particular the step that says that the second solution within the hole is a valid solution to the generally covariant field equation. I think my understanding of general covariance is at fault here. I'll summarise the argument as described by Norton: 1) g(x) is a solution in the hole in one coordinate system...ok 2) g'(x') is the same solution in another coordinate system...fine 3) g'(x), gained by using the function from 2 with the first coord system args, is a different gravitational field....fine 4) g'(x) is a solution of the field equations (what!?) How can he just say that g'(x) is a solution to the field equations? I can understand that the field equations are generally covariant and therefore take the same form in different coordinate systems. But I don't understand that a solution explicitly expressed in terms of one coordinate system can take the same form and be a solution in a different coord system. This is a rough paraphrase of my question... A generally covariant defintion of the circle is a curve equidistant from some point. 1)A solution in one coord system is x^2 + y^2 = 25 2)The same solution in another coord system is r=5 3)The equation x=5 is a different curve to 1) 4) The equation x=5 is a solution of the definition of a circle !? How can 4) be stated?