transformation, but it is a solution nevertheless. Here is the problem that disturbed Einstein so much: if these coordinates systems differ only after there are then two solutions; they have the same initial conditions but they impose different geometries after . On the basis of this observation Einstein spent three years searching for non-generally covariant field equations in a frantic race against Hilbert. To be more accurate, Einstein conceived of a situation where the matter distribution is known everywhere outside some closed region of spacetime devoid of matter, the hole. Then the field equations together with the boundary conditions supposedly enable the metric field to be determined inside the hole. One takes the and coordinates to differ inside the hole but agree outside of it. The argument then proceeds as in the above paragraph. As these two solutions have the same functional form, they assume the same values; they just assume them at different places. Therefore, one solution is obtained from the other by actively dragging the metric function over the spacetime manifold into the new configuration. This is known as a diffeomorphism, sometimes called an active diffeomorphism by physicists to distinguish it from coordinate transformations (passive diffeomorphisms). Einstein failed to find non-generally covariant field equations only to return to the hole argument and resolve it. It basically involved accepting that these two solutions are physically equivalent by claiming that how the metric is localized over the spacetime manifold is physically irrelevant and that individual spacetime points defined in terms of spacetime coordinates have no physical meaning in and of themselves (this is the source of the problem for manifold substantialism). To provide meaning to 'location', Einstein generalized the situation given in the above paragraphs by introducing two particles; then physical points (inside the hole) can be defined in terms of their coinciding world lines. This works because matter gets dragged across together with the metric under active diffeomorphisms. Without the introduction of these particles one would not be able to define physical spacetime points (within the hole); see the quotes of Einstein given below in the section 'Einstein's resolution'. Meaning of coordinate invariance For the philosophically inclined, there is still some subtlety. If the metric components are considered the dynamical variables of General Relativity, the condition that the equations are coordinate invariant doesn't have any content by itself. All physical theories are invariant under coordinate transformations if formulated properly.