value surface, but which doesn't affect the initial surface or any points at infinity. Then you can conclude that the generally covariant field equations do not determine the future uniquely, since this new coordinate transformed metric is an equally valid solution of the same field equations in the original coordinate system. So the initial value problem has no unique solution in general relativity. This is also true in electrodynamics—since you can do a gauge transformation which will only affect the vector potential tomorrow. The resolution in both cases is to use extra conditions to fix a gauge. Disputing the above version of Einstein's hole argument Einstein's derivation of the gravitational field equations was delayed because of the hole argument which he created in 1913. However the problem was not as given in the section above. By 1912, the time Einstein started what he called his "struggle with the meaning of the coordinates", he already knew to search for tensorial equations as these are unaffected by coordinate change. He had already found the form of the gravitational field (namely as a tetrad or frame field or metric ), and the equations of motion of matter in a given gravitational field (which follow from maximizing the proper time given by ). It is evident that this is invariant under coordinate transformations. What disturbed him was a consequence of his principle of general covariance and arises from the following. General covariance states that the laws of physics should take the same mathematical form in all reference frames and hence all coordinate systems and so the differential equation that are the field equations of the gravitational field should take the same mathematical form in all coordinates systems. In other words, given two coordinate systems, say coordinates and coordinates, one has exactly the same differential equation to solve in both, except in one the independent variable is and in the other the independent variable is . This implies that as soon as one finds a metric function in the coordinate system that solves the field equations, one can simply write down the very same function but replace all the 's with 's, which solves the field equations in the coordinate system. As these two solutions have the same functional form but belong to different coordinate systems they impose different spacetime geometries. Note that this second solution is not related to the first through a coordinate