In general relativity, the hole argument is an apparent paradox that much troubled Albert Einstein while developing his famous field equations.
Some philosophers of physics take the argument to raise a problem for manifold substantialism, a doctrine that the manifold of events in spacetime is a "substance" which exists independently of the metric field defined on it or the matter within it. Other philosophers and physicists disagree with this interpretation, and view the argument as a confusion about gauge invariance and gauge fixing instead.
Einstein's hole argument
In a usual field equation, knowing the source of the field, and the boundary conditions, determines the field everywhere. For example, if we are given the current and charge density and appropriate boundary conditions, Maxwell's equations determine the electric and magnetic fields. They do not determine the vector potential though, because the vector potential depends on an arbitrary choice of gauge.
Einstein noticed that if the equations of gravity are generally covariant, then the metric cannot be determined uniquely by its sources as a function of the coordinates of spacetime. As an example: consider a gravitational source, such as the Sun. Then there is some gravitational field described by a metric g(r). Now perform a coordinate transformation r r' where r' is the same as r for points which are inside the Sun but r' is different from r outside the Sun. The coordinate description of the interior of the Sun is unaffected by the transformation, but the functional form of the metric g' for the new coordinate values outside the Sun is changed. Due to the general covariance of the field equations, this transformed metric g' is also a solution in the untransformed coordinate system.
This means that one source, the Sun, can be the source of many seemingly different metrics. The resolution is immediate: any two fields which only differ by such a "hole" transformation are physically equivalent, just as two different vector potentials which differ by a gauge transformation are physically equivalent. Then all these mathematically distinct solutions are not physically distinguishable — they represent one and the same physical solution of the field equations.
There are many variations on this apparent paradox. In one version, consider an initial value surface with some data and find the metric as a function of time. Then perform a coordinate transformation which moves points around in the future of the initial value surface, but which doesn't affect the initial surface or any points at infinity. The conclusion may be 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.