In general relativity, the hole argument is an apparent paradox that much troubled Albert Einstein while he was developing his field equations.
For example, if we are given the current and charge density and appropriate boundary conditions, Maxwell's equations determine the electric and magnetic fields.
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.
Due to the general covariance of the field equations, this transformed metric g' is also a solution in the untransformed coordinate system.
In one version, consider an initial value surface with some data and find the metric as a function of time.
Einstein's derivation of the gravitational field equations was delayed because of the hole argument which he created in 1913.
), and the equations of motion of matter in a given gravitational field (which follow from maximizing the proper time given by
[4] General covariance states that the laws of physics should take the same mathematical form in all reference frames (accelerating or not) and hence all coordinate systems, so the differential equations that are the field equations of the gravitational field should take the same mathematical form in all coordinates systems.
coordinate system that solves the field equations, one can simply write down the very same function but replace all the
On the basis of this observation, Einstein spent three years searching for non-generally covariant field equations in a frantic race against Hilbert.
Therefore, one solution is obtained from the other by actively dragging the metric function over the spacetime manifold into the new configuration.
Einstein failed to find non-generally covariant field equations only to return to the hole argument and resolve it.
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.
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'.
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.
This description is a metric tensor at every point, or a connection which defines which nearby vectors are parallel.
If one takes the point of view that coordinate invariance is trivially true, the principle of coordinate invariance simply states that the metric itself is dynamical and its equation of motion does not involve a fixed background geometry.
While two gravitational fields that differ by an active diffeomorphism look different geometrically, after the trajectories of all the particles are recalculated, their interactions manifestly define 'physical' locations with respect to which the gravitational field takes the same value under all active diffeomorphisms.
[6] (Note that if the two metrics were related to each other by a mere coordinate transformation the world lines of the particles would not get transposed; this is because both these metrics impose the same spacetime geometry and because world lines are defined geometrically as trajectories of maximum proper time — it is only with an active diffeomorphism that the geometry is changed and trajectories altered.)
Einstein believed that the hole argument implies that the only meaningful definition of location and time is through matter.
According to this insight, the physical content of any theory is exhausted by the catalog of the spacetime coincidences it licenses.
"[4] This is the true meaning[clarification needed] of the saying "The stage disappears and becomes one of the actors"; space-time as a 'container' over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the world.
Loop quantum gravity physicists regard background independence as a central tenet in their approach to quantizing gravity – a classical symmetry that ought to be preserved by the quantum theory if we are to be truly quantizing geometry (=gravity).
One immediate consequence is that LQG is UV-finite because small and large distances are gauge equivalent as one can replace one metric function for another related to the first by an active diffeomorphism.
[8] The direct proof of finiteness of canonical LQG in the presence of all forms of matter has been provided by Thiemann.
that loop quantum gravity violates background independence by introducing a preferred frame of reference ('spin foams').
[citation needed] Perturbative string theory (in addition to a number of non-perturbative formulations) is not 'obviously' background independent, because it depends on boundary conditions at infinity, similarly to how perturbative general relativity is not 'obviously' background dependent.
However some sectors of string theory admit formulations in which background independence is manifest, including most notably the AdS/CFT.
It is believed that string theory is background independent in general, even if many useful formulations do not make it manifest.