III.1.3 The Principle of General Covariance
Because they are based on tensors, the equations of motion in General Relativity are not related to the coordinate system. However, the components of the tensors in some basis are treated as space-time fields which are dependent on the chosen coordinate system, and because the action is scalar it is invariant under coordinate transformations. Since all coordinates systems are artifacts of the human mind, the laws of physics must have the same form in any reference frame. This is called general, or diffeomorphism, covariance, or invariance.
This General Covariance Principle was assumed by Einstein as an extension of the Principle of Relativity. However, in contrast with the global space-time symmetries of Special Relativity, the invariance of the form of the laws, under transformations of the coordinates depending smoothly on arbitrary functions of space and time, is a local symmetry with diffeomorphism group. This extension of the concept of continuous symmetry from global to local symmetries led to Noether s theorem which connects between global symmetries and conservation laws.
In Special Relativity, the Minkowski metric possesses a global Lorentz symmetry, and the full isometry group is the Poincar group which is an extension of the Lorentz group that also includes translations. Similarly, the theory of isometries in General Relativity on fixed backgrounds is described by Killing vector fields which form a Lie algebra that generates the group of isometries, but this group is non-trivial only in special cases. The full symmetry of General Relativity is the diffeomorphism group of the manifold which is the group of all continuous and differentiable mappings of the manifold to itself, but this group is different for manifolds of different topologies. For example, for the Schwarzschild metric, the corresponding Killing group is a subgroup of the Poincar group. Nevertheless, if one searches for the symmetry group that leaves a general metric of the Einstein equation invariant one finds that this group contains only the identity transformation. In contrast to all other fundamental theories of physics, General Relativity is not based on some kind of symmetry, but it is a result of the Equivalence Principle.