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General Covariance

The fundamental principle of General Relativity stating that the laws of physics must be the same in all inertial frames of reference, expressed mathematically as tensors transforming covariantly.

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The statement of the theorem

The principle of General Covariance requires that the physical laws, expressed by an action

S

or field equations, remain invariant under arbitrary coordinate transformations

x^{\mu} \to x'^{\mu}

. Mathematically, this means that the equations must be formulated using tensors

T^{\mu...}_{...}

that transform according to the Jacobian of the transformation, ensuring that the action integral remains invariant:

\delta S = \int \mathcal{L} \sqrt{-g} d^4x \implies \delta S = 0