general covariance

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General covariance is the statement that a physical theory, e.g., Einstein's theory of gravity, is expressed mathematically in a form that is invariant under arbitrary changes of coordinate systems, i.e., diffeomorphisms. For instance, in Special relativity, in the absence of gravity, the worldline x^\alpha(\tau)\, of a particle moving freely through spacetime is described by an inertial observer as

\frac{d^2 x^\alpha}{d\tau^2} = 0\,.

Whereas this equation is specially covariant (invariant under Lorentz transformations), it does not hold true for an accelerating observer, and must be replaced by

\frac{d^2x^\alpha}{d\tau^2} + \Gamma^{\alpha}_{~\mu \nu }\frac{dx^\mu }{d\tau}\frac{dx^\nu }{d\tau} = 0\,,

which is generally covariant, and true for all observers.

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