General covariance

General covariance is the principle that physical laws can be expressed in a form unchanged under arbitrary smooth changes of coordinates. It is distinct from requiring that all coordinate transformations represent physical symmetries.

Tensor: A mathematical object whose components transform systematically under changes of coordinates or basis. Tensor equations retain their form when coordinate components change.
General relativity: A theory of gravity in which spacetime geometry is dynamical and matter follows its curvature. General covariance is a formal feature of its equations, not the whole physical content of the theory.
Albert Einstein: The physicist who developed special and general relativity and made major contributions to quantum theory. Einstein used general covariance as a guiding requirement while developing general relativity.
Arnowitt–Deser–Misner formalism: A Hamiltonian formulation of general relativity that decomposes spacetime into space and time. Its split makes the constraints associated with coordinate freedom explicit.
Diffeomorphism: A smooth, invertible map between manifolds whose inverse is also smooth. General covariance concerns expressing laws consistently under smooth coordinate changes.
Background independence: The property of a theory whose spacetime geometry is not fixed in advance as an unchanging structure. Unlike general covariance, it concerns whether spacetime geometry itself is dynamical.
Marcel Grossmann: The mathematician and physicist who helped Einstein apply differential geometry to gravitation. Grossmann introduced Einstein to mathematical tools needed for generally covariant gravitational equations.
Cosmological perturbation theory: The study of small departures from a homogeneous cosmological model using perturbative methods. Coordinate freedom creates gauge choices that must be controlled when defining perturbations.
Einstein field equations: Equations relating spacetime curvature to the distribution of energy and momentum. Their tensor form is unchanged by arbitrary smooth coordinate transformations.
Gauge symmetry: A redundancy in a theory’s mathematical description that relates configurations representing the same physical state. It clarifies why coordinate freedom need not be a physical symmetry between distinct states.

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