Principle of covariance

The principle that physical laws should be expressed in equations whose form remains unchanged under coordinate transformations. It guides the use of coordinate-independent formulations, especially in general relativity.

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Coordinate system: A scheme that assigns numbers to locations or events so they can be described mathematically. Covariance concerns how physical equations are represented when this scheme changes.
Covariant derivative: A derivative that accounts for how vector and tensor components vary across curved spaces. It lets differential equations remain tensorial in curved spacetime.
General relativity: A theory of gravitation in which spacetime geometry is dynamical and shaped by matter and energy. Its equations are formulated covariantly under arbitrary coordinate transformations.
Special relativity: A theory of space, time, and motion in inertial frames, with the speed of light invariant. Its standard formulation emphasizes Lorentz transformations, not arbitrary coordinate transformations.
Coordinate transformation: A rule that converts coordinates describing the same point or event from one coordinate system to another. The principle specifies how equations should behave when coordinates are transformed.
Metric tensor: A tensor that defines distances, angles, and spacetime intervals in a geometric space. It enters covariant equations and relates coordinate components to geometric measurements.
Equivalence principle: The principle that gravitational effects locally resemble effects of acceleration, in specified physical regimes. It helped motivate describing gravity through spacetime geometry rather than as an ordinary force.
Lorentz covariance: The property that equations retain their form under Lorentz transformations between inertial frames. It imposes a narrower transformation requirement than general covariance.
Tensor: A geometric object whose components transform according to defined rules under changes of coordinates. Tensor equations provide a standard way to express covariant physical laws.
Christoffel symbols: Coordinate-dependent coefficients that describe how coordinate basis vectors vary across a manifold. They appear in covariant derivatives and compensate for changing coordinates and bases.