What does covariant mean?
"Covariant" is a technical term used chiefly in mathematics and physics to describe quantities or equations whose behavior under a change of coordinates follows a systematic, predictable rule. When a coordinate system is transformed — say, from Cartesian to polar coordinates — a covariant object changes in step with the basis vectors themselves, preserving the underlying meaning of an equation. This property lies at the heart of tensor analysis and was central to Einstein's formulation of general relativity, where he demanded that the laws of physics take a generally covariant form, holding true for every observer regardless of their frame of reference. Beyond this specialized sense, "covariant" can more loosely describe things that vary together in a corresponding way. It stands in contrast with "contravariant," which describes objects transforming by the inverse rule, and with "invariant," describing what does not change at all. Related forms include the noun "covariance" and the verb "covary."
(Mathematics and physics) Transforming under a change of coordinates in the same manner as the basis vectors; (more generally) varying in a corresponding or dependent way with something else.
"The field equations are covariant under all smooth transformations of the coordinates."
Almost exclusively used in technical scientific writing.
(Mathematics and physics) A quantity, especially a tensor component, that is covariant rather than contravariant.
"In this notation, indices placed low indicate covariant components."
Usually encountered in plural form, e.g. "covariants," within tensor analysis.
Used only when "covariant" functions as a noun in mathematical contexts referring to quantities with covariant transformation properties.
"The tensor's covariants transform according to the Jacobian of the coordinate change."
Einstein made 'general covariance' the guiding principle of general relativity — the idea that the laws of physics must look the same no matter who observes them.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of covariant
Derived from the prefix "co-" (from Latin "cum," meaning "together") combined with "variant," from Latin "variare" ("to change") via "varius" ("changing, diverse"). The term arose in nineteenth-century mathematics alongside the development of tensor calculus, gaining prominence through its use in early twentieth-century relativity theory. It shares its Latin root with words such as "vary," "various," and "variance."
Related word forms
How covariant is actually used
"Covariant" belongs almost exclusively to formal scientific and mathematical register; it is rarely used outside technical contexts. In statistics, "covariance" (the derived noun for joint variability of random variables) is far more common than "covariant" itself. The contrast with "contravariant" and "invariant" is essential to its precise meaning in tensor analysis and relativity.
Easily confused with covariant
A covariant quantity transforms in the same manner as the coordinate basis vectors, whereas a contravariant quantity transforms in the opposite (inverse) manner.
An invariant quantity remains unchanged under a transformation, while a covariant quantity changes but does so systematically according to a defined rule.