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In physics, the '''principle of covariance''' emphasizes formulation of physical laws using only those physical quantities the measurements of which the observers in different [[frame of reference|frames of reference]] could unambiguously correlate.

Mathematically, the physical quantities must transform ''covariantly'', that is, under a certain [[group representations|representation]] of the [[group (mathematics)|group]] of [[coordinate transformations]] between admissible frames of reference of the physical theory.<ref name=Post>E.J.Post,
''Formal Structure of Electromagnetics: General Covariance and Electromagnetics'', Dover publications</ref> This group is referred to as the [[covariance group]].

Principle of covariance does not require '''invariance''' of the physical laws under the group of admissible transformations although in most cases the equations are actually invariant. However, in the theory of [[weak interactions]] the equations are not invariant under reflections (but are, of course, still covariant).

== Covariance in Newtonian mechanics ==
In [[Newtonian mechanics]] the admissible frames of reference are [[inertial frames]] with relative velocities much smaller than the [[speed of light]]. Time is then absolute and the transformations between admissible frames of references are [[Galilean transformations]] which (together with rotations, translations, and reflections) form the [[Galilean group]]. The covariant physical quantities are [[Euclidean space|Euclidean]] scalars, [[Euclidean vector|vectors]], and [[tensors]]. An example of a covariant equation is [[Newton's second law]],
:<math>
m\frac{d\vec{v}}{dt}=\vec{F},
</math>
where the covariant quantities are the mass <math>m</math> of a moving body (scalar), the momentum <math>\vec{p}</math> of the body (vector), the force <math>\vec{F}</math> acting on the body, and the invariant time <math>t</math>.

== Covariance in special relativity ==
In [[special relativity]] the admissible frames of reference are all inertial frames. The transformations between frames are the [[Lorentz transformations]] which (together with the rotations, translations, and reflections) form the [[Poincaré group]]. The covariant quantities are four-scalars, [[four-vectors]] etc., of the [[Minkowski space]] (and also more complicated objects like [[bispinor]]s and others). An example of a covariant equation is the [[Lorentz force]] equation of motion of a charged particle in an electromagnetic field (a generalization of the second Newton's law)
:<math>
m\frac{du^a}{ds}=qF^{ab}u_b,
</math>
where <math>m</math> and <math>q</math> are the mass and charge of the particle (invariant 4-scalars); <math>ds</math> is the [[invariant interval]] (4-scalar); <math>u^a</math> is the [[4-velocity]] (4-vector); and <math>F^{ab}</math> is the [[electromagnetic tensor|electromagnetic field strength tensor]] (4-tensor).

== Covariance in general relativity ==
{{Empty section|date=July 2010}}

== See also ==
* [[Principle of relativity]]
* [[General covariance]]

== References ==
<references />

{{DEFAULTSORT:Principle Of Covariance}}
[[Category:Concepts in physics]]

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