In mathematics and theoretical physics, an **invariant** is a property of a system which remains unchanged under some transformation.

## Contents

## Examples

In the current era, the immobility of Polaris (the North Star) under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.

For example the rule describing Newton's force of gravity between two chunks of matter is the same whether they are in this galaxy or another (translational invariance in space). It is also the same today as it was a million years ago (translational invariance in time). The law does not work differently depending on whether one chunk is east or north of the other one (rotational invariance). Nor does the law have to be changed depending on whether you measure the force between the two chunks in a railroad station, or do the same experiment with the two chunks on a uniformly moving train (principle of relativity).

— David Mermin:It's About Time - Understanding Einstein's Relativity, Chapter 1

Another example of a physical invariant is the speed of light under a Lorentz transformation^{[1]} and time under a Galilean transformation. Such spacetime transformations represent shifts between the reference frames of different observers.

By Noether's theorem invariance of the action of a physical system under a continuous symmetry represents a fundamental conservation law. For example, invariance under translation leads to conservation of momentum, and invariance in time leads to conservation of energy.

Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching from rectangular coordinates to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant.

## Importance

Invariants are important in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants.

Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics, and are frequently used in electromagnetism, special relativity, and general relativity.

## See also

- Casimir operator
- Charge (physics)
- Conservation law
- Conserved quantity
- General covariance
- Eigenvalues and eigenvectors
- Invariants of tensors
- Killing form
- Physical constant
- Symmetry (physics)
- Weyl transformation

## References

**^**French, A.P. (1968).*Special Relativity*. W. W. Norton & Company. ISBN 0-393-09793-5.