In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and theoretical physics

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, an invariant is a property of a system which remains unchanged under some transformation

Transformation (mathematics)

In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

.

Examples

In the current era, the immobility of polaris

Polaris

Polaris |Alpha]] Ursae Minoris, commonly North Star or Pole Star, also Lodestar) is the brightest star in the constellation Ursa Minor. It is very close to the north celestial pole, making it the current northern pole star....

 (the North Star) under the diurnal motion

Diurnal motion

Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle. The time for...

 of the celestial sphere

Celestial sphere

In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...

 is a classical illustration of physical invariance.
Another examples of a physical invariant is the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

 under a Lorentz transformation

Lorentz transformation

In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

  and time

Time

Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

 under a Galilean transformation

Galilean transformation

The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view...

. Such spacetime transformations represent shifts between the reference frames of different observers, and so by Noether's theorem

Noether's theorem

Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

 invariance under a transformation represents a fundamental conservation law

Conservation law

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....

. For example, invariance under translation leads to conservation of momentum, and invariance in time leads to conservation of energy

Conservation of energy

The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

.

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

Symmetry in physics

In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

 and invariants.

Covariance and contravariance generalize the mathematical properties of invariance

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 in tensor mathematics

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

, and are frequently used in electromagnetism

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, special relativity

Special relativity

Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, and general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

.

See also

• General covariance
  General covariance
  In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
  
  
• Invariant (mathematics)
  Invariant (mathematics)
  In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
  
  
• Physical constant
  Physical constant
  A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...
  
  
• Rotational invariance
  Rotational invariance
  In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument...
  
  
• Translational invariance (applied to space and time)
• Principle of Relativity
  Principle of relativity
  In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
  
  is invariance with respect to constant velocity.