Covariance (physics)

Covariance has two different but closely related meanings in physics . On the one hand, in the tensor calculus, there is the distinction between covariant and contravariant quantities, and on the other hand there is the covariance of theories or their underlying equations.

Covariant and contravariant

• A transformation behavior is called covariant , in which the base vectors and the vectors (quantities) represented therein transform in the same way.
• A transformation behavior is called contravariant if the base vectors and the vectors (quantities) represented therein transform in different ways.

The covariant transformation behavior guarantees the shape retention of equations when changing the reference system ( coordinate system ) or with group transformations. These statements also apply to the tensor notation.

A theory or equation is covariant with respect to a group transformation if the form of the equations remains unchanged after the occurring quantities have been subjected to one of the transformations of the group (see also invariance ).

Examples of covariance

Under Galileo transformations, the acceleration and the force transform in the Newtonian equations of motion in the same sense as the position vectors . Therefore Newton's equations of motion and thus classical mechanics are covariant with respect to the group of Galileo transformations.

In the same sense, the Einstein equations of gravity in general relativity are covariant under any (nonlinear smooth) coordinate transformations.

Likewise, the Dirac equation of quantum electrodynamics is covariant under the group of linear Lorentz transformations .

The left side of the Klein-Gordon equation for a scalar field does not change under Lorentz transformations, it is more specifically invariant or scalar .

Tensor calculus

Transform in the tensor calculus

• the covariant parts of a tensor like the coordinates of a linear form
• the contravariants like the coordinate tuples of a position vector.

As a result, co- and contravariant quantities are zero after a transformation if and only if they were zero before the transformation.

notation

• The coordinates of covariant vectors (or of linear forms) are written with lower indices ${\displaystyle a_{m}}$
• The coordinates of contravariant vectors, which transform linearly like the coordinates of the position vector, are written with upper indices . ${\displaystyle a^{m}}$

According to Einstein's summation convention , every term in an equation must have the same index.

Mathematical representation

In a narrower sense of the word, covariant in mathematical physics describes quantities that transform like differential forms. These covariant quantities form a vector space on which a group of linear transformations act. ${\displaystyle P}$ ${\displaystyle {\mathcal {V}}}$

The set of linear mappings of the covariant quantities into the real numbers

${\displaystyle Q:P\mapsto Q(P)\in \mathbb {R} \ ,\quad Q(a\,P+b\,{\tilde {P}})=a\,Q(P)+b\,Q({\tilde {P}})}$

forms the vector space to be dual . Let us write the transformed, covariant quantities with a matrix as ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}^{*}}$ ${\displaystyle P^{\prime }}$ ${\displaystyle \Lambda }$

${\displaystyle P^{\prime }=\Lambda \,P\,,}$

then defines the contravariant or contradictory law of transformation of the dual space ${\displaystyle Q^{\prime }(P^{\prime })=Q(P)}$

${\displaystyle Q^{\prime }=\Lambda ^{-1\,{\text{T}}}\,Q\,.}$

Because of

${\displaystyle (\Lambda _{2})^{-1\,{\text{T}}}\,(\Lambda _{1})^{-1\,{\text{T}}}=(\Lambda _{2}\,\Lambda _{1})^{-1\,{\text{T}}}}$

the contravariant transformation of the same group link as the covariant transformation is sufficient.

Tensors of the times the tensor product of the times the tensor product of hot covariant fold and fold contravariantly. ${\displaystyle u}$ ${\displaystyle {\mathcal {V}}^{*}}$ ${\displaystyle o}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle u}$ ${\displaystyle o}$

In index notation, the index position with indices below and above makes it clear whether it is the components of a covariant or a contravariant vector,

${\displaystyle P_{m}^{\prime }=\sum _{n}\Lambda _{m}{}^{n}\,P_{n}\ ,\quad Q^{\prime \,m}=\sum _{r}(\Lambda ^{-1\,{\text{T}}})^{m}{}_{r}\,Q^{\,r}\,.}$

The calculation steps show that applies ${\displaystyle Q^{\prime }(P^{\prime })=Q(P)}$

${\displaystyle {\begin{aligned}Q^{\prime }(P^{\prime })&=\sum _{m}Q^{\prime \,m}\,P_{m}^{\prime }\\&=\sum _{mnr}(\Lambda ^{-1\,{\text{T}}})^{m}{}_{r}\,Q^{\,r}\,\Lambda _{m}{}^{n}\,P_{n}\\&=\sum _{mnr}\Lambda ^{-1}{}_{r}{}^{m}\,\Lambda _{m}{}^{n}\,Q^{\,r}\,P_{n}\\&=\sum _{nr}\delta _{r}{}^{n}\,Q^{\,r}\,P_{n}\\&=\sum _{n}Q^{\,n}\,P_{n}\\&=Q(P)\,.\end{aligned}}}$

Index drag

Is the contravariant transformation law equivalent to the covariant and applies to all of the transformation group ${\displaystyle \Lambda }$

${\displaystyle \Lambda ^{-1\,{\text{T}}}=\eta \,\Lambda \,\eta ^{-1}}$

with an invertible , symmetrical matrix , then the transformation group is due to ${\displaystyle \eta =\eta ^{\text{T}}}$

${\displaystyle \Lambda ^{\text{T}}\,\eta \,\Lambda =\eta }$

a subgroup of the orthogonal group that leaves the symmetrical bilinear form invariant. Then define a contravariant vector if is a covariant vector. In index notation, the components are written as abbreviated ${\displaystyle (P,{\tilde {P}})=P^{\text{T}}\,\eta \,{\tilde {P}}}$ ${\displaystyle \eta \,P}$ ${\displaystyle P}$ ${\displaystyle \eta \,P}$

${\displaystyle P^{m}=\sum _{n}\eta ^{mn}\,P_{n}\,.}$

Then the opposite applies

${\displaystyle P_{m}=\sum _{n}\eta ^{-1}{}_{mn}\,P^{n}\,.}$

This relationship between the components of the covariant vector and the contravariant vector is called index pulling or raising or lowering . ${\displaystyle P}$ ${\displaystyle \eta P}$

Is the contravariant transformation law equivalent to the covariant and applies to all of the transformation group ${\displaystyle \Lambda }$

${\displaystyle \Lambda ^{-1\,{\text{T}}}=\epsilon \,\Lambda \,\epsilon ^{-1}}$

with an invertible, antisymmetric matrix , then the transformation group is due to ${\displaystyle \epsilon =-\epsilon ^{\text{T}}}$

${\displaystyle \Lambda ^{\text{T}}\,\epsilon \,\Lambda =\epsilon }$

a subgroup of the symplectic group that leaves the antisymmetric bilinear form invariant. Then define a contravariant vector if is a covariant vector. In index notation you can abbreviate the components of ${\displaystyle \langle P,{\tilde {P}}\rangle =P^{\text{T}}\,\epsilon \,{\tilde {P}}}$ ${\displaystyle \epsilon \,P}$ ${\displaystyle P}$ ${\displaystyle \epsilon \,P}$

${\displaystyle P^{m}=\sum _{n}\epsilon ^{mn}\,P_{n}}$

write. Then the opposite applies

${\displaystyle P_{m}=\sum _{n}\epsilon ^{-1}{}_{mn}\,P^{n}\,.}$

This relationship between the components of the covariant vector and the contravariant vector defines the indexing of vectors that transform under the symplectic group. ${\displaystyle P}$ ${\displaystyle \epsilon P}$

See also

• Covariant derivative
• Minkowski room

Books

• Peter Szekeres, A Course in Modern Mathematical Physics , Cambridge University Press, New York, 2004 ISBN 0-521-82960-7

Web links

• Norbert Dragon, keywords and additions to calculation methods in physics (PDF; 1.9 MB)

Individual evidence

1. ↑ James Bjorken and Sidney Drell: Relativistische Quantenmechanik , BI-Wissenschaftsverlag, Mannheim, 1990, (BI-Hochschultaschenbuch Volume 98), ISBN 3-411-00098-8 , Chapter 2.