Lorentz covariance

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The mathematical term Lorentz covariance is a property of the underlying manifold of a system that is examined within the framework of the theory of relativity .

On manifolds is "covariant" and "contra-variant" refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors can be Lorentz covariant quantities.

The local Lorentz covariance, which results from the general theory of relativity, refers to the Lorentz covariance, which is only applied locally at each point in an infinitesimal region of spacetime. There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance.

definition

Lorentz covariance is understood to mean two different but closely related meanings:

• A physical quantity is called a Lorentz covariant if it transforms under a given representation of the Lorentz group . According to the representation theory of the Lorentz group, such quantities can be scalars , four-vectors, four-tensors or spinors . In particular, a Lorentz covariant scalar, which is also invariant under Lorentz transformations , is called a Lorentz scalar.

• An equation is called a Lorentz covariant if it can be expressed in Lorentz covariant quantities. The key property of such equations is that they hold in any inertial frame when they hold in an inertial frame . This condition is a requirement according to the principle of relativity .

Examples

In general, the transformation behavior of a Lorentz tensor can be identified by its rank, which is the number of free indices it has. A non-indexed quantity is a scalar, a simply-indexed quantity is a vector. Some special tensors that appear in physics are listed below.

The sign convention of the Minkowski metric is used throughout the article. ${\displaystyle \eta =\operatorname {diag} (1,-1,-1,-1)}$

Lorentz angelfish

• The proper time is a Lorentz scalar: . ${\displaystyle \mathrm {d} \tau ^{2}={\frac {\mathrm {d} s^{2}}{c^{2}}}}$

Quad vectors

• The Viererort is a Lorentz covariant four-vector: . ${\displaystyle x^{\alpha }}$ ${\displaystyle x^{\alpha }=(ct,x,y,z)}$
• The four-velocity is also a covariant four-vector: . ${\displaystyle u^{\mu }}$ ${\displaystyle u^{\mu }={\frac {\mathrm {d} x^{\mu }}{\mathrm {d} \tau }}}$
• The four-fold gradient is a Lorentz-covariant four-vector: ${\displaystyle \partial ^{\alpha }}$ ${\displaystyle \partial ^{\alpha }=\left(-{\frac {1}{c}}{\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right).}$
• The four-potential of electrodynamics is a Lorentz covariant four-vector: . ${\displaystyle A^{\mu }=\left({\frac {\phi }{c}},{\vec {A}}\right)}$

Quad tensors

In addition to four-scalars and four-vectors, there are also Lorentz covariant four-tensors. Examples are:

• The Minkowski metric (the metric of a flat space) . ${\displaystyle \eta _{\mu \nu }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$
• The field strength tensor of electrodynamics: . ${\displaystyle F_{\mu \nu }:=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$
• The dual field strength tensor of electrodynamics: ${\displaystyle {\hat {F}}^{\mu \nu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }F_{\mu \nu }}$

Individual evidence

1. ↑ Albert Einstein: On the electrodynamics of moving bodies . Vieweg + Teubner Verlag, Wiesbaden 1922, p. 26-50 .