Tensor density

In physics , the term “ tensor density” was introduced by Hermann Weyl in order to capture the “difference between quantity and intensity , insofar as it has a physical meaning”: “ The tensors are the intensity and the tensor densities are the quantities ”. After Weyl a tensor assigns a coordinate system , a tensor field in such a way to that it at a change of coordinates with the absolute value of the Jacobian is multiplied. A tensor density of level zero is therefore a scalar density whose integral yields an invariant according to the transformation theorem .

More generally, a weighted tensor density is defined by multiplying by a power of the amount of the functional determinant. The weight is the exponent to this power. (On the other hand, Weyl uses the term tensor (density) with weight in a different meaning: the weight is the exponent in the power of the calibration ratio with which the metric is multiplied when rescaling .). A different definition uses the functional determinant instead of its amount. For even weight, both definitions match. For odd weight, the terms tensor density and pseudotensor density are interchanged, because pseudotensors or pseudotensor densities are multiplied by the sign of the functional determinant. The first definition is used below. (Another variant differs in the sign of the weight.)

definition

A tensor density of weight assigns coordinates to a tensor , with the relationship under a coordinate change ${\ displaystyle G}$ ${\ displaystyle x}$ ${\ displaystyle {\ mathfrak {T}} (x)}$ ${\ displaystyle x \ mapsto x '}$

{\ displaystyle {\ mathfrak {T}} (x ') = \ left | \ det {\ frac {\ partial x} {\ partial x'}} \ right | ^ {G} {\ mathfrak {T}} ( x)}

applies. Let the tensor components with respect to the coordinates be . Then the following transformation law applies when changing coordinates: ${\ displaystyle x}$ ${\ displaystyle {\ mathfrak {T}} _ {\ mu _ {1} \ cdots \ mu _ {m}} ^ {\ nu _ {1} \ cdots \ nu _ {n}}}$

{\ displaystyle {\ mathfrak {T}} {'} _ {\ rho _ {1} \ cdots \ rho _ {m}} ^ {\ sigma _ {1} \ cdots \ sigma _ {n}} = \ left | \ det {\ frac {\ partial x} {\ partial x '}} \ right | ^ {G} {\ frac {\ partial x ^ {\ mu _ {1}}} {\ partial x' ^ {\ rho _ {1}}}} \ cdots {\ frac {\ partial x ^ {\ mu _ {m}}} {\ partial x '^ {\ rho _ {m}}}} {\ frac {\ partial x '^ {\ sigma _ {1}}} {\ partial x ^ {\ nu _ {1}}}} \ cdots {\ frac {\ partial x' ^ {\ sigma _ {n}}} {\ partial x ^ {\ nu _ {n}}}} {\ mathfrak {T}} _ {\ mu _ {1} \ cdots \ mu _ {m}} ^ {\ nu _ {1} \ cdots \ nu _ {n }}}

Examples

A zero weight tensor density is a common tensor field.

Let it be the amount of the determinant of the component matrix of the metric tensor (or more generally of a two-fold covariant tensor). Then, by the product theorem for determinants, a scalar density of weight 2 and a scalar density of weight 1. If a tensor, then a tensor density is weight . Conversely, any tensor density of weight can be written as such a product by putting. ${\ displaystyle g = | \ det {(g _ {\ mu \ nu})} |}$ ${\ displaystyle g}$ ${\ displaystyle {\ sqrt {g}}}$ ${\ displaystyle T}$ ${\ displaystyle {\ mathfrak {T}} = {\ sqrt {g}} ^ {G} T}$ ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle T = {\ sqrt {g}} ^ {(- G)} {\ mathfrak {T}}}$

An example of a pseudotensor density of weight −1 is the Levi-Civita tensor .

Individual evidence

↑ ^a ^b Hermann Weyl: Space - Time - Matter . Lectures on general relativity. 6th edition. Springer, Berlin / Heidelberg / New York 1970, ISBN 3-540-05039-6 , pp. 110 . (Tensor density with weight: p. 127.)
↑ ^a ^b Ernst Schmutzer: Relativistic Physics . Classic theory. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig 1986, LCCN 75-401751 , A I. § 14. Tensordichten, p. 132 . (Pseudotensors: p. 121.)
^ ^A ^b Hans Stephani: Relativity . An Introduction to Special and General Relativity. 3. Edition. Cambridge University Press, Cambridge, UK 2004, ISBN 0-521-81185-6 , pp. 119 .
^ Bernard F. Schutz: Geometrical methods of mathematical physics . Cambridge University Press, Cambridge 1980, ISBN 0-521-23271-6 , pp. 128 .
↑ Steven Weinberg: Gravitation and cosmology . Principles and applications of the general theory of relativity. John Wiley & Sons, New York 1972, ISBN 0-471-92567-5 , pp. 98 .

literature

Erwin Schrödinger : The structure of space-time. Edited and translated by Jürgen Audretsch. Reprographic reprint of the 1987 edition. Wissenschaftliche Buchgesellschaft, Darmstadt 1993, ISBN 3-534-02282-3 (English original title: Space-Time Structure ).

[Weyl-1] Hermann Weyl: Space - Time - Matter . Lectures on general relativity. 6th edition. Springer, Berlin / Heidelberg / New York 1970, ISBN 3-540-05039-6 , pp. 110 . (Tensor density with weight: p. 127.)

[Schmutzer-2] Ernst Schmutzer: Relativistic Physics . Classic theory. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig 1986, LCCN 75-401751 , A I. § 14. Tensordichten, p. 132 . (Pseudotensors: p. 121.)

[Stephani-3] A ^b Hans Stephani: Relativity . An Introduction to Special and General Relativity. 3. Edition. Cambridge University Press, Cambridge, UK 2004, ISBN 0-521-81185-6 , pp. 119 .

[Schutz-4] Bernard F. Schutz: Geometrical methods of mathematical physics . Cambridge University Press, Cambridge 1980, ISBN 0-521-23271-6 , pp. 128 .

[Weinberg-5] Steven Weinberg: Gravitation and cosmology . Principles and applications of the general theory of relativity. John Wiley & Sons, New York 1972, ISBN 0-471-92567-5 , pp. 98 .