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tensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another (see classical treatment of tensors), except that it is additionally multiplied or ''weighted'' by a power of the Jacobian determinant of the coordinate transition function or its absolute value. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.
==Definition==
Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article. Other authors classify them differently, into the types called even tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd.
Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-''reversing'' coordinate transformations. Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-''preserving'' coordinate transformations.
In this article we have chosen the convention that assigns a weight of +2 to the determinant of the metric tensor expressed with covariant indices. With this choice, classical densities, like charge density, will be represented by tensor densities of weight +1. Some authors use a sign convention for weights that is the negation of that presented here.〔E.g. pp 98. The chosen convention involves in the formulae below the Jacobian determinant of the inverse transition , while the opposite convention considers the forward transition resulting in a flip of sign of the weight.〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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