indices on a tensor density with weight again yields a tensor density of weight Using (2) and (3) one sees that raising and lowering indices using the metric tensor (weight 0) leaves the weight unchanged. Matrix inversion and matrix determinant of tensor densities If is a non-singular matrix and a rank-two tensor density of weight with covariant indices then its matrix inverse will be a rank-two tensor density of weight − with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant. If is a rank-two tensor density of weight with covariant indices then the matrix determinant will have weight where is the number of space-time dimensions. If is a rank-two tensor density of weight with contravariant indices then the matrix determinant will have weight The matrix determinant will have weight General relativity Relation of Jacobian determinant and metric tensor Any non-singular ordinary tensor transforms as where the right-hand side can be viewed as the product of three matrices. Taking the determinant of both sides of the equation (using that the determinant of a matrix product is the product of the determinants), dividing both sides by and taking their square root gives When the tensor is the metric tensor, and is a locally inertial coordinate system where diag(−1,+1,+1,+1), the Minkowski metric, then −1 and so where is the determinant of the metric tensor Use of metric tensor to manipulate tensor densities Consequently, an even tensor density, of weight W, can be written in the form where is an ordinary tensor. In a locally inertial coordinate system, where it will be the case that and will be represented with the same numbers. When using the metric connection (Levi-Civita connection), the covariant derivative of an even tensor density is defined as For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely to the expression that would be appropriate for the covariant derivative of an ordinary tensor. Equivalently, the product rule is obeyed where, for the metric connection, the covariant derivative of any function of is always zero, Examples The expression is a scalar density. By the convention of this article it has a weight of +1. The density of electric current (for example, is the amount of electric charge crossing the 3-volume element divided by that element — do not use the metric in this calculation) is a contravariant vector