Tensor density
Encyclopedia
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.
Note that these classifications elucidate the different ways that tensor densities transform under the somewhat pathological 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.
Some authors use a sign convention for weights that is the negation of that presented here. In this article we have chosen the convention that assigns a weight of +2 rather than −2 to the determinant of the metric tensor expressed with covariant indices.
where
is the order-2 tensor density in the 
coordinate system, 
is the transformed tensor density in the 
coordinate system; and we use the Jacobian determinant. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when W is an integer. (However, see even and odd tensor densities below.)
We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-2 pseudotensor density of weight W transforms as
where sgn( ) is a function that returns +1 when its argument is positive or −1 when its argument is negative.
When W is an even integer the above formula for an (authentic) tensor density can be rewritten as
This formula has the advantage of being well defined even when W is an arbitrary real number.
Similarly, when W is an odd integer the formula for an (authentic) tensor density can be rewritten as
This formula has the advantage of being well defined even when W is an arbitrary real number.
If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is +1.
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 T 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 
.
, 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 which 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,
is a scalar density. By the convention of this article it has a weight of +1, though, e.g., Weinberg uses a convention that gives it a weight of −1.
The density of electric current
(e.g., 
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 density of weight +1. It is often written as 
, where 
is an absolute tensor.
The density of Lorentz force
(i.e., the linear momentum transferred from the electromagnetic field to matter within a 4-volume element 
divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.
The determinant
of the metric tensor, g = det(gμν), is an (even) authentic scalar density of weight +2.
In N-dimensional space-time, the Levi-Civita symbol
may be regarded as either a rank-N covariant (odd) authentic tensor density of weight −1 (εα1…αN) or a rank-N contravariant (odd) authentic tensor density of weight +1 (εα1…αN). Notice that the Levi-Civita symbol (so regarded) does not obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that
but in general relativity, where
is always negative, this is never equal to 
.
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
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 transform under the somewhat pathological 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.
Some authors use a sign convention for weights that is the negation of that presented here. In this article we have chosen the convention that assigns a weight of +2 rather than −2 to the determinant of the metric tensor expressed with covariant indices.
Tensor and pseudotensor densities
For example, a mixed rank-2 (authentic) tensor density of weight W transforms aswhere
is the order-2 tensor density in the coordinate system, is the transformed tensor density in the coordinate system; and we use the Jacobian determinant. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when W is an integer. (However, see even and odd tensor densities below.)We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-2 pseudotensor density of weight W transforms as
where sgn( ) is a function that returns +1 when its argument is positive or −1 when its argument is negative.
Even and odd tensor densities
The transformations for even and odd tensor densities have the benefit of being well defined even when W is not an integer. Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2.When W is an even integer the above formula for an (authentic) tensor density can be rewritten as
This formula has the advantage of being well defined even when W is an arbitrary real number.
Similarly, when W is an odd integer the formula for an (authentic) tensor density can be rewritten as
This formula has the advantage of being well defined even when W is an arbitrary real number.
Weights of zero and one
A tensor density of any type that has weight zero is also called an absolute tensor. An (even) authentic tensor density of weight zero is also called an ordinary tensor.If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is +1.
Multiplication of tensor densities
A product of tensor densities of any types will have a weight equal to the sum of the weights of the factors. A product of authentic tensor densities and pseudotensor densities will be an authentic tensor density when an even number of the factors are pseudotensor densities; it will be a pseudotensor density when an odd number of the factors are pseudotensor densities. Similarly, a product of even tensor densities and odd tensor densities will be an even tensor density when an even number of the factors are odd tensor densities; it will be an odd tensor density when an odd number of the factors are odd tensor densities.Relation of Jacobian determinant and metric tensor
A non-singular ordinary tensor transforms aswhere 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 givesWhen the tensor T is the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
,
, and is a locally inertial coordinate system where diag(−1,+1,+1,+1), the Minkowski metric, then −1 and sowhere
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 formwhere
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
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
), the covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
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 which 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, though, e.g., Weinberg uses a convention that gives it a weight of −1. The density of electric current
(e.g., 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 density of weight +1. It is often written as , where is an absolute tensor.The density of Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
(i.e., the linear momentum transferred from the electromagnetic field to matter within a 4-volume element divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1. The determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the metric tensor, g = det(gμν), is an (even) authentic scalar density of weight +2.
In N-dimensional space-time, the Levi-Civita symbol
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
may be regarded as either a rank-N covariant (odd) authentic tensor density of weight −1 (εα1…αN) or a rank-N contravariant (odd) authentic tensor density of weight +1 (εα1…αN). Notice that the Levi-Civita symbol (so regarded) does not obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that
but in general relativity, where
is always negative, this is never equal to .See also
- relative scalar
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