Invariant Integrals and Tensor Densities

 

We would like to be able to integrate a quantity over a particular range of coordinate values in such a way that the integrand gives the same value in any other generalized coordinate system. If the integrand is a pure scalar quantity, then this is easily achieved because of the way that scalar quantities transform. Let  be such a pure scalar quantity. The transformation law that it obeys when a new primed coordinate system is introduced is as follows:

 

(9.1)

 

 

Say that we are considering the scalar quantity  at two distinct points . We can sum the scalar function evaluated at the two distinct points in a new primed coordinate system such that the following relation holds:

 

(9.2)

 

Here  are the same points in the new coordinate system. Unlike tensors of higher rank, a scalar field can be evaluated at two different points and still be a scalar field.  There should be no problem making this behavior hold if we go to infinitesimal sums.

(9.3)

When we are integrating over coordinate ranges we want the following integral invariance to hold.

 

(9.4)  

 

The word 'Quantity' is meant to represent a tensor of some general type. However, with integrands of this pattern we run into a problem in that what we are integrating is not necessarily a pure tensor quantity. We know that the 4-D volume element  transforms according to the following Jacobian relation

 

(9.5)

.

 

This relation implies that the differential element  transforms in a funny way. It's not transforming like a scalar quantity and it's not transforming like a vector quantity. This differential element transforms according to a rule that's similar to tensors but is different in that a power of the transformation Jacobian comes into the transformation. A new set of geometric quantities called Tensor Densities can be defined in an analogous manner to tensors but the transformations involve powers of the transformation Jacobian. A general definition of the tensor density can be written in the following way.

 

Tensor density: A tensor density,  , of weight  transforms like a tensor except that the W^th power of the Jacobian appears as a factor with the pattern shown below.

 

(9.6)

 

Since the differential element  transforms according to equation (9.5) with the pattern

 

(9.7)

 

then  must be a scalar density of weight . The integrand in (9.4) will be a scalar only if the factor labelled 'Quantity' is a tensor density of weight . To make integrals be independent of the coordinates, the integrand is multiplied by the square root of the metric determinant as shown in the following expression.

                                                                       

 

(9.8)  

 

This works since the metric transforms according to

(9.9)

Since the right hand side of this equation is essentially the product of three matrices multiplied together, we can use the rule for the product of matrix determinants to give

(9.10)

The value g is negative for an indefinite metric so when we take the square root of this relation we insert a minus sign and the result is

 

(9.11)

Thus  is a scalar density of weight . We then see that the integral invariance given by (9.8) works since we have made

(9.12)

and therefore (9.3) must be realized.

 

The covariant derivative of a tensor density has the following pattern

 

 

(9.13)  

 

For example, the covariant derivative of a vector density  has the form

 

 

(9.14) 

 

For the special case when  this leads to the important divergence equation

 

(9.15)

 

In terms of a tensor density formed from multiplying a tensor  by , this divergence expression becomes

(9.16)

 

 

It can be shown that the metric determinant, which acts as a scalar density of weight 2, satisfies the following relations.

 

 

(9.17)

and

(9.18)

 

Relations (9.16), (9.17), and (9.18) turn out to be of great use in the Lagrangian formulation of general relativity.

 

 

 

 

 

 

15/10/2002 3:07 PM