Tensors

A Manifold is a set of points that locally looks like a bit of Euclidean n-dimensional space. N real coordinates may label the points in this manifold. The manifold can support a differentiable structure. This means that the functions involved in changes of coordinates are differentiable. A manifold differs from a surface in that it can stand alone as a mathematical structure. A surface, on the other hand, is going to be embedded in a higher dimensional Euclidean space.

Example: Compare a 2-sphere  (surface of a 3-D sphere) manifold with the fundamental Euclidean plane  manifold.

compared to

Even though the 2-sphere is a compact manifold, locally it has Euclidean character. Hence, locally it's no different from the  manifold, which goes off to infinity. The topology of the structures is not relevant since a topological statement is a global statement.

Although parts of manifolds can be covered by coordinate systems it is not the case that you can get a suitable coordinate system to cover the entire manifold. What we want is a coordinate system to cover the manifold uniquely. If it doesn't do so then the coordinate system is said to be a degenerate coordinate system. An example of a degenerate coordinate system is the 2-D polar coordinates . When , the variable  is indeterminate. Despite this shortcoming, polar coordinates are used often since they are very useful in certain situations.

To get around problems with coordinate systems at various points of a manifold we adopt a strategy of using more than one set of coordinates. We use the minimum number of distinct set of coordinates that will successfully cover all of the points of the manifold.

The manifold is said to be 'patched' by the different coordinate sets. The set of all necessary coordinate patches is called an atlas. Since manifolds are differentiable structures it is possible to go from one coordinate system patch to another by using overlapping regions of the patches.

In general relativity we want to make global statements about physics independent of the local coordinate patches that are necessary to cover the 4-D spacetime manifold. In order to do this Einstein used tensors to represent physical quantities. Equations written in terms of tensors automatically maintain the same pattern amongst the tensor quantities no matter what coordinate system is being referred to.

We will now introduce the mathematical technology necessary for writing equations in terms of tensors. Consider the following change of coordinates in a n-dimensional manifold.

 (2.1)

This change of coordinates can be written more succinctly as

 (2.2)

where  denote the n functions , and  represents the old coordinates . To get information about how one coordinate system changes with the other we can construct the Jacobian transformation matrix, which gives the following.

 (2.3)

The Jacobian, formed from this matrix, is defined as the determinant of the matrix in (2.3) :

 (2.4)

We assume that for the range of coordinates that we are considering, J is well defined. In that case the inverse transformation can be solved for. It follows from the product rule for determinants that, if we define the Jacobian of the inverse transformation by

 (2.5)

then .

Contravariant tensors

Consider two neighbouring points in the manifold: P and Q . Let these points have coordinates  and , respectively.  The two points define an infinitesimal displacement or infinitesimal vector .  The vector is not to be regarded as free, but as being attached to the point P

The components of this vector in the  -coordinate system, are  which are connected to the  by

 (2.6)

A contravariant tensor of rank 1 is a set of quantities, written  in the  -coordinate system, associated with a point P, which transforms under a change of coordinates according to

 (2.7)

where the transformation matrix is evaluated at P.  The infinitesimal vector  is a special case of (2.7) where the components  are infinitesimal. An example of a vector with finite components is provided by the tangent vector  to the curve .  It is important to distinguish between the actual geometric object like the tangent vector in the following diagram (depicted by an arrow)

and its representation in a particular coordinate system, like the n numbers  in the  -coordinate system and the different numbers  in the  -coordinate system.

We now generalize the definition (2.7) to obtain contravariant tensors of higher rank.  A contravariant tensor of rank 2 is a set of n2 quantities associated with a point P, denoted by  in the  -coordinate system, which transform according to

 (2.8)

The quantities  are the components in the  -coordinate system, the transformation matrices are evaluated at P, and the law involves two dummy indices  and .  An example of such a quantity is provided by the product , say, of two contravariant vectors  and .  The definition of third and higher-order contravariant tensors proceeds in an analogous manner.  An important case is a tensor of rank 0, called a scalar or scalar invariant , which, at P, transforms according to

 (2.9) .

Covariant and mixed tensors

As in the last section we begin by considering the transformation of a prototype quantity.  Let

 (2.10)

be a real-valued function on the manifold, i.e. At every point P in the manifold  produces a real number.  We also assume that  is continuous and differentiable, so that we can obtain the differential coefficients .

Now, using the inverse of the coordinate transformation, the equation (2.10) can be written equivalently as

 (2.11)

Differentiating this with respect to , using the function-of-a-function rule, we obtain

 (2.12)

Then changing the order of the terms, the dummy index, and the free index (from  to  ) gives

 (2.13)

This is the prototype equation for covariant tensors. Notice that it involves the inverse transformation matrix .  Thus, a covariant tensor of rank 1 is a set of quantities, written  in the  -coordinate system, associated with a point P, which transforms according to

 (2.14)

Again, the transformation matrix occurring is assumed to be evaluated at P.  Similarily, we define a covariant tensor of rank 2 by the transformation law

 (2.15)

and so on for higher-rank tensors.  Note the conversion that contravariant tensors have raised indices whereas covariant tensors have lowered indices.  (The way to remember this is that co goes below.)  The fact that the differentials  transform as a contravariant vector explains the convention that the coordinates themselves are written as  rather than , although note that it is only the differentials and not the coordinates which have tensorial character.

We can go on to define mixed tensors in the obvious way.  For example, a mixed tensor of rank 3- one contravariant rank and two covariant rank-- satisfies

 (2.16)

If a mixed tensor has contravariant rank p and covariant rank q, then it is said to have type (p , q).

We now come to the reason why tensors are important in mathematical physics.  Let us illustrate the reason by way of an example.  Suppose we find in one coordinate system that two tensors, Xab and Yab say, are equal, i.e.

 (2.17)

Let us multiply both sides by the matrices  and  and take the implied summations to get

 (2.18)

Since  and  are both covariant tensors of rank 2 it follows that .  In other words, the equation (2.17) holds in any other coordinate system.  In short, a tensor equation that holds in one coordinate system necessarily holds in all coordinate systems.  Thus, although we need to introduce coordinate systems for convenience in tackling particular problems, we wish to work with tensorial equations that are coordinate-independent so that our physics relationships remain unchanged.