Let us consider a general coordinate transformation:

n is the dimension of space.

We have:

In analogy with this we say that V^a is a contravariant vector if under (1) it transforms like (2), i.e.

Similarly, let us consider the gradient of a scalar function. We say that A(x) is a scalar function if:A'(x') = A(x). It follows that:

In analogy with this transformation law, we say that U_a is a covariant vector if under a change of variables (1), we have:

Tensor Product

Consider two covariant vectors, U_a,V_a. We have:

We say that a set of functions T_ab that transforms under (1) as (6) is a covariant tensor of rank 2. That is:

We call S_ab = U_aV_b the tensorial product of the two covariant vectors U_a and V_b.

Generally we says that a set of functions T

The tensor product of two arbitrary tensors U

is a (p+r) covariant and (q+s) contravariant tensor.

Contraction

is a scalar.

NOTATION(Einstein): Repeated indices in a monomial are summed from 1 to n.

Proof:

In general the contraction of an upper index with a lower index in a p-covariant and q-contravariant tensor produces a (p-1) covariant and (q-1) contravariant tensor.

Notice that contraction of different indices, produces different tensors.

Some important tensors

Kronecker delta:δ^a_b is a one covariant, one contravariant tensor. Moreover it is an invariant tensor, because it has the same components in all coordinate systems.

Proof:

Symmetric and antisymmetric tensors.

Consider T^c_ab a tensor. Then: S^c_ab = T^c_ab + T^c_ba and A^c_ab = T^c_ab - T^c_ba are tensors, called the symmetric and antisymmetric component of T^c_ab.

Proof:

The proof is similar for the antisymmetric part.Notice that this operation can be applied to any pair of indices. Iterative application of the operation will produce tensors that will form representations of the permutation group S_n

Pseudotensors

We define a pseudotensor of weight r a set of functions Db1….bqa1….ap that under (1) transform as follows:

where J is the jacobian of the transformation (1). i.e.

r=0, tensor

r=1, pseudotensor o tensor density

r=-1, tensor capacity

The Levi-Civita symbol

Here sgn is the sign of the permutation in brackets. If some of the indices are repeated, it gives zero.

Use the determinant identity:

or

It follows that ε^a₁….a_n is a pseudotensor of weight 1.

Similarly, we can prove that:

so ε_a1….anis a pseudotensor of weight -1.

An important identity

We have that:

Proof:

We must have: i₁ = σ(1)….,i_n = σ(n) and j₁ = λ(1),….j_n = λ(n)for some permutations σ and λ, otherwise the determinant vanishes (either two columns or two rows are equal). Then, we get:

Covariant Derivative

In cartesian coordinates, the partial derivative of a tensor is a tensor. This is no longer true in a general space. In order to define parallel transport we must provide for more structure.

We assume the space is provided with a set of n³ functions Γⁱ_jk which transformation properties under (1) will be defined below:

Let Aⁱ be a contravariant vector. We define the covariant derivative of Aⁱ by:

NOTATION: A ⁱ _,j =

We want the covariant derivative to be a tensor under (1). That is:

comparing LHS and RHS, we get:

The set of quantities Γⁱ_jk that transforms under (1) according to (19) is called an affine connection and the spaces provided with it, spaces with affine connection.

We postulate that the covariant derivative satisfies Leibnitz's rule:

Consider Aⁱ and B_j, then S = AⁱB_i is a scalar. For a scalar, the covariant derivative coincides with the ordinary partial derivative. We have:

That is:

This is true for any Aⁱ. It follows that:

In general:

Torsion and Curvature

Let us consider a space with an affine connection Γ

We have:

T is a tensor, called the torsion tensor.

Additionally, consider:

where:

is the curvature tensor.