Notation question from differential geometry and PDE Let Greek indices denote indices that range over $0,1,\dots,d$. I'm looking at $\mathbb R^d \times \mathbb R$ (here I'm looking at the wave equation, so there are space and time indices). Let $g$ be a function. What do the notations $\partial^\alpha g \partial_\alpha g$ and $\partial_\beta g\partial^\beta g$ mean and how are they different? I know that the repeated indices means that you are summing over all values of the indices, so $\partial^\alpha g \partial_\alpha g = \sum_{\alpha=0}^d \partial^\alpha g \partial_\alpha g$ and similarly $\partial_\beta g\partial^\beta g = \sum_{\beta=0}^d \partial_\beta g\partial^\beta g$. But what is the difference between these two? It has something to do with the Minkowski metric. 
One of these is probably $-g_t^2 + |\nabla g|^2$, but what is the other one?
 A: Basically $\partial^\beta$ are the components of the dual vector to $\partial_\beta$. In Minkowski space the sum $\partial_\beta g\partial^\beta g$ is indeed $-g_t^2 + |\nabla g|^2$. 
And because $\alpha$ and $\beta$ are dummy variables the two sums $\partial_\beta g\partial^\beta g$ and $\partial^\alpha g\partial_\alpha g$ are the same, the product of functions is commutative.
When we talk about the dual of $\partial_\beta$ we mean as follows:
Let's say that we are given a smooth manifold of dimension n, if we look at the tangent space of a point p in the manifold we can prove that it is isomorphic to the derivation space which is spanned by $\partial_\beta$ therefore we can treat the tangent space as the vector space spanned by $\partial_\beta$ which has dimension n.
For this vector space we can define the dual space as in linear algebra which is spanned by $\partial^\beta$ which satisfy $\partial^\alpha(\partial_\beta) = \delta^\alpha{}_\beta$.
If you have a metric g (or a pseudo metric as in general relativity), which means the tangent space is an inner product space, then the components of dual vectors are related by the formula: 
$$v_\alpha=g_{\alpha\beta} v^\beta, v^\alpha=g^{\alpha\beta} v_\beta$$
where subscript components are for the dual vector and superscript for the vector in the tangent space.
