Generally speaking if you have a tensor $T$ on a manifold, and if you have a collection (of usually coordinate) vector fields $e_1, \cdots, e_n$ the "index notation" for $T$ is (lets assume for a moment $T$ is bilinear):
$$T_{ij} = T(e_i,e_j)$$
meaning $T_{ij}$ is a real-valued function for all $i$, and $j$. $T_{ij}$ is defined wherever the vector fields $\{ e_i : i = 1,2,\cdots n\}$ are defined. On a manifold with a metric (meaning an inner product on every tangent space), it is typical to define
$$g_{ij} = \langle e_i, e_j \rangle$$
where $\langle \cdot, \cdot \rangle$ is the inner product on the tangent spaces.
If the tensor takes something other than two vectors as input, for example the Riemann curvature tensor is sometimes thought of as a bilinear function from the tangent space to the space of skew-adjoint linear transformations of that tangent space, i.e. at every point $p$ of the manifold it is bilinear $T_p N \oplus T_p N \to Hom(T_p N, T_p N)$ taking values in the skew-adjoint maps (with respect to the inner product). So given $e_i, e_j \in T_p N$, $R(e_i,e_j)$ is a linear functional on the tangent space, so you could express $R(e_i,e_j)(e_k)$ as a linear combination of vectors in the dual space $T^*_p N$. The standard basis vectors of the dual space (corresponding to the collection $\{e_i\}$) is typically denoted $e_1^*, \cdots, e_n^*$. So you write $R(e_i,e_j)(e_k) = \sum_l R^l_{ijk}e^*_l$, and call $R^l_{ijk}$ the Riemann tensor "in coordinates".
In case any of this is unfamiliar, $e^*_j(e_i) = 1$ only when $i=j$ and $e^*_j(e_i) = 0$ otherwise. Or "in coordinates" $e^*_j(e_i) = \delta_{ij}$.
I think many intro general relativity textbooks explain this fairly well nowadays. When I was an undergraduate I liked:
- A First Course in. General Relativity. Second Edition. Bernard F. Schutz.
Best Answer
The Hodge star operator belongs to the subject of multilinear algebra, or perhaps exterior algebra. (The Wikipedia page on "exterior algebra" is probably the more helpful of the previous two.) Typically, one first encounters the Hodge star in a course on calculus on manifolds, or a course on Riemannian geometry.
If you're interested in a having a clear mathematical understanding of the Hodge star, then you'll need to know (at minimum) what the following concepts mean:
Now, suppose you understand the above words. If $V$ is an $n$-dimensional real vector space --- so you can think of $V$ as $\mathbb{R}^n$ --- equipped with an inner product $\langle \cdot, \cdot \rangle$ and an orientation, then one can define (but I won't define it here) the Hodge star operator $$\ast \colon \Lambda^k(V^*) \to \Lambda^{n-k}(V^*).$$ Note that if $\alpha \in \Lambda^k(V^*)$ is a $k$-form on $V = \mathbb{R}^n$, then $\ast \alpha \in \Lambda^{n-k}(V^*)$ is an $(n-k)$-form on $V = \mathbb{R}^n$.
Standard Example: Taking $V = \mathbb{R}^3$ with its standard inner product and orientation, letting $\{e_1, e_2, e_3\}$ be the standard basis, and $\{e^1, e^2, e^3\}$ be the dual basis of $V^* = (\mathbb{R}^3)^*$, then: \begin{align*} \ast e^1 & = e^2 \wedge e^3 \\ \ast e^2 & = e^3 \wedge e^1 \\ \ast e^3 & = e^1 \wedge e^2 \end{align*}
Remark: It turns out that the the gradient and divergence operators (in $\mathbb{R}^n$), as well as the curl operator (in $\mathbb{R}^3$), can all be expressed in terms of three concepts: the Hodge star operator, the exterior derivative, and the musical isomorphisms. Using those formulas, one can derive a similar formula for the Laplacian on $\mathbb{R}^n$.
Now, if you are interested in the Laplacian in even greater generality --- e.g., on a (possibly curved) Riemannian $n$-manifold $M$ --- then of course you'll need to know more words, such as:
When one speaks of the Hodge star operator on a Riemannian manifold $M$, what one really means is the Hodge star operator $\ast$ on the tangent spaces $T_pM$ of $M$. In other words, the Hodge star operator at a point $p \in M$ is defined to be the Hodge star operator on the vector space $V = T_pM$.