Tensor index notation with e.g. square brackets

Question

I want to learn playing with indices and some notation in General relativity. But in every book just is used this notation. I know upper and lower but I don"t know the meaning of some combination of these indices, like when we have 3 indices in both side of the letter in bracket (e.g of the form $\partial_{[a}F_{bc]}$).

Could you please explain me or suggest a book that say almost everything , for example, when we need to put 2 indices in a tensor, one up and one of them down, ...?

Answer 1

Here's various things used in index notation :

• Types of indices :
  • Greek indices for spacetime indices (tensor indices)
  • Lower-case latin indices for
    1. spacelike components
    2. local Lorentz components
    3. group components (for gauge indexes)
  • Upper case latin indexes for spinor indexes
  • Dotted upper case latin indices for conjugate spinor indexes. Generally speaking, dotted indexes indicate complex conjugate indexes.
• Upper index : Denotes the components of a vector, or a basis of dual vectors. Example : $V^\mu$ is the components of a vector, $dx^\mu$ is a basis of dual vectors, $T_\mu dx^\mu$ is a dual vector.
• Lower index : Denotes the components of a dual vector, or a basis of vectors. Example : $T_\mu$ is the components of a dual vector, $\partial x_\mu$ is a basis of vectors, $V^\mu \partial x_\mu$ is a vector.
• Parenthesis : Denotes the symmetrization of a tensor with respect to those indices. That is, for $n$ indices, $$T_{(\alpha\beta\gamma...)} = \frac{1}{n!}\sum_{p \in \mathrm{permutations}} T_{p(\alpha\beta\gamma...)}.$$ Examples : $T_{(\alpha\beta)} = \frac{1}{2} (T_{\alpha\beta} + T_{\beta\gamma})$, $p^{(i}q^{j)}=\frac12(p^iq^j+p^jq^i)$. Note, in particular, that the parenthesis notation can span indices over multiple tensors.
• Brackets : Denotes the antisymmetrization of a tensor with respect to those indices. That is, for $n$ indices, $$T_{[\alpha\beta\gamma...]} = \frac{1}{n!}\sum_{p \in \mathrm{permutations}}(-1)^{n_p} T_{p(\alpha\beta\gamma...)} $$ where $n_p$ indicates the number of single permutations of $p$. Example : $$T_{[\alpha\beta\gamma]} = \frac{1}{6} (T_{\alpha\beta\gamma} - T_{\gamma\beta\alpha} + T_{\beta\gamma\alpha} - T_{\alpha\gamma\beta} + T_{\gamma\alpha\beta} - T_{\beta\alpha\gamma})$$ Like the parenthesis notation, it can span over multiple tensors. Example : $f_{[\alpha}g_{\beta]}=\frac12(f_\alpha g_\beta - f_\beta g_\alpha)$
• Comma : Denotes the partial derivative with respect to this component. Example : ${V^\alpha}_{,\beta} = \partial_\beta V^\alpha$
• Semicolon : Denotes the covariant derivative with respect to this component. Example : ${V^\alpha}_{;\beta} = \partial_\beta V^\alpha + {\Gamma^{\alpha}}_{\beta\gamma} V^\gamma$

Less commonly used :

• Pipe : Denotes the covariant derivative with respect to local Lorentz indices. Example : ${V^a}_{\vert i} = \partial_i V^a + {\omega^a}_{bi} V^b$
• Colon : Denotes the covariant derivative with the Levi-Civita connection. Example : ${V^\alpha}_{:\beta} = \sum_{\alpha,\beta,\gamma} (\partial_\alpha V^\alpha + \{^\alpha_{\beta\gamma}\} V^\gamma)$

Answer 2

Interpret indices $[ijk]$ as a "determinant", \begin{eqnarray} [ijk] \rightarrow \left| \begin{matrix} i & j & k \\ i & j & k \\ i & j & k \\ \end{matrix} \right|. \end{eqnarray} Expand above indices, \begin{eqnarray} [ijk] \rightarrow ijk + jki + kij - ikj - jik -kji. \end{eqnarray} Set them to a tensor $T_{[ijk]}$ (including signs) and divide by normalization factor, so we get \begin{eqnarray} T_{ [ijk] } = \frac{1}{3!} \left(T_{ijk} + T_{jki} + T_{kij} - T_{ikj} - T_{jik} - T_{kji} \right). \end{eqnarray}

I think it can be done because of antisymmetrization structure of a determinant.