## Index juggling ### Raising and lowering of indices In tensor calculus, the **raising and lowering of indices** is a notational operation that does not change the underlying geometric object, but re-expresses it using the **metric tensor**: $ T^j = T_i Z^{ij}, \quad T_i = T^j Z_{ij} $ These transformations are reversible and do not introduce new content. The reshuffling of index positions in such transformations is often referred to as **index juggling**. ### Dot notation When a tensor carries **both upper and lower indices**, the structural identity of the tensor is not fully captured by variance alone. This becomes especially problematic when: 1. Index raising/lowering is performed; 2. The result still contains both upper and lower indices; 3. Specific values (e.g. $T^1_2$) are assigned to indices, making their symbolic identity indistinguishable. We use **dot notation** to track the original **slot** that each index occupied in the tensor. Dot notation is applied **only** when: - A tensor has both upper and lower indices; - Index juggling (raising or lowering) is involved; - We need to distinguish structurally different but notationally similar forms. > [!Example] > Consider a tensor $T^j_i$. If we raise and lower both indices: > $ > T^k_l = T^j_i Z_{il} Z^{jk} > $ > the resulting tensor $T^k_l$ may correspond to different original structures. Once we assign specific values like $T^1_2$, it's ambiguous whether "1" was originally the upper index or the lower. > > To resolve this, we write: > $ > T^{\cdot k}_l = T^i_{\cdot j} Z_{il} Z^{jk} > $ > indicating that $k$ originated from the **second slot** in the original tensor (that is, $j$ in $T^j_i$), preserving the structural identity through index juggling. > > This transformation can be broken into two steps: > 1. Lowering the first index: > $ > T_{lj} = T^i_{\cdot j} Z_{il} > $ > 2. Raising the second index: > $ > T^{\cdot k}_l = T_{lj} Z^{jk} > $ > [!Note] > Special cases: > - **Symmetric tensors**: If $T_{ij} = T_{ji}$, then $T^i_j = T^j_i$, and no ambiguity arises even without dot notation. > - **Christoffel symbols**: Conventionally written as $\Gamma^i_{jk}$, where the order is fixed by definition, so dot notation is usually unnecessary. ### Equivalence of $\delta^i_j$ and $Z_{ij}$ Objects related by index juggling are considered equivalent. In this sense, the Kronecker delta $\delta^i_j$ and the metric tensor $Z_{ij}$ (and its inverse $Z^{ij}$) are interchangeable through index transformations: $ \delta^i_j Z_{ik} = Z_{jk}, \quad Z_{ij} Z^{jk} = \delta^k_i, \quad \delta^i_j Z^{jk} = Z^{ik} $ Thus, $ \delta^i_j = Z_{jk} Z^{ik}, \quad Z_{ij} = \delta^k_j Z_{ik}, \quad Z^{ij} = \delta^i_k Z^{kj}. $ Some texts write $Z_{ij}$ as $\delta_{ij}$, $Z^{ij}$ as $\delta^{ij}$, and $\delta^i_j$ as $Z^i_j$. ### The effect of index juggling on tensor notation Index juggling allows contractions with the metric tensor to be absorbed into index placement. For example, the inner product $ \mathbf{U} \cdot \mathbf{V} = Z_{ij} U^i V^j $ can simply be written as $ \mathbf{U} \cdot \mathbf{V} = U_i V^i \quad \text{or} \quad U^i V_i. $ This hides the metric but not the computation. Even in a simple-looking expression like $ \mathbf{U} \cdot \mathbf{V} = U_1 V^1 + U_2 V^2 + U_3 V^3, $ the covariant components $U_i$ still depend on $Z_{ij}$. Index juggling compresses notation, but does not reduce computational cost.