Tensor

Definition

A tensor is a multilinear map: \[T: V^* \times \cdots \times V^* \times V \times \cdots \times V \to \mathbb{R}\]

• \((p,q)\)-tensor: \(p\) contravariant, \(q\) covariant indices
• Rank: \(p + q\)

Tensor Product

\[(T \otimes S)_{i_1 \ldots i_p j_1 \ldots j_q}^{k_1 \ldots k_r} = T_{i_1 \ldots i_p}^{k_1 \ldots k_r} S_{j_1 \ldots j_q}^{k_{r+1} \ldots k_{r+s}}\]

Contraction

Raise/lower indices with metric \(g_{\mu\nu}\): \[T^\mu_{\ \nu} = g^{\mu\alpha} T_{\alpha\nu}\]

Sum over repeated indices (Einstein notation).

Components

\[T^{i_1 \ldots i_p}_{j_1 \ldots j_q}\]

Transformation law: \[T'^{i_1 \ldots i_p}_{j_1 \ldots j_q} = \frac{\partial x'^{i_1}}{\partial x^{k_1}} \cdots \frac{\partial x^{l_q}}{\partial x'^{j_q}} T^{k_1 \ldots k_p}_{l_1 \ldots l_q}\]

Symmetry

• Symmetric: \(T_{ij} = T_{ji}\)
• Antisymmetric: \(T_{ij} = -T_{ji}\)
• Antisymmetric tensors: \(\varepsilon^{\mu\nu\rho\sigma}\) (Levi-Civita)