Basic Tensor Notation

Tensors are typically represented using indices to denote their components:

1. Tensor Symbols

   T, \mathbf{T}, \mathsf{T}, \mathcal{T}, \mathbb{T}

   Different ways to represent tensor symbols in LaTeX.

2. Tensor Components with Indices

   T^{i}_{j}, \quad T^{ij}_{k}, \quad T^{i_1 i_2 \ldots i_n}_{j_1 j_2 \ldots j_m}

   Tensor components with superscript (contravariant) and subscript (covariant) indices.

3. Tensor Rank

   \ ext{Rank } (m,n) \ ext{ tensor: } T^{i_1 i_2 \ldots i_m}_{j_1 j_2 \ldots j_n}

   A tensor of rank (m,n) has m contravariant indices and n covariant indices.

Einstein Summation Convention

The Einstein summation convention is commonly used with tensors, where repeated indices imply summation:

1. Implicit Summation

   A^i B_i = \sum_{i=1}^n A^i B_i

   When an index appears once as a superscript and once as a subscript, summation is implied.

2. Matrix Multiplication as Tensor Contraction

   C^i_j = A^i_k B^k_j = \sum_{k=1}^n A^i_k B^k_j

   Matrix multiplication expressed as tensor contraction using Einstein notation.

3. Free Indices

   D^{ij}_k = A^i_l B^j_m C^{lm}_k

   Indices that appear only once (i, j, k) are free indices and represent components of the resulting tensor.

Common Tensors in Physics

1. Metric Tensor

   g_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

   The Minkowski metric tensor used in special relativity.

2. Kronecker Delta

   \delta^i_j = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}

   The Kronecker delta is a rank (1,1) tensor that acts as an identity operator.

3. Levi-Civita Symbol

   \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{if any index is repeated} \end{cases}

   The Levi-Civita symbol is used for cross products and determinants.

Tensor Operations

1. Tensor Addition

   C^{ij}_k = A^{ij}_k + B^{ij}_k

   Tensors of the same rank can be added component-wise.

2. Tensor Contraction

   A^i_i = \sum_{i=1}^n A^i_i

   Contraction of a tensor by setting a contravariant and covariant index equal.

3. Tensor Product

   C^{ij}_{kl} = A^i_k \otimes B^j_l

   The tensor product combines two tensors into a higher-rank tensor.

Raising and Lowering Indices

The metric tensor can be used to raise or lower indices:

1. Raising an Index

   A^{\mu} = g^{\mu\nu}A_{\nu}

   Using the inverse metric tensor to raise an index.

2. Lowering an Index

   A_{\mu} = g_{\mu\nu}A^{\nu}

   Using the metric tensor to lower an index.

Covariant Derivatives

1. Christoffel Symbols

   \Gamma^{\lambda}_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}\left(\frac{\partial g_{\rho\mu}}{\partial x^{\nu}} + \frac{\partial g_{\rho\nu}}{\partial x^{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\rho}}\right)

2. Covariant Derivative of a Vector

   \nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu\lambda}V^{\lambda}

   The covariant derivative generalizes the partial derivative to curved spaces.

3. Covariant Derivative of a Covector

   \nabla_{\mu}V_{\nu} = \partial_{\mu}V_{\nu} - \Gamma^{\lambda}_{\mu\nu}V_{\lambda}

Advanced Tensor Notation

• R^{\rho}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}_{\mu\sigma} + \Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma} - \Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}

  The Riemann curvature tensor.

• R_{\mu\nu} = R^{\lambda}_{\mu\lambda\nu}

  The Ricci tensor, a contraction of the Riemann tensor.

• R = g^{\mu\nu}R_{\mu\nu}

  The Ricci scalar, a contraction of the Ricci tensor.

• G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R

  The Einstein tensor, used in general relativity.