Tensor Calculus

Tensors are the mathematical objects that transform covariantly under coordinate transformations, making them ideal for formulating physical laws that hold in all reference frames.

1. Index Notation and Einstein Summation

Einstein Summation Convention

When an index appears twice in a single term (once as superscript, once as subscript), sum over all values:

$$a^\mu b_\mu = \sum_{\mu=0}^{3} a^\mu b_\mu = a^0b_0 + a^1b_1 + a^2b_2 + a^3b_3$$

Free vs Dummy Indices

Dummy indices: Appear twice, summed over (can be renamed)

$$a^\mu b_\mu = a^\nu b_\nu = a^\alpha b_\alpha$$

Free indices: Appear once, must match on both sides of equation

$$A^\mu_\nu B^\nu_\rho = C^\mu_\rho$$

Here $\nu$ is dummy (summed), $\mu$ and $\rho$ are free.

2. Vectors and Covectors

Contravariant Vectors (Upper Index)

Transform like differentials $dx^\mu$:

$$V'^\mu = \frac{\partial x'^\mu}{\partial x^\nu}V^\nu$$

Example: velocity $u^\mu = dx^\mu/d\tau$, momentum $p^\mu$

Covariant Vectors (Lower Index)

Transform like gradients $\partial_\mu = \partial/\partial x^\mu$:

$$\omega'_\mu = \frac{\partial x^\nu}{\partial x'^\mu}\omega_\nu$$

Example: gradient of scalar $\nabla_\mu\phi = \partial_\mu\phi$

Raising and Lowering Indices

The metric tensor $g_{\mu\nu}$ converts between upper and lower indices:

$$V_\mu = g_{\mu\nu}V^\nu, \quad V^\mu = g^{\mu\nu}V_\nu$$

where $g^{\mu\nu}$ is the inverse metric: $g^{\mu\lambda}g_{\lambda\nu} = \delta^\mu_\nu$

3. General Tensors

Definition

A tensor of type $(p,q)$ has $p$ upper indices and $q$ lower indices:

$$T'^{\mu_1\cdots\mu_p}_{\nu_1\cdots\nu_q} = \frac{\partial x'^\mu_1}{\partial x^{\alpha_1}}\cdots\frac{\partial x'^\mu_p}{\partial x^{\alpha_p}}\frac{\partial x^{\beta_1}}{\partial x'^\nu_1}\cdots\frac{\partial x^{\beta_q}}{\partial x'^\nu_q}T^{\alpha_1\cdots\alpha_p}_{\beta_1\cdots\beta_q}$$

Examples

• Scalar (0,0): $\phi$ (invariant under transformations)
• Vector (1,0): $V^\mu$
• Covector (0,1): $\omega_\mu$
• Metric (0,2): $g_{\mu\nu}$
• Electromagnetic field (0,2): $F_{\mu\nu}$
• Riemann tensor (1,3): $R^\rho_{\sigma\mu\nu}$

Tensor Rank

The rank is $p + q$. In $n$ dimensions, a rank-$r$ tensor has $n^r$ components. Example: in 4D spacetime, $R^\rho_{\sigma\mu\nu}$ has $4^4 = 256$ components (reduced to 20 by symmetries).

4. Tensor Operations

Addition

Tensors of the same type can be added component-wise:

$$(A + B)^\mu_\nu = A^\mu_\nu + B^\mu_\nu$$

Tensor Product (Outer Product)

Multiply all components of two tensors:

$$(A \otimes B)^{\mu\nu}_{\rho\sigma} = A^\mu_\rho B^\nu_\sigma$$

Creates a tensor of type $(p_1+p_2, q_1+q_2)$ from types $(p_1,q_1)$ and $(p_2,q_2)$.

Contraction

Sum over one upper and one lower index:

$$T^\mu_{\mu\nu} = \sum_\mu T^\mu_{\mu\nu}$$

Reduces rank by 2. Example: trace of matrix $\text{tr}(M) = M^\mu_\mu$

Symmetrization and Antisymmetrization

$$T_{(\mu\nu)} = \frac{1}{2}(T_{\mu\nu} + T_{\nu\mu}) \quad \text{(symmetric part)}$$
$$T_{[\mu\nu]} = \frac{1}{2}(T_{\mu\nu} - T_{\nu\mu}) \quad \text{(antisymmetric part)}$$

5. The Metric Tensor

Definition and Properties

The metric $g_{\mu\nu}$ defines the geometry of spacetime. It is:

• Symmetric: $g_{\mu\nu} = g_{\nu\mu}$
• Non-degenerate: $\det(g) \neq 0$
• Defines inner product: $V \cdot W = g_{\mu\nu}V^\mu W^\nu$

Line Element

Infinitesimal proper time/distance:

$$ds^2 = g_{\mu\nu}dx^\mu dx^\nu$$

Minkowski Metric

Flat spacetime in special relativity:

$$\eta_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \quad \text{(signature } -,+,+,+ \text{)}$$

Inverse Metric

$$g^{\mu\lambda}g_{\lambda\nu} = \delta^\mu_\nu$$

where $\delta^\mu_\nu$ is the Kronecker delta (1 if $\mu=\nu$, 0 otherwise).

6. Covariant Derivative

Why We Need It

Ordinary partial derivatives $\partial_\mu V^\nu$ don't transform as tensors in curved space. We need a covariant derivative that does.

Definition for Vectors

$$\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda}V^\lambda$$
$$\nabla_\mu V_\nu = \partial_\mu V_\nu - \Gamma^\lambda_{\mu\nu}V_\lambda$$

where $\Gamma^\lambda_{\mu\nu}$ are Christoffel symbols (connection coefficients).

General Tensors

For a general tensor, add one $+\Gamma$ term for each upper index and one $-\Gamma$ term for each lower:

$$\nabla_\rho T^\mu_\nu = \partial_\rho T^\mu_\nu + \Gamma^\mu_{\rho\sigma}T^\sigma_\nu - \Gamma^\sigma_{\rho\nu}T^\mu_\sigma$$

Christoffel Symbols

In terms of the metric (Levi-Civita connection):

$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}\left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}\right)$$

Symmetric in lower indices: $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$

7. Parallel Transport and Geodesics

Parallel Transport

A vector $V^\mu$ is parallel transported along a curve $x^\mu(\lambda)$ if:

$$\frac{DV^\mu}{D\lambda} = \frac{dx^\nu}{d\lambda}\nabla_\nu V^\mu = 0$$

Expanding:

$$\frac{dV^\mu}{d\lambda} + \Gamma^\mu_{\nu\rho}\frac{dx^\nu}{d\lambda}V^\rho = 0$$

Geodesic Equation

The straightest possible path—a curve whose tangent vector is parallel transported:

$$\frac{d^2x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\rho}\frac{dx^\nu}{d\lambda}\frac{dx^\rho}{d\lambda} = 0$$

Free particles follow geodesics in spacetime.

8. Important Tensors in Physics

Stress-Energy Tensor $T_{\mu\nu}$

Describes energy, momentum, and stress. For perfect fluid:

$$T_{\mu\nu} = (\rho + p)u_\mu u_\nu + pg_{\mu\nu}$$

Electromagnetic Field Tensor $F_{\mu\nu}$

$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}$$

Riemann Curvature Tensor $R^\rho_{\sigma\mu\nu}$

Measures spacetime curvature:

$$R^\rho_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$$

9. Practical Tips

1. Check Index Balance

Every term in an equation must have the same free indices in the same positions.

2. Avoid Index Collisions

Don't use the same dummy index three or more times in one term.

3. Remember Symmetries

Use symmetry properties to reduce calculations: $g_{\mu\nu} = g_{\nu\mu}$, $F_{\mu\nu} = -F_{\nu\mu}$

4. Matrix vs. Tensor

Matrices are representations of (1,1) tensors in a specific basis. Tensor equations hold in all bases.

Mathematics