Tensor Operations

Tensors are the natural language of modern physics — from the stress tensor in continuum mechanics to the Riemann curvature tensor in general relativity.

Transformation Law

A rank-2 contravariant tensor transforms as:

$T'^{ij} = \Lambda^i_{;k}\, \Lambda^j_{;l}\, T^{kl}$

where $\Lambda$ is the transformation matrix (Jacobian). This is just a double matrix multiplication.

The Metric Tensor

The metric tensor $g_{ij}$ encodes the geometry of space. It raises and lowers indices:

$T_i = g_{ij}\, T^j, \qquad T^i = g^{ij}\, T_j$

where $g^{ij}$ is the inverse metric. In flat 3D Cartesian space $g_{ij} = \delta_{ij}$ (identity).

Polar Coordinates (2D)

In polar coordinates $(r, \theta)$, the metric is:

$g = \begin{pmatrix} 1 & 0 \\ 0 & r^2 \end{pmatrix}$

The inverse is $g^{rr} = 1$, $g^{\theta\theta} = 1/r^2$.

Christoffel Symbols

The Christoffel symbols encode how basis vectors change across the manifold:

$\Gamma^k_{;ij} = \frac{1}{2}\, g^{kl}\bigl(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}\bigr)$

For 2D polar coordinates, the non-zero Christoffel symbols are:

$\Gamma^r_{;\theta\theta} = -r, \qquad \Gamma^\theta_{;r\theta} = \Gamma^\theta_{;\theta r} = \frac{1}{r}$

All other components vanish. These give rise to the fictitious forces (centripetal, Coriolis) in rotating frames.

Index Raising

Given a covariant tensor $T_{ij}$ and the inverse metric $g^{ij}$, we raise the first index:

$T^i_{;j} = g^{ik}\, T_{kj}$

Implementation

• tensor_contract(A, B) — matrix multiply two 2D lists (works for any square size)
• metric_polar(r) — returns the 2x2 polar metric as a list of lists
• christoffel_polar_r_theta_theta(r) — returns $\Gamma^r_{;\theta\theta} = -r$
• christoffel_polar_theta_r_theta(r) — returns $\Gamma^\theta_{;r\theta} = 1/r$
• raise_index(T_lower, g_inv) — computes $T^i_{;j} = g^{ik} T_{kj}$

def metric_polar(r):
    return [[1.0, 0.0], [0.0, r**2]]

def christoffel_polar_r_theta_theta(r):
    return -r