A scalar is a single number. A vector is an ordered list of numbers that transforms in a specific way when you change coordinates. A tensor generalises this to objects with multiple indices, each transforming according to well-defined rules.

Tensors are the language of general relativity because they express physical laws in a form that’s valid in any coordinate system — which is exactly what you need when spacetime itself is curved.

From vectors to tensors

You already know two kinds of object:

• A scalar $f$ is unchanged by coordinate transformations: $f’ = f$
• A vector $v^i$ transforms as $v’^i = \frac{\partial x’^i}{\partial x^j} v^j$ — each component mixes according to the Jacobian of the coordinate change

A tensor of rank $(p, q)$ has $p$ upper indices and $q$ lower indices, each transforming with its own Jacobian factor:

$$T’^{i_1 \ldots i_p}{}{j_1 \ldots j_q} = \frac{\partial x’^{i_1}}{\partial x^{a_1}} \cdots \frac{\partial x’^{i_p}}{\partial x^{a_p}} \frac{\partial x^{b_1}}{\partial x’^{j_1}} \cdots \frac{\partial x^{b_q}}{\partial x’^{j_q}} \; T^{a_1 \ldots a_p}{}{b_1 \ldots b_q}$$

This looks fearsome, but the pattern is simple: each upper index gets a $\partial x’/\partial x$ factor, each lower index gets a $\partial x/\partial x’$ factor. The indices “know how to translate themselves.”

Upper and lower: two kinds of index

• Upper indices (contravariant): $v^i$. Transform like coordinate differentials $dx^i$.
• Lower indices (covariant): $w_i$. Transform like partial derivatives $\partial/\partial x^i$.

A covector (or one-form) $w_i$ eats a vector and produces a scalar: $w_i v^i = $ number. This is a generalisation of the dot product.

Important tensors

Contraction

Contracting a tensor means setting one upper and one lower index equal and summing (Einstein convention). This reduces the rank by 2:

$$T^i{}_i = T^0{}_0 + T^1{}_1 + T^2{}_2 + T^3{}_3$$

The result is coordinate-independent. Contraction is the tensor generalisation of the trace of a matrix.

Tensor fields

A tensor field assigns a tensor to every point on a manifold. The metric tensor $g_{ij}(x)$ is a (0,2) tensor field — at each point, it takes two vectors and returns a number (their “generalised dot product”). Vector fields, scalar fields, and all the objects of general relativity are tensor fields.

The power of tensors: if a tensor equation holds in one coordinate system, it holds in all coordinate systems. Write the laws of physics as tensor equations, and they’re automatically valid for every observer — accelerating, rotating, or falling freely through curved spacetime.