Math Refresher: Index Notation and Tensors

Relativity mixes time and space into a single object, a four-vector. To write equations that work in every frame, the site uses index notation: upper and lower indices, summed in pairs. This page builds that notation in the smallest form needed.

1. Why index notation

Group time and position into one object with four entries,

$$x^{\mu }=(ct,x,y,z),\mu =0,1,2,3.$$

The index $\mu$ selects which entry; $x^0=ct$ and $x^1=x$. A four-vector is any quantity whose four entries transform together under a Lorentz transformation, the same way $x^{\mu }$ does. Four-momentum is one example, $p^{\mu }=(E/c,\mathbf{p})$.

2. Einstein summation

When an index appears once as a superscript and once as a subscript in the same expression, sum it over its four values. This is Einstein summation notation:

$$A^{\mu }{B}_{\mu }=A^0{B}_0+A^1{B}_1+A^2{B}_2+A^3{B}_3.$$

The summed index is a dummy index; its name does not matter, so $A^{\mu }{B}_{\mu }=A^{\nu }{B}_{\nu }$. An index left unsummed is a free index and must match on both sides of any equation.

3. The metric

To form the spacetime interval, time and space enter with opposite signs. The Minkowski metric encodes those signs:

$${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1).$$

It has entries ${\eta }_{00}=+1$, ${\eta }_{11}={\eta }_{22}={\eta }_{33}=-1$, and zeros elsewhere. Contracting the position four-vector with itself through the metric gives

$$x^{\mu }{x}_{\mu }={\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }=c^2{t}^2-x^2-y^2-z^2,$$

which is the invariant interval. The metric is the geometric price of mixing time and space: it replaces the ordinary dot product's all-plus signs.

4. Covariant and contravariant indices

An index can sit up or down, and the two positions record two different transformation rules.

An upper index marks a contravariant vector. Its components transform with the Lorentz matrix $\Lambda$ itself, exactly like a coordinate displacement:

$$d{x}^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }d{x}^{\nu }.$$

Positions, velocities, and momenta are contravariant. The name records that these components transform contrary to the coordinate axes, so that the arrow they describe stays the same under a change of frame.

A lower index marks a covariant vector, also called a covector. Its components transform with the inverse matrix,

$$A_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{A}_{\nu }.$$

The standard example is the gradient. By the chain rule the derivative ${\partial }_{\mu }=\partial /\partial x^{\mu }$ transforms as ${\partial }_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{\partial }_{\nu }$, so it is covariant: its components go with the axes. A covector is a rule that takes a vector and returns a number, and it transforms so that this number stays the same.

The two rules are inverse for a reason. Contracting a contravariant vector with a covariant one,

$$A^{\mu }{B}_{\mu },$$

mixes one $\Lambda$ with one ${\Lambda }^{-1}$, which cancel, leaving the same value in every frame. That cancellation is what makes the contraction a scalar.

The metric converts between the two kinds:

$$A_{\mu }={\eta }_{\mu \nu }{A}^{\nu },A^{\mu }={\eta }^{\mu \nu }{A}_{\nu }.$$

Lowering leaves the time component alone and flips the sign of the spatial ones, $x_{\mu }=(ct,-x,-y,-z)$; the inverse metric ${\eta }^{\mu \nu }$ has the same diagonal entries as ${\eta }_{\mu \nu }$.

5. Contraction and Lorentz scalars

Summing a matched upper and lower index is a contraction. Contracting two four-vectors gives

$$A^{\mu }{B}_{\mu }=A^0{B}_0-\mathbf{A}\cdot \mathbf{B},$$

a single number. A fully contracted expression is a Lorentz scalar: it has the same value in every inertial frame. The invariant interval $s^2=x^{\mu }{x}_{\mu }$, the squared mass $m^2=p^{\mu }{p}_{\mu }$, and the plane-wave phase $\varphi =k^{\mu }{x}_{\mu }$ are all scalars built this way.

A tensor generalizes the rule: it carries several indices, each upper one transforming like a contravariant vector and each lower one like a covariant vector. The metric itself is a tensor with two lower indices. The rest of the site uses only four-vectors and contractions; the full tensor law returns only for the field transformations on QFT.

Previous: Linear Algebra

Next: Groups and Symmetry