Foundations — Metric tensor — raising - lowering indices

This page assumes nothing. If the parent note wrote a symbol like $V^{\mu }$, $g_{\mu \nu }$, or ${\delta }^{\mu }{}_{\nu }$ and you weren't sure what each little mark meant, this is where we build every one of them, in an order where each piece leans only on the piece before it. Read top to bottom.

0. What a "vector" and a "basis" actually are

To turn the arrow into numbers we need reference arrows to measure against. Those reference arrows are the basis.

Any arrow $\mathbf{V}$ can be built by taking some amount of ${\mathbf{e}}_1$ plus some amount of ${\mathbf{e}}_2$. Those "amounts" are the numbers we call components. But — crucial for this whole topic — the reference arrows need not be at right angles, and need not have length 1. That freedom is exactly why two flavours of components appear.

1. The Greek index $\mu$ and the dummy index — a counter, nothing more

The parent page is full of letters like $\mu$ (mu), $\nu$ (nu), $\alpha$, $\beta$. They look scary. They are not.

This is the single biggest reading skill for the parent page. When you meet $g_{\mu \nu }{V}^{\nu }$, the $\nu$ is repeated (down in $g$, up in $V$), so it is secretly ${\sum }_{\nu }{g}_{\mu \nu }{V}^{\nu }$, and $\mu$ is the one free label left over.

2. Upper vs lower index: the two flavours of "amount"

Now the star confusion of the topic: $V^{\mu }$ (index up) versus $V_{\mu }$ (index down).

Full detail of the "shadow" flavour lives in Dual space and covectors; the "recipe" flavour is standard vector building.

3. The dot product — the machine that measures

Before we can define the metric we need the one operation everything is built on.

Two facts we will lean on hard:

• Length from dot product: $\mathbf{A}\cdot \mathbf{A}=|\mathbf{A}{|}^2\cos 0=|\mathbf{A}{|}^2$. So the dot product of an arrow with itself is its length squared. This is why the metric ends up controlling lengths.
• Bilinearity: the dot product is "linear in each slot" — you can pull numbers out and split sums: $(a\mathbf{A})\cdot (b\mathbf{B})=ab(\mathbf{A}\cdot \mathbf{B}),(\mathbf{A}+\mathbf{B})\cdot \mathbf{C}=\mathbf{A}\cdot \mathbf{C}+\mathbf{B}\cdot \mathbf{C}.$

Every "$=$" in the parent's length derivation is one of these two facts, nothing more. Deeper background: Inner product spaces.

4. Building the metric $g_{\mu \nu }$ from pieces you now own

Now everything clicks together. The metric is defined as the table of all dot products of the reference arrows.

Since a dot product is a number, $g_{\mu \nu }$ is a grid of numbers — a matrix. In 2D: $g=\left(\begin{array}{cc}{\mathbf{e}}_1\cdot {\mathbf{e}}_1& {\mathbf{e}}_1\cdot {\mathbf{e}}_2\\ {\mathbf{e}}_2\cdot {\mathbf{e}}_1& {\mathbf{e}}_2\cdot {\mathbf{e}}_2\end{array}\right).$

• The diagonal entries $g_{11},g_{22}$ are lengths-squared of each reference arrow (an arrow dotted with itself).
• The off-diagonal entries $g_{12}=g_{21}$ measure how tilted the two axes are ($\cos \vartheta$ times lengths). If the axes are perpendicular, $\cos 90^{\circ }=0$, so off-diagonals vanish.

This "grid of pairwise dot products" is exactly what the Line element and ds^2 and Minkowski spacetime pages encode — the $d{s}^2$ formula is the metric written out.

5. The Kronecker delta ${\delta }^{\mu }{}_{\nu }$ and the inverse metric $g^{\mu \nu }$

Two more symbols the parent uses without ceremony.

6. Putting the reading skills together

Take the parent's key line and read it with everything above: $V_{\mu }=g_{\mu \nu }{V}^{\nu }.$

• $V_{\mu }$ — the shadow number in slot $\mu$ (free index, so this holds for each $\mu$).
• $g_{\mu \nu }$ — the measuring grid, row $\mu$, column $\nu$.
• $V^{\nu }$ — the recipe numbers.
• Repeated $\nu$ (down in $g$, up in $V$) ⇒ sum over $\nu$.

So $V_1=g_{11}{V}^1+g_{12}{V}^2$: take row 1 of the grid, dot it against the recipe list. That is exactly matrix-times-vector. The whole topic is "multiply your number list by the right grid."

Tensors of higher rank (two or more indices) obey the same reading rules, one index at a time — see Tensors — definition and transformation laws and, for how the metric's derivatives build further machinery, Christoffel symbols.

Equipment checklist

You are ready for Metric tensor — raising - lowering indices once you can answer each of these without peeking:

Prerequisite Map

Connections

• Metric tensor — raising - lowering indices
• Tensors — definition and transformation laws
• Dual space and covectors
• Line element and ds^2
• Minkowski spacetime
• Christoffel symbols
• Inner product spaces
• Einstein summation convention