Metric tensor — raising - lowering indices: Question bank

Before we start, one shared vocabulary reminder so no symbol is unearned:

Spot the error

Each line contains a mistake. Name it, then reveal the fix.

∥ V ∥^{2} = \sum_{μ} (V^{μ})^{2}

in every coordinate system."

Wrong. The invariant norm is

∥ V ∥^{2} = g_{μν} V^{μ} V^{ν}

; the naive sum-of-squares only works when

g_{μν} = δ_{μν}

A^{μ} B^{μ}

is the dot product of

A

and

B

Illegal contraction — both indices are up. The invariant is

A^{μ} B_{μ} = g_{μν} A^{μ} B^{ν}

, summing one up with one down.

"To raise an index, contract with

g_{μν}

You must contract with the inverse

g^{μν}

. The down-indexed

g_{μν}

lowers; using it to raise is dimensionally and algebraically wrong unless

g

is its own inverse.

V_{θ} = g_{θ θ} V^{θ} = r^{2} V^{θ}

, therefore

V^{θ} = r^{2} V_{θ}

The raise step is inverted. Since

g^{θ θ} = 1/ r^{2}

, we get

V^{θ} = g^{θ θ} V_{θ} = V_{θ} / r^{2}

, not

r^{2} V_{θ}

T^{μν} = g_{μν} T^{μν}

lowers both indices at once."

An index can only appear twice (once up, once down) if it is summed; here

μ, ν

appear as both free and dummy — a notation clash. Lower one index at a time with a fresh dummy:

T^{μ}_{ν} = g_{ν α} T^{μα}

"Since

g^{μα} g_{α ν} = δ^{μ}_{ν}

, the metric equals its own inverse."

That relation just says they are matrix inverses;

g^{μν} = g_{μν}

only when

g^{2} = 1

, e.g. Minkowski

diag (- 1, 1, 1, 1)

, but not for polar coordinates.

"In

g_{μν} = diag (- 1, 1, 1, 1)

, the norm of

U^{μ} = (2, 1, 0, 0)

2^{2} + 1^{2} = 5

The time part carries a minus sign:

U_{μ} U^{μ} = (- 1) (2)^{2} + (1) (1)^{2} = - 4 + 1 = - 3

. Forgetting the signature is the classic slip.

"The metric is what defines vectors."

No — vectors exist without a metric. The metric adds geometry (lengths, angles, and the up/down translation). Vectors and their dual covectors are defined first; the metric merely connects them.

Why questions

Answer in one or two sentences, then reveal.

Why do we even need two kinds of components?

Because a general basis is not orthonormal: measuring a vector by counting basis steps (

V^{μ}

) and by projecting onto basis directions (

V_{μ}

) give different numbers, and both are useful.

Why is the metric defined as dot products of basis vectors specifically?

Because the squared length

g_{μν} V^{μ} V^{ν}

falls out of expanding

V \cdot V = (V^{μ} e_{μ}) \cdot (V^{ν} e_{ν})

; the dot products

e_{μ} \cdot e_{ν}

are exactly what is needed to get lengths from components.

Why must the inverse metric raise, rather than the metric itself?

Lowering is multiplication by

g_{μν}

; raising must undo it, and undoing a matrix multiply requires the matrix inverse

g^{μν}

Why does the summed (dummy) index always appear once up and once down?

A valid contraction pairs a vector slot with a dual slot (Dual space and covectors); this is the only combination that produces a coordinate-independent scalar, guaranteed by the Einstein summation convention.

Why does the

θ

-component get a factor

1/ r^{2}

when raised in polar coordinates?

The basis vector

e_{θ}

has length

r

, so

g_{θ θ} = r^{2}

; its inverse

g^{θ θ} = 1/ r^{2}

compensates for that stretched basis vector when converting

V_{θ} \to V^{θ}

Why do high-school physics classes never mention covariant vs contravariant?

They work exclusively in orthonormal Cartesian coordinates where

g_{μν} = δ_{μν}

, so

V^{μ} = V_{μ}

and the distinction is invisible.

Why is the invariant dot product coordinate-independent even though

V^{μ}

and

g_{μν}

each change with coordinates?

The changes cancel:

g_{μν}

transforms oppositely to the pair

V^{μ} V^{ν}

, leaving the scalar

g_{μν} A^{μ} B^{ν}

fixed — that is the defining property of a genuine scalar.

Why do the Christoffel symbols involve derivatives of

g_{μν}

and not

g_{μν}

itself?

Raising/lowering uses the metric's values, but describing how the basis (and hence components) changes from point to point requires how the metric changes — its derivatives.

Edge cases

Handle the degenerate / limiting situations. Reveal after reasoning.

What happens to raising/lowering when

g_{μν} = δ_{μν}

(flat orthonormal)?

The operations become the identity:

V_{μ} = V^{μ}

. This is the only regime where the distinction disappears, which is why it's the beginner's default.

What if the metric is singular (determinant zero) — can you still raise indices?

No. A zero determinant means

g^{μν}

does not exist, so there is no way to undo lowering; the geometry is degenerate and lengths/angles are not well defined there.

For a null (lightlike) vector in Minkowski space, what is its norm?

Exactly zero:

g_{μν} V^{μ} V^{ν} = 0

even though

V^{μ} \neq = 0

. An indefinite metric allows non-zero vectors with zero length — impossible in Euclidean space.

Take the zero vector

V^{μ} = (0, 0, \dots)

. What are its lowered components?

All zero:

V_{μ} = g_{μν} \cdot 0 = 0

regardless of the metric. The zero vector is the same in every dialect.

In 2D polar coordinates, what happens to

g^{θ θ} = 1/ r^{2}

r \to 0

It blows up (

\to \infty

), signalling the coordinate singularity at the origin — the

θ

direction is ill-defined at a point, not a failure of the geometry itself.

If a vector points purely along one orthonormal axis, does lowering change it?

Only by the diagonal entry for that axis: e.g. in Minkowski a purely-time vector

U^{μ} = (2, 0, 0, 0)

lowers to

U_{μ} = (- 2, 0, 0, 0)

— one flipped sign, others untouched.

What does

g_{μν} V^{μ} V^{ν} = 0

with

V \neq = 0

tell you about the space?

The metric is indefinite (like Minkowski spacetime); in a positive-definite inner product space the norm vanishes only for the zero vector.

Recall Quick self-audit before you leave

If you can't state why a contraction needs one up and one down index, and why raising needs the inverse metric, re-read the parent note's derivations — every trap above is one of those two rules in disguise.

Connections

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

Question bank — Metric tensor — raising - lowering indices

True or false — justify