Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.

## Contents

* 1 Vectors, covectors and the metric
* 1.1 Mathematical formulation
* 1.2 Raising and lowering vectors and covectors
* 1.3 Example in Minkowski spacetime
* 1.4 Coordinate free formalism
* 2 Tensors
* 2.1 Examples
* 2.2 Example of raising and lowering
* 2.3 General rank
* 3 See also
* 4 References

## Vectors, covectors and the metric

### Mathematical formulation

Mathematically vectors are elements of a vector space V over a field K, and for use in physics V is usually defined with K=ℝ or ℂ. Concretely, if the dimension n=dim(V) of V is finite, then, after making a choice of basis, we can view such vector spaces as ℝn or ℂn.

The dual space is the space of linear functionals mapping V->K. Concretely, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V*:=Hom(V,K), so that α∈V* is a linear map α:V->K.

Then under a choice of basis {ei}, we can view vectors v∈V as an Kn vector with components vi (vectors are taken by convention to have indices up). This picks out a choice of basis {ei} for V*, defined by the set of relations ei(ej)=δji.

For applications, raising and lowering is done using a structure known as the (pseudo-)metric tensor (the 'pseudo-' refers to the fact we allow the metric to be indefinite). Formally, this is a non-degenerate, symmetric bilinear form

g:V×V->K a bilinear form
g(u,v)=g(v,u) for all u,v∈V (Symmetric)
∀v∈V s.t. v≠0→,∃u∈V such that g(v,u)≠0 (Non-degenerate)

In this basis, it has components g(ei,ej)=gij, and can be viewed as a symmetric matrix in Matn×n(K) with these components. The inverse metric exists due to non-degeneracy and is denoted gij, and as a matrix is the inverse to gij.

### Raising and lowering vectors and covectors

Raising and lowering is then done in coordinates. Given a vector with components vi, we can contract with the metric to obtain a covector:

gijvj=vi

and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:

gijαj=αi.

This process is called raising the index.

Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in the metric and inverse metric tensors being inverse to each other (as is suggested by the terminology):

gijgjk=gkjgji=δik=δki

where δji is the Kronecker delta or identity matrix.

Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia. Possible metrics on real space are indexed by signature (p,q). This is a metric associated to n=p+q dimensional real space. The metric has signature (p,q) if there exists a basis (referred to as an orthonormal basis) such that in this basis, the metric takes the form (gij)=diag(+1,⋯,+1,−1,⋯,−1) with p positive ones and q negative ones.

The concrete space with elements which are n-vectors and this concrete realization of the metric is denoted ℝp,q=(ℝn,gij), where the 2-tuple (ℝn,gij) is meant to make it clear that the underlying vector space of ℝp,q is ℝn: equipping this vector space with the metric gij is what turns the space into ℝp,q.

Examples:

* ℝ3 is a model for 3-dimensional space. The metric is equivalent to the standard dot product.
* ℝn,0=ℝn, equivalent to n dimensional real space as an inner product space with gij=δij. In Euclidean space, raising and lowering is not necessary due to vectors and covector components being the same.
* ℝ1,3 is Minkowski space (or rather, Minkowski space in a choice of orthonormal basis), a model for spacetime with weak curvature. It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space.

Well-formulated expressions are constrained by the rules of Einstein summation: any index may appear at most once and furthermore a raised index must contract with a lowered index. With these rules we can immediately see that an expression such as

gijviuj

is well formulated while

gijviuj

is not.

### Example in Minkowski spacetime

The covariant 4-position is given by

Xμ=(−ct,x,y,z)

with components:

X0=−ct,X1=x,X2=y,X3=z

(where x,y,z are the usual Cartesian coordinates) and the Minkowski metric tensor with metric signature (− + + +) is defined as

ημν=ημν=(−1000010000100001)

in components:

η00=−1,ηi0=η0i=0,ηij=δij(i,j≠0).

To raise the index, multiply by the tensor and contract:

Xλ=ηλμXμ=ηλ0X0+ηλiXi

then for λ = 0:

X0=η00X0+η0iXi=−X0

and for λ = j = 1, 2, 3:

Xj=ηj0X0+ηjiXi=δjiXi=Xj.

So the index-raised contravariant 4-position is:

Xμ=(ct,x,y,z).

This operation is equivalent to the matrix multiplication

(−1000010000100001)(−ctxyz)=(ctxyz).

Given two vectors, Xμ and Yμ, we can write down their (pseudo-)inner product in two ways:

ημνXμYν.

By lowering indices, we can write this expression as

XμYμ.

What is this in matrix notation? The first expression can be written as

(X0X1X2X3)(−1000010000100001)(Y0Y1Y2Y3)

while the second is, after lowering the indices of Xμ,

(−X0X1X2X3)(Y0Y1Y2Y3).

### Coordinate free formalism

It is instructive to consider what raising and lowering means in the abstract linear algebra setting.

We first fix definitions: V is a finite-dimensional vector space over a field K. Typically K=ℝ or ℂ.

ϕ is a non-degenerate bilinear form, that is, ϕ:V×V->K is a map which is linear in both arguments, making it a bilinear form.

By ϕ being non-degenerate we mean that for each v∈V, there is a u∈V such that

ϕ(v,u)≠0.

In concrete applications, ϕ is often considered a structure on the vector space, for example an inner product or more generally a metric tensor which is allowed to have indefinite signature, or a symplectic form ω. Together these cover the cases where ϕ is either symmetric or anti-symmetric, but in full generality ϕ need not be either of these cases.

There is a partial evaluation map associated to ϕ,

ϕ(⋅,−):V->V*;v↦ϕ(v,⋅)

where ⋅ denotes an argument which is to be evaluated, and − denotes an argument whose evaluation is deferred. Then ϕ(v,⋅) is an element of V*, which sends u↦ϕ(v,u).

We made a choice to define this partial evaluation map as being evaluated on the first argument. We could just as well have defined it on the second argument, and non-degeneracy is also independent of argument chosen. Also, when ϕ has well defined (anti-)symmetry, evaluating on either argument is equivalent (up to a minus sign for anti-symmetry).

Non-degeneracy shows that the partial evaluation map is injective, or equivalently that the kernel of the map is trivial. In finite dimension, the dual space V* has equal dimension to V, so non-degeneracy is enough to conclude the map is a linear isomorphism. If ϕ is a structure on the vector space sometimes call this the canonical isomorphism V->V*.

It therefore has an inverse, ϕ−1:V*->V, and this is enough to define an associated bilinear form on the dual:

ϕ−1:V*×V*->K,ϕ−1(α,β)=ϕ(ϕ−1(α),ϕ−1(β)).

where the repeated use of ϕ−1 is disambiguated by the argument taken. That is, ϕ−1(α) is the inverse map, while ϕ−1(α,β) is the bilinear form.

Checking these expressions in coordinates makes it evident that this is what raising and lowering indices means abstractly.

## Tensors

We will not develop the abstract formalism for tensors straightaway. Formally, an (r,s) tensor is an object described via its components, and has r components up, s components down. A generic (r,s) tensor is written

Tμ1⋯μrν1⋯νs.

We can use the metric tensor to raise and lower tensor indices just as we raised and lowered vector indices and raised covector indices.

### Examples

* A (0,0) tensor is a number in the field 𝔽.
* A (1,0) tensor is a vector.
* A (0,1) tensor is a covector.
* A (0,2) tensor is a bilinear form. An example is the metric tensor gμν.
* A (1,1) tensor is a linear map. An example is the delta, δμν, which is the identity map, or a Lorentz transformation Λμν.

### Example of raising and lowering

For a (0,2) tensor,[1] twice contracting with the inverse metric tensor and contracting in different indices raises each index:

Aμν=gμρgνσAρσ.

Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:

Aμν=gμρgνσAρσ

Let's apply this to the theory of electromagnetism.

The contravariant electromagnetic tensor in the (+ − − −) signature is given by[2]

Fαβ=(0−Exc−Eyc−EzcExc0−BzByEycBz0−BxEzc−ByBx0).

In components,

F0i=−Fi0=−Eic,Fij=−εijkBk

To obtain the covariant tensor Fαβ, contract with the inverse metric tensor:

Fαβ=ηαγηβδFγδ=ηα0ηβ0F00+ηαiηβ0Fi0+ηα0ηβiF0i+ηαiηβjFij

and since F00 = 0  and F0i = − Fi0, this reduces to

Fαβ=(ηαiηβ0−ηα0ηβi)Fi0+ηαiηβjFij

Now for α = 0, β = k = 1, 2, 3:

F0k=(η0iηk0−η00ηki)Fi0+η0iηkjFij=(0−(−δki))Fi0+0=Fk0=−F0k

and by antisymmetry, for α = k = 1, 2, 3, β = 0:

Fk0=−Fk0

then finally for α = k = 1, 2, 3, β = l = 1, 2, 3;

Fkl=(ηkiηl0−ηk0ηli)Fi0+ηkiηljFij=0+δkiδljFij=Fkl

The (covariant) lower indexed tensor is then:

Fαβ=(0ExcEycEzc−Exc0−BzBy−EycBz0−Bx−Ezc−ByBx0)

This operation is equivalent to the matrix multiplication

(−1000010000100001)(0−Exc−Eyc−EzcExc0−BzByEycBz0−BxEzc−ByBx0)(−1000010000100001)=(0ExcEycEzc−Exc0−BzBy−EycBz0−Bx−Ezc−ByBx0).

### General rank

For a tensor of order n, indices are raised by (compatible with above):[1]

gj1i1gj2i2⋯gjninAi1i2⋯in=Aj1j2⋯jn

and lowered by:

gj1i1gj2i2⋯gjninAi1i2⋯in=Aj1j2⋯jn

and for a mixed tensor:

gp1i1gp2i2⋯gpningq1j1gq2j2⋯gqmjmAi1i2⋯inj1j2⋯jm=Ap1p2⋯pnq1q2⋯qm

We need not raise or lower all indices at once: it is perfectly fine to raise or lower a single index. Lowering an index of an (r,s) tensor gives a (r−1,s+1) tensor, while raising an index gives a (r+1,s−1) (where r,s have suitable values, for example we cannot lower the index of a (0,2) tensor.)

## See also

* Ricci calculus
* Einstein notation
* Metric tensor
* Musical isomorphism
* Dual space § Bilinear products and dual spaces

## References

1. ↑ 1.0 1.1 Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. New York: McGraw Hill. ISBN 0-07-033484-6.
2. ↑ NB: Some texts, such as: Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4. , will show this tensor with an overall factor of −1. This is because they used the negative of the metric tensor used here: (− + + +), see metric signature. In older texts such as Jackson (2nd edition), there are no factors of c since they are using Gaussian units. Here SI units are used.

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Tensors

Glossary of tensor theory

Scope|

| Mathematics|

* coordinate system
* multilinear algebra
* Euclidean geometry
* tensor algebra
* dyadic algebra
* differential geometry
* exterior calculus
* tensor calculus

|

* Physics
* Engineering

* continuum mechanics
* electromagnetism
* transport phenomena
* general relativity
* computer vision

Notation|

* index notation
* multi-index notation
* Einstein notation
* Ricci calculus
* Penrose graphical notation
* Voigt notation
* abstract index notation
* tetrad (index notation)
* Van der Waerden notation

Tensor
definitions|

* tensor (intrinsic definition)
* tensor field
* tensor density
* tensors in curvilinear coordinates
* mixed tensor
* antisymmetric tensor
* symmetric tensor
* tensor operator
* tensor bundle
* two-point tensor

Operations|

* tensor product
* exterior product
* tensor contraction
* transpose (2nd-order tensors)
* raising and lowering indices
* Hodge star operator
* covariant derivative
* exterior derivative
* exterior covariant derivative
* Lie derivative

Related
abstractions|

* dimension
* basis
* vector, vector space
* multivector
* covariance and contravariance of vectors
* linear transformation
* matrix
* spinor
* Cartan formalism
* differential form
* exterior form
* connection form
* geodesic
* manifold
* fiber bundle
* Levi-Civita connection
* affine connection

Notable tensors|

| Mathematics|

* Kronecker delta
* Levi-Civita symbol
* metric tensor
* nonmetricity tensor
* Christoffel symbols
* Ricci curvature
* Riemann curvature tensor
* Weyl tensor
* torsion tensor

|

Physics|

* moment of inertia
* angular momentum tensor
* spin tensor
* Cauchy stress tensor
* stress–energy tensor
* EM tensor
* gluon field strength tensor
* Einstein tensor
* metric tensor (GR)

Mathematicians|

* Leonhard Euler
* Carl Friedrich Gauss
* Augustin-Louis Cauchy
* Hermann Grassmann
* Gregorio Ricci-Curbastro
* Tullio Levi-Civita
* Jan Arnoldus Schouten
* Bernhard Riemann
* Elwin Bruno Christoffel
* Woldemar Voigt
* Élie Cartan
* Hermann Weyl
* Albert Einstein

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