In
mathematics and
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, raising and lowering indices are operations on
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s which change their
type. Raising and lowering indices are a form of
index manipulation in tensor expressions.
Vectors, covectors and the metric
Mathematical formulation
Mathematically vectors are elements of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field
, and for use in physics
is usually defined with
or
. Concretely, if the dimension
of
is finite, then, after making a choice of
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
, we can view such vector spaces as
or
.
The
dual space is the space of
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s mapping
. 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
, so that
is a linear map
.
Then under a choice of basis
, we can view vectors
as an
vector with components
(vectors are taken by convention to have indices up). This picks out a choice of basis
for
, defined by the set of relations
.
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
:
:
:
In this basis, it has components
, and can be viewed as a symmetric matrix in
with these components. The inverse metric exists due to non-degeneracy and is denoted
, and as a matrix is the inverse to
.
Raising and lowering vectors and covectors
Raising and lowering is then done in coordinates. Given a vector with components
, we can contract with the metric to obtain a covector:
:
and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:
:
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):
:
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
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
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if ''A'' is the symmetric matrix that defines the quadra ...
. Possible metrics on real space are indexed by signature
. This is a metric associated to
dimensional real space. The metric has signature
if there exists a basis (referred to as an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
) such that in this basis, the metric takes the form
with
positive ones and
negative ones.
The concrete space with elements which are
-vectors and this concrete realization of the metric is denoted
, where the 2-tuple
is meant to make it clear that the underlying vector space of
is
: equipping this vector space with the metric
is what turns the space into
.
Examples:
*
is a model for 3-dimensional space. The metric is equivalent to the standard
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
.
*
, equivalent to
dimensional real space as an inner product space with
. In Euclidean space, raising and lowering is not necessary due to vectors and covector components being the same.
*
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
:
is well formulated while
:
is not.
Example in Minkowski spacetime
The covariant
4-position
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
is given by
:
with components:
:
(where ,, are the usual
Cartesian coordinates) and the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
tensor with
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
(− + + +) is defined as
:
in components:
:
To raise the index, multiply by the tensor and contract:
:
then for :
:
and for :
:
So the index-raised
contravariant 4-position is:
:
This operation is equivalent to the matrix multiplication
:
Given two vectors,
and
, we can write down their (pseudo-)inner product in two ways:
:
By lowering indices, we can write this expression as
:
What is this in matrix notation? The first expression can be written as
:
while the second is, after lowering the indices of
,
:
Coordinate free formalism
It is instructive to consider what raising and lowering means in the abstract linear algebra setting.
We first fix definitions:
is a finite-dimensional vector space over a field
. Typically
or
.
is a non-degenerate bilinear form, that is,
is a map which is linear in both arguments, making it a bilinear form.
By
being non-degenerate we mean that for each
, there is a
such that
:
In concrete applications,
is often considered a structure on the vector space, for example an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
or more generally a
metric tensor which is allowed to have indefinite signature, or a symplectic form
. Together these cover the cases where
is either symmetric or anti-symmetric, but in full generality
need not be either of these cases.
There is a partial evaluation map associated to
,
:
where
denotes an argument which is to be evaluated, and
denotes an argument whose evaluation is deferred. Then
is an element of
, which sends
.
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
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
has equal dimension to
, so non-degeneracy is enough to conclude the map is a linear isomorphism. If
is a structure on the vector space sometimes call this the canonical isomorphism
.
It therefore has an inverse,
and this is enough to define an associated bilinear form on the dual:
:
where the repeated use of
is disambiguated by the argument taken. That is,
is the inverse map, while
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
tensor is an object described via its components, and has
components up,
components down. A generic
tensor is written
:
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
.
* 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
* A (1,1) tensor is a linear map. An example is the delta,
, which is the identity map, or a Lorentz transformation
Example of raising and lowering
For a (0,2) tensor,
twice contracting with the inverse metric tensor and contracting in different indices raises each index:
:
Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:
:
Let's apply this to the theory of electromagnetism.
The
contravariant electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
in the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
is given by
[NB: Some texts, such as: , 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 mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
. In older texts such as Jackson (2nd edition), there are no factors of since they are using Gaussian units
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
. Here SI units are used.
:
In components,
:
To obtain the
covariant tensor , contract with the inverse metric tensor:
:
and since and , this reduces to
:
Now for , :
:
and by antisymmetry, for , :
:
then finally for , ;
:
The (covariant) lower indexed tensor is then:
:
This operation is equivalent to the matrix multiplication
:
General rank
For a tensor of order , indices are raised by (compatible with above):
:
and lowered by:
:
and for a mixed tensor:
:
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
tensor gives a
tensor, while raising an index gives a
(where
have suitable values, for example we cannot lower the index of a
tensor.)
See also
*
Ricci calculus
*
Einstein notation
*
Metric tensor
*
Musical isomorphism
*
References
{{tensors
*