Abstract index notation

The index notation is a shape to represent tensors in writing that finds application especially in physics and sometimes also in the mathematical branch of differential geometry.

In its more common form of the tensor notation are in particular coordinates. With the abstract index notation, however, Tensors are called coordinate -independent, where the notation indicates the type of the tensor and can represent contractions and covariant differentiations coordinate- free. The abstract index notation was introduced by Roger Penrose.

Most commonly, this notation is made ​​in the context of general relativity, their formulation in the form of tensors. Even some modern texts on special relativity use this notation, and in the context of gauge theories it is also found in quantum field theory. This notation is particularly suitable for calculations in local coordinates, thus in physics is much more common than in mathematics.

There are two basic forms of notation. In one place the tensors with indices of elements of the tensors in local coordinates dar. In this variant, the Einstein summation convention is used to perform contractions or track formations. The second possibility is the abstract tensor notation. In this, the indices are no longer show the components in coordinates, but are only symbols that indicate the rank of the tensor.

• 4.1 Examples of level 1
• 4.2 Examples of Level 2

Tensors

In differential geometry, the geometry of curved spaces is investigated, which are described by so-called differentiable manifolds. These manifolds at each point P allow the definition of a d-dimensional real vector space which is referred to as the tangent in that point. If the manifold is embedded in a higher-dimensional space, corresponding to the tangent space exactly the d- dimensional hyper- surface that contacts the manifold at the point p and there is tangential to her. The dual space of the tangent space is called the cotangent space.

The elements of a tensor product of k copies of the Kotangentialraums and l copies of the tangent space are called (k, l) - tensors respectively. So you are multilinear mappings, the k elements of the tangent space and l elements of Kotangentialraums mapped to a real number. A (k, l ) tensor is a diagram depicting any point of the manifold to a (k, l )-tensor.

The coordinate representations of tensor fields must meet a specific transformation behavior under changing cards pictures, so local diffeomorphisms.

Index notation

Subscript notation write the arguments in which the tensor is linear, not by means of a clip but by an argument indices. These indices are raised or subscript, depending on whether the argument of the tangent or cotangent is the. A (k, l) - tensor T with arguments from the tangent space and from the cotangent space is therefore listed as:

The index notation based on the fact that tensors are multilinear mappings and therefore the arguments in which they are linear, fulfill a distributive and commute with the multiplication by scalars. This means that, for example, with r and s are real numbers and and from the tangent space can be used instead and thus can be expected to continue as with numbers.

If you understand the above formula as coordinate notation, it is the summation convention easy to understand. This notation, however, can also interpret coordinates freely, with the position of the indices only describes what type tensor is present, ie being above indices denote copies of the tangent space and below indices copies of Kotangentialraums. The sign of the tensor is omitted in this notation, that is consecutively written tensors are interpreted as tensor product. Once an up - and once index below a contraction is understood analogously to the canonical pairing, which is not principally dependent on base.

Tensors in physics

A tensor is, in the parlance of physics, an equivalence class of triples (B, S, T ), consisting of

• The length of the signature S, which also indicates the number of arguments of the image T, is level of the tensor.
• T for the imaging operation, the usual notation is not used, but similar effects to ( and historically old ) Index notation, said indices after the function icon can be arranged in alternative upper and lower T. What indexes above and which are written below, indicates the signature. It has an entry h or t in the signature for the high and low points of the corresponding index.
• Two triple ( B, S, T ) and (B ', S', T ') designate the same tensor, if the structure is consistent, i.e., S = S', and the components of T and T ' on the coordinate change matrix A are connected. That is, B = B = A, when the base is viewed as a column vector of the basis vectors, and A is an n × n matrix. The transformation behavior then has the following form

Relationship to the geometric tensor

A class of equivalent Darstellungstripel (B, S, T ) denotes the coordinates of a display element from the Tensorproduktraum

Where the vector space and the dual vector space of linear forms. The element itself is then the sum

With a base vector, and an element of the dual basis.

Examples of the coordinate notation

Examples of Stage 1

Example of a contravariant parameter is the column vector of the coordinates of a position vector, so as triples (B, ( h), x). Contravariant sizes have always agreed superscripts. The range of the indices always corresponds according to its origin of the base and thus has a number, corresponding to the dimension of the area.

Under a Basis-/Koordinatenwechsel B = B = A, the vector transforms as

The invariant geometric object is the vector

In the relativistic space-time coordinates are as

Specified.

For example a co-variant parameter is the row vector of the coordinates of a one - form, i.e., a linear functionals, or a triple (W (t), α ). Covariate variables have agreed always subscripts. They transform themselves by definition,

The invariant geometric object is the covector

In the relativistic space-time coordinates are as

Specified.

Analogous to the multiplication of a row with a column vector in defining the use of a linear functional on a vector:

The last notation uses the Einstein summation convention, which states that summing over the same named indexes if the one on the bottom and the other is above. This is also called, somewhat inaccurately, by the scalar product of a co- and a contravariant vector.

It is easy to check that this actually is a scalar, that is, a tensor transformationsinvarianten 0 Level:

Examples of Level 2

It very often takes place a description contravariant coordinates in covariant, that is, a transformation of a vector into a 1- form and vice versa. This is called high places or down Indexes.

This is made possible by a metric tensor g, a tensor of rank (0,2) with double covariant coordinates. That it corresponds to the tuples (B i (t, t), g) and the transformation rule

In general, requires that the metric tensor symmetrical - g ( x, y) = g ( y, x) respectively - and is non-degenerate, i.e., there must be an inverse symmetrical tensor g -1 of the stage (2, 0) indicates which contravariant coordinates has, so that:

The inverse of the metric tensor is also called its contravariant form.

The adjoint 1-form of the position vector x then has the " lowered " coordinates

The application of the adjoint 1-form on the position vector

Is a square image that maps the position vector of a real number.

The vector was an expressed by the Cartesian coordinates x, y, z and the time coordinate ct.

In the special theory of relativity and in Minkowski space is the coordinate matrix of the metric tensor with diagonal entries ( 1, -1, -1, -1) on the diagonal, it will be admitted as Koordinaten-/Basistransformationen only Lorentz transformations, which make these normal form of the metric tensor unchanged. The corresponding adjoint covariant vector is in these coordinates:

It follows that. It should be noted that the apparent simplicity of this formula is a complex construction hides: The vector x is expressed in two different coordinate representations is already being received in one of the metric tensor. Although the usual coordinate representation of a scalar product has the same complexity, but does not hide this.

Through the contravariant and covariant notation are representations in the form ( x, y, z, ict ) with the imaginary unit i avoided, as they were formerly in use and are still used in some textbooks.

In addition to their use in the special theory of relativity allows direct transition to the general case.

Abstract indices and Tensorräume

A general homogeneous tensor is an element of an arbitrarily often repeated tensor product of vector spaces V and V *, such as:

Now each factor receives in this tensor a label using a Latin letter in superscript position, if it is a contra-variant factor (ie, V) is or in a lowered position, if it is a covariant factor ( the dual space V *). Thus, the product is as

Respectively

Representable.

It is important to be aware that these terms represent the same object. Thus tensors of this type are represented by the following equivalent expressions:

Contraction

Whenever in the tensor product of vector spaces V and V * is a covariant and a contravariant factor occurs, there is a track associated with it. For example, is

The trace of the first two vector spaces. and

The track of the first and fifth vector space. This track operations can be in the abstract index notation be represented as follows:

Zopfabbildungen ( Braiding Map)

For each tensor exist so-called Zopfabbildungen. For example swaps the Zopfabbildung

The two Tensorfaktoren (ie ). Zopfabbildungen stand in a unique relationship to the symmetric group, by exchanging the Tensorfaktoren. With the Zopfabbildung is known that σ the permutation applies to the Tensorfaktoren.

Zopfabbildungen are important in differential geometry. For example, can the Bianchi identity expressed by the fact. Here R is the Riemann curvature tensor, which is regarded as tensor in. The first Bianchi identity reads:

In the abstract index notation, the arrangement of the indices is fixed (usually ordered lexicographically ). Thus, a Zopfabbildung are represented by interchanging the indices. For example, the Riemann curvature tensor in the abstract index notation:

The Bianchi identity is to