Tensor Analysis-I - Sharif
Transcript of Tensor Analysis-I - Sharif
But one thing is certain, in all my life I have never labored nearly as hard, andI have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. - Einstein
Vector Space
β’ A vector space ππ,πΉπΉ is defined as a set of vectors ππ and scalars πΉπΉ,that satisfy the familiar axioms of vector space structure.
β’ If any vector in the space can be expanded with a set ππππ ππ=1ππ of
vectors, we say that this set is a basis if each ππππ & ej are mutually linearly independent. ππ if finite is unique and this unique number is denoted as the dimension of ππ i.e. dim ππ = ππ.
β’ In the context of relativity as we will see, we will always work with finite dimensional vector spaces, so our mathematics would be rather simple and precise.
Vector Space
β’ For a basis ππππ ππ=1ππ we can expand each vector in unique way with
certain coefficients, π£π£ππ:
π£π£ = βπ£π£ππππππ β‘ π£π£ππππππ
The Dual Space
β’ A linear functional on V is a linear map ππ:ππ β πΉπΉ. The set ππβ of all linear functionals on V is called the dual space of V.
β’ One can easily realize that this space is also a vector space.β’ It is customary to write β¨π£π£,πΉπΉβ© or β¨ππ, π£π£β© to denote ππ(π£π£). When written
this way it is called the natural pairing or dual pairing between V and ππβ. Elements of ππβare often called covectors.
The Dual Space
β’ If ππππ ππ=1ππ is a basis of V, there is a canonical dual basis or cobasis
ππππ ππ=1ππ
of ππβ, defined by ππππ ,ππππ = πΏπΏππππ , where πΏπΏππ
ππ is the Kroneckerdelta.
β’ any element ππ β ππβ can be expanded in terms of the dual basis,
where ππππ β πΉπΉ are scalars.ππ = ππππππππ
The Dual Space
β’ From the argument in the last page dim(ππβ) = dim(ππ) so these to spaces are isomorphic, but it seems that our isomorphy depends on our choice of basis ππππ ππ=1
ππ , for this reason we say that these spaces are not isomorphic in a natural way (there is an exception, that is the case where we have an innerproduct as will be defined later in these slides. In that case π π π π πππ π π π πΏπΏπππΏπΏπΏπΏπΏπΏ guarantees the naturalness of this isomorphy.).
β’ On the other hand ππββ & ππ are isomorphic naturally.
Inner product spaces
β’ So far everything we have said about vector spaces and linear maps works in essentially the same way regardless of the underlying field πΉπΉ, but there are are a few places where the underlying field matters and here we consider one of them.
β’ Although in SR we are concerned with real numbers, here we express the definitions for complex numbers to be more general.
Inner product spaces
β’ Thus, let πΉπΉ be a subfield of πΆπΆ, and let ππ be a vector space over πΉπΉ. A sesquilinear form on V is a map ππ βΆ ππ Γ ππ β πΉπΉ satisfying the following two properties.
1. linear on the second entry: ππ(π’π’,πΏπΏπ£π£ + ππππ) = πΏπΏππ(π’π’, π£π£) + ππππ(π’π’,ππ)2. Hermitian: ππ(π£π£,π’π’) = ππ π’π’, π£π£β’ Note that these two properties together imply that g is antilinear on
the first entry.β’ If πΉπΉ is a subfield of real numbers ππ would be bilinear and symmetric,
so a real sesquilinear form is in fact a symmetric bilinear form.
Inner product spaces
β’ If the sesquilinear form g is3. (3) nondegenerate, so that ππ(π’π’, π£π£) = 0 for all π£π£ implies π’π’ = 0,
then it is called an inner product.β’ A vector space equipped with an inner product is called an inner
product space.
Inner product spaces
β’ By Hermiticity ππ(π’π’,π’π’) is always a real number. We may thus distinguish some important subclasses of inner products. If ππ(π’π’,π’π’) β₯0 (respectively, ππ(π’π’,π’π’) β€ 0) then g is nonnegative definite (respectively, nonpositive definite).
β’ A nonnegative definite (respectively, nonpositive definite) inner product satisfying the condition that ππ(π’π’,π’π’) = 0 implies π’π’ = 0 is positive definite (respectively, negative definite).
Inner product spaces
β’ A positive definite or negative definite inner product is always nondegenerate but the converse is not true.
Tensor product
β’ We define the tensor product space of two vector spaces, ππ,ππ by the space ππ βππ = π π π π πΏπΏππ ππππ β ππππ ij
. And I hope that you are familiar with the definition of the tensor product of basis vectors in finite dimensions from linear algebra!
β’ We will use the term tensor to describe the general element of ππ βππ.
Dual representation of tensor product
β’ Every pair of vectors (π£π£,ππ) where π£π£ β ππ,ππ β ππ defines a bilinear map ππβ Γ ππβ β πΉπΉ, by setting
β’ Then one can define for any element in A β ππ βππ, a map ππβ Γππβ β πΉπΉ by setting:
β’ One can also show that this map is one-to-one(exercise!). So this identification gives a dual representation of ππ βππ as bilinear maps.
(π£π£,ππ) βΆ (ππ,Ο) β Ο(π£π£)Ο(ππ).
π΄π΄ = π΄π΄ππππππππ β ππππ β π΄π΄ ππ,ππ = π΄π΄ππππππ ππππ ππ(ππππ)
Multilinear maps and tensor spaces of type (r, s)
β’ From now on I work in real vector spaces, we refer to any multilinearmap
as a tensor of type (r, s) on V. The integer r β₯ 0 is called the contravariant degree and s β₯ 0 the covariant degree of T . I will denote this space by ππ(ππ,π π ).
ππ:ππβ Γ ππβ Γ β―Γ ππβ Γ ππ Γ ππ Γ β―Γ ππ β π π
ππ π π
Multilinear maps and tensor spaces of type (r, s)
β’ If T is a tensor of type (r, s) and S is a tensor of type (p, q) then define T β S, called their tensor product, to be the tensor of type (r + p, s + q) defined by,
β’ This product generalizes the definition of tensor products of vectors and convectors that we introduced earlier.
ππ β ππ ππ1, β¦ ,ππππ ,ππ1, β¦ ,ππππ,π’π’1, β¦ ,π’π’π π , π£π£1, β¦ , π£π£ππ= ππ ππ1, β¦ ,ππππ ,π’π’1, β¦ ,π’π’π π ππ ππ1, β¦ ,ππππ, π£π£1, β¦ , π£π£ππ
Multilinear maps and tensor spaces of type (r, s)
β’ You are advised to show as an exercise that tensor products,
form a basis for ππ(ππ,π π ). β’ then one can write:
ππππ1 ββ―β ππππππ β ππππ1 ββ―β πππππ π , π π ππ , ππππ = 1, β¦ ,ππ
βππ β ππ ππ,π π : ππ = ππππ1β¦πππ π ππ1β¦ππππππππ1 ββ―β ππππππ β ππππ1 ββ―β πππππ π
ππππ1β¦πππ π ππ1β¦ππππ= ππ(ππππ1 , β¦ ,ππππππ , ππππ1 , β¦ , πππππ π )
Multilinear maps and tensor spaces of type (r, s)
β’ Let {ππππ } and {ππππβ²} be two bases of V related by
β’ Then one can see that the dual basis transforms by ππππ = π΄π΄ππ
ππππππβ², ππππβ² = π΄π΄β1 ππππππππ
ππβ²ππ = π΄π΄ππππππππ
Multilinear maps and tensor spaces of type (r, s)
β’ Then for a general tensor we have:
β’ Then we can see (exercise):ππ = ππππ1β¦πππ π
ππ1β¦ππππππππ1 ββ―β ππππππ β ππππ1 ββ―β πππππ π = πππππβ²1β¦ππβ²π π ππβ²1β¦ππβ²πππππππβ²1 ββ―β πππππβ²ππ β πππππβ²1 ββ―β ππβ²ππβ²π π
πππππβ²1β¦ππβ²π π ππβ²1β¦ππβ²ππ = π΄π΄ππ1
ππ1β² β¦π΄π΄ππππππππβ² π΄π΄β1 ππ1β²
ππ1 β¦ π΄π΄β1 πππ π β²πππ π ππππ1β¦πππ π
ππ1β¦ππππ
Index Notation, a physicistβs way of looking at tensors!
β’ Physicists simply define tensors as objects that transform the way we explained in the last slide. This definition is practical but imprecise and one should always keep in mind what one really means by a tensor.
Operations on tensors
β’ Contraction: The process of tensor product creates tensors of higher degree from those of lower degrees, We now describe an operation that lowers the degree of tensor.
β’ Firstly, consider a mixed tensor ππ = ππππππ ππππ β ππππ of type (1, 1). Its contraction is defined to be a scalar denoted πΆπΆ11ππ , given by
πΆπΆ11ππ = ππ (ππππ , ππππ) = ππ (ππ1 , ππ1) +Β·Β·Β· + ππ (ππππ, ππππ).
Operations on tensors
β’ Although a basis of V and its dual basis have been used in this definition, it is independent of the choice of basis
β’ More generally, for a tensor T of type (r,s) with both r > 0 and s > 0 one can define its (p, q)-contraction (1 β€ p β€ r, 1 β€ q β€ s) to be the tensor πΆπΆππ
ππππ of type (r β 1,s β 1) defined by
ππ ππβ²ππβ², ππππβ²
β² = π΄π΄ππππβ²π΄π΄ππβ²β1ππππ ππππ , ππππ = ππ(ππππ , ππππ)
πΆπΆππππππ ππ1, β¦ ,ππππβ1, π£π£1, β¦ , π£π£π π β1
= οΏ½ππ=1
ππ
ππ ππ1,β¦,ππππβ1, ππππ ,ππππ+1, β¦ ,ππππβ1, π£π£1, β¦ , π£π£ππβ1, ππππ , π£π£ππ+1, β¦ , π£π£π π β1 .
Operations on tensors
β’ One can then easily find the contracted components of a tensor,
πΆπΆππππππ
ππ1β¦πππ π β1
ππ1β¦ππππβ1 = ππππ1β¦ππππβ1ππ ππππ+1β¦πππ π β1ππ1β¦ππππβ1ππ ππππ+1β¦ππππβ1
Operations on tensors
β’ Raising and lowering indices: Finally we define one last notion which is widely used in relativity. Let V be a real inner product space with metric tensor ππ = ππππππ ππππ β ππππ , then one can see:
π’π’. π£π£ = πππππππ’π’πππ’π’ππ = πΆπΆ11πΆπΆ22ππ β π’π’ β π£π£π’π’ππ = πππππππ’π’ππ = πΆπΆ21 ππ β π’π’ππππ = ππππππππππ = πΆπΆ12(ππβ1 βππ)