Tensors as multilinear forms\\ Handout ``Methoden der Theoretischen Physik-\"Ubungen''

Tensors as multilinear forms
Handout ``Methoden der Theoretischen Physik-Übungen''

Karl Svozil

Abstract

Tensors are defined as multilinear forms on vector spaces

1 Notation

Consider the vector space \Bbb R^D of dimension D, a basis \mathfrak B={e₁,e₂,¼,e_D} consisting of D basis vectors e_i, and n arbitrary vectors x₁,x₂,¼,x_n Î \Bbb R^D with vector components Xⁱ₁,Xⁱ₂,¼,Xⁱ_n Î \Bbb R.

Tensor fields define tensors in every point of \Bbb R^D separately. In general, with respect to a particular basis, the components of a tensor field depend on the coordinates.

We adopt Einstein's summation convention to sum over equal indices (one pair with a superscript and a subscript). Sometimes, sums are written out explicitly.

In what follows, the notations `` x·y'', `` (x,y)'' and `` áx | yñ'' will be used synonymously for the scalar product. Note, however, that the notation `` x·y'' may be a little bit misleading; e.g. in the case of the ``pseudo-Euclidean'' metric diag(+,+,+,¼,+,-).

For a more systematic treatment, see for instance Klingbeil [1] and Dirschmid [2]. .

2 Multilinear form

A multilinear form

a:\Bbb Rⁿ ® \Bbb R
(1)
is a map satisfying

a( x₁,x₂,¼, A x¹_i+ B x²_i,¼,x_n)

=

Aa( x₁,x₂,¼, x¹_i,¼,x_n)

+Ba( x₁,x₂,¼, x²_i,¼,x_n)
(2)

for every one of its (multi-)arguments.

3 Covariant tensors

A tensor of rank n

a:\Bbb Rⁿ ® \Bbb R

(3)

is a multilinear form

a( x₁,x₂,¼,x_n) =

D
å
i₁=1

D
å
i₂=1

D
å
i_n=1

X^i_i₁ X^i₂₂¼X^i_n_na( e_i₁,e_i₂,¼,e_{i_n}).

(4)

The

A_{i₁i₂¼i_n}=a( e_i₁,e_i₂,¼,e_{i_n})

(5)

are the components of the tensor a with respect to the basis \mathfrak B.

3.0.1 Question: how many components are there?

Answer: Dⁿ.

3.0.2 Question: proof that tensors are multilinear forms.

Answer: by insertion.

a( x₁,x₂,¼, Ax¹_j+Bx_j²,¼,x_n)

D
å
i₁=1

D
å
i₂=1

D
å
i_n=1

X^i_i₁X^i₂₂¼[A(X¹)^i_j_j+B(X²)^i_j_j]¼X^i_n_na( e_i₁,e_i₂,¼,e_{i_j},¼,e_{i_n})

D
å
i₁=1

D
å
i₂=1

D
å
i_n=1

X^i_i₁X^i₂₂¼(X¹)^i_j_j¼X^i_n_na( e_i₁,e_i₂,¼,e_{i_j},¼,e_{i_n})

D
å
i₁=1

D
å
i₂=1

D
å
i_n=1

X^i_i₁X^i₂₂¼(X²)^i_j_j¼X^i_n_na( e_i₁,e_i₂,¼,e_{i_j},¼,e_{i_n})

A a( x₁,x₂,¼, x¹_j,¼,x_n)+B a( x₁,x₂,¼, x_j²,¼,x_n)

3.1 Basis transformations

Let \mathfrak B and \mathfrak B¢ be two arbitrary bases of \Bbb R^D. Then ervery vector e¢_i of \mathfrak B¢ can be represented as linear combination of basis vectors from \mathfrak B:

e¢_i=

D
å
j=1

a_i^j e_j, i=1,¼, D .

(6)

(Formally, we may treat e¢_i and e_i as scalar variables e¢_i and e_j, respectively; such that a_i^j = [(¶ e¢_i)/(¶ e_j)].)

Consider an arbitrary vector x Î \Bbb R^D with components Xⁱ with respect to the basis \mathfrak B and X¢ⁱ with respect to the basis \mathfrak B¢:

x = D
å
i=1
Xⁱ e_i = D
å
i=1
X¢ⁱ e¢_i.
(7)
Insertion into (6) yields

x = D
å
i=1
Xⁱ e_i = D
å
i=1
X¢ⁱ e¢_i = D
å
i=1
X¢ⁱ D
å
j=1
a_i^j e_j = D
å
i=1
é
ë D
å
j=1
a_i^jX¢ⁱ ù
û e_j.
(8)
A comparison of coefficient yields the transformation laws of vector components

X^j = D
å
j=1
a_i^jX¢ⁱ.
(9)
The matrix a={a_i^j} is called the transformation matrix. In terms of the coordinates X^j, it can be expressed as

a_i^j = ¶X^j
¶X¢_i
.
(10)

A similar argument using

e_i= D
å
j=1
a¢_i^j e¢_j, i=1,¼, D
(11)
yields the inverse transformation laws

X¢^j = D
å
j=1
a¢_i^jXⁱ.
(12)
(Again, formally, we may treat e¢_i and e_i as scalar variables e¢_i and e_j, respectively; such that a¢_i^j = [(¶e_i)/(¶ e¢_j)].) Thereby,

e_i= D
å
j=1
a¢_i^j e¢_j = D
å
j=1
a¢_i^j D
å
k=1
a_j^k e_k = D
å
j=1
D
å
k=1
[a¢_i^j a_j^k] e_k,
(13)
which, due to the linear independence of the basis vectors e_i od \mathfrak B, is only satisfied if

a¢_i^j a_j^k = d_i^k or a¢a=\Bbb I.
(14)
That is, a¢ is the inverse matrix of a. In terms of the coordinates X^j, it can be expressed as

a¢_i^j = ¶X¢^j
¶X_i
.
(15)

3.2 Transformation of Tensor components

Because of multilinearity (!) and by insertion into (6),

a( e¢_j₁,e¢_j₂,¼,e¢_{j_n}) = a æ
è D
å
i₁=1
a_j₁^i₁ e_i₁, D
å
i₂=1
a_j₂^i₂ e_i₂,¼, D
å
i_n=1
a_{j_n}^i_n e_{i_n} ö
ø

= D
å
i₁=1
D
å
i₂=1
¼ D
å
i_n=1
a_j₁^i₁a_j₂^i₂¼a_{j_n}^i_n a( e_i₁,e_i₂,¼,e_{i_n})
(16)

or

A¢_{j₁j₂¼j_n} = D
å
i₁=1
D
å
i₂=1
¼ D
å
i_n=1
a_j₁^i₁a_j₂^i₂¼a_{j_n}^i_n A_{i₁ i₂¼i_n}.
(17)

4 Contravariant tensors

4.1 Definition of contravariant basis

Consider again a covariant basis \mathfrak B={e₁,e₂,¼,e_D} consisting of D basis vectors e_i. We shall define now a contravariant basis \mathfrak B^*={e¹,e²,¼,e^D} consisting of D basis vectors eⁱ by the requirement that the scalar product obeys

d_i^j = e_i·e^j º (e_i,e^j) º áe_i | e^jñ = ì
ï
í
ï
î

1 if i=j

0 if i ¹ j
.
(18)
To distinguish elements of the two bases, the covariant vectors are denoted by subscripts, whereas the contravariant vectors are denoted by superscripts. The last term e_i·e^j º (e_i,e_j) º áe_i | e_jñ recalls different notations of the scalar product.

The entire tensor formalism developed so far can be applied to define contravariant tensors as multinear forms

b:\Bbb Rⁿ ® \Bbb R
(19)
by

b( x¹,x²,¼,xⁿ) = D
å
i₁=1
D
å
i₂=1
¼ D
å
i_n=1
X_i₁¹ X_i₂²¼X_{i_n}ⁿb( e^i₁,e^i₂,¼,e^i_n).
(20)
The

B^{i₁i₂¼i_n}=b( e^i₁,e^i₂,¼,e^i_n)
(21)
are the components of the contravariant tensor b with respect to the basis \mathfrak B^*.

4.2 Connection between the transformation of covariant and contravariant entities

Because of linearity, we can make the Ansatz

e¢^j=
å
i
b_i^jeⁱ,
(22)
where {b_i^j}=b is the transformation matrix associated with the contravariant basis. How is b related to a, the transformation matrix associated with the covariant basis?

By exploiting (18) one can find the connection between the transformation of covariant and contravariant basis elements and thus tensor components.

d_i^j = e¢_i·e¢^j=(a_i^ke_k)·(b_l^je^l)=a_i^kb_l^j e_k·e^l=a_i^kb_l^j d_k^l = a_i^kb_k^j,
(23)
and

b=a^-1 = a¢.
(24)
The entire argument concerning transformations of covariant tensors and components can be carried through to the contravariant case. Hence, the contravariant components transform as

b( e¢^j₁,e¢^j₂,¼,e¢^j_n) = b æ
è D
å
i₁=1
a¢^j₁_i₁ e^i₁, D
å
i₂=1
a¢^j₂_i₂ e^i₂,¼, D
å
i_n=1
a¢^j_n_{i_n} e^i_n ö
ø

= D
å
i₁=1
D
å
i₂=1
¼ D
å
i_n=1
a¢^j₁_i₁a¢^j₂_i₂¼a¢^j_n_{i_n} b( e^i₁,e^i₂,¼,e^i_n)
(25)

or

B¢^{j₁j₂¼j_n} = D
å
i₁=1
D
å
i₂=1
¼ D
å
i_n=1
a¢^j₁_i₁a¢^j₂_i₂¼a¢^j_n_{i_n} B^{i₁ i₂¼i_n}.
(26)

5 Orthonormal bases

For orthonormal bases,

d_i^j = e_i·e^jÛe_i = eⁱ ,

(27)

and thus the two bases are identical

\mathfrak B=\mathfrak B^*

(28)

and formally any distinction between covariant and contravariant vectors becomes irrelevant. Conceptually, such a distinction persists, though.

6 Invariant tensors and physical motivation

7 Metric tensor

Metric tensors are defined in metric vector spaces. A metric vector space (sometimes also refered to as ``vector space with metric'' or ``geometry'') is a vector space with inner or scalar product.

This includes (pseudo-) Euclidean spaces with indefinite metric. (I.e., the distance needs not be positive or zero.)

7.1 Definition inner or scalar product

The scalar or inner product, is a symmetric bilinear functional \Bbb R^D×\Bbb R^D® \Bbb R such that

(x+y,z)=(x,z)+(y,z) for all x,y,z Î \Bbb R^D;

(x,y+z)=(x,y)+(x,z) for all x,y,z Î \Bbb R^D;

(ax,y)=a(x,y) for all x,y Î \Bbb R^D, a Î \Bbb R;

(x,ay)=a(x,y) for all x,y Î \Bbb R^D, a Î \Bbb R;

(x,y)=(y,x) for all x,y Î \Bbb R^D

Axioms 1 and 3 assert that the scalar product is linear in the first variable. Axioms 2 and 4 assert that the scalar product is linear in the second variable. Axiom 5 asserts the bilinear function is symmetric.

7.2 Definition metric

A metric is a functional \Bbb R^D® \Bbb R with the following properties

||x -y || = 0 Û x = y,

||x- y|| = ||x- y|| (symmetry),

||x-z|| £ ||x- y||+ ||y- z|| (triangle inequality).

7.3 Construction of a metric from a scalar product by metric tensor

The metric tensor is defined by the scalar product

g_ij=e_i·e_j º (e_i, e_j) º áe_i, e_jñ.
(29)
and

g^ij=eⁱ·e^j º (eⁱ, e^j) º áeⁱ, e^jñ.
(30)
Likewise,

gⁱ_j=eⁱ·e_j º (eⁱ, e_j) º áeⁱ, e_jñ = dⁱ_j .
(31)
Note that it easy to change a covarant tensor into a contravariant and vice versa by the application of a metric tensor. This can be seen as follows. Because of linearity, any contravariant basis vector eⁱ can be written as a linear sum of covariant basis vectors:

eⁱ=A^ije_j.
(32)
Then,

g^ik = eⁱ·e^k = (A^ije_j)·e^k=A^ij(e_j·e^k)=A^ijd_j^k=A^ik
(33)
and thus

eⁱ=g^ije_j
(34)
and

e_i=g_ije^j.
(35)

Question: Show that, for orthonormal basis, the metric tensor can be represented as a Kronecker delta function in all basis (form invariant); i.e., d_ij,dⁱ_j,d_i^j,d^ij.

Question: Why is g a tensor? Show its multilinearity.

7.4 What can the metric tensor do for us?

Most often it is used to raise/lower the indices; i.e., to change from contravariant to covariant and conversely from covariant to contravariant.

In the previous section, the metric tensor has been derived from the scalar product. The converse is true as well. The metric tensor represents the scalar product between vectors: let x=Xⁱe_i Î \Bbb R^D and y=Y^je_j Î \Bbb R^D be two vectors. Then ("^T" stands for the transpose),

x·y º (x,y) º áx | yñ = Xⁱ e_i·Y^j e_j = XⁱY^j e_i·e_j = XⁱY^j g_ij = X^T g Y.
(36)

It also characterizes the length of a vector: in the above equation, set y=x. Then,

x·x º (x,x) º áx | xñ = XⁱX^j g_ij º X^T g X,
(37)
and thus

||x|| =
Ö

XⁱX^j g_ij

=
Ö

X^T g X

.
(38)

The square of an infinitesimal vector ds = {dxⁱ} is

(d s)² = g_ijdxⁱdx^j = dx^T g dx.
(39)

Question: Prove that ||x|| mediated by g is indeed a metric.

7.5 Transformation of the metric tensor

Insertion into the definitions and coordinate transformations (10) and (15) yields

g_ij=e_i·e_j = a¢^l_ie¢_l·a¢^m_je¢_m = a¢^l_i a¢^m_j e¢_l·e¢_m = a¢^l_i a¢^m_j g¢_lm = ¶X¢^l
¶Xⁱ
¶X¢^m
¶X^j
g¢_lm.
(40)

Conversely,

g¢_ij=e¢_i·e¢_j = a^l_ie_l·a^m_je_m = a^l_i a^m_j e_l·e_m = a^l_i a^m_j g_lm = ¶X^l
¶X¢ⁱ
¶X^m
¶X¢^j
g_lm.
(41)

If the geometry (i.e., the basis) is locally orthonormal, g_lm=d_lm, then g¢_ij=[(¶X^l)/(¶X¢ⁱ)][(¶X_l)/(¶X¢^j)].

7.6 Examples

For a more systematic treatment, see for instance Snapper&Troyer [3].

7.6.1 D-dimensional Euclidean space

g º {g_ij}=diag (

1,1,¼,1
D times
)
(42)

One application in physics is quantum mechanics, where D stands for the dimension of a complex Hilbert space. All definitions can be easily adopted to accommodate the complex numbers. E.g., axiom 5 of the scalar product becomes (x,y)=(x,y)^*, where `` ^* '' stands for complex conjugation. Axiom 4 of the scalar product becomes (x,ay)=a^* (x,y).

7.6.2 Lorentz plane

g º {g_ij}=diag (1,-1)
(43)

7.6.3 Minkowski space of dimension D

In this case the metric tensor is called the Minkowski metric and is often denoted by ``h'':

h º {h_ij}=diag (

1,1,¼,1
D-1 times
,-1)
(44)

One application in physics is the theory of special relativity, where D=4. Alexandrov's theorem states that the mere requirement of the preservation of zero distance (i.e., lightcones), combined with bijectivity of the transformation law yields the Lorentz transformations ([4,5,6,7,8] are original articles reviewed in [9,10]; see also [11] for an elementary proof).

7.6.4 Negative Euclidean space of dimension D

g º {g_ij}=diag (

-1,-1,¼,-1
D times
)
(45)

7.6.5 Artinian four-space

g º {g_ij}=diag (+1,+1,-1 ,-1)
(46)

7.6.6 General relativity

In general relativity, the metric tensor g is linked to the energy-mass distribution. There, it appears as the primary concept when compared to the scalar product. In the case of zero gravity, g is just the Minkowski metric (often denoted by ``h'') diag (1,1,1,-1) corresponding to ``flat'' space-time.

The best known non-flat metric is the Schwarzschild metric

g º æ
ç
ç
ç
ç
ç
ç
ç
è

(1-2m/r)^-1

0

0

0
0

r²

0

0
0

0

r²sin² q

0
0

0

0

- ( 1-2m/r)
ö
÷
÷
÷
÷
÷
÷
÷
ø
(47)
with respect to the spherical space-time coordinates r,q,f,t.

7.6.7 Computation of the metric tensor of the ball

Consider the transformation from the standard orthonormal three-dimensional ``cartesian'' coordinates X₁=x, X₂=y, X₃=z, into spherical coordinates X₁¢=r, X₂¢=q, X₃¢=j. In terms of r,q, j, the cartesian coordinates can be written as X₁=r sinqcosj º X₁¢sinX₂¢cosX₃¢, X₂=r sinqsinj º X₁¢sinX₂¢sinX₃¢, X₃=r cosq º X₁¢cosX₂¢. Furthermore, since we are dealing with the cartesian orthonormal basis, g_ij=d_ij; hence finally

g¢_ij =

¶X^l

¶X¢ⁱ

¶X_l

¶X¢^j

º diag(1,r²,r²sin² q),

(48)

and

(ds)² = (dr)²+r²(dq)²+r²sin² q(dj)².

(49)

The expression (ds)² = (dr)²+r²(dj)² for polar coordinates (D=2) is obtained by setting q = p/4 and dq = 0.

7.6.8 Computation of the metric tensor of the Moebius strip

Parameter representation of the Moebius strip:

F(u,v) =

æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

(1+vcos(

))sinu

(1+vcos(

))cosu

vsin(

)

ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø

(50)

with u Î [0,2p] represents the position of the point on the circle, and v Î [-a,a] a > 0, where 2a is the ``width'' of the Moebius strip.

F_v

=

¶F
¶v
=\allowbreak æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

cos 1
2
usinu

cos 1
2
ucosu

sin 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø
(51)
F_u

=

¶F
¶u
=\allowbreak æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

- 1
2
vsin 1
2
usinu+ æ
è 1+vcos 1
2
u ö
ø cosu

- 1
2
vsin 1
2
ucosu- æ
è 1+vcos 1
2
u ö
ø sinu

1
2
vcos 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø
(52)

( ¶F
¶v
)^T ¶F
¶du

=

\allowbreak æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

cos 1
2
usinu

cos 1
2
ucosu

sin 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø T

æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

- 1
2
vsin 1
2
usinu+ æ
è 1+vcos 1
2
u ö
ø cosu

- 1
2
vsin 1
2
ucosu- æ
è 1+vcos 1
2
u ö
ø sinu

1
2
vcos 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø

=

- 1
2
æ
è cos 1
2
usin²u ö
ø vsin 1
2
u- 1
2
æ
è cos 1
2
ucos²u ö
ø vsin 1
2
u

+ 1
2
æ
è sin 1
2
u ö
ø vcos 1
2
u=\allowbreak 0
(53)

( ¶F
¶v
)^T ¶F
¶v

=

\allowbreak æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

cos 1
2
usinu

cos 1
2
ucosu

sin 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø T

æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

cos 1
2
usinu

cos 1
2
ucosu

sin 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø

=

cos² 1
2
usin²u+cos² 1
2
ucos²u+sin² 1
2
u=\allowbreak 1
(54)

( ¶F
¶u
)^T ¶F
¶u

=

\allowbreak æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

- 1
2
vsin 1
2
usinu+ æ
è 1+vcos 1
2
u ö
ø cosu

- 1
2
vsin 1
2
ucosu- æ
è 1+vcos 1
2
u ö
ø sinu

1
2
vcos 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø T

æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
è

- 1
2
vsin 1
2
usinu+ æ
è 1+vcos 1
2
u ö
ø cosu

- 1
2
vsin 1
2
ucosu- æ
è 1+vcos 1
2
u ö
ø sinu

1
2
vcos 1
2
u
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø

=

1
4
v²sin² 1
2
usin²u+cos²u+2( cos²u) vcos 1
2
u+( cos²u)v²cos² 1
2
u

+ 1
4
v²sin² 1
2
ucos²u+sin²u+2( sin²u) vcos 1
2
u+( sin²u) v²cos² 1
2
u

+ 1
4
v²cos² 1
2
u = \allowbreak 1
4
v²+v²cos² 1
2
u+1+2vcos 1
2
u

=

(1+vcos( u
2
))²+ 1
4
v²
(55)

Thus the metric tensor is given by

g¢_ij = ¶X^s
¶X¢ⁱ
¶X^t
¶X¢^j
g_st = = ¶X^s
¶X¢ⁱ
¶X^t
¶X¢^j
d_st º æ
ç
è

F_u·F_u
F_v·F_u

F_v·F_u
F_v·F_v
ö
÷
ø =diag æ
è (1+vcos( u
2
))²+ 1
4
v² , 1 ö
ø .
(56)

8 Invariant tensors and physical motivation

What makes some touples (or matrix, or tensor components in general) of numbers or scalar functions a tensor? It is the interpretation of the scalars as tensor components with respect to a particular basis. In another basis, if we were talking about the same tensor, the tensor components; i.e., the numbers or scalar functions would be different.

The tensor components are scalars and can thus be treated as scalars. For instance, due to commutativity and associativity, one can exchange their order. (Notice, though, that this is generally not the case for differential operators such as ¶_i=¶/ ¶xⁱ.)

A form invariant tensor with respect to certain transformations is a tensor which retains the same functional form if the transformations are performes; i.e., if the basis changes accordingly. That is, numbers are mapped into the same numbers (not just any numbers). Functions remain the same but with the new parameter components as arguement. For instance; 4® 4 and f(X₁,X₂,X₃)®f(X¢₁,X¢₂,X¢₃). If a tensor is invariant with respect to one transformation, it need not be invariant with respect to another transformation, or with respect to changes of the scalar product; i.e., the metric.

Nevertheless, totally symmetric (antisymmetric) tensors remain totally symmetric (antisymmetric) in all cases:

A_{i₁i₂ ¼i_si_t¼i_n} = A_{i₁i₂ ¼i_ti_s¼i_n}

Þ

A¢_{j₁i₂ ¼j_s j_t¼j_n} = a_j₁^i₁a_j₂^i₂¼a_{j_s}^i_sa_{j_t}^i_t¼a_{j_n}^i_n A_{i₁ i₂¼i_s i_t¼i_n}

       = a_j₁^i₁a_j₂^i₂¼a_{j_s}^i_sa_{j_t}^i_t¼a_{j_n}^i_n A_{i₁ i₂¼i_t i_s¼i_n}

       = a_j₁^i₁a_j₂^i₂¼a_{j_t}^i_ta_{j_s}^i_s¼a_{j_n}^i_n A_{i₁ i₂¼i_t i_s¼i_n}

       = A¢_{j₁i₂ ¼j_t j_s¼j_n}
(57)
A_{i₁i₂ ¼i_si_t¼i_n} = -A_{i₁i₂ ¼i_ti_s¼i_n} =

Þ

A¢_{j₁i₂ ¼j_s j_t¼j_n} = a_j₁^i₁a_j₂^i₂¼a_{j_s}^i_sa_{j_t}^i_t¼a_{j_n}^i_n A_{i₁ i₂¼i_s i_t¼i_n}

       = -a_j₁^i₁a_j₂^i₂¼a_{j_s}^i_sa_{j_t}^i_t¼a_{j_n}^i_n A_{i₁ i₂¼i_t i_s¼i_n}

       = -a_j₁^i₁a_j₂^i₂¼a_{j_t}^i_ta_{j_s}^i_s¼a_{j_n}^i_n A_{i₁ i₂¼i_t i_s¼i_n}.

       = -A¢_{j₁i₂ ¼j_t j_s¼j_n}
(58)

In physics, it would be nice if the natural laws could be written into a form which does not depend on the particular reference frame or basis used. Form invariance thus is a gratifying physical feature, reflecting the symmetry against changes of coordinated and bases. Therefore, physicists tend to be crazy to write down everything in a form invariant manner. One strategy to accomplishe this to start out with form invariant tensors and compose everything from them. This method guarantees form invarince (at least in the 0'th order).

9 Some tricks

There are some tricks which are commonly used. Here, some of them are enumerated:

Indices which appear as internal sums can be renamed arbitrarily (provided their name is not already taken by some other index).

With the euclidean metric, d_ii=D.

¶Xⁱ /¶X^j=dⁱ_j.

With the euclidean metric, ¶Xⁱ /¶Xⁱ=D.

For D=3 and the euclidean metric, the Grassmann identity holds:
e_ijke_klm = d_ild_jm-d_imld_jl.

For D=3 and the euclidean metric,
|a×b | = Ö{e_ijke_ista_ja_sb_kb_t} = Ö{|a|²|b|²-(a·b)²}=Ö{det(

a·a
a·b

a·b
b·b
)} = |a||b|sinq_ab.

Let u,v º X₁¢,X₂¢ be two parameters associated with an orthonormal cartesian basis {(0,1),(1,0)} and let F:(u,v)® \Bbb R³ be a mapping from some area of \Bbb R² into a twodimensional surface of \Bbb R³. Then the metric tensor is given by g_ij = [(¶F^k)/(¶X¢ⁱ)][(¶F^m)/(¶X¢^j)] d_km.

10 Some common misconceptions

10.1 Confusion between component representation and ``the real thing''

Given a particular basis, a tensor is uniquely characterized by its components. However, without reference to a particular basis, any components are just blurbs.

Example (wrong!): a rank-1 tensor (i.e., a vector) is given by (1,2).

Correct: with respect to the basis {(0,1),(1,0)}, a rank-1 tensor (i.e., a vector) is given by (1,2).

10.2 A matrix is a tensor

See the above section.

Example (wrong!): A matrix is a tensor of rank 2.

Correct: with respect to the basis {(0,1),(1,0)}, a matrix represents a rank-2 tensor. The matrix components are the tensor components.

10.3 Decomposition of tensors

Although a tensor of rank n transforms like the tensor product of n tensors of rank 1, not all rank-n tensors can be decomposed into a single tensor product of n tensors of rank 1.

Nevertheless, any rank-n tensor can be decomposed into the sum of Dⁿ tensor products of n tensors of rank 1.

10.4 Form invariance of tensors

Although form invariance is a gratifying feature, a tensor (field) needs not be form invariant. Indeed, while

S º æ
ç
ç
ç
è

x₂²

-x₁x₂
- x₁x₂

x₁²
ö
÷
÷
÷
ø
(59)
is a form invariant tensor field with respect to the basis {(0,1),(1,0)} and orthogonal transformations (rotations around the origin)

æ
ç
ç
ç
è

cosj

sinj
-sinj

cosj
ö
÷
÷
÷
ø ,
(60)

T º æ
ç
ç
ç
è

x₂²

x₁x₂
x₁x₂

x₁²
ö
÷
÷
÷
ø
(61)
is not (please verify). This, however, does not mean that T is not a respectable tensor field; its just not form invariant under rotations.

Note that the tensor product of form invariant tensors is again a form invariant tensor.

References

[1]: Ebergard Klingbeil. Tensorrechnung für Ingenieure. Bibliographisches Institut, Mannheim, 1966.
[2]: Hans Jörg Dirschmid. Tensoren und Felder. Springer, Vienna, 1996.
[3]: Ernst Snapper and Robert J. Troyer. Metric Affine Geometry. Academic Press, New York, 1971.
[4]: A. D. Alexandrov. On Lorentz transformations. Uspehi Mat. Nauk., 5(3):187, 1950.
[5]: A. D. Alexandrov. A contribution to chronogeometry. Canadian Journal of Math., 19:1119-1128, 1967.
[6]: A. D. Alexandrov. Mappings of spaces with families of cones and space-time transformations. Annali die Matematica Pura ed Applicata, 103:229-257, 1967.
[7]: A. D. Alexandrov. On the principles of relativity theory. In Classics of Soviet Mathematics. Volume 4. A. D. Alexandrov. Selected Works, pages 289-318. 1996.
[8]: H. J. Borchers and G. C. Hegerfeldt. The structure of space-time transformations. Communications in Mathematical Physics, 28:259-266, 1972.
[9]: Walter Benz. Geometrische Transformationen. BI Wissenschaftsverlag, Mannheim, 1992.
[10]: June A. Lester. Distance preserving transformations. In Francis Buekenhout, editor, Handbook of Incidence Geometry. Elsevier, Amsterdam, 1995.
[11]: Karl Svozil. Conventions in relativity theory and quantum mechanics. Foundations of Physics, 32:479-502, 2002. e-print arXiv:quant-ph/0110054.

File translated from T_EX by T_TH, version 3.02.
On 22 Oct 2002, 16:42.

Tensors as multilinear forms Handout ``Methoden der Theoretischen Physik-Übungen''

Karl Svozil

Abstract

1 Notation

2 Multilinear form

3 Covariant tensors

3.0.1 Question: how many components are there?

3.0.2 Question: proof that tensors are multilinear forms.

3.1 Basis transformations

3.2 Transformation of Tensor components

4 Contravariant tensors

4.1 Definition of contravariant basis

4.2 Connection between the transformation of covariant and contravariant entities

5 Orthonormal bases

6 Invariant tensors and physical motivation

7 Metric tensor

7.1 Definition inner or scalar product

7.2 Definition metric

7.3 Construction of a metric from a scalar product by metric tensor

7.4 What can the metric tensor do for us?

7.5 Transformation of the metric tensor

7.6 Examples

7.6.1 D-dimensional Euclidean space

7.6.2 Lorentz plane

7.6.3 Minkowski space of dimension D

7.6.4 Negative Euclidean space of dimension D

7.6.5 Artinian four-space

7.6.6 General relativity

7.6.7 Computation of the metric tensor of the ball

7.6.8 Computation of the metric tensor of the Moebius strip

8 Invariant tensors and physical motivation

9 Some tricks

10 Some common misconceptions

10.1 Confusion between component representation and ``the real thing''

10.2 A matrix is a tensor

10.3 Decomposition of tensors

10.4 Form invariance of tensors

References

Tensors as multilinear forms
Handout ``Methoden der Theoretischen Physik-Übungen''