Classification of second order symmetric tensors in the Lorentz metric

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Classification of second order symmetric tensors in the

Lorentz metric

Applied Mathematics, Link¨opings Universitet Hampus Hjelm Andersson LiTH - MAT - EX - - 2010 / 21 - - SE

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Matematiska Institutionen 581 83 LINK ¨OPING SWEDEN 9 June 2010 x x http://www.ep.liu.se LiTH - MAT - EX - - 2010 / 21 - - SE

Classification of second order symmetric tensors in the Lorentz metric

Hampus Hjelm Andersson

This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is in-troduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor. Sammanfattning Abstract F¨orfattare Author Titel Title

URL f¨or elektronisk version

Serietitel och serienummer Title of series, numbering

ISSN 0348-2960 ISRN ISBN Spr˚ak Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats ¨ Ovrig rapport Avdelning, Institution Division, Department Datum Date

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Abstract

This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.

Keywords: Indefinite linear algebra, Lorentz metric, Minkowski space, sym-metric tensors, Segre type, classification of canonical form.

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Acknowledgements

I would like to thank my supervisor Göran Bergqvist. My opponent Björn Morén deserves my thanks as well, and anyone else who wants to be mentioned here.

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Introduction

This text is written as a bachelor of science final thesis at Link¨opings universitet by Hampus Hjelm Andersson with G¨oran Bergqvist as supervisor and examiner in LA_{TEX in 2010.}

The main purpose with this bachelor thesis was to make an introduction to the concept of how symmetric tensors in the Lorentz metric can be character-ized and the use of indefinite algebra to do so. This will be done as a litterature review of two books in particular [1, 2] but some results from other books [3, 4] will also be used. The majority of results and notation will however be taken from [2]. This method was chosen since the only other possibility would be to create the results oneself, but since I wasn’t familiar with this subject before the start of the project and considering the relatively short time period, that was not really a possibility at all. One reason for choosing this subject is that the Lorentz metric is commonly used in general relativity. The indefinite lin-ear algebra is also used in many applications such as differential and difference equations with constant coefficients, and algebraic Riccati equations and many other topics. Another good reason is that the indefinite linear algebra is a gen-eralisation of the definite algebra and thus gives a more complete description of the subject. In indefinite linear algebra the inner product is defined differently, this gives rise to new interesting cases where vectors can be orthogonal to them-selves and the union of a subspace and its orthogonal complement doesn’t have to span the space. The amount of theory about the general indefinite algebra is quite extensiv so I will mostly be focusing on the special case with the Lorentz inner product. So with this new inner product some of the old results are no longer true, I will especially look more at the canonical form of real symmetric tensors who will have four different characterizations instead of one as it was in the definite case.

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Chapter 2

Introductory Theory

2.1 The indefinite inner product

In classical linear algebra orthogonality, length, angle and many other concepts are dependent on the definite inner product which acts on the space. In indef-inite algebra however the defindef-inite inner product is replaced with an indefindef-inite one. Here the standard inner product is written as (.., ..) and

(x, y) =

n

X

k=1

xkyk

where x, y ∈ Cn. The axioms that define the indefinite inner product is quite similar to those of the standard inner product as the following definition shows. Definition 1. A function [.., ..] on Cn_{× C}n

to C is said to be an indefinite inner product if the following conditions are satisfied

i) [αw1+ βw2, w3] = α[w1, w3] + β[w2, w3]

for all α, β ∈ C and w1, w2, w3∈ Cn

ii) [w1, w2] = [w2, w1] for all w1, w2∈ Cn

iii) if [w1, w] = 0 for all w ∈ Cn then w1= 0

The thing to notice here is (iii) which is a new formulation of the ”old” definition which said (w, w) ≥ 0 for all w ∈ C and (w, w) = 0 ⇐⇒ w = 0. A consequence of this is that a vector now can be orthogonal to or have a negativ inner product by itself.

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4 Chapter 2. Introductory Theory

The orthogonal complement has also a new definition which says that

W⊥_{= {x ∈ C}n: [x, w] = 0 ∀w ∈ W }.

Here the only difference is that (.., ..) is replaced with [.., ..] An operator with a matrix A as representative will be called H-selfadjoint iff

[Aw1, w2] = [w1, Aw2] ∀w1, w2∈ Cn

It no longer holds that A needs to be equal to its conjugated transpose. However now we get a new condition for A to be H-selfadjoint which states that HA = A∗H were A* is the conjugated transpose to A in the usual sense.

A canonical form is a pair (J, P ) such that these are in the equvivalence class of H and an H-selfadjoint matrix A and for this pair there exists T such that

A = T−1J T, H = T∗P T where J is in Jordan normal form and P hermitian.

2.1.1 The Lorentz inner product

In the remaining of this report a different notation will be used, that coincides with the notaion of the book that most of the results are taken from [2]. Instead of writing [x, y] = (Hx, y) = (x, Hy) this will now be denoted by Habxayb= xbyb

where a summation over respective index is understood and for x, y ∈ Cn_{, x}a_{, y}b

are the components of x, y. Here Habxa = xb will be used to simplify things

and also S c

a Hcb= Sab. To simplify things further xayb+xbyais written 2x(ayb).

The Lorentz inner product η is defined by η(x, y) = ηabxayb and ηab =

diag(-1,1,1,1). So with notation as above

Hab= ηab=     −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1    

The standard basis with the Lorentz inner product will be called t, x, y, z and is as follows t =     −1 0 0 0     , x =     0 1 0 0     , y =     0 0 1 0     , z =     0 0 0 1    

Here tata = −1 and taxa = taya = taza = 0 and x, y, z are still orthonormal

as in classical linear algebra. Sometimes though it is usefull to express things in a different basis. One basis that will be used frequently is l, n, x, y which is defined by l =t + z√ 2 = 1 √ 2     −1 0 0 1    

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2.2. Minkowski space 5 n =−t + z√ 2 = 1 √ 2     1 0 0 1    

and x, y are the same as before. Here lala= nana= 0 and lana = 1 and of cours

l, n⊥ x, y respectivly. The vectors l, n are called null vectors and this basis is called a null basis (which will be explained in the section of Minkowski space). However sometimes this won’t be enough and one will be compelled to switch basis. Then its convenient to introduce a new complex basis l, n, m, ¯m where m = x+iy√

2 and ¯m = x−iy_√

2 . Then using a null rotation on this new basis l, n, m, ¯m

will be a changed to a new basis l0, n0, m0, ¯m0 where l0= Al, m0= eiθ(m − A ¯B) n0 = A−1n + Bm + ¯B ¯m − AB ¯Bl

(A, θ ∈ R, A > 0, B ∈ C) (2.2)

This rotation preserves the properties of the null basis i.e l0

al0a= n0an0a= 0 and

l_a0n0a = 1. We also have that l0_am0a = l0_am¯0a _{= n}0

am0a = n0am¯0a = m0am0a =

¯ m0

am¯0a= 0 and m0am¯0a = 1. The effect on x, y in this new basis can be seen

by noting that

x0+ iy0= (cosθ + isinθ)(x + iy − A ¯B) =⇒

=⇒ x

0_{= xcosθ − ysinθ −}√_2ARe(eiθ_B)¯ y0 = xsinθ + ycosθ −√2AIm(eiθ_B)¯

2.2 Minkowski space

Here a short introduction to Minkowski space will be given. A Minkowski space is a 4-dimensional real vector space with the Lorentz inner product. The Minkowski space could be seen as the tangent space of a Minkowski space-time at a point p with the inner product η(p). A Minkowski space-time is the result of regarding R4 as a 4-dimensional smooth manifold with the global smooth Lorentz-metric η. From now on however and throughout the rest of this report it will be assumed that there exists a point p where we have a Minkowski space. In Minkowski space a non-zero member x will be called spacelike, timelike or null depending on whether η(x, x) is positiv, negativ or zero. The subsets of R4 will be denoted be S, T and N and consist of spacelike, timelike or null members respectively. From this the Minkowski space can be decomposed in a disjoint

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An example of a null basis is the one mentioned above where f1 = l, f2 =

n, f3 = x, f4 = y. This can be illustrated by the following picture, if we omit

the y axis (since it is difficult to illustrate 4-dimensional spaces).

This kind of picture where one can see the null directions is often referred to as a light-cone in applications such as general relativity.

All 2-dimensional subspaces of Minkowski space can be disjointly discom-posed into S2∪ T2∪ N2 where each one of these are called either spacelike,

timelike or null 2-spaces. Every spacelike 2-space contain no null vectors and the timelike spaces contain two distinct null directions whereas the null 2-spaces only have one null direction. In the same manner the 3-spaces can be charac-terized with either zero, two (and therfore infinitely many) or one distinct null directions if they are spacelike, timelike or null 3-spaces. The null direction of a null space is called principal null direction and a timelike space contains two principal null directions. If V is a null space and spanned by orthogonal vec-tors then one of the vecvec-tors must necessarily span the principal null direction. The orthogonal complement of a timelike 2-space is spacelike and the other way around also holds. If V is null then the orthogonal complement V⊥ is null as well and intersect with the same principal null direction. So a new thing with this indefinite algebra is that although V and V⊥ are orthogonal 2-spaces V ⊕ V⊥does not span the Minkowski space.

Here will follow some examples of the different spaces and their orthogonal com-plement in the Minkowski space.

2.2.1 Some examples of subspaces

S1can be spanned by for example x, y, z respectively and T1by t whereas N1can

be spanned by many vectors such as t − x, t + y, t + z(=√2l), t − z(=√2n), . . .. If S1 is spanned by z then the orthogonal complement is timelike and spanned

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2.3. Segre type 7

by t, x, y. The orthogonal complement to a null 1-space N1spanned by say t − z

is itself null and spanned by t − z, x, y. If one have S2 to be spanned by x, y

then the orthogonal complement S₂⊥ = T2 is timelike and spanned by t, z and

the two principal null directions are t + z, t − z. When a 2-space is null say N2

spanned by t + z, x then its orthogonal complement is null as well and spanned by t + z, y and both these have t + z as the principal null direction. Here it’s obvious that N2⊕N2⊥does not span the Minkowski space since there is no linear

combination of vectors from N2⊕ N2⊥ that span the direction t − z.

2.3 Segre type

A known fact about matrices is that they can be either diagonalizable or at least ”almost” diagonalizable into its Jordan form, see appendix. Here a new notation called Segre type is introduced to simplify the writing of Jordan forms. The Segre type includes almost all the information as the Jordan form except the eigenvalue, which won’t be that important anyway since the theorems where this notation is used are for arbitrary matrices. The following matrix

                        α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 β 1 0 0 0 0 0 0 0 0 0 0 0 0 0 β 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 δ                        

has Segre type {(421)(2)(22)(1)}, were the parentheses indicates the transition between different eigenvalues. It is convenient to omit the brackets of single digits, hence the above Segre type is written {(421)2(22)1}. If the matrix has a timelike eigenvector for example



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2.4 Existence of invariant 2-spaces

An invariant 2-space W of Sab(where Sbais selfadjoint with respect to the Lorentz

inner product) is a space with the property that if va ∈ W then va_Sb a ∈ W .

Later it will be used that if S is a second order symmetric tensor then there exist at least one invariant 2-space. This can be shown by first assuming that there are only real eigenvalues to the Jordan form of Sb

a. From this one gets

that the only possible Jordan forms contain at least two different eigenvectors and hence there exists an invariant 2-space. This procedure will be shown later in section 3.3. If however there is a complex eigenvalue α + iβ to W , then there must exist a complex vector u + iv whose real and imaginary parts span an invariant 2-space of W . To see this assume that S(u + iv) = (α + iβ)(u + iv), then the real part is S(u) = αu − βv and the imaginary S(v) = αv + βu and hence there must exist an invariant 2-space to Sb

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Chapter 3

Symmetric tensors

The main part of this thesis will concern the classification of second order sym-metric tensors. These are commonly used in physics such as general relativity together with the Lorentz metric, later on there will be a short introduction to this. In classical linear algebra with the definite inner product the theorem concerning the classification of the symmetric tensor S is the spectral theorem which says that S is symmetric iff there exist an ON basis of eigenvectors, which is equivalent to S being diagonalizable. If S is diagonalizable then its canonical form can be decomposed as

Sab= p1xaxb+ p2yayb+ p3zazb+ p4wawb

where S : R4

7→ R4 _{and the eigenvalues p}

1, p2, p3, p4 ∈ R, this can be found

in appendix and [3]. The proof for this uses the fact that only the zero vector is orthogonal to itself. Since this is no longer true with an indefinite metric a different theorem with proof is necessary. Because of the possibility of null vectors this theorem will be more comprehensive then the former. In fact it will turn out to exist more then one form for S to be decomposed into.

3.1 Properties of invariant 2-spaces

First some properties of the invariant 2-spaces of a symmetric tensor will be necessary to find the complete description of S, it is also good to remember that a space-time is a 4-dimensional manifold with the Lorentz metric.

Theorem 3.1.1. Let M be a space-time and S 6= 0 a second order symmetric tensor at a point p ∈ M then

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10 Chapter 3. Symmetric tensors

Proof. We begin with noting that S can be represented along a real null tetrad (l, n, x, y) at a point p as

Sab= 2S1l(anb)+ S2lalb+ S3nanb+ 2S4l(axb)+ 2S5l(ayb)+

+2S6n(axb)+ 2S7n(ayb)+ 2S8x(ayb)+ S9xaxb+ S10yayb (3.1)

with S1, . . . , S10∈ R

i)Assume first that W is a spacelike invariant 2-space of S which is spanned by x and y. Then if x is mapped by S it produces the following equation

Sabxb= S4la+ S6na+ S8ya+ S9xa

But since W is an invariant 2-space spanned by x and y, S mapped by x cannot be a linear combination of l and n so S4_{= S}6_{= 0. By a similar manner with y}

one gets that S5= S7= 0. The result of this is that S can be represented as

Sab= 2S1l(anb)+ S2lalb+ S3nanb+ 2S8x(ayb)+ S9xaxb+ S10yayb

from this and the fact that (bxb+ cyb) · (dlb+ enb) = 0, b, c, d, e ∈ C follows

the result that the orthogonal complement of W is as well an invariant 2-space spanned by l and n.

If however W is timelike one can use almost the same argument as above but for (l − n) =√2t and x instead, some modifications are however necessary.

Sab(l − n)b= S1(la− na) − S2la+ S3na− S4xa− S5ya+ S6xa+ S7ya

Sabxb= S4la+ S6na+ S8ya+ S9xa

Since W is an invariant 2-space the following must be true S2 _{= S}3_{, S}5 ₌

0, S7= 0 from the first equation and S4= −S6, S8= 0 from the second. Then from (3.1) with these conditions S is equal to

Sab= 2S1l(anb)+ S2lalb+ S2nanb+ 2S4l(axb)− 2S4n(axb)+ S9xaxb+ S10yayb

Here l + n =√2z and y span the orthogonal complement to W . Because of Sab(l + n)b= S1la+ S1na+ S2la+ S2na+ S4xb− S4xb=

= (S1+ S2)(l + n)a

Sabyb= S10ya

the space that l + n and y span is an invariant 2-space.

In the case when W is null it can be taken to be spanned by l and x. If we do the same calculations as above then

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3.1. Properties of invariant 2-spaces 11

and

Sablb= S1la+ S3na+ S6xa+ S7ya (3.3)

using lblb= 0.

From (3.2) we have that S6_{= S}8_{= 0 and from (3.3) S}3_{= S}7_{= 0 which follows}

from the fact that W is spanned by l and x. So now S is reduced to

Sab= 2S1l(anb)+ S2lalb+ +2S4l(axb)+ 2S5l(ayb)+ S9xaxb+ S10yayb

and from this one realises that l and y span an invariant 2-space for S, since any linear combination of l and y mapped by S is itself a linear combination of l and y. Futhermore from (3.3) with S3_{= S}6_{= S}7_{= 0 ⇒ S}

ablb = S1lb it is

easily seen that l is a null eigenvector of S.

(ii) In the last sentence of (i) it was confirmed that if W is a null invariant 2-space then there exist a null eigenvector to S. On the other hand if S admits a null eigenvector say l then Sablb = S1lb+ S3nb+ S6xb+ S7yband this implies

that S3_{= S}6_{= S}7_{= 0 and S is then represented by}

Sab= 2S01l(anb)+ S02lalb+ 2S04l(axb)+ 2S05l(ayb)+ 2S08x(ayb)+ S09xaxb+ S010yayb

By a null rotation S08= 0 which is most easily realised by noting that S09xaxb+

2S08x(ayb)+ S010yayb can be written xa S09 S08 S08 S010 yb

and since x and y are spacelike this matrix can be diagonalized according to the spectral theorem from classical linear algebra. In this new basis

Sab= 2S1l(anb)+ S2lalb+ 2S4l(axb)+ 2S5l(ayb)+ S9xaxb+ S10yayb

and it follows that l and x span a null invariant 2-space.

(iii) If S admits a spacelike invariant 2-space say W , spanned by x and y, then in (3.1) S4_{= S}5_{= S}6_{= S}7_{= 0 by the same reasoning as in the first part of (i).}

But S8_{= 0 is also true if one use the same reasoning as in the latter part of (ii).}

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3.2 The classification of symmetric tensors

The complete algebraic description of S at a point p can now be given with the following theorem

Theorem 3.2.1. If M is a spacetime and S 6= 0 a second order symmetric tensor at p, then S can be decompsed into one of the following canonical forms along a real null tetrad (l, n, x, y) at p.

Sab= 2p1l(anb)+ p2(lalb+ nanb) + p3xaxb+ p4yayb (3.4)

Sab= 2p1l(anb)+ γlalb+ p3xaxb+ p4yayb (3.5)

Sab= 2p1l(anb)+ 2l(axb)+ p1xaxb+ p2yayb (3.6)

Sab= 2p1l(anb)+ p2(lalb− nanb) + p3xaxb+ p4yayb (3.7)

here pk ∈ R and γ = ±1 in (3.5) and p26= 0 in (3.7). The eigenvectors in

(3.4) can be choosen as t = l−n_√ 2, z =

l+n_√

2, x, y with eigenvalues p1− p2, p1+

p2, p3, p4 respectively. The Segre type for (3.4) is {1, 111} and S is therefore

diagonalizable. For (3.5) l, x, y are eigenvectors and p1, p2, p3 its eigenvalues.

Here the Segre type is {211}. Equation (3.6) has only two eigenvectors, these can be taken as l, y with eigenvalues p1, p2 and the Segre type is {31}. The

only decomposition that have non-real eigenvectors is (3.7) which has two and these are l ± in with eigenvalues (p1± ip2) and two other real eigenvectors are

x, y with p3, p4as eigenvalues. All of the Segre types that have been mentioned

above can also occur in its degenerated forms.These two theorems are much the same as in [2]. If we write (3.4) in the standard basis t, x, y, z we get the familiar form of

Sab= (p2− p1)tatb+ (p1+ p2)nanb+ p3xaxb+ p4yayb

which is a diagonalization of S.

Proof. Assume first that S admits a null eigenvector l and construct a null tetrad (l, n, x, y) at p .Then equation (3.1) decomposes with S3_{= S}6_{= S}7_{= 0}

as a consequence of (3.3), in addition S8_{= 0 as in (ii) of Theorem 3.1.1. So S}

is represented by

Sab= 2S1l(anb)+ S2lalb+ 2S4l(axb)+ 2S5l(ayb)+ S9xaxb+ S10yayb (3.8)

and four different cases now arises.

a) For the case when S9_{6= S}1_{6= S}10 _{one can get in a new tetrad S}4_{= S}5_{= 0}

using a null rotation with A = 1 and θ = 0. With A = 1, θ = 0 and B = α + iβ α, β ∈ R (2.2) transforms into

l0 = l B = α + iβ m0 = m − Bl m = x+iy√ 2 n0 = n + Bm + Bm − BBl =⇒ n0 = n +√2αx −√2βy − (α2+ β2)l x0_√+iy0 2 = x+iy_√ 2 − (α − iβ)l =⇒ x0 = x −√2αl y0= y +√2βl

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3.2. The classification of symmetric tensors 13 Sab= S01(la0n0b+ n0alb0) + S02l0al0b+ S04(la0x0b+ xa0l0b) + S05(la0y0b+ ya0lb0) + S09x0ax0b+ S010ya0y0b= = S01(la(nb+ √ 2αxb− √ 2βyb− (α2+ β2)lb) + (na+ √ 2αxa− √ 2βya− (α2+ β2)la)lb)+ S02lalb+ S04(la(xb− √ 2αlb) + (xa− √ 2αla)lb) + S05(la(yb+ √ 2βlb) + (ya+ √ 2βla)lb)+ S09(xa− √ 2αla)(xb− √ 2αlb) + S010(ya+ √ 2βla)(yb+ √ 2βlb)

Now it is convenient to factorise with respect to 2l(anb), lalb, 2l(axb), 2l(ayb), xaxb, yayb

as the goal is to show that α and β can be choosen so that the coefficients in front of 2l(axb)and 2l(ayb) can be made equal to zero.

Sab= (lanb+ nalb)(S01)+ + lalb(−2S01(α2+ β2) + S02− 2 √ 2S04α + 2√2S05β + 2S09α2+ 2S010β2)+ + (laxb+ xalb)( √ 2S01α + S04−√2S09α)+ + (layb+ yalb)(− √ 2S01β + S05+√2S010β)+ + xaxbS09+ yaybS010 (3.9) S4= (√2S01α + S04−√2S09α) = 0 S 01 6=S09 ⇐⇒ α = S 04 √ 2(S09_{− S}01₎ (3.10) S5= (−√2S01β + S05+√2S010β) = 0 S 01_6=S010 ⇐⇒ β = S 05 √ 2(S01_{− S}010₎ (3.11) Choose α and β in this way so that S4_{= S}5_{= 0 in this new basis. From}

this we have that S takes the form of

Sab= 2S1l(anb)+ S2lalb+ S9xaxb+ S10yayb

where S1= S01,

S2= (−2S01(α2+ β2) + S02− 2√2S04α + 2√2S05β + 2S09α2+ 2S010β2) S9= S09, S10= S010

in (3.9) which is equation (3.5). If S2 6= 0 the Segre type is {211} or {2(11)} (where the latter occurs when S9 _{= S}10_{). If however S}2 _{= 0 it}

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which is a consequence of (3.10) since S1_{= S}9_{. But now it is possible}

to set S2 = 0 instead with the same rotation. To see this look at the coefficient in front of lalbin (3.9) with S1= S9and S5= 0. This gives the

following equation −2S1(α2+ β2) + S2− 2√2S4α + 2S1α2+ 2S10β2= 0 which hold if we choose α = S2

2√2S4 and β = 0 hence S

2 _{= 0 in this new}

basis. So S is then decomposed as

Sab= 2S1l(anb)+ 2S4l(axb)+ S9xaxb+ S10yayb

again with a null rotation (θ = 0, B = 0) one can get that S4_{= 1. So the}

transformation used is l0 = Al B = 0 m0 = m m = x+iy√ 2 n0 = A−1n x0_√+iy0 2 = x+iy_√ 2 =⇒ x0 = x y0 = y Sab= S1(l0an0b+ n0al0b) + S 4_(l0 ax0b+ x0alb0) + S 9_x0 ax0b+ S 10_y0 ayb0 = = S1(AlaA−1nb+ A−1naAlb) + S4(Alaxb+ xaAlb) + S9xaxb+ S10yayb= = \A = 1 S4\ = 2S 1_l (anb)+ 2l(axb)+ S9xaxb+ S10yayb

which is the same form as (3.6) with the Segre type {31} which can be deduced by assuming that there exists an eigenvector like αl + βn + γx and deriving a contradiction.

c) S1= S106= S9_{is in essence the same as above because of the symmetry of}

S10 and S9.

d ) S1_{= S}10_{= S}9_{. If S}4 _{= S}5 _{= 0 one basically gets the same as in a) with}

Segre type {(211)} or {(1, 111)} and equation (3.5) or (3.4) depending on whether S2_{6= 0 or S}2_{= 0. If one of S}4_{, S}5_{is non-zero then a null rotation}

can be used to get the Segre typ (31) as in b). If both are non-zero then set w = S₂2l + S4x + S5y and the resulting equation is of type (3.6), still with Segre type (31).

However if there does not exist any null eigenvectors to S then theorem 3.1.1 (ii) tells us that there cannot exist a null invariant 2-space. Since S must have an invariant 2-space it is either timelike or spacelike. Now from theorem 3.1.1 (iii) follows that there must exist an orthogonal pair of spacelike eigenvectors say x and y. After constructing a null tetrad (l, n, x, y) at a point p and using the fact that S maps x and y onto x and y respectively one gets that S4_{= S}5₌

S6_{= S}7_{= S}8_{= 0 and S}2_{6= 0 6= S}3_{(or else null eigenvectors would be allowed).}

To see this assume S2= 0 now

Sab= 2S1l(anb)+ S3nanb+ S9xaxb+ S10yayb (3.12)

and

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3.3. Alternative approach for deducing the canonical expressions of S 15

hence n is a null eigenvector, so S2 _{and S}3 _{cannot be zero. If one now uses a}

null rotation with (θ = B = 0) the following turns out to be true l0 = Al n0 = A−1n x0 = x y0 = y Sab= S01(l0an 0 b+ n0alb0) + S 02_l0 al 0 b+ S03n0an 0 b+ S09x0ax 0 b+ S 010_y0 ay 0 b= = S01(AlaA−1nb+ A−1naAlb) + S02AlaAlb+ S03A−1naA−1nb+ S09xaxb+ S010yayb= = \A = ±4 r |S 03 S02|, S 01 _{= S}1_{, S}09_{= S}9_{, S}010_{= S}10_{, \ =} = 2S1l(anb)± √ S02_S03_l alb± √ S02_S03_n anb+ S9xaxb+ S10yayb

so A can be choosen so that |S2_{| = |S}3_{|. If S}2 _{= S}3 _{then S is on the form}

of (3.4) with Segre type {1, 111}, {1, 1(11)}, {1, (111)} depending on whether (S1_{6= S}9_{6= S}10_{), (S}1_{= S}9_{6= S}10_{) or (S}1_{= S}9_{= S}10_{) and if S}2 _{= −S}3 _{then this}

leads to (3.7) with Segre type {z ¯z11} or {z ¯z(11)}.

The fact that γ = ±1 in (3.5) which is mentioned in the beginning is achieved with a null rotation (θ = B = 0) in exactly the same manner as in the last part of the proof.

3.3 Alternative approach for deducing the

canon-ical expressions of S

There is a more implicit way of showing the different types of canonical form that can exist. If we assume that there are only real eigenvalues to S b

a then the

Jordan forms that are possible for S b a are i)     p1 1 0 0 0 p1 1 0 0 0 p1 1 0 0 0 p1     ii)  p1 1 0 0 

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16 Chapter 3. Symmetric tensors iv)     p1 1 0 0 0 p1 0 0 0 0 p2 0 0 0 0 p3     v)     p1 0 0 0 0 p2 0 0 0 0 p3 0 0 0 0 p4    

here p1, p1, p1, p1 ∈ R and can be equal. So the Segre types for these matrixes

are {4}, {22}, {31}, {211} and {1111} with the possibility of degeneracy. The fact that S is symmetric can now be used to exclude two of these forms. If S is symmetric then S b

a gcb must also be symmetric. So if gcb is taken as an

arbitrary symmetric matrix one gets some conditions on gcb that will make the

Segre types of {4}, {22} impossible with the Lorentz inner product, which the following calculations show.

    p1 1 0 0 0 p1 1 0 0 0 p1 1 0 0 0 p1         g00 g01 g02 g03 g01 g11 g12 g13 g02 g12 g22 g23 g03 g13 g23 g33     =     p1g00+ g01 p1g01+ g11 p1g02+ g12 p1g03+ g13 p1g01+ g02 p1g11+ g12 p1g12+ g22 p1g13+ g23 p1g02+ g03 p1g12+ g13 p1g22+ g23 p1g23+ g33 p1g03 p1g13 p1g23 p1g33    

This matrix should be symmetric and thus the following must hold.

g02= g11 g03= g12 g13= 0 g13= g22 g23= 0 g33= 0 =⇒ gcb=     g00 g01 g11 g03 g01 g11 g03 0 g11 g03 0 0 g03 0 0 0    

So detgcb = (g03)4 and hence not -1 which is the determinant of the Lorentz

inner product.Therefore {4} cannot be a Segre type to S. This also concludes the statement in section 2.4 that the Jordan form have at least two eigenvectors, since all other Jordan forms have at least two eigenvectors.

If one does in the same manner for {22} the results are     p1 1 0 0 0 p1 0 0 0 0 p2 1 0 0 0 p2         g00 g01 g02 g03 g01 g11 g12 g13 g02 g12 g22 g23 g03 g13 g23 g33     =

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3.3. Alternative approach for deducing the canonical expressions of S 17     p1g00+ g01 p1g01+ g11 p1g02+ g12 p1g03+ g13 p1g01 p1g11 p1g12 p1g13 p2g02+ g03 p2g12+ g13 p2g22+ g23 p2g23+ g33 p2g03 p2g13 p2g23 p2g33    

Here two cases are possible, either p1= p2 or p16= p2.

If p16= p2 then because of symmetry

g11= 0 p1g02+ g12= p2g02+ g03 ⇐⇒ g03= g02(p1− p2) + g12 p1g03+ g13= p2g03 ⇐⇒ g13= g03(p2− p1) p1g12= p2g12+ g13 p16=p2 ⇐⇒ g12= g13 p1− p2 g13=g03(p2−p1) ⇐⇒ g12= −g03 p1g13= p2g13 p16=p1 ⇐⇒ g13= 0 g33= 0 =⇒ g03= 0 =⇒ g12= 0 =⇒ g02= 0 =⇒ gcb=     g00 g01 0 0 g01 0 0 0 0 0 g22 g23 0 0 g23 0    

and here detgcb= (g23)2+ (g01)2 which shows that {22} is not a possible Segre

type when p16= p2. If however p1= p2then

g11= 0 g13= 0 g03= g12 g33= 0 =⇒ gcb=     g00 g01 g02 g03 g01 0 g03 0 g02 g03 g22 g23 g03 0 g23 0    

So detgcb= . . . = (g01g23−(g03)2)2which never will be equal to -1. The possible

Segre types are then {31}, {211} and {1111}. For the case when S has Segre type {31} one can do the same calculations as above and get that

=⇒ gcb=     g00 g01 g11 g03 g01 g11 0 0 g11 0 0 0 g03 0 0 g33    

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3.4 Applications in physics

In general relativity the Lorentz metric is used together with a 4-dimensional, smooth, connected manifold, called the spacetime. Equations that restrict the Lorentz metric g are then required so that the gravitational field can be inter-preted by g. These are called Einstein field equations and are given by

Rab−

1

2Rgab+ Λgab= 8πTab

where Rabis called the Ricci tensor and R the Ricci scalar, Λ the cosmological

constant and Tab is called the energy-momentum tensor which is symmetric so

Tab= Tba. Often one can assume that Λ = 0 and the equations are then written

as

Rab−

1

2Rgab≡ Gab= 8πTab

where Gab is also known as the Einstein tensor. If vaccum is assumed the

equations are reduced further to one of the two equvivalent statetments Rab= 0, Gab= 0

Both the energy-momentum tensor and Ricci tensor are second order symmetric tensors at a specific point and the results from the previous section can therefore be applied to these.

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Chapter 4

Conclusions and discussion

4.1 Conclusions and discussion

The characterization of the symmetric tensor in the Lorentz metric had already been done in [2]. What I did was to summerize all the concepts for this subject and to make the proof much more detailed for both the theorem about invariant 2-spaces and the characterization of tensors, since these were rather concise in [2]. The results in theorem 3.2.1 can be generalized to the case when we have one timelike and arbitrarily many spacelike vectors in the standard basis. This will not affect the decomposition of S much, the only change would be that terms of p5wawb, . . . , pnvavb would be added after p4yayb in theorem 3.2.1.

If one would like to continue on these subjects there is plenty to choose from, one could for example try to make the characterization for some other H or just a general H. Another possibility would be to look more deeply at the applications in physics or of indefinite linear algebra. One could also just go deeper into indefinite linear algebra and its differences from definite linear algebra.

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Bibliography

[1] Israel Gohberg, Peter Lancaster, Leiba Rodman (2005), Indefinite Linear Algebra and Applications, Birkh¨auser Verlag

[2] G S Hall (2004), Symmetries and Curvature Structure in General Relativity, World Scientific Publishing

[3] Ulf Janfalk (2007) Linj¨ar algebra, Department of Mathematics, Link¨opings Universitet

[4] Sergei Treil (2010) Linear Algebra Done Wrong,

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Appendix A

Complementary theory

A.1 Jordan Form

Theorem 1. If A ∈ Cn×n _{then there is a block diagonal}

J =      Jn1(α) 0 · · · 0 0 Jn2(β) · · · 0 .. . ... . .. 0 0 0 0 Jnk(ω)      ∈ Cn×n

such that A = T J T−1 (where T represents a change of basis). Here J is called the Jordan canonical form for the operator A. This decomposition is uniquely determined by A up to a permutation of the diagonal blocks. Here the diagonal blocks are defined by

Jni(α) =      α 1 · · · 0 0 α · · · 0 .. . ... . .. 1 0 0 0 α      ∈ Cni×ni

i = 1, . . . , k and α, . . . , ω are the eigenvalues of J, where some of them may coincide. For a proof of this theorem se [4].

A.2 The Spectral theorem for positive definite

metrics

Theorem 2. If E is a finite dimensional euclidean space, dim E = n, and F : E → E a linear transformation then

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24 Appendix A. Complementary theory

Proof. The proof will be done by induction.

If dimE = 1 then every vector 6= 0 will be an eigenvector and if normalized then it is an orthonormal basis for E. Assume that the theorem is true for all euclidean spaces of dim E = n ≤ p, p ≥ 1. Then if dim E = p + 1 and F is represented by the p + 1 × p + 1 matrix A, one gets accordingly to the fundamental theorem of algebra that there is at least one complex root to the equation det(A − λE) = 0, this root is real according to lemma 7.3.9 in [3] and therefore F has an eigenvector. This eigenvector can be normalized and called u1. Set E1 = {u ∈ E : (u, u1) = 0} and define a transformation F1 : E1→ E1

as F1(u) = F (u) ∀u ∈ E1. For this to be a meaningfull definition it must hold

that F (u) ∈ E1 as soon as u ∈ E1. This is what lemma 7.3.8 in [3] says. So

from lemma 7.3.9 in [3] dim E1= dim E − 1 = p + 1 − 1 = p. It follows that F1

is a symmetric transformation from E1to E1. Since dim E1= p there exist an

orthonormal basis to E1according to the induction assumption. We also know

that the eigenvectors to F1 are eigenvectors to F since F = F1 on E1. These

are all orthogonal to u1 and hence span all of E together with u1.

Remark: note that this method does not work with the indefinite inner product since u1 and E1 no longer have to span the entire space.

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c

Classification of second order symmetric tensors in the Lorentz metric