Asymptotiska egenskaper för Lanczosspinoren

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Asymptotic properties of the Lanczos

spinor

Applied Mathematics, Link¨opings Universitet. Thomas B¨ackdahl

LITH-MAT-EX--03/15--SE

Examensarbete 20p

Level: D

Supervisor: Fredrik Andersson

Department of Mathematics Applied Mathematics

Link¨oping University Examiner: Magnus Herberthson

Department of Mathematics Applied Mathematics

Link¨oping University Link¨oping 2003-10-29

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Matematiska Institutionen 581 83 LINK ¨OPING SWEDEN 2003-10-28 × × http://www.ep.liu.se/exjobb/mai/2003/tm/015/

Asymptotiska egenskaper f¨or Lanczosspinoren Asymptotic properties of the Lanczos spinor

Thomas B¨ackdahl

Asymptotically flat spaces are widely studied because it is one natural way of de-scribing an isolated system in general relativity. In this thesis we study what happens to the Lanczos potential at spacelike infinity in such spacetimes. By transforma-tions of the Weyl-Lanczos equation, we derive expressions for the limiting equatransforma-tions on both the timelike unit hyperboloid, and the timelike unit cylinder. Finally the Newman-Penrose formalism is used to get a component version of the equations.

Lanczos, Weyl, general relativity, spacelike infinity, tensor, spinor, Newman-Penrose formalism Nyckelord Keyword Sammanfattning Abstract F¨orfattare Author Titel Title

URL f¨or elektronisk version

Serietitel och serienummer Title of series, numbering

ISSN 0348-2960 ISRN LITH-MAT-EX--03/15--SE 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

Asymptotically flat spaces are widely studied because it is one natural way of describing an isolated system in general relativity. In this thesis we study what happens to the Lanczos potential at spacelike infinity in such spacetimes. By transformations of the Weyl-Lanczos equation, we derive expressions for the limiting equations on both the timelike unit hyperboloid, and the timelike unit cylinder. Finally the Newman-Penrose formalism is used to get a component version of the equations.

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Acknowledgements

A year ago I didn’t know anything about general relativity so first I want to thank Rolf Riklund for introducing me to the world of relativ-ity. My supervisors Fredrik Andersson and Magnus Herberthson need a lot of thanks. They have spent a lot of time and effort teaching me the necessary tools, discussing problems and correcting errors. Special thanks to my opponent Erik Tellgren for helping me make this thesis as good as possible. I also want to thank God, without him there wouldn’t be a universe to study.

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Chapter 1 Introduction

In this thesis we study some asymptotically flat spacetimes. Intu-itively this means that far away the average density vanishes, or in other words, all matter and radiation is confined inside a space-tube. It is not known if the universe is asymptotically flat or not. This means that there may be a finite average matter or radiation density far away, or that the universe may be finite. The study of asymptoti-cally flat spacetimes is important either way because it is the best way of describing an isolated system. The fields in a system can be very hard to calculate. Therefore one often studies the leading term, that is the part of the fields that dominate at large distances. Even the equations emerging from these simplifications can be difficult to solve. Therefore one may want to study potential equations instead because they are sometimes easier to solve, or can give additional physical un-derstanding. This applies both to electromagnetism and gravitation. Electromagnetism is much easier so we begin our study there to see how things work. Then we try to do the same things for gravitation using the Weyl-Lanczos equation. Finally we express these equations in form of components in the Newman-Penrose formalism. First of all though we introduce the necessary tools i.e. tensors, spinors and con-formal rescalings. In this thesis we use the conventions from Penrose & Rindler [8].

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Chapter 2 Tensors

To understand spinors one needs to know what tensors are. Later we also need to do some tensor calculations. Therefore it is convenient to begin with an introduction of tensors. This chapter is strongly influenced by Chapter 2 in Tobias Olsson’s master’s thesis [7], which is a more thorough but still quick introduction to tensors.

2.1 Manifolds and coordinates

Tensors live on mathematical entities called manifolds. Therefore one needs to know a little about manifolds to be able to define tensors. Here we just give a quick introduction, for a complete definition of a manifold see [11] or [1]. An n-dimensional manifold M is a set that locally looks like Rn_{. For example the sphere S}2_{is a manifold because}

it shares the local properties of R2. Otherwise a map would not be of much use to an orienteer. Observe that global properties may differ considerably; R2 is flat, non-compact and has infinite area, whereas S2 is curved, compact and has finite area.

The local similarity of the manifold and Rn is defined by a coordi-nate system. This coordicoordi-nate system may not always be valid globally. For example we cannot make a world map that works everywhere. We will always have problems with at least one point, usually one of the poles. In order to have a manifold structure there should exist a set of overlapping coordinate systems that cover the manifold.

When one consider curved geometries there is in general no natural global vector space structure. Therefore we define tangent spaces Vp

at every point p, where Vp is an ordinary vector space. Observe that

in general one cannot naturally identify Vp and Vq when p 6= q. To

simplify the notation one usually writes V instead of Vp when it is

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2.2 What are tensors? 3

Figure 2.1: Tangent space at point p.

We define tangent vectors from the notion of directional derivatives because there is no other structure to relate to.

Let (xµ)n_µ=1be a coordinate-system in a neighbourhood of a point p ∈ M which we assume to be C∞i.e., we assume that the coordinate functions are C∞with respect to each other in all overlapping regions. Let h ∈ C∞ be a function defined in a neighbourhood of p. Later we need to relax the C∞ condition at some points on the manifold. We define tangent vectors Xµat p by differentiating h at p,

Xµ(h) =

∂h ∂xµ(p).

Since the choice of h is not important we usually write Xµ=

∂ ∂xµ.

In [11] it is shown that {Xµ; µ = 1...n} is a basis for V , called the

coordinate-basis associated with (xµ)n₁. Thus for all v ∈ V there exist numbers vµ, µ = 1, ..., n such that

v = n X 1 vµXµ, that is v(h) = n X 1 vµ ∂h ∂xµ(p), ∀h.

The numbers vµ are called v:s components in the coordinate-basis {X_µ, µ = 1, ..., n}. A vector field, v, on a manifold M is a usually smooth assignment of tangent vectors at every point p ∈ M . Often vector fields are called simply vectors.

2.2 What are tensors?

Let V be any finite-dimensional vector space over the real numbers (R). Consider the vector space, V∗, of linear maps f : V → R

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4 Tensors

with addition and multiplication defined in the natural way. We call V∗ the dual vector space to V , and its elements are called dual vec-tors. If (v1, ..., vn) is a basis of V then we can define basis elements

v1∗_{, ..., v}n∗ _{∈ V}∗ _by

vµ∗(vν) = δµν, (2.2.1)

where δνµ is the Kronecker delta.

We now define a tensor. A tensor, T , of type (k, l) over V is a multi-linear map T : V∗× . . . × V∗ | {z } k × V × . . . × V | {z } l → R. (2.2.2)

Given k dual vectors and l vectors, T produces a real number. Hence with the above definition, a tensor of type (0, 1) is a dual vector and a tensor of type (1, 0) is an element of V∗∗. But as V is reflexive there is a natural isomorphism between V and V∗∗. Therefore we treat it as a vector. A tensor of type (1, 1) is an ordinary linear map.

Now, let V be the tangent space at a point p ∈ M . Given a coordinate system we can construct a coordinate basis _∂x∂1, ...,_∂x∂n in V and we denote the dual basis of V∗ as dx1, ..., dxn. If we change coordinate system, the components ¯Tν¯1...¯νk

¯

µ1...¯µl of a tensor T in the new basis are related to the components Tν1...νk

µ1...µl in the old basis by the tensor transformation law,

¯ Tν¯1...¯νk ¯ µ1...¯µl = n X ν1...µl=1 Tν1...νk µ1...µl ∂ ¯xν¯1 ∂xν1 . . . ∂xµl ∂ ¯xµ¯l. (2.2.3) A smooth assignment of a tensor over Vp to each point p in the

manifold M is called a tensor field.

If we have a tensor T of type (k, l) we can define an operator, contraction, as a map defined as

contraction(T ) =

n

X

k=1

T (..., vk∗, ...; ..., vk, ...),

where {vk}n1 is a basis of V and {vk∗}n1 is its dual basis. The

contrac-tion is independent of the basis. Note that a contraccontrac-tion of a tensor of type (1, 1) is just the trace of the linear map. Also note that we in general get different contractions if we contract over different pairs of ’slots’ in the tensor.

2.2.1 Abstract index notation

We shall use a notation, the abstract index notation. A tensor of type (k, l) will be denoted by a stem letter followed by k contravariant

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2.2 What are tensors? 5

(upper) and l covariant (lower), small letters. For example Tabc def g

denotes a tensor of type (3, 4). One should not confuse this with ordinary index notation, where Tαβγδζη are the components of the

tensor in some basis. We write the contraction of a tensor using the same letter on the contracted indices, e.g. Wacbdc is a tensor of type

(2, 1). The tensor product of two tensors Tab and Scde is denoted

Ta bScde.

We can also use the index notation to express symmetries of a ten-sor. To symmetrize over some indices means that one takes the arith-metic mean of all permutations of the indices. For antisymmetrization just change sign of all odd permutations. For example we define

T_(abc)= 1

3!(Tabc+ Tacb+ Tcab+ Tcba+ Tbca+ Tbac), (2.2.4) and similarly

T[abc]=

1

3!(Tabc− Tacb+ Tcab− Tcba+ Tbca− Tbac), (2.2.5) Note that for any tensor of type (0, 2) Tab = T(ab)+ T[ab]. If one needs

to exclude some indices from the symmetrization we surround them with bars (| · |) like

T(a|bc|d)=

1

2!(Tabcd+ Tdbca). (2.2.6)

2.2.2 The metric tensor

We now introduce the metric tensor. A metric is supposed to tell us the squared distance between two events in space-time. A metric, gab, should be a symmetric non-degenerate tensor of type (0, 2). Note

that it does not have to be positive definite as in ordinary Riemannian geometry.

Given a metric g at a point we can always find an orthonormal basis such that gab is diagonal with ±1 on the diagonal. The number

of + and − signs occurring, does not depend on which basis is used and is called the signature of the metric. In ordinary Riemannian geometry the signature is therefore (+ . . . +). This signature is called Riemannian signature. A space-time is defined to be a manifold with a metric with signature (+−−−) or (−+++). We call this Lorentzian signature. In this thesis we will only use the signature (+ − −−).

If we contract the metric with an ordinary vector va we get the dual vector vb = gabva. We also define the inverse of gab called the

dual metric gab as

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6 Tensors

If we contract gab _{with a dual vector we get an ordinary vector. This}

gives rise to a method of raising and lowering indices. Tabcd= gaegcfTebfd

2.3 Curvature

A derivative operator, covariant derivative, on a manifold is a map which takes each differentiable tensor field of type (k, l) to a new tensor field of type (k, l+1). We see that the operator increases the number of covariant indices by one. That is why we call it a covariant derivative operator and denote it ∇a even though it’s not a dual vector.

A covariant derivative must satisfy the following properties • Linearity: For all tensors A and B of type (k, l) and α, β ∈ R,

∇_d(αAa1...ak b1...bl+ βB a1...ak b1...bl) = α∇dAa1...akb1...bl+ β∇dB a1...ak b1...bl (2.3.1) • The Leibniz property: For all tensors A of type (k, l) and B of

type (m, n) we have ∇a(Ab1...bkc1...clB d1...dm e1...en) = (∇aA b1...bk c1...cl)B d1...dm e1...en + Ab1...bk c1...cl(∇aB d1...dm e1...en). (2.3.2) • Consistency with the notion of tangent vectors as directional derivatives on scalar fields: For all smooth functions f and all vector fields ta we have

ta(f ) = ta∇af. (2.3.3)

These properties are sufficient for an operator to be a covariant deriva-tive, but they do not make the covariant derivative unique. One there-fore adds two more properties for the natural covariant derivative:

• Torsion free: For all differentiable functions f from a manifold M to R,

∇_a∇_bf = ∇b∇af. (2.3.4)

• Compatible with the metric:

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2.3 Curvature 7

After some calculations we get

∇_atb= ∂atb− Γabctc, (2.3.6)

where

Γbca= 1₂gad(∂bgdc+ ∂cgdb− ∂dgbc) (2.3.7)

is the Christoffel symbol, and ∂a is the coordinate derivative.

∂af = n X µ=1 ∂f ∂xµ(dx µ₎ a

2.3.1 The Maxwell tensor

In general relativity the electromagnetic field is described by an anti-symmetric tensor Fab. If we choose our units and our basis in a good

way Fab gets the following form:

    0 Ex Ey Ez −Ex ₀ _−Bz _By −Ey _Bz ₀ _−Bx −Ez _−By _Bx ₀     (2.3.8)

where E and B are the electric and magnetic fields. The electric and magnetic potentials can be gathered in a tensor Ab such that

Fab= 2∇[aAb]. (2.3.9)

The dual of Fab is defined by Fab∗ = 12abcdFcd, where abcd is the

alternating tensor. The dual is basically the Maxwell tensor with E and B interchanged. The charge and current can be combined in a tensor Jb called the current vector.

With these tensors we can write the Maxwell equations in tensor form:

∇_aFab = µ0Jb (2.3.10)

∇_aF∗ab= 0 (2.3.11)

2.3.2 The Riemann tensor

Since the covariant derivative is not in general commutative, we need a commutation relation:

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8 Tensors

where Rabcdis a tensor defined by the metric and is called the curvature

tensor or the Riemann tensor. It is given by

Rabcd= ∂aΓcbd− ∂bΓcad+ ΓcbeΓead− ΓecaΓebd. (2.3.13)

This tensor is important in the theory of gravitation because it de-scribes the curvature of the manifold and therefore the gravitational field.

It can be shown that the Riemann tensor has the following sym-metries:

Rabcd = R[ab][cd]

Rabcd = Rcdab

R[abc]d = 0

(2.3.14)

From the Riemann tensor we define another tensor called the Ricci tensor by

Rab = Racbc. (2.3.15)

We contract once more and get the Ricci scalar or the scalar curvature:

R = Raa (2.3.16)

2.3.3 The Weyl tensor

The Weyl tensor, Cabcd, is defined in n > 2 dimensions,

Cabcd = Rabcd− 2 n − 2(ga[cRd]b− gb[cRd]a) + 2 (n − 1)(n − 2)(ga[cgd]b)R. (2.3.17)

Together with the Ricci tensor it holds the same information as the Riemann tensor. It possesses the same symmetries as the Riemann tensor but it is also completely trace-free. In this thesis we only study the four dimensional case because our spacetime is four dimensional.

2.3.4 The Lanczos tensor

In electromagnetism it is often useful to study the electromagnetic potential because it sometimes makes the equations easier to solve or it gives physical insight. The same can be done in the theory of gravitation. It can be shown [2] that in four dimensions there exists a potential Labc (analogous to the electromagnetic potential Aa) for the

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2.3 Curvature 9

equation though is quite complicated in tensor form. The relation between Cabcd and Labc is given by

Cabcd = ∇[dL|ab|c]+ ∇[bL|cd|a]+4₃ga[cgd]b∇fLefe

− g_a[cH|b|d]− ga[cHd]b− gb[dH|a|c]− gb[dHc]a

(2.3.18)

where

Hdb= ∇eLbed− ∇dLbee. (2.3.19)

The symmetries of the Lanczos tensor are Labc= L[ab]c and L[abc]= 0.

These can be derived from the symmetries of the Weyl tensor. As in the electromagnetic case we have a gauge freedom, so we can alter the Lanczos tensor without changing the Weyl tensor. To simplify things we impose an algebraic gauge condition Labb = 0. A differential gauge

condition can also be imposed, which we will come back to in the spinor chapter. This gauge condition is given by ∇c_L

abc= ζab where

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Chapter 3 Spinors

In this chapter we introduce spinors. In this thesis we use spinors because certain equations are much simpler in this formalism. For a more complete definition of spinors se [11] or [8].

3.1 What are spinors?

Spinors are quite similar to tensors, but instead of working with two four dimensional real vector spaces V and V∗, spinors work with four two-dimensional complex vector spaces S, S∗, ¯S and ¯S∗. S is a two dimensional complex vector space associated to a point p ∈ M . S∗ is the dual of S i.e. S∗= {α : S → C; α is linear}. In the same way ¯S∗ = {α : S → C; α is anti-linear} and ¯S = {α : S∗ → C; α is anti-linear}. Anti-linear means α(aξ + bζ) = ¯aα(ξ) + ¯bα(ζ) for all ξ, ζ ∈ S and a, b ∈ C. A spinor is then defined as a multi-linear map taking its arguments from S, ¯S, S∗ and ¯S∗, to C. A spinor T of type (p, q; r, s) is therefore a multi-linear map

T : S∗× · · · × S∗ | {z } p × S × · · · × S | {z } q × ¯S∗× · · · × ¯S∗ | {z } r × ¯S × · · · × ¯S | {z } s → C. (3.1.1) The order of indices is important in the same sense as for tensors. The order of unprimed and primed indices doesn’t matter though, because we don’t allow index substitutions that change primed to un-primed indices and vice versa. The abstract index notation and con-traction works the same way as with tensors but we use capital letters for spinor indices. The complex conjugate of a spinor σA is denoted

¯ σA0.

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3.2 Infeld-van der Waerden symbols 11

3.2 Infeld-van der Waerden symbols

A spinor ξAA0 is called hermitian if it satisfies ξAA0 = ¯ξAA0, in other words it is real. There is an isomorphism between the space of all hermitian spinors with 2 indices and the real 4-dimensional tangent space. This isomorphism is called the Infeld-van der Waerden symbol, and it is denoted σBB_a 0. Thus we have

TA1A01...AnA 0 n _{= σ}A1A 0 1 a1 . . . σ AnA0n an T a1...an _(3.2.1) where TA1A01...AnA 0

n_{is called the spinor equivalent of the tensor T}a1...an_. Usually we only write TA1A01...AnA0n ↔ Ta1...an_.

3.3 The metric spinor

As with tensors we have

ζAB = ζ(AB)+ ζ[AB].

Since S is only 2 dimensional, the space of antisymmetric spinors with 2 indices is one-dimensional. Therefore we choose an antisymmetric spinor AB = −BA. The pair (S, AB) is then called a spin-space.

The spinor AB is then used to raise and lower indices in the same

way as the metric in the case of tensors. The order of the indices is important so we use the convention

ζA= ζBBA ζB= BAζA.

where ACis defined by ACBC = BA, where BAis the two-dimensional

Kronecker delta. The complex conjugate ¯A0_B0 we usually denote _A0_B0. It is easily shown that

gab↔ ABA0_B0.

3.4 Spinor algebra

The identity

A[BCD] = 0 (3.4.1)

follows from the fact that a spinor will vanish if one antisymmetrizes over more indices than the number of dimensions. From this identity it follows that

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12 Spinors

which is equivalent to

ABCD = ACBD− ADBC. (3.4.3)

Contracted with an arbitrary spinor ξCD it yields

ξAB− ξBA= ABξCC (3.4.4)

so that

ξAB = ξ(AB)+12ABξC

C_. _(3.4.5)

3.4.1 Symmetric spinors

Symmetry and trace-freeness are equivalent concepts in spinor formal-ism. We see that every spinor which is symmetric in two indices is trace-free in those indices. Let ϕABC1...Cn = ϕ(AB)C1...Cn, then

ϕAAC1...Cn = AB_ϕ ABC1...Cn = − BA_ϕ BAC1...Cn = −ϕB B C1...Cn , thus ϕAAC1...Cn = 0. It is also true that every trace-free spinor is symmetric because if one contract (3.4.3) with a spinor ϕCDE1...En where ϕCCE1...En = 0 one gets

ϕABE1...En− ϕBAE1...En = ABϕC

C

BAE1...En = 0. (3.4.6) This yields

ϕABE1...En = ϕ(AB)E1...En+ ϕ[AB]E1...En = ϕ(AB)E1...En. (3.4.7)

3.5 The Maxwell spinor

It is more convenient for us to work in spinor form. Therefore we introduce spinor equivalents i.e. Fab ↔ FABA0_B0, A_a ↔ A_AA0 and Jb ↔ JBB0_{. F}

ab is antisymmetric, thus

FABA0_B0 = −F_BAB0_A0 (3.5.1) which implies

2FABA0_B0 = F_ABA0_B0− F_BAB0_A0 = F_ABA0_B0− F_BAA0_B0

+ FBAA0_B0− F_BAB0_A0 = _ABF_CC_A0_B0+ _A0_B0F_ABC0C 0

. (3.5.2) Since FABA0_B0 = ¯F_A0_B0_AB, this means that one can write

FABA0_B0 = 1

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3.5 The Maxwell spinor 13

where ϕAB = FABA0A 0

is symmetric and thus trace-free. The spinor ϕAB is called the Maxwell spinor.

Sometimes one needs a spinor equivalent of the dual Maxwell tensor F_ab∗ = 1₂abcdFcd. From Pirani [9] we get the relation

abcd↔ AA0_BB0CC 0_DD0 = i(ÃC˜BDÃ0D 0 ˜ B0C 0 − ÃD˜BCÃ0C 0 ˜ B0D 0 ). (3.5.4) With the help of this relation we can get a spinor equivalent of F∗.

F_ab∗ ↔ i

4((ABϕ¯B0A0+ B0A0ϕAB) − (BAϕ¯A0B0 + A0B0ϕAB))

= ₂i(ABϕ¯A0_B0 − _A0_B0ϕ_AB)

(3.5.5)

3.5.1 Maxwell equation

To see the power of spinors we express the Maxwell equations in spinor formalism. We begin with the tensor version (2.3.10). The spinor equivalent of the Maxwell tensor is then used to obtain

µ0JBB0 =∇_AA0ACA 0_C0 FCC0_BB0 =1₂CAC0A0∇_AA0(ϕ_CB_C0_B0 + ¯ϕ_C0_B0_CB) =1₂∇C_B0ϕ_CB+1 2∇ C0 Bϕ¯C0_B0 (3.5.6) and also 0 =∇AA0ACA 0_C0 F_CC∗ 0_BB0 =iCAC0A0∇_AA0( ¯ϕ_C0_B0_CB− ϕ_CB_C0_B0) =i∇C0Bϕ¯C0_B0 − i∇C_B0ϕ_CB. (3.5.7)

Together we get one single equation, which is the Maxwell equation in spinor formalism:

∇A_B0ϕ_AB = µ₀J_BB0. (3.5.8)

3.5.2 The potential equation

Now we want a spinor equivalent of the potential equation (2.3.9). 2∇(AA 0 AB)A0 = ∇_AA 0 ABA0 + ∇_BA 0 AAA0 = −A0B0∇_AA0A_BB0 − B 0_A0 ∇_BB0A_AA0 = −A0B0(∇AA0A_BB0− ∇_BB0A_AA0) = −A0B0FABA0_B0 = −A0B0A0_B01 2ϕAB− A0B0 AB1₂ϕ¯A0_B0 = −ϕAB−1₂ABϕ¯A0A 0 = −ϕAB (3.5.9)

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14 Spinors

From this we conclude that

ϕAB = 2∇A0_(AA_B)A 0

. (3.5.10)

3.6 Spinors in curved space-time

In spinor formalism we also need to study curved spacetimes and its associated covariant derivative. Most things are similar to the tensor case.

3.6.1 Covariant derivative operator for spinors

As with tensor formalism we need a covariant derivative in order to make equations invariant with respect to coordinate changes. A co-variant derivative has to follow the following rules:

• Linearity: For all Spinors S and T ,

∇_AA0(S_......+ T_......) = ∇_AA0S_......+ ∇_AA0T_....... (3.6.1) • Leibniz rule: For all spinors S and T ,

∇AA0(S_......T_......) = (∇_AA0S_......)T_......+ S_......(∇_AA0T_......). (3.6.2) • Commute with complex conjugation:

∇_AA0S_......= ∇_AA0S_....... (3.6.3) As with the tensor case we usually impose the following:

• Torsion free: For differentiable functions f from a manifold M to C,

(∇AA0∇_BB0 − ∇_BB0∇_AA0)f = 0. (3.6.4) • Compatible with the metric:

∇_AA0_BC = 0 (3.6.5)

3.7 The Weyl-Lanczos equation

To be able to rewrite the Weyl-Lanczos equation in spinor formalism we need the spinor equivalent to the Weyl tensor, which is given by:

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3.7 The Weyl-Lanczos equation 15

where ΨABCD is called the Weyl spinor. ΨABCD is completely

sym-metric and thus trace-free.

The spinor equivalent to the Lanczos tensor is given by:

Labc↔ LABCC0_A0_B0+ ¯L_A0_B0_C0_C_AB (3.7.2) where LABCC0 = L_(ABC)C0 is called the Lanczos spinor. The symme-tries follow from the symmesymme-tries given in section 2.3.4, including the algebraic gauge condition. There is also a differential gauge condition given by

ζBC = ∇AA0LA_BCA 0

, (3.7.3)

where ζBC is an arbitrary spinor.

The Weyl-Lanczos equation is considerably simpler in spinor for-malism. With the previously defined spinors ΨABCD and LABCA0 the Weyl-Lanczos equation simply becomes:

ΨABCD= 2∇(AA

0

LBCD)A0. (3.7.4) This explains why one often works with spinors when one is dealing with the Lanczos potential. We notice the close similarity with the electromagnetic potential equation (3.5.10).

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Chapter 4 Conformal structure

4.1 Asymptotic entities

When studying asymptotic entities like physical fields far away, it is quite natural to study these fields along different curves that go to infinity. In this chapter we would like to formalize this intuitive idea and develop the necessary tools. Intuitively one wants the limits along asymptotically similar curves to be the same. Other things to consider are that the limits can become infinite or zero. This is not satisfactory if we want to study the leading term. Therefore we may need to multiply with some scalar field to cause the limit to be finite. We also have to impose some kind of regularity on the curves. In general relativity we even have different kinds of curves; e.g. spacelike, timelike and lightlike curves.

As a first step in our formalisation we will use conformal rescal-ings to bring infinity to a surface or a point. After that limits can be taken along curves ending there. The simplest example of a con-formal rescaling is to reflect points of a plane in the unit circle. The mapping can be described by R = 1_r and Φ = φ when one uses planar polar coordinates (r, φ) in the first plane and (R, Φ) in the conformally rescaled plane. We complete the conformally rescaled plane by adding the origin, which represents infinity in the first plane. This is a way of bringing infinity to a point, in this case the origin. This mapping maps circles centred at the origin into circles centred at the origin. It also maps rays from the origin into rays ending at the origin. This indicates that angles are preserved. If one takes limits of fields along curves that go to infinity, in a nice way, in the first plane, one sees that the limit, if it exists, may depend on the direction of the curve. In the same way the limit along curves that end at the origin in the rescaled plane may depend on the direction of the curve.

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4.2 Conformal rescalings 17

As an example we study the field f = cos φ arctan r. After a re-flection in the unit circle it becomes ˜f = cos Φ arctan_R1. The limit is then taken along a ray (Φ constant) ending at the origin. This gives limR→0f = lim˜ R→0cos Φ arctan_R1 = π₂ cos Φ, and we see that the limit

is different along different rays.

This leads to the concept of direction dependent tensors and direc-tion dependent differentiable structures. The idea of reflecting points in the unit circle can be generalised to reflecting in the unit sphere in R3 and reflecting in the unit timelike hyperboloid in a flat spacetime.

4.2 Conformal rescalings

A not too formal definition of a conformal rescaling is that, for a manifold (M, gab), one replaces the metric gab with a metric ˜gab such

that ˜gab= Ω2gab for some smooth function Ω > 0. In spinor form this

reads ˜AB = ΩAB. The function Ω is called the conformal factor. If

M , gab and Ω satisfies some regularity conditions one can add certain

boundary points to M , corresponding to asymptotic regions of M . The larger manifold is denoted fM . Notice that the regularity of fM and Ω can be lower at the new points.

In this thesis we are especially interested in rescalings of asymp-totically flat spacetimes. Here we are mainly interested in the point i0 added to the rescaled manifold representing spacelike infinity. Loosely speaking this is the point where non-trapped spacelike curves end.

In the rescaled spacetime we can introduce the points where non-trapped light-like curves start and end. We denote this set by I (pro-nounced ’scri’ which is short for ’script-i’) and add these points to M . I is called the null infinity and is often split in two parts. I− is the past pointing part representing the points where light-like curves start. I+ is in the same way the future pointing part representing the points where non-trapped light-like curves end. There are also timelike infinities but we disregard them in this thesis.

4.2.1 A flat example

As an example one may study a flat space with an ordinary spherical coordinate system (t, r, θ, φ). The conformal rescaling is supposed to bring spacelike infinity to a finite boundary point. Therefore we choose

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18 Conformal structure

to reflect in the unit timelike hyperboloid, Thus:

T = t r2_{− t}2 = Ωt R = r r2_{− t}2 = Ωr Θ = θ Φ = φ (4.2.1)

with Ω = _r2_−t1 2. We also get R2− T2 = 1

r2_{− t}2 = Ω. This means that

the conformal rescaling transforms a timelike hyperboloid r2_{− t}2 _{= α}

to a timelike hyperboloid R2− T2 _{= α}−1_{. See figure 4.1.}

Figure 4.1: The conformal rescaling.

4.2.2 Change of derivative operator

When we have a covariant derivative we usually demand that it should be compatible with the metric, i.e. ∇agbc = 0 or in spinor formalism

∇_AA0_BC = 0. This means that we want to be able to raise and lower indices on both sides of a derivative operator in the same way. When we rescale our system such that ˜gab = Ω2gab or ˜AB = ΩAB,

we need a new covariant derivative operator such that e∇_ag˜bc = 0 or

e

∇AA0˜_CB = 0. The transformation from ∇_a to e∇_a one can get from Wald [11], but we will not use it here.

For spinors, Penrose & Rindler [8], page 356 gives us a relation between e∇AA0 and ∇_AA0. e ∇_AA0χP ...S 0_... B...F0_...= ∇_AA0χP ...S 0_... B...F0_...− Υ_BA0χP ...S 0_... A...F0_...− · · · − Υ_AF0χP ...S 0_... B...A0_...− . . . + APΥXA0χX...S 0_... B...F0_...+ · · · + _A0S 0 ΥAX0χP ...X 0_... B...F0_... + . . . (4.2.2) where ΥAA0 = Ω−1∇_AA0Ω. (4.2.3)

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4.3 Direction dependent limits 19

4.2.3 Conformally invariant fields

From Wald [11] we get that the Weyl tensor with one index raised is conformally invariant, that is eCabcd = Cabcd. This leads to the fact

that the Weyl spinor is also conformally invariant. This can be seen by writing the spinor equivalent of eCabcd = Cabcd:

A0B0_C0_D0ΨA_BCD+ A_B_CDΨ¯A 0 B0_C0_D0 =˜A0B0˜_C0_D0ΨeA_BCD+ ˜A_B˜_CDΨe¯ A0 B0_C0_D0 (4.2.4) so that A0B0_C0_D0AEΨ_EBCD + A_B_CDA 0_E0 ¯ ΨE0_B0_C0_D0 =A0B0Ω_C0_D0Ω−1AEΨe_EBCD+ A_BΩ_CDΩ−1A 0_E0_¯ e ΨE0_B0_C0_D0 =A0B0_C0_D0AEΨe_EBCD + A_B_CDA 0_E0¯ e ΨE0_B0_C0_D0 (4.2.5) or e ΨABCD = ΨABCD (4.2.6)

Here the last equation is obtained by contracting with C0D0.

4.3 Direction dependent limits

As noted above we need to define direction dependent limits at a point. More precisely we define what it means that a function is regularly direction dependent at a point with respect to a coordinate system. We use a definition from Herberthson [6]. The definition is a little bit more technical and harder to grasp than the other definitions presented earlier.

Let M be a manifold which is C∞ everywhere except at a point p, where it is C1. Let f be a function which is direction dependent at p, and let (U, Ψ) be a coordinate system containing p, with coordinates xi so that xi(p) = 0. In terms of these coordinates, let F be constant on rays from the origin, so that limΨ(p)(f ◦ Ψ−1) = F (η). We write

F (η) = limpf where the limit is taken along a C1curve that ends in p

and has tangent vector η at p. f is then said to be regularly direction dependent (with respect to this coordinate system) if, for all m ≥ 0, 1 ≤ k ≤ m, 1 ≤ ik, jk≤ 4, lim Ψ(p)(x i1 ∂ ∂xj1) . . . (xim∂x∂jm)(f ◦ Ψ −1_{) = (x}i1 ∂ ∂xj1) . . . (xim∂x∂jm)F(η) (4.3.1)

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Intuitively this means that f is regularly direction dependent if all asymptotic information (to the first order) is contained in its limit.

As an example of a regularly direction dependent function on R2, we study f = cos φ arctan1_r. We see that it is regularly direction dependent by comparing with the definition.

As an example of a non regularly direction dependent function we study f = r sin1_r. We see that limr→0f = 0 but limr→0x∂f_∂x doesn’t

even exist (it would have to be zero). Therefore f is not regularly direction dependent.

To transfer this definition from a coordinate system to the mani-fold, we need some regularity on the manifold. Our definition should be independent of the choice of coordinate system, but we also want our atlas to be as large as possible. This leads to C>1 regularity for M at p and that the definition above is independent of the choice of coordinate system, as long as the coordinate systems satisfies some extra regularity conditions [6].

We denote a regularly direction dependent function (or tensor field) f as f ∈ C>−1or C>−1(p). We define a tensor to be regularly direction dependent if all its components in a coordinate basis are regularly direction dependent. With the notation f ∈ C>nwe mean that f ∈ Cn and f ’s (n + 1):st derivatives are C>−1.

One can naturally identify the directions with rays from the limit point. Therefore we define our direction dependent fields as the limit along the ray that passes through the field point. Due to this definition we can talk about direction dependent fields in the tangent space not only in the space of directions. We also note that the direction dependent fields are constant along rays.

4.3.1 Asymptotically flat spacetimes

In this thesis we will only study asymptotically flat spacetimes. Be-cause a formal definition is quite cumbersome and technical we only look at the highlights. For a full definition se Wald [11].

Let ( fM , ˜gab) be a manifold with a metric. A conformal

transfor-mation and addition of I and i0 gives ( fM , ˜gab) where ˜gab = Ω2gab for

some Ω > 0. The relevant conditions for M to be asymptotically flat are then Ω = 0 on I+, I−, i0 e ∇_aΩ 6= 0 on I+, I− lim i0 ∇ea∇ebΩ = −2˜gab.

Also the condition imposed on ˜gab is that it is C>0 at i0. Physically

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4.3 Direction dependent limits 21

tube.

4.3.2 Asymptotic properties of the curvature

and derivatives

In this section we investigate some useful facts emerging from the def-inition above. Let r ≥ 0 be C>0 such that r(p) = 0 and e∇ar(˜η) 6= 0

for spacelike directions ˜η from i0. Since the metric ˜gab ∈ C>0 it

follows that the Christoffel symbols eΓi

jk are all C>−1. Furthermore

xm ∂_∂xnΓe_jkior equivalently r_∂x∂nΓe_jkiare also regularly direction depen-dent. By studying the definition of the Riemann tensor we therefore see that

r eRabcd, r eCabcd, r eRab, r eR (4.3.2)

are all regularly direction dependent.

From the definition of asymptotically flat spacetimes it follows [6], with the definition of r above, that for every direction ˜η, Ω ∼ r2 and therefore that √ Ω eRabcd, √ Ω eCabcd, √ Ω eRab, √ Ω eR (4.3.3)

are all regularly direction dependent. Some small calculations give that also r eΨABCD and

√

Ω eΨABCD are regularly direction dependent.

We will now derive another important result. Let ua∈ C>−1 and

let r be defined as above. Furthermore, let x1, x2, x3, x4 be coordinates in a neighbourhood of i0, such that the metric at i0 takes the form of i0 takes the form of the Minkowski metric. Let ∂abe their coordinate

derivative. Now consider

r e∇aub= r(∂aub− eΓabcuc). (4.3.4)

Because eΓabc is regularly direction dependent reΓabc vanishes when

taking the limit. We therefore have lim

i0 r e∇aub = lim_i0 r(∂aub− eΓab

c_u

c) = lim

i0 r∂aub. (4.3.5) This means that the left hand side exists and it can be evaluated using the coordinate derivative of asymptotic Minkowski coordinates. This makes it possible for us to do our calculations in the tangent space at i0 with the limit fields constant on rays, thus we may, in this sense, regard the spacetime as flat at i0.

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4.3.3 Projection

To study these direction dependent limits at i0, we represent the direc-tions there by points on a timelike hypersurface in the tangent space to i0. The limiting fields can then be projected onto this hypersurface. See figure 4.2. We will use two different hypersurfaces, the timelike unit hyperboloid and the timelike unit cylinder. We multiply our ten-sor fields in the rescaled spacetime with a scalar field f . If we choose f such that lim_i0∇˜_af is normal to timelike hyperboloids (e.g. f =

√ Ω), then we prefer to project the field quantities in the direction of the position vector field onto the timelike unit hyperboloid. If we on the other hand choose f such that lim_i0∇˜_af is normal to some timelike cylinder, we prefer to project the field quantities in the direction of the position vector field onto that cylinder.

Figure 4.2: Direction dependent limits.

4.4 Regularity at i

0

Now consider a general spacetime, which is asymptotically flat at spacelike infinity. This means that the rescaled manifold is C>1 at i0. The structure tells us that there is a tangent space to i0. Let {T, X, Y, Z} be an asymptotically Minkowski coordinate system in a neighbourhood of i0, and denote the induced coordinates in the tangent space by the same letters. By differentiation of Ω one then obtains the metric ˜gab with regularity C>0. The Minkowski

coordi-nate system can then be transformed to a spherical coordicoordi-nate system {T, R, Θ, Φ} in the usual way. We are dealing with asymptotically flat spaces and therefore e∇a∇e_bΩ = −2˜g_ab at i0. If one makes the choice Ω = R2− T2 _{= X}2_{+ Y}2_{+ Z}2_{− T}2 _{it follows that e}_∇

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4.5 The timelike unit hyperboloid 23

i0_{, because}

e

∇_bΩ = ˜∂bΩ = 2X(dX)b+ 2Y (dY )b+ 2Z(dZ)b− 2T (dT )b (4.4.1)

and we therefore obtain e ∇_a(X(dX)b) = X e∇a(dX)b+ (dX)a(dX)b = (dX)a(dX)b+ X ˜∂a(dX)b | {z } 0 − XΓabc(dX)c | {z } →0 → (dX)a(dX)b. (4.4.2) Thus, e ∇_a∇e_bΩ → 2(dX)_a(dX)_b+ 2(dY )_a(dY )_b+ 2(dZ)_a(dZ)_b− 2(dT )_a(dT )_b = −2gab (4.4.3) Therefore we choose Ω = R2− T2 _{at i}0_.

4.5 The timelike unit hyperboloid

A hypersurface is a said to be timelike if it has at least one timelike tangent vector at every point. If one then chooses the normal to be the position vector field, one gets a timelike hyperboloid. Later we will study the potential equations on the timelike unit hyperboloid Ω = R2− T2 _{= 1 in the tangent space to i}0_{. In the rest of this chapter}

we only work in the tangent space and thus use a flat metric. Here we derive some important properties of this hypersurface. The surface unit normal ˜ηa can be acquired from:

˜ ηa= e ∇_a(R2_{− T}2₎ − q − e∇b(R2− T2) e∇b(R2− T2) = q −2R(dR)a+ 2T (dT )a −(2R(dR)_b− 2T (dT )_b)(−2R(_∂R∂ )b_{− 2T (} ∂ ∂T)b) = −√ R R2_−T2(dR)a+ T √ R2_−T2(dT )a= 1 √ Ωx˜a (4.5.1)

where ˜xa= T (dT )a− R(dR)a is the position co-vector field. We also

notice that e ∇a √ Ω = e∇a p R2_{− T}2₌ ∂ ∂R p R2_{− T}2_(dR) a+_∂T∂ p R2_{− T}2_{(dT )} a = √ R R2_−T2(dR)a− T √ R2_−T2(dT )a= −˜ηa. (4.5.2)

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We can also get a useful identity ˜ηA0

AηÃ0_B= 1 2ÃB by noting that ˜ ξAB =˜ηA 0 AηÃ0_B= ˜η_B0_Bη˜B 0 A= −˜ηB 0 Bη˜B0_A= − ˜ξ_BA. Hence 2 ˜ξAB = ˜ξAB− ˜ξBA= ÃBξCC = ÃB˜CDξ˜CD =ÃB˜CDη˜C 0 Cη˜C0_D = −˜_ABη˜_C0_Cη˜C 0_C = ÃB By analogy we get ˜ηAA0η˜_AB0 = 1 2Ã0B0, and similarly ÃB = 2˜ηA 0 AηÃ0B and Ã0B 0 = 2˜ηAA0η˜_AB 0 .

4.5.1 Differential structure

We begin by studying the position covector field. ˜ xb= T (dT )b− X(dX)b− Y (dY )b− Z(dZ)b (4.5.3) We then differentiate: e ∇_ax˜b= e∂ax˜b= (dT )a_∂T∂ + (dX)a_∂X∂ + (dY )a_∂Y∂ + (dZ)a_∂Z∂ · (T (dT )_b− X(dX)_b− Y (dY )_b− Z(dZ)_b) =(dT )a(dT )b− (dX)a(dX)b− (dY )a(dY )b− (dZ)a(dZ)b =˜gab, (4.5.4) since (dT )a, (dX)a, (dY )a and (dZ)a are all constant.

We can then use the fact that ˜xa=

√ Ω˜ηa. ˜ gab = e∇ax˜b= e∇a( √ Ω˜ηb) = ˜ηb∇e_a √ Ω +√Ω e∇_aη˜b. (4.5.5)

We define a derivative operator on the space of directions by ∂AA0 = lim

i0 √

Ω e∇_AA0, (4.5.6)

and in the limit we get ˜

gab = −˜ηbη˜a+ ∂aη˜b.

We can then define

hab = ∂aη˜b = ˜gab+ ˜ηaη˜b, (4.5.7)

which in spinor form reads

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4.5 The timelike unit hyperboloid 25

which gives us the following relation: ˜ ηA0B∂A 0 Aη˜B 0 B= ˜ηA0B˜_ABÃ 0_B0 + ˜ηA0Bη˜_AA 0 ˜ ηBB 0 = ˜ηAB 0 +1₂ÃBη˜BB 0 = 3₂ηÃB 0 (4.5.9)

For an arbitrary direction ˜ηAA0 we get: ˜

ηCC0∂CC0η˜_AA0 = ˜ηCC 0

(˜CA˜C0_A0 + ˜η_CC0η˜_AA0) = ˜η_AA0 − ˜η_AA0 = 0 Let F (˜η) = limi0f be the limit of some arbitrary regularly direction dependent scalar field. We then see that since (the extension of) F is constant along rays, the derivative along these rays, i.e. ˜ηa∂aF must

be zero.

Let vb(˜η) = F1(˜η)(dT )b+ F2(˜η)(dX)b+ F3(˜η)(dY )b+ F4(˜η)(dX)b

be a co-vector field with all Fi(˜η) regularly direction dependent. We

then get that ˜

ηa∂avb(˜η) = F1(˜η)˜ηa∂a(dT )b+ F2(˜η)˜ηa∂a(dX)b+ F3(˜η)˜ηa∂a(dY )b

+ F4(˜η)˜ηa∂a(dZ)b,

(4.5.10) since each ˜ηa∂aFi= 0.

Futhermore we see that √Ω e∇a(dT )b = −

√

ΩΓabc(dT )c → 0, and

analogously with (dX)b, (dY )b and (dZ)b. From this we get

˜

ηa∂avb(˜η) = 0. (4.5.11)

It is easily seen that it is also valid for tensors. In other words ˜ηa∂a

is the null operator acting on limits of regularly direction dependent tensor fields.

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Chapter 5 Asymptotic

electromagnetism

To be able to see more clearly what we do later with the Weyl-Lanczos equation, we will first study how the potential equation for electromag-netism behaves at spacelike infinity. First we do a conformal rescaling that brings i0 to the origin. Then we take the direction dependent limit in the direction described by points on the unit timelike hyper-boloid.

5.1 Conformal rescaling

We do a conformal rescaling of our spacetime. The conformally rescaled entities we denote by the corresponding symbol with a tilde. Let ˜

AA0 = Ω_AA0 and Υ_AA0 = Ω−1∇_AA0Ω = Ω−1∇e_AA0Ω. We derive a transformed version of the electromagnetic potential equation:

ϕAB= 2∇A0_(AA_B)A 0 = 2 e∇_A0_(AA_B)A 0 + 2ΥA0_(BA_A)A 0 − 2A0A 0 ΥX0_(AA_B)X 0 = 2 e∇_A0_(AA_B)A 0 − 2Υ_A0_(BA_A)A 0 (5.1.1) ϕAB = 2 e∇A0_(AA_B)A 0 − 2Ω−1( e∇_A0_(BΩ)A_B)A 0 = 2ΩΩ−2 Ω e∇_A0_(AA_B)A 0 − ( e∇_A0_(AΩ)A_B)A 0 = 2Ω e∇_A0_(A Ω−1A_B)A0= 2Ω e∇_A0_(A ˜ A0B0A_B)B0 (5.1.2)

Choosing eFab = Fab gives Ω ˜ϕAB = ϕAB, because of (3.5.3). Hence

˜ ϕAB = 2 e∇A0_(A Ω−1A_B)A0. If we choose eAAA0 = A_AA0 we get ˜ ϕAB = 2 e∇A0_(AAe_B)A 0 . (5.1.3)

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5.2 Projection 27

This is expected because Fab and Aa are differential forms which are

invariant under conformal rescalings.

5.2 Projection

Since the limits of our spinors will be regularly direction dependent, we describe these direction dependencies by projecting onto the unit timelike hyperboloid. We choose this surface because we want an orthogonal projection in the direction of the position vector field. To get finite quantities in the limit we need to multiply with some scalar field that goes to zero fast enough when approaching i0. As motivated in section 4.3.3, we choose the scalar field to be Ωq with some positive constant q.

η~ a

Figure 5.1: The unit timelike hyperboloid with normal ˜ηa.

It is natural [3] to study potentials where limi0 √

Ω eAaand limi0Ω eF_ab exists. Therefore we define

AAA0 = lim i0 √ Ω eAAA0 ϕ_AB = lim i0 Ω ˜ϕAB (5.2.1)

and use that

∂AA0 = lim i0 √ Ω e∇_AA0 ˜ ηAA0 = − lim i0 ∇eAA 0 √ Ω. From (5.1.3) one obtains

Ω ˜ϕAB = 2Ω e∇A0_(A(√1 Ω √ Ω eA_B)A0) = 2 √ Ω e∇_A0_(A( √ Ω eAB)A 0 ) + 2Ω √ Ω eA(BA 0 e ∇_A)A0√1 Ω = 2 √ Ω e∇_A0_(A( √ Ω eAB)A 0 ) − 2 √ Ω eA(BA 0 e ∇_A)A0 √ Ω. (5.2.2)

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28 Asymptotic electromagnetism

The limit equation then becomes ϕ_AB = 2∂_A0_(AA_B)A 0

+ 2˜η_A0_(AA_B)A 0

. (5.2.3)

5.3 Splitting the potential

One may want to split the potential into a tangential and a normal part. Therefore we let Ab= Φ˜ηb+ Πb, where we have put Φ = −Abη˜b

and Πb= Ab− Φ˜ηb so that ˜ηbΠb = 0 and let

DAK = ˜ηA 0 (K∂A)A0 Π†AK = ˜ηA 0 KΠAA0 A†AK = ˜ηA 0 KAAA0 = ˜ηA 0 K(Φ˜ηAA0+ Π_AA0) = Φ˜ηA0Kη˜AA0 + Π† AK = Π † AK − 1 2Φ˜AK (5.3.1)

This means that DAK is the tangential derivative of the hyperboloid.

Using the motivation from Section 4.5.1 we observe that ˜ ηA0[K∂A]A0 = 1 2˜KAη˜ A0 C∂CA0 = −1 2˜KAη˜ CC0 ∂CC0 (5.3.2) is the null operator for all the fields we are considering. We also notice that AAA0 = 2˜ηC_A0η˜_CB

0

AAB0 = 2˜η_A0CA†

AC, and also that Π † AB

is symmetric because it is trace-free; Π†_AA= ˜ηAA0ΠAA0 = 0. ˜ ηBA 0 ∂AA0A† C B_{= ˜}_ηA0 B∂AA0(˜ηB 0_B ACB0) = ˜ηA0BACB0∂_AA0η˜B 0_B + ˜ηA0Bη˜B 0_B ∂AA0A_CB0 = 3₂η˜AB 0 ACB0 + ˜B 0_A0 ∂AA0A_CB0 (5.3.3) This gives ˜ηBA 0 ∂AA0A† CB = 32ACB0η˜A B0 −∂_AA0A_CA 0 . This relation can then be used to get an expression for ϕ_AB.

ϕ_AB = 2∂A0_(AA_B)A 0 + 2˜ηA0_(AA_B)A 0 = −2˜ηCA 0 ∂_A0_(AA † B) C + 3AB0_(Bη˜_A)B 0 − 2A_A0_(Bη˜_A)A 0 = −2˜ηCA 0 ∂_A0_(AΠ † B) C _{+ ˜}_η CA 0 ∂A0_(A(Φ˜_B)C) + A† (BA) = −2˜ηCA 0 ∂_A0_(AΠ † B) C _{+ D} (AB)Φ + Π † (AB)+ 1 2Φ ˜(AB) | {z } 0 = −2D_C(AΠ†_B)C+ Π†_AB+ DABΦ (5.3.4)

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Chapter 6 Asymptotic gravitation

6.1 Conformal rescaling of the W-L

equa-tions

In this section we conformally rescale the Weyl-Lanczos equations and rewrite them in a form suitable for the timelike unit hyperboloid.

ΨABCD= 2∇(AA 0 LBCD)A0 = −2∇_A0_(AL_BCD)A 0 = −2 e∇_A0_(AL_BCD)A 0 − 2Υ_A0_(BL_ACD)A 0 − 2Υ_A0_(CL_ABD)A 0 − 2Υ_A0_(DL_ABC)A 0 + 2A0A 0 ΥX0_(AL_BCD)X 0 = −2 e∇_A0_(AL_BCD)A 0 − 2Υ_A0_(AL_BCD)A 0 , (6.1.1) where again ΥAA0 = Ω−1∇_AA0Ω = Ω−1∇e_AA0Ω. This gives

ΨABCD= −2 e∇A0_(AL_BCD)A 0 − 2Ω−1∇e_A0_(AΩ LBCD)A 0 = −2Ω−1Ω e∇_A0_(AL_BCD)A 0 +∇e_A0_(AΩ L_BCD)A0 = −2Ω−1∇e_A0_(A ΩLBCD)A 0 (6.1.2) Combine this with eΨABCD = ΨABCD and define eLBCDA

0 = LBCDA 0 . We then get e ΨABCD= −2Ω−1∇e_A0_(A Ω eLBCD)A 0 . (6.1.3)

Note that it is not possible to define eL in such a way that the form of Weyl-Lanczos equation is completely preserved. We note that with this definition Ω eLABCA0 is a Lanczos potential of Ω eΨ_ABCD in the rescaled spacetime.

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30 Asymptotic gravitation

Since √Ω eΨABCD is regularly direction dependent, but eΨABCD is

not we want to rewrite this equation as: √ Ω eΨABCD= −2 √ Ω e∇_A0_(ALe_BCD)A 0 − 2Ω−12Le_(BCDA 0 e ∇_A)A0Ω = −2√Ω e∇_A0_(ALe_BCD)A 0 − 4eL_(BCDA0∇e_A)A0 √ Ω (6.1.4) since e∇√Ω = 1₂Ω−12∇Ω._e

6.1.1 Conformal rescaling of the gauge-condition

The Lanczos differential gauge ζBC transforms as follows:

ζBC = ∇AA0LA_BCA 0 = e∇_AA0LA_BCA 0 − _AAΥXA0LX_BCA 0 + ΥBA0LA_ACA 0 + ΥCA0LA_BAA 0 − _A0A 0 ΥAX0LA_BCX 0 = e∇_AA0LA_BCA 0 − 4Υ_AA0LA_BCA 0 = e∇_AA0LA_BCA 0 − 4LA_BCA0Ω−1∇e_AA0Ω (6.1.5) We also have eLBCDA 0 = LBCDA 0 which gives e LABCA 0 = ˜AELe_EBCA 0 = Ω−1AELEBCA 0 = Ω−1LABCA 0 . This yields ζBC = e∇AA0(Ω eLA_BCA 0 ) − 4 eLABCA 0 e ∇_AA0Ω = Ω e∇_AA0LeA_BCA 0 − 3eLABCA 0 Ω−1∇e_AA0Ω (6.1.6)

If we then define ˜ζBC = Ω−1ζBC one obtains

˜ ζBC = e∇AA0LeA_BCA 0 − 3eLABCA 0 Ω−1∇e_AA0Ω = e∇_AA0LeA_BCA 0 − 6eLABCA 0 Ω−12∇_e_AA0 √ Ω, (6.1.7) or written differently √ Ω ˜ζBC = √ Ω e∇_AA0LeA_BCA 0 − 6eLABCA 0 e ∇_AA0 √ Ω. (6.1.8)

6.2 Projection on hypersurfaces

We only study Lanczos potentials that are regularly direction depen-dent at i0 after a conformal rescaling and a multiplication with some scalar field. Note that we do not know that such potentials exist in general. However in some special cases such potentials have been

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6.2 Projection on hypersurfaces 31

found (see [4]). In this section we project on the unit timelike hyper-boloid in the same way as in the electromagnetic case. We get finite regularly direction dependent limits if we define (see [6])

ΨABCD = lim i0 √ Ω eΨABCD ∂AA0 = lim i0 √ Ω e∇_AA0 (6.2.1)

and we assume the same thing for the following limits LBCDA 0 = lim i0 ˜ LBCDA 0 ζ_BC = lim i0 √ Ω ˜ζBC. (6.2.2)

Furthermore we observe that ˜ ηAA0 = − lim i0 ∇eAA 0 √ Ω.

As in the electromagnetic case, these limits give us tensor fields defined on the unit timelike hyperboloid in the tangent space of i0. We can then extend these to the whole tangent space by demanding that they are constant on rays from i0. That these limits are sometimes finite can be seen by taking the limit on some known solutions from [4]. Inserting this into (6.1.4) and (6.1.8) gives

ΨABCD = −2∂A0_(AL_BCD)A 0 + 4L_(BCDA0η˜_A)A0 (6.2.3) and ζ_BC = ∂AA0LA_BCA 0 + 6LABCA 0 ˜ ηAA0. (6.2.4)

6.2.1 Projection onto the Cylinder

If we instead want to study the equations on the timelike unit cylinder, we multiply by the scalar field R instead of√Ω. The relation between R and Ω is given by R2 Ω = R2 R2_{− T}2 = 1 1 − a2

where a = _RT. From equation (4.5.1) we get that ˜ ηa= − 1 √ 1 − a2(dR)a+ a √ 1 − a2(dT )a. (6.2.5)

To get finite limits we need to multiply with some scalar field that tends to zero when approaching i0. Now we choose the scalar field, as motivated in section 4.3.3, to be Rq for some constant q > 0.

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Figure 6.1: The timelike unit cylinder with normal R.

Therefore we define ΨABCD = lim i0 R eΨABCD ζ_BC = lim i0 R ˜ζBC ∂AA0 = lim i0 R e∇AA 0 (6.2.6)

and observe that

(dR)AA0 = lim

i0 ∇eAA 0R. The limit for ˜LBCDA

0

is defined as in the previous section. All these direction dependent limits are taken along the direction of rays from the cylinder to i0. The limiting value is then prescribed to the entire ray in the tangent space.

We multiply (6.1.4) with √R Ω. R eΨABCD = −2R e∇A0_(ALe_BCD)A 0 −_√ 4 1−a2Le(BCD A0 e ∇_A)A0( √ Ω) (6.2.7) After taking the limit one then gets

ΨABCD = −2∂A0_(AL_BCD)A 0 +√ 4 1−a2L(BCD A0_η_˜ A)A0 (6.2.8) and multiplying (6.1.8) by √R

Ω and taking the limit yields

ζ_BC = ∂AA0LA_BCA 0 + √6 1−a2L A BCA 0 ˜ ηAA0. (6.2.9) These limits are not perfect because ΨABCD is zero at the null

infinities (I). The null infinities are given by the surfaces where a = ±1. To get finite values in these directions one may define

ˆ ΨABCD = lim i0 R3 ΩΨe_ABCD= 1 1−a2ΨABCD ˆ LBCDA 0 = lim i0 R2 ΩL˜BCD A0 ₌ 1 1−a2LBCDA 0 (6.2.10)

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6.3 Splitting the Lanczos spinor 33

After a multiplication of (6.1.3) with R_Ω3 we get

R3 ΩΨe_ABCD= −2R 3 Ω2∇e_A0_(A Ω2Ω−1Le_BCD)A 0 = −2R3∇e_A0_(A R−2 R_Ω2Le_BCD)A 0 − 2R_Ω33Le_(BCDA 0 e ∇_A)A0Ω2 = −2R e∇_A0_(A R2 Ω Le_BCD)A 0 − 2R5 ΩLe_(BCDA 0 e ∇_A0_)AR−2 − 8R3_Ω32 Ω3 Le_(BCDA 0 e ∇_A)A0 √ Ω = −2R e∇_A0_(A R2 Ω Le_BCD)A 0 + 4R_Ω2Le_(BCDA 0 e ∇_A0_)AR −_√8 1−a2 R2 Ω Le_(BCDA 0 e ∇_A)A0 √ Ω (6.2.11) After taking the limit we get

ˆ ΨABCD = −2∂A0_(ALˆ_BCD)A 0 +4 ˆL_(BCDA0(dR)_A)A0+√8 1−a2Lˆ(BCD A0_η_˜ A)A0 (6.2.12)

6.3 Splitting the Lanczos spinor

Again we study the projection onto the timelike unit hyperboloid. We split the potential in a tangential and a normal part. Finally we express the equations in terms of the derivative operators defined by (5.3.1). Let

L†ABCD = ˜ηA

0

DLABCA0

= 1₄(L†_ABCD+ L†_ABDC+ L†_ACDB+ L†_BCDA)

+1₄(L†_ABCD− L†_ABDC) +1₄(L_ABCD† − L†_ACDB) +1₄(L†_ABCD− L†_BCDA) = L†_(ABCD)+1₄˜ADL†_BCEE +1₄˜BDL†_ACEE+1₄˜CDL†_ABEE

= L†_(ABCD)−1₂˜D(AL † BC)E E (6.3.1) We see that ΦAB = L † ABEE = ˜ηEE 0

LABEE0, i.e. the part of L which is aligned to the normal of the timelike hyperboloid.

We use a temporary notation ∂†_KA = ˜ηKA

0 ∂A0_A and get ∂†E(AL†BCD)E = ˜ηA 0_E ∂A0_(Aη˜B 0 |E|LBCD)B0 = ˜ηA0Eη˜B0E∂A0_(AL_BCD)B0 + ˜ηA 0_E LB0_(BCD∂_A0_A)η˜B 0 E = ˜A0B0∂A0_(AL_BCD)B0 −3 2η˜(A B0 LBCD)B0 = ∂_A0_(AL_BCD)A 0 −3 2L † (ABCD) (6.3.2)

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where we have used equation (4.5.9) in the third equality. From these expressions we can derive an expression for Ψ in terms of unprimed spinors. We start from equation (6.2.3).

ΨABCD = −2∂A0_(AL_BCD)A 0

+ 4L_(BCDA0η˜_A)A0

= −2∂†E_(AL†_BCD)E− 3L†_(ABCD)+ 4L†_(ABCD) = −2∂†E(AL†BCD)E+ L†(ABCD)

(6.3.3)

We define Π†_ABCD = L†_(ABCD)and ΦAB = L†_ABEE, so that L†_ABCD =

Π†_ABCD−1

2˜D(AΦBC) and get:

ΨABCD = −2∂†E(A(Π † BCD)E− 3 4˜B|E|ΦCD)) + Π † ABCD = −2∂†E(AΠ † BCD)E + 3 2∂ † (ABΦCD)+ Π † ABCD = −2DE(AΠ † BCD)E+ 3 2D(ABΦCD)+ Π † ABCD (6.3.4) where DAK = ˜ηA 0 (K∂A)A0.

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Chapter 7 Asymptotic fields in

NP-formalism

In this chapter we derive equations for the Lanczos potential on the cylinder similar to (6.2.8) in component form using Newman-Penrose formalism. We begin by defining the following derivative operators

D = e∇₀₀0 = õAoÃ 0 e ∇_AA0 = la∇e_a ∆ = e∇₁₁0 = ˜ιA˜ιA 0 e ∇_AA0 = na∇e_a δ = e∇₀₁0 = õA˜ιA 0 e ∇_AA0 = ma∇e_a ¯ δ = e∇₁₀0 = ˜ιAoÃ 0 e ∇_AA0 = ¯ma∇e_a (7.0.1)

and the spin coefficients

κ = maDla ε = 1₂(naDla+ maD ¯ma) π = − ¯maDna ρ = maδl¯a α = 1₂(naδl¯a+ ma¯δ ¯ma) λ = − ¯maδn¯ a σ = maδla β = 1₂(naδla+ maδ ¯ma) µ = − ¯maδna τ = ma∆la γ = 1₂(na∆la+ ma∆ ¯ma) ν = − ¯ma∆na (7.0.2) Where la↔ õAoÃ0 na↔ ˜ιA˜ιA0 ma↔ õA˜ιA0 m¯a↔ ˜ιAõA0 (7.0.3) so that lana= 1, mam¯a= −1 and all the other inner products of the

tetrad vectors vanish. A tetrad with these properties is called a null tetrad.

7.1 Choice of tetrad

We want to choose a null tetrad in the transformed system. The prob-lem is that the rescaled system is not flat, so it is hard to find a null

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36 Asymptotic fields in NP-formalism

tetrad that is valid everywhere. However our system is asymptoti-cally flat at i0. Therefore the errors we introduce when choosing a C>−1-tetrad which is only a null tetrad at i0, will vanish at i0. We may investigate these facts by letting ηab be the Minkowski metric,

and (La, Na, Ma, ¯Ma) be a null tetrad field. Then define hab ∈ C>0

such that ˜gab = ηab + hab is the metric for our transformed system.

Then it can be shown that [6] there is a null tetrad (la_{, n}a_{, m}a_{, ¯}_ma_{) in}

the Minkowski spacetime such that the difference between the tetrads is C>0 and tends to zero when approaching i0. This means that all quantities formed from (la_{, n}a_{, m}a_{, ¯}_ma_{) equals the corresponding}

quan-tities formed from the true null tetrad (La, Na, Ma, ¯Ma) to first or-der. Hence we can perform all calculations in the Minkowski tetrad (la_{, n}a_{, m}a_{, ¯}_ma_).

We can therefore choose la=( ∂ ∂T) a_{− (} ∂ ∂R) a √ 2 n a₌(∂T∂ ) a_{+ (} ∂ ∂R) a √ 2 ma=( ∂ ∂Θ) a₋ i sin Θ( ∂ ∂Φ) a √ 2R m¯ a₌(∂Θ∂ ) a₊ i sin Θ( ∂ ∂Φ) a √ 2R (7.1.1)

This tetrad is used in the rest of this thesis in the transformed system. Because we can use the Minkowski metric we asymptotically get

la= (dT )a+ (dR)a √ 2 na= (dT )a− (dR)a √ 2 ma= − R (dΘ)a− i sin Θ(dΦ)_√ a 2 m¯a= − R (dΘ)a+ i sin Θ(dΦ)_√ a 2 . (7.1.2) By making the ansatz ˜ηa= bla+ cna one may express the

normal-ized position co-vector field ˜ηain this tetrad. The coefficients b and c

is obtained from equation (4.5.1). b = √ T − R 2√R2_{− T}2 = a − 1 √ 2√1 − a2 c = √ T + R 2√R2_{− T}2 = a + 1 √ 2√1 − a2 (7.1.3)

Notice also that ˜ηa is orthogonal to ma and ¯ma.

7.2 Weyl-Lanczos on the cylinder

From page 44 in Andersson [2] one gets a Newman-Penrose formulation of the Weyl-Lanczos equation. By replacing Ψ with Ω eΨ and L with Ω eL one obtains the equation in the transformed system. These equations

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7.2 Weyl-Lanczos on the cylinder 37

can be derived in the same way as we will derive the equations for the gauge condition. Ω 2Ψe0 = δ(Ω eL0) − D(Ω eL4) − ( ¯α + 3β − ¯π)Ω eL0+ 3σΩ eL1 + (3ε − ¯ε + ¯ρ)Ω eL4− 3κΩeL5 2Ω eΨ1 = 3δ(Ω eL1) − 3D(Ω eL5) − ¯δ(Ω eL4) + ∆(Ω eL0) − (3γ + ¯γ + 3µ − ¯µ)Ω eL0− 3( ¯α + β − ¯π − τ )Ω eL1+ 6σΩ eL2 + (3α − ¯β + 3π + ¯τ )Ω eL4+ 3(ε − ¯ε + ¯ρ − ρ)Ω eL5− 6κΩeL6 Ω eΨ2 = δ(Ω eL2) − D(Ω eL6) − ¯δ(Ω eL5) + ∆(Ω eL1) − νΩ eL0 − (2µ − ¯µ + γ + ¯γ)Ω eL1− ( ¯α − β − ¯π − 2τ )Ω eL2+ σΩ eL3+ λΩ eL4 + (α − ¯β + 2π + ¯τ )Ω eL5− (ε + ¯ε − ¯ρ + 2ρ)Ω eL6− κΩeL7 2Ω eΨ3 = δ(Ω eL3) − D(Ω eL7) − 3¯δ(Ω eL6) + 3∆(Ω eL2) − 6νΩ eL1 + 3(¯µ − µ + γ − ¯γ)Ω eL2− ( ¯α − 3β − 3τ − ¯π)Ω eL3 + 6λΩ eL5− 3(α + ¯β − ¯τ − π)Ω eL6− (3ε + ¯ε − ¯ρ + 3ρ)Ω eL7 Ω 2Ψe₄ = ∆(Ω eL₃) − ¯δ(Ω eL₇) − 3ν eL₂+ (¯µ + 3γ − ¯γ)Ω eL₃+ 3λΩ eL₆ − (3α + ¯β − ¯τ )Ω eL7 (7.2.1) where e Ψ0 = õAo˜Bo˜CõDΨe_ABCD e Ψ1 = õAo˜Bo˜C˜ιDΨe_ABCD e Ψ2 = õAo˜B˜ιC˜ιDΨe_ABCD e Ψ3 = õA˜ιB˜ιC˜ιDΨe_ABCD e Ψ4 = ˜ιA˜ιB˜ιC˜ιDΨe_ABCD (7.2.2) and e L0 = õAo˜Bo˜C¯õA 0 e LABCA0 Le₄= õAõBõC¯˜ιA 0 e LABCA0 e L1 = õAo˜B˜ιCo¯Ã 0 e LABCA0 Le₅= õAõB˜ιC¯˜ιA 0 e LABCA0 e L2 = õA˜ιB˜ιCo¯Ã 0 e LABCA0 Le₆= õA˜ιB˜ιC¯˜ιA 0 e LABCA0 e L3 = ˜ιA˜ιB˜ιCo¯Ã 0 e LABCA0 Le₇= ˜ιA˜ιB˜ιC¯˜ιA 0 e LABCA0. (7.2.3)

Multiplying with Ω−12 _{and using the identities} 1 √ ΩD (Ωf ) = √ ΩDf + 2f D √ Ω √1 Ωδ (Ωf ) = √ Ωδf + 2f δ √ Ω 1 √ Ω∆ (Ωf ) = √ Ω∆f + 2f ∆√Ω √1 Ωδ (Ωf ) =¯ √ Ω¯δf + 2f ¯δ√Ω (7.2.4)

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38 Asymptotic fields in NP-formalism

gives for the eΨ2 equation:

√ Ω eΨ2 = √ Ω δ eL2− D eL6− ¯δ eL5+ ∆ eL1− ν eL0− (2µ − ¯µ + γ + ¯γ) eL1 − ( ¯α − β − ¯π − 2τ ) eL2+ σ eL3+ λ eL4+ (α − ¯β + 2π + ¯τ ) eL5 − (ε + ¯ε − ¯ρ + 2ρ) eL6− κeL7 + 2eL2δ √ Ω − 2 eL6D √ Ω − 2eL5δ¯ √ Ω + 2 eL1∆ √ Ω (7.2.5) To get a form suitable for the cylinder we multiply with √R

Ω = 1 √ 1−a2 and get: R eΨ2 = R δ eL2− D eL6− ¯δ eL5+ ∆ eL1− ν eL0− (2µ − ¯µ + γ + ¯γ) eL1 − ( ¯α − β − ¯π − 2τ ) eL2+ σ eL3+ λ eL4+ (α − ¯β + 2π + ¯τ ) eL5 − (ε + ¯ε − ¯ρ + 2ρ) eL6− κeL7 +√_1−a1 2 2 eL2δ √ Ω − 2 eL6D √ Ω − 2eL5δ¯ √ Ω + 2 eL1∆ √ Ω (7.2.6) The other equations are handled similarly.

Define ΨABCD= lim i0 R eΨABCD D = lim_i0 RD LBCDA 0 = lim i0 ˜ LBCDA 0 ∆ = lim i0 R∆ δ = lim i0 Rδ ¯ δ = lim i0 R¯δ (7.2.7)

and again observe that ˜

ηa= − lim i0 ∇ea

√ Ω

Also define all boldface spin coefficients as before but with lim_i0R e∇_AA0 instead of e∇AA0. This gives for the Ψ₂ equation:

Ψ2 = δL2− DL6− ¯δL5+ ∆L1− νL0− (2µ − ¯µ + γ + ¯γ)L1 − ( ¯α − β − ¯π − 2τ )L2+ σL3+ λL4+ (α − ¯β + 2π + ¯τ )L5 − (ε + ¯ε − ¯ρ + 2ρ)L6− κL7+√_1−a1 2 − 2L2m a_η_˜ a+ 2L6laηã + 2L5m¯aηã− 2L1naηã (7.2.8) We then use the following identities:

la_η_˜ a √ 1−a2 = 1 √ 2(1−a) ma_η_˜ a √ 1−a2 = 0 na_η_˜ a √ 1−a2 = −1 √ 2(1+a) ¯ ma_η_˜ a √ 1−a2 = 0 (7.2.9)

Asymptotiska egenskaper för Lanczosspinoren

Asymptotic properties of the Lanczos

spinor

Acknowledgements

Contents

Chapter 1

Introduction

Chapter 2

Tensors

2.1

Manifolds and coordinates

2.2

What are tensors?

2.2.1

Abstract index notation

2.2.2

The metric tensor

2.3

Curvature

2.3.1

The Maxwell tensor

2.3.2

The Riemann tensor

2.3.3

The Weyl tensor

2.3.4

The Lanczos tensor

Chapter 3

Spinors

3.1

What are spinors?

3.2

Infeld-van der Waerden symbols

3.3

The metric spinor

3.4

Spinor algebra

3.4.1

Symmetric spinors

3.5

The Maxwell spinor

3.5.1

Maxwell equation

3.5.2

The potential equation

3.6

Spinors in curved space-time

3.6.1

Covariant derivative operator for spinors

3.7

The Weyl-Lanczos equation

Chapter 4

Conformal structure

4.1

Asymptotic entities

4.2

Conformal rescalings

4.2.1

A flat example

4.2.2

Change of derivative operator

4.2.3

Conformally invariant fields

4.3

Direction dependent limits

4.3.1

Asymptotically flat spacetimes

4.3.2

Asymptotic properties of the curvature

and derivatives

4.3.3

Projection

4.4

Regularity at i

4.5

The timelike unit hyperboloid

4.5.1

Differential structure

Chapter 5

Asymptotic