On the Pursuit of Generalizations for the Petrov Classification and the
Goldberg-Sachs Theorem

Carlos Batista

Doctoral Thesis

Universidade Federal de Pernambuco, Departamento de Física

Supervisor: Bruno Geraldo Carneiro da Cunha

Brazil - November - 2013

Thesis presented to the graduation program of the Physics Department of
Universidade Federal de Pernambuco as part of the duties to obtain the degree
of Doctor of Philosophy in Physics.

Examining Board:
Prof. Amilcar Rabelo de Queiroz (IF-UNB, Brazil)

Prof. Antônio Murilo Santos Macêdo (DF-UFPE, Brazil)

Prof. Bruno Geraldo Carneiro da Cunha (DF-UFPE, Brazil)

Prof. Fernando Roberto de Luna Parisio Filho (DF-UFPE, Brazil)

Prof. Jorge Antonio Zanelli Iglesias (CECs, Chile)

Abstract

The Petrov classification is an important algebraic classification for the
Weyl tensor valid in 4-dimensional space-times. In this thesis such
classification is generalized to manifolds of arbitrary dimension and
signature. This is accomplished by interpreting the Weyl tensor as a linear
operator on the bundle of $p$-forms, for any $p$, and computing the Jordan
canonical form of this operator. Throughout this work the spaces are assumed
to be complexified, so that different signatures correspond to different
reality conditions, providing a unified treatment. A higher-dimensional
generalization of the so-called self-dual manifolds is also investigated.

The most important result related to the Petrov classification is the
Goldberg-Sachs theorem. Here are presented two partial generalizations of such
theorem valid in even-dimensional manifolds. One of these generalizations
states that certain algebraic constraints on the Weyl “operator” imply the
existence of an integrable maximally isotropic distribution. The other version
of the generalized Goldberg-Sachs theorem states that these algebraic
constraints imply the existence of a null congruence whose optical scalars
obey special restrictions.

On the pursuit of these results the spinorial formalism in 6 dimensions was
developed from the very beginning, using group representation theory. Since
the spinors are full of geometric significance and are suitable tools to deal
with isotropic structures, it should not come as a surprise that they provide
a fruitful framework to investigate the issues treated on this thesis. In
particular, the generalizations of the Goldberg-Sachs theorem acquire an
elegant form in terms of the pure spinors.

Keywords: General relativity, Weyl tensor, Petrov classification,
Integrability, Isotropic distributions, Goldberg-Sachs theorem, Spinors,
Clifford algebra.

This thesis is based on the following published articles:

$\bullet$ Carlos Batista, Weyl tensor classification in four-dimensional
manifolds of all signatures, General Relativity and Gravitation 45 (2013),
785\.

$\bullet$ Carlos Batista, A generalization of the Goldberg-Sachs theorem and
its consequences, General Relativity and Gravitation 45 (2013), 1411.

$\bullet$ Carlos Batista and Bruno G. Carneiro da Cunha, Spinors and the Weyl
tensor classification in six dimensions, Journal of Mathematical Physics 54
(2013), 052502\.

$\bullet$ Carlos Batista, On the Weyl tensor classification in all dimensions
and its relation with integrability properties, Journal of Mathematical
Physics 54 (2013), 042502.

Acknowledgments

In order for such a long work, lasting almost five years, to succeed it is
unavoidable to have the aid and the support of a lot of people. In this
section I would like to sincerely thank to everybody that contributed in some
way to my doctoral course.

I want to acknowledge my supervisor, Bruno Geraldo Carneiro da Cunha, for the
sensitivity in suggesting a research project that fully matches my
professional tastes. I also thank for all advise he gave me during our
frequent meetings. It is inspiring to be supervised by such a wise scientist
as Professor Bruno. Finally I thank for the freedom and the continued support
he provided me, so that I could follow my own track. I take the chance to
acknowledge all other Professors from UFPE that contributed to my education,
particularly the Professors Antônio Murilo, Sérgio Coutinho, Henrique Araújo
and Liliana Gheorghe, whose knowledge and commitment have inspired me.

In the same vein, I thank to all the mates as well as to the staff of the
physics department. Specially, I thank to my doctorate fellow Fábio Novaes
Santos for all the times he patiently helped me, thank you very much. I also
would like to mention my friends Carolina Cerqueira, Danilo Pinheiro, Diego
Leite and Rafael Alves, who contributed for a more pleasant environment in the
physics department. I acknowledge the really qualified and efficient work of
the graduation secretary Alexsandra Melo as well as the friendship and support
of the under-graduation secretary Paula Franssinete.

Finally, and most importantly, I would like to thank for the unconditional
support of all my family. Particularly, I thank to my mother, Ana Lúcia, and
to my sister, Natália Augusta, for always encouraging me to study, since my
childhood, as well as stimulating my vocation. I also thank to my parents in
law, Guilherme e Lúcia Helena, for taking responsibility on the construction
of my house, what allowed me to proceed using my whole time to study. To
conclude, I want to effusively and repeatedly thank to my wife, Juliana. In
addition for her being my major inspiration, she supports me and encourages me
like no one else. There are no words to say how much I am glad for having her
besides me. I love you, my wife!!

During my Ph.D. I received financial support from CAPES (Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior) and CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico). It is worth mentioning that I
really appreciate doing what I enjoy, study physics and mathematics, on my own
country and still be paid for this. I will do my best, as I always tried to,
in order for my work as a researcher and as a Professor, in the future, to
return this investment.

> _“The black holes of nature are the most perfect macroscopic objects there
> are in the universe: the only elements in their construction are our
> concepts of space and time. And since the general theory of relativity
> provides only a single unique family of solutions for their descriptions,
> they are the simplest objects as well.”_
>
>
> Subrahmanyan Chandrasekhar
> (The Mathematical Theory of Black Holes)

###### Contents

  1. 0 Motivation and Outline
  2. I Review and Classical Results
    1. 1 Introducing General Relativity
      1. 1.1 Gravity is Curvature
      2. 1.2 Riemannian Geometry, the Formalism of  Curved Spaces
      3. 1.3 Geodesics
      4. 1.4 Symmetries and Conserved Quantities
      5. 1.5 Einstein’s Equation
      6. 1.6 Differential Forms
      7. 1.7 Cartan’s Structure Equations
      8. 1.8 Distributions and Integrability
      9. 1.9 Higher-Dimensional Spaces
    2. 2 Petrov Classification, Six Different Approaches
      1. 2.1 Weyl Tensor as an Operator on the Bivector Space
      2. 2.2 Annihilating Weyl Scalars
      3. 2.3 Boost Weight
      4. 2.4 Bel-Debever and Principal Null Directions
      5. 2.5 Spinors, Penrose’s Method
      6. 2.6 Clifford Algebra
      7. 2.7 Interpreting the PNDs
      8. 2.8 Examples
      9. 2.9 Other Classifications
    3. 3 Some Theorems on Petrov Types
      1. 3.1 Shear, Twist and Expansion
      2. 3.2 Goldberg-Sachs
      3. 3.3 Mariot-Robinson
      4. 3.4 Peeling Property
      5. 3.5 Symmetries
  3. II Original Research
    1. 4 Generalizing the Petrov Classification and the Goldberg-Sachs Theorem to All Signatures
      1. 4.1 Null Frames
      2. 4.2 Generalized Petrov Classification
        1. 4.2.1 Euclidean Signature
        2. 4.2.2 Lorentzian Signature
        3. 4.2.3 Split Signature
        4. 4.2.4 Annihilating Weyl Scalars
      3. 4.3 Generalized Goldberg-Sachs Theorem
      4. 4.4 Geometric Consequences of the Generalized Goldberg-Sachs Theorem
        1. 4.4.1 Complex Manifolds
        2. 4.4.2 General Results
        3. 4.4.3 Euclidean Signature
        4. 4.4.4 Lorentzian Signature
        5. 4.4.5 Split Signature
    2. 5 Six Dimensions Using Spinors
      1. 5.1 From Vectors to Spinors
        1. 5.1.1 A Null Frame
        2. 5.1.2 Clifford Algebra in 6 Dimensions
        3. 5.1.3 Isotropic Subspaces
      2. 5.2 Other Signatures
      3. 5.3 An Algebraic Classification for the Weyl Tensor
      4. 5.4 Generalized Goldberg-Sachs
        1. 5.4.1 Lorentzian Signature
      5. 5.5 Example, Schwarzschild in 6 Dimensions
    3. 6 Integrability and Weyl Tensor Classification in All Dimensions
      1. 6.1 Algebraic Classification for the Weyl Tensor
        1. 6.1.1 Inner Product of $p$-forms
        2. 6.1.2 Even Dimensions
        3. 6.1.3 An Elegant Notation
      2. 6.2 Integrability of Maximally Isotropic Distributions
      3. 6.3 Optical Scalars and Harmonic Forms
      4. 6.4 Generalizing Mariot-Robinson and Goldberg-Sachs Theorems
    4. 7 Conclusion and Perspectives
    5. A Segre Classification and its Refinement
    6. B Null Tetrad Frame
    7. C Clifford Algebra and Spinors
    8. D Group Representations

###  Chapter 0 Motivation and Outline

The so called Petrov classification is an algebraic classification for the
Weyl tensor of a 4-dimensional curved space-time that played a prominent role
in the development of general relativity. Particularly, it helped on the
search of exact solutions for Einstein’s equation, the most relevant example
being the Kerr metric. Furthermore, such classification contributed for the
physical understanding of gravitational radiation. There are several theorems
concerning this classification, they associate the Petrov type of the Weyl
tensor with physical and geometric properties of the space-time. Probably the
most important of these theorems is the Goldberg-Sachs theorem, which states
that in vacuum the Weyl tensor is algebraically special if, and only if, the
space-time admits a shear-free congruence of null geodesics. It was because of
this theorem that Kinnersley was able to find all type $D$ vacuum solutions
for Einstein’s equation, an impressive result given that such equation is
highly non-linear.

Since the Petrov classification and the Goldberg-Sachs theorem have been of
major importance for the study of 4-dimensional Lorentzian spaces, it is quite
natural trying to generalize these results to manifolds of arbitrary dimension
and signature. This is the goal of the present thesis. In what follows the
Petrov classification will be extended to all dimensions and signatures in a
geometrical approach. Moreover, there will be presented few generalizations of
the Goldberg-Sachs theorem valid in even-dimensional spaces. The relevance of
this work is enforced by the increasing significance of higher-dimensional
manifolds in physics and mathematics.

This thesis was split in two parts. The part I shows the classical results
concerning the Petrov classification and its associated theorems, while part
II presents the work developed by the present author during the doctoral
course. In chapter 1 the basic tools of general relativity and differential
geometry necessary for the understanding of this thesis are reviewed. It is
shown that gravity manifests itself as the curvature of the space-time and it
is briefly discussed the relevance of higher-dimensional manifolds. Chapter 2
shows six different routes to define the Petrov classification. In addition,
the so called principal null directions are interpreted from the physical and
geometrical points of view. Chapter 3 presents some of the most important
theorems concerning the Petrov classification, as the Goldberg-Sachs, the
Mariot-Robinson and the Peeling theorems. In chapter 4 the Petrov
classification is generalized to 4-dimensional spaces of arbitrary signature
in a unified approach, with each signature being understood as a choice of
reality condition on a complex space. Moreover, it is shown that this
generalized classification is related to the existence of important geometric
structures. Chapter 5 develops the spinorial formalism in 6 dimensions with
the aim of uncovering results that are hard to perceive by means of the
standard vectorial approach. In particular, the spinorial language reveals
that the Weyl tensor can be seen as an operator on the space of 3-vectors,
which is exploited in order to classify this tensor. It is also proved an
elegant partial generalization of the Goldberg-Sachs theorem making use of the
concept of pure spinors. An algebraic classification for the Weyl tensor valid
in arbitrary dimension and signature is then developed in chapter 6, where it
is also proved two partial generalizations of the Goldberg-Sachs theorem valid
in even-dimensional manifolds. Finally, chapter 7 discuss the conclusions and
perspectives of this work.

Some background material is also presented in the appendices. Appendix A
introduces a classical algebraic classification for square matrices called the
Segre classification and defines a refinement for it. Such refined
classification is used throughout the thesis. Appendix B describes what a null
tetrad is. The formal treatment of Clifford algebra and spinors is addressed
in appendix C, where some pedagogical examples are also worked out. Finally,
appendix D introduces and give some examples of the basics concepts on group
representation theory.

##  Part I Review and Classical Results

###  Chapter 1 Introducing General Relativity

Right after Albert Einstein arrived at his special theory of relativity, in
1905, he noticed that the Newtonian theory of gravity needed to be modified.
Newton’s theory predict that when a gravitational system is perturbed the
effect of such perturbation is immediately felt at all points of space, in
other words the gravitational interaction propagates with infinite velocity.
This, however, is in contradiction with one of the main results of special
relativity, that no information can propagate faster than light. Moreover,
according to Einstein’s results energy and mass are equivalent, which implies
that the light must feel the gravitational attraction, in disagreement with
the Newtonian gravitational theory.

It took long 10 years for Einstein to establish a relativistic theory of
gravitation, the General Theory of Relativity. In spite of the sophisticated
mathematical background necessary to understand this theory, it turns out that
it has a beautiful geometrical interpretation. According to general
relativity, gravity shows itself as the curvature of the space-time. Such
theory has had several experimental confirmations, notably the correct
prediction of Mercury’s perihelion precession and the light deflection. In
particular, it is worth noting that the GPS technology strongly relies on the
general theory of relativity.

The aim of the present chapter is to describe the basic tools of general
relativity necessary in the rest of the thesis. Readers already familiar with
such theory are encouraged to skip this chapter. Throughout this thesis it
will be assumed that repeated indices are summed, the so-called Einstein
summation convention. The symmetrization and anti-symmetrization of indices
are respectively denoted by round and square brackets. So that, for instance,
$T_{(\mu\nu)}=\frac{1}{2}(T_{\mu\nu}+T_{\nu\mu})$ and
$L_{[\mu\nu\rho]}=\frac{1}{6}(L_{\mu\nu\rho}+L_{\nu\rho\mu}+L_{\rho\mu\nu}-L_{\nu\mu\rho}-L_{\rho\nu\mu}-L_{\mu\rho\nu})$.

####  1.1 Gravity is Curvature

According to the special theory of relativity we live in a four-dimensional
flat space-time endowed with the metric:

$ds^{2}\,=\,\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,=\,dt^{2}\,-\,dx^{2}\,-\,dy^{2}\,-\,dz^{2}\,,$

where $\\{x^{\mu}\\}=\\{t,x,y,z\\}$ are cartesian coordinates. Note that if we
make a Poincaré transformation,
$x^{\mu}\mapsto\Lambda^{\mu}_{\phantom{\mu}\nu}x^{\nu}+a^{\mu}$, where
$a^{\mu}$ is constant and
$\eta_{\rho\sigma}\Lambda^{\rho}_{\phantom{\rho}\mu}\Lambda^{\sigma}_{\phantom{\sigma}\nu}=\eta_{\mu\nu}$,
then the metric remains invariant. Physically, performing a Poincaré
transformation means changing from one inertial frame to another, which should
not change the Physics. But, in addition to the inertial coordinates we are
free to use any coordinate system of our preference. For example, in a
particular problem it might be convenient to use spherical coordinates on the
space. The procedure of changing coordinates is simple, for example, if
$g_{\mu\nu}$ is the metric on the coordinate system $\\{x^{\mu}\\}$ then using
new coordinates, $\\{x^{\prime\mu}\\}$, we have:

$g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,=\,g_{\mu\nu}\,\left(\frac{\partial
x^{\mu}}{\partial x^{\prime\rho}}dx^{\prime\rho}\right)\,\left(\frac{\partial
x^{\nu}}{\partial
x^{\prime\sigma}}dx^{\prime\sigma}\right)\;\Rightarrow\;g^{\prime}_{\rho\sigma}\,=\,\frac{\partial
x^{\mu}}{\partial x^{\prime\rho}}\frac{\partial x^{\nu}}{\partial
x^{\prime\sigma}}\,g_{\mu\nu}\,.$

Where $g^{\prime}_{\rho\sigma}$ are the components of the metric on the
coordinates $\\{x^{\prime\mu}\\}$. In general, if
$T^{\mu_{1}\ldots\mu_{p}}_{\phantom{\mu_{1}\ldots\mu_{p}}\nu_{1}\ldots\nu_{q}}$
are the components of a tensor $\boldsymbol{T}$ on the coordinate system
$\\{x^{\mu}\\}$, then its components on the coordinates $\\{x^{\prime\mu}\\}$
are:

$T^{\prime\mu_{1}\ldots\mu_{p}}_{\phantom{\mu_{1}\ldots\mu_{p}}\nu_{1}\ldots\nu_{q}}\,=\,\left(\frac{\partial
x^{\prime\mu_{1}}}{\partial x^{\rho_{1}}}\ldots\frac{\partial
x^{\prime\mu_{p}}}{\partial x^{\rho_{p}}}\right)\,\left(\frac{\partial
x^{\sigma_{1}}}{\partial x^{\prime\nu_{1}}}\ldots\frac{\partial
x^{\sigma_{q}}}{\partial
x^{\prime\nu_{q}}}\right)\,T^{\rho_{1}\ldots\rho_{p}}_{\phantom{\rho_{1}\ldots\rho_{p}}\sigma_{1}\ldots\sigma_{q}}\,.$
(1.1)

So far so good. But there is one important thing whose transformation under
coordinate changes is non trivial, the derivative. Let $V^{\mu}$ be the
components of a vector on the coordinate system $\\{x^{\mu}\\}$. Then it is a
simple matter to prove that $\partial_{\nu}V^{\mu}$ does not transform as a
tensor under a general coordinate change. Nevertheless, after some algebra, it
can be proved that defining

$\Gamma^{\mu}_{\nu\rho}\,\equiv\,\frac{1}{2}\,g^{\mu\sigma}\left(\partial_{\nu}g_{\rho\sigma}+\partial_{\rho}g_{\nu\sigma}-\partial_{\sigma}g_{\nu\rho}\right)\,,$
(1.2)

with $g^{\mu\nu}$ being the inverse of $g_{\mu\nu}$ and $\partial_{\nu}$ being
the partial derivative with respect to the coordinate $x^{\nu}$, then the
combination

$\displaystyle\nabla_{\nu}\,V^{\mu}\,\equiv\,\frac{\partial V^{\mu}}{\partial
x^{\nu}}\,+\,\Gamma^{\mu}_{\nu\rho}\,V^{\rho}\,=\,\partial_{\nu}\,V^{\mu}\,+\,\Gamma^{\mu}_{\nu\rho}\,V^{\rho}$
(1.3)

does transform as a tensor. The object $\Gamma^{\mu}_{\nu\rho}$, called
Christoffel symbol (it is not a tensor), serves to correct the non-tensorial
character of the partial derivative. The operator $\nabla_{\nu}$ is called the
covariant derivative, it has the remarkable property that when acting on a
tensor it yields another tensor. Its action on a general tensor is, for
example,

$\nabla_{\nu}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\mu_{2}\mu_{3}}\,=\,\partial_{\nu}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\mu_{2}\mu_{3}}\,+\,\Gamma^{\mu_{1}}_{\nu\sigma}\,T^{\sigma}_{\phantom{\sigma}\mu_{2}\mu_{3}}-\Gamma^{\sigma}_{\nu\mu_{2}}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\sigma\mu_{3}}-\Gamma^{\sigma}_{\nu\mu_{3}}\,T^{\mu_{1}}_{\phantom{\mu_{1}}\mu_{2}\sigma}\,.$
(1.4)

Using this formula it is straightforward to prove that
$\nabla_{\rho}g_{\mu\nu}=0$, so the metric is covariantly constant. Since
coordinates are physically meaningless we should always work with tensorial
objects, because they are invariant under coordinate changes. Therefore, we
should only use covariant derivatives instead of partial derivatives. Although
they seem awkward, the covariant derivatives are, actually, quite common. For
instance, in 3-dimensional calculus it is well-known that the divergence of a
vector field in spherical coordinates looks different than in cartesian
coordinates, this happens because we are implicitly using the covariant
derivative.

Now comes a puzzle. From the physical point of view one might expect that no
reference frame is better than another, all of them are equally arbitrary. In
particular, the concept of acceleration is relative, since according to the
classical Einstein’s mental experiment (Gedankenexperiment) gravity and
acceleration are locally indistinguishable, the so-called equivalence
principle. In spite of this, Minkowski space-time has an infinite class of
privileged frames, the cartesian frames (also called inertial frames). From
the geometrical point of view these frames are special because the Christoffel
symbols, $\Gamma^{\mu}_{\nu\sigma}$, vanish identically in all points. But, as
just advocated, the existence of these preferred frames is not a reasonable
assumption. Therefore, we conclude that the space-time should not admit the
existence of a frame such that $\Gamma^{\mu}_{\nu\sigma}$ vanishes in all
points. Geometrically this implies that the space-time is curved! Somebody
could argue that the inertial frames represent non-accelerated observers and,
therefore, may exist. But our universe is full of mass everywhere, which
implies that the gravitational field is omnipresent. Using then the
equivalence principle we conclude that all objects are accelerated, so that it
is nonsense to admit the existence of globally non-accelerated frames. Now we
might wonder ourselves: If the space-time is not flat then why has special
relativity been so successful? The reason is that in every point of a curved
space-time we can always choose a reference frame such that
$g_{\mu\nu}=\eta_{\mu\nu}$ and $\Gamma^{\mu}_{\nu\sigma}=0$ at this point.
Hence, special relativity is always valid locally.

Another natural question that emerges is: What causes space-time bending? Let
us try to answer this. In special relativity a free particle moves on straight
lines, which are the geodesics of flat space-time. Analogously, on a curved
space-time the free particles shall move along the geodesics. Thus, no matter
the peculiarities of a particle, if it is free it will follow the geodesic
path compatible with its initial conditions of position and velocity. This
resembles gravity, which, due to the equality of the inertial and
gravitational masses, is such that all particles with the same initial
condition follow the same trajectory. For example, a canon-ball and a feather
both acquire the same acceleration under the gravitational field. Therefore,
it is reasonable to say that the gravity bends the space-time. There is
another path which leads us to the same conclusion. In line with Einstein’s
elevator experiment, gravity is locally equivalent to acceleration. Now
suppose we are in a reference frame such that $\Gamma^{\mu}_{\nu\sigma}=0$,
then if this referential is accelerated it is simple matter to verify that the
Christoffel symbol will be different from zero. Thus acceleration is related
to the non-vanishing of $\Gamma^{\mu}_{\nu\sigma}$. Furthermore, the lack of a
coordinate system such that $\Gamma^{\mu}_{\nu\sigma}=0$ in all points of the
space-time implies that the space-time is curved. So that we arrive at the
following relations:

$\textrm{Gravity}\;\;\longleftrightarrow\;\;\textrm{Acceleration}\;\;\longleftrightarrow\;\;\Gamma^{\mu}_{\nu\sigma}\neq
0\;\;\longleftrightarrow\;\;\textrm{Curvature}\,,$

which again leads us to the conclusion that gravity causes the curvature of
the space-time. This is the main content of the General Theory of Relativity.

In the standard model of particles the fundamental forces of nature are
transmitted by bosons: photons carry the electromagnetic force, $W$ and $Z$
bosons communicate the weak interaction and gluons transmit the strong nuclear
force. In the same vein, the gravitational interaction might be carried by a
boson, dubbed the graviton. Indeed, heuristically speaking, since the emission
of a particle of non-integer spin changes the total angular momentum of the
system111For instance, suppose that a particle has integer spin and then emits
a fermion. So, by the rule of angular momenta addition (see eq. (D.3) in
appendix D), it follows that its angular momentum after the emission is a
superposition of non-integer values. Therefore it must have changed. it
follows that interactions carried by fermions are generally incompatible with
the existence of static forces [1]. Now comes the question: What are the mass
and the spin of the graviton? Since the gravitational force has a long range
(energy goes as $1/r$) it follows that the mass must be zero, just as the mass
of the photon. Moreover, since the graviton is a boson its spin must be
integer. One can prove that it must be different from zero, since a scalar
theory of gravitation predicts that the light is not affected by gravity [2],
which contradicts the experiments and the fact that energy and mass are
equivalent. The spin should also be different from one, since the interaction
carried by a massless particle of spin one is the electromagnetic force which
can be both attractive and repulsive, whereas gravity only attracts. It turns
out that the graviton has spin 2. Indeed, in [1] it is shown how to start from
the theory of a massless spin 2 particle on flat space-time and arrive at the
general theory of relativity. For a wonderful introductory course in general
relativity see [3]. More advanced texts are available at [4, 5]. Historical
remarks and interesting philosophical thoughts can be found in [6].

####  1.2 Riemannian Geometry, the Formalism of
Curved Spaces

In order to make calculations on general relativity it is of fundamental
importance to get acquainted with the tools of Riemannian geometry. The intent
of the present section is to briefly introduce the bare minimum concepts on
such subject necessary for the understanding of this thesis.

Roughly, an $n$-dimensional manifold $M$ is a smooth space such that locally
it looks like $\mathbb{R}^{n}$. For example, the 2-sphere is a 2-dimensional
manifold, since it is smooth and if we look very close to some patch of the
spherical surface it will look like a flat plane (the Earth surface is round,
but for its inhabitants it, locally, looks like a plane). More precisely, a
manifold of dimension $n$ is a topological set such that the neighborhood of
each point can be mapped into a patch of $\mathbb{R}^{n}$ by a coordinate
system in a way that the overlapping neighborhoods are consistently joined [4,
7]. Now imagine curves passing through a point $p$ belonging to the surface of
the 2-sphere. The possible directions that these curves can take generate a
plane, called the tangent space of $p$. Generally, associated to each point
$p\in M$ of an $n$-dimensional manifold we have a vector space of dimension
$n$, denoted by $T_{p}M$ and called the tangent space of $p$. A vector field
$\boldsymbol{V}$ is then a map that associates to every point of the manifold
a vector belonging to its tangent space. The union of the tangent spaces of
all points of a manifold $M$ is called the tangent bundle and denoted by $TM$.
A vector field is just an element of the tangent bundle.

Now, suppose that we introduce a coordinate system $\\{x^{\mu}\\}$ in the
neighborhood of $p\in M$ and let $\boldsymbol{V}$ be a vector field in this
neighborhood. Denoting by $V^{\mu}$ the components of $\boldsymbol{V}$ on such
coordinate system then it is convenient to use the following abstract
notation:

$\boldsymbol{V}\,=\,V^{\mu}\,\frac{\partial\,}{\partial
x^{\mu}}\,\equiv\,V^{\mu}\,\partial_{\mu}\,.$

This is useful because when we make a coordinate transformation,
$x^{\mu}\mapsto x^{\prime\mu}$, and use the chain rule to transform the
partial derivative we find that the components of the vector field change just
as displayed in (1.1). Therefore, the vector fields on a manifold can be
interpreted as differential operators that act on the space of functions over
the manifold. Furthermore, the partial derivatives $\\{\partial_{\mu}\\}$
provide a basis for the tangent space at each point, forming the so-called
coordinate frame. For example, on the 2-sphere we can say that
$\\{\partial_{\theta},\partial_{\phi}\\}$ is a coordinate frame, where
$\theta$ is the polar angle while $\phi$ denotes the azimuthal angle.

A metric $\boldsymbol{g}$ is a symmetric non-degenerate map that act on two
vector fields and gives a function over the manifold. In this thesis it will
always be assumed that the manifold is endowed with a metric, hence the pair
$(M,\boldsymbol{g})$ will sometimes be called the manifold. In particular,
note that the Minkowski manifold is $(\mathbb{R}^{4},\eta_{\mu\nu})$. The
components of the metric on a coordinate frame are denoted by
$g_{\mu\nu}=\boldsymbol{g}(\partial_{\mu},\partial_{\nu})$. By conveniently
choosing a coordinate frame, we can always manage to put the matrix
$g_{\mu\nu}$ in a diagonal form such that all slots are $\pm 1$ at some
arbitrary point $p\in M$, $g_{\mu\nu}\mapsto
g^{\prime}_{\mu\nu}=\operatorname{diag}(1,1,\ldots,-1,-1,\ldots)$. The modulus
of the metric trace when it is in such diagonal form is called the signature
of the metric and denoted by $s$, $s=|\Sigma_{\mu}\,g^{\prime}_{\mu\mu}|$.
Denoting by $n$ the dimension of the manifold then if $s=n$ the metric is said
to be Euclidean, for $s=(n-2)$ the signature is Lorentzian and if $s=0$ the
metric is said to have split signature. In Riemannian geometry it is customary
to low and raise indices using the metric, $g_{\mu\nu}$, and its inverse,
$g^{\mu\nu}$.

The partial derivative of a scalar function,
$\partial_{\mu}f\equiv\nabla_{\mu}f$, is a tensor. But, as discussed in the
preceding section, when acting on tensors this partial derivative must be
replaced by the covariant derivative, defined on equations (1.2) and (1.4). In
the formal jargon, this tensorial derivative is called a connection.
Particularly, the connection defined by (1.2) and (1.4) is named the Levi-
Civita connection. The covariant derivative share many properties with the
usual partial derivative, it is linear and obey the Leibniz rule. However,
these two derivatives also have a big difference: while the partial
derivatives always commute, the covariant derivatives generally do not. More
precisely it is straightforward to prove that:

$\displaystyle(\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu})\,V^{\rho}\,=\,R^{\rho}_{\phantom{\rho}\sigma\mu\nu}\,V^{\sigma}\,,$
(1.5) $\displaystyle
R^{\rho}_{\phantom{\rho}\sigma\mu\nu}\,\equiv\,\partial_{\mu}\Gamma^{\rho}_{\sigma\nu}-\partial_{\nu}\Gamma^{\rho}_{\sigma\mu}+\Gamma^{\rho}_{\kappa\mu}\Gamma^{\kappa}_{\sigma\nu}-\Gamma^{\rho}_{\kappa\nu}\Gamma^{\kappa}_{\sigma\mu}\,.$
(1.6)

The object $R^{\rho}_{\phantom{\rho}\sigma\mu\nu}$ is called the Riemann
tensor. Although its definition was made in terms of the non-tensorial
Christoffel symbols, $R^{\rho}_{\phantom{\rho}\sigma\mu\nu}$ is indeed a
tensor, as the left hand side of equation (1.5) is a tensor. The Riemann
tensor is also called the curvature tensor, because it measures the curvature
of the manifold222Actually it measures the curvature of the tangent bundle..
In particular, a manifold is flat if, and only if, the Riemann tensor
vanishes. Defining
$R_{\rho\sigma\mu\nu}=g_{\rho\kappa}R^{\kappa}_{\phantom{\kappa}\sigma\mu\nu}$
then, after some algebra, it is possible to prove that this tensor has the
following symmetries.

$R_{\rho\sigma\mu\nu}=R_{[\rho\sigma][\mu\nu]}\,\,;\;R_{\rho\sigma\mu\nu}=R_{\mu\nu\rho\sigma}\,\,;\;R_{\rho[\sigma\mu\nu]}=0\,\,;\;\nabla_{[\kappa}R_{\rho\sigma]\mu\nu}=0$
(1.7)

Particularly, the last two symmetries above are called Bianchi identities.
There are other important tensors that are constructed out of the Riemann
curvature tensor:

$\displaystyle
R_{\mu\nu}\,\equiv\,R^{\rho}_{\phantom{\rho}\mu\rho\nu}\quad;\quad
R\,\equiv\,g^{\mu\nu}R_{\mu\nu}\,=\,R^{\nu}_{\phantom{\nu}\nu}$ $\displaystyle
C_{\rho\sigma\mu\nu}\equiv
R_{\rho\sigma\mu\nu}-\frac{2}{n-2}\left(g_{\rho[\mu}R_{\nu]\sigma}-g_{\sigma[\mu}R_{\nu]\rho}\right)+\frac{2}{(n-1)(n-2)}R\,g_{\rho[\mu}g_{\nu]\sigma}\,.$

These tensors are respectively called Ricci tensor, Ricci scalar and Weyl
tensor. The Ricci tensor is symmetric, while the Weyl tensor has all the
symmetries of equation (1.7) except for the last one, the differential Bianchi
identity. The Weyl tensor will be of central importance in this piece of work,
since the main goal of this thesis is to define an algebraic classification
for this tensor and relate such classification with integrability properties.
The Weyl tensor has two landmarks: it is traceless,
$C^{\rho}_{\phantom{\rho}\sigma\rho\nu}=0$, and it is invariant under
conformal transformations, i.e., if we transform the metric as
$g_{\mu\nu}\mapsto\Omega^{2}g_{\mu\nu}$ then the tensor
$C^{\rho}_{\phantom{\rho}\sigma\mu\nu}$ remains invariant.

####  1.3 Geodesics

Given two points $p_{1}$ and $p_{2}$ on a manifold $(M,\boldsymbol{g})$, the
trajectory of minimum length connecting these points is called a geodesic. If
$x^{\mu}(\tau)$ is a curve joining these points, with
$x^{\mu}(\tau_{i})=p_{i}$, then its length is given by:

$\Delta(\tau_{1},\tau_{2})\,=\,\int_{\tau_{1}}^{\tau_{2}}\,\sqrt{g_{\mu\nu}\,\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}\,\,d\tau\,.$

Note that $\Delta$ is invariant under the change of parametrization of the
curve. Let us exploit this freedom adopting the arc length,
$s(\tau)\equiv\Delta(\tau_{1},\tau)$, as the curve parameter. Then performing
a standard variational calculation we find that the curve of minimum length
connecting $p_{1}$ and $p_{2}$ satisfies the following differential equation
known as the geodesic equation:

$\frac{d^{2}x^{\rho}}{ds^{2}}\,+\,\Gamma^{\rho}_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}\,=\,0$
(1.8)

Note that using cartesian coordinates on the Minkowski space we have that
$\Gamma^{\rho}_{\mu\nu}=0$, so that eq. (1.8) implies that the geodesics of
flat space are straight lines, as it should be. Using equations (1.3) and
(1.8) we find that the geodesic equation can be elegantly expressed by:

$T^{\mu}\,\nabla_{\mu}\,T^{\nu}\,=\,0\,,\quad
T^{\mu}\equiv\frac{dx^{\mu}}{ds}\,.$ (1.9)

Note that the vector field $T^{\mu}$ is tangent to the curve. If instead of
the arc length parameter, $s$, we have used another parameter $\tau$, we would
have found the equation $N^{\mu}\nabla_{\mu}N^{\nu}=fN^{\nu}$, where
$N^{\mu}\equiv\frac{dx^{\mu}}{d\tau}$ and $f$ is some function. The parameters
$\tau^{\prime}$ such that $f=0$ are called affine parameters. It is simple
matter to verify that the affine parameters are all linearly related to the
arc length, $\tau^{\prime}=a\,s+b$ with $a\neq 0$ and $b$ being constants.
Physically, the arc length $s$ of a time-like curve (geodesic or not)
represents the proper time of the observer following this curve. In general
relativity, free massive particles follow time-like geodesics, whereas free
massless particles describe null geodesics. It is worth remarking that here a
particle is said to be free when the only force acting on it is the
gravitational force.

In order to gain some intuition on the formalism introduced so far, let us go
back to the example of the $2$-sphere. Let $S$ be a sphere of radius $r$
embedded on the 3-dimensional Euclidean space $\mathbb{R}^{3}$, as depicted in
figure 1.1. The metric of the 3-dimensional space is
$ds^{2}=dx^{2}+dy^{2}+dz^{2}$. Then, the points on the sphere can be locally
labeled by the coordinates $\theta$ and $\phi$ related to the cartesian
coordinates by $x=r\sin\theta\cos\phi$, $y=r\sin\theta\sin\phi$ and
$z=r\cos\theta$. Inserting these expressions in the 3-dimensional metric and
assuming that $r$ is constant we are led to the metric of the $2$-sphere,
$ds^{2}=r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}$. Once we have this
metric we can compute its associated curvature by means of equation (1.6). In
particular, the Ricci scalar is found to be $R=2/r^{2}$. So, the bigger the
radius the smaller the curvature. Now, let $\boldsymbol{V}$ be a vector field
tangent to the sphere, $\boldsymbol{V}\cdot\hat{\boldsymbol{r}}=0$. Where the
dot denotes the inner product of $\mathbb{R}^{3}$. Then, the covariant
derivative of $\boldsymbol{V}$ along some curve tangent to the sphere is just
the projection of the ordinary derivative of $\boldsymbol{V}$ along this curve
onto the tangent planes of the sphere, see figure 1.1. For instance, the
covariant derivative of $\boldsymbol{V}$ along the great circle
$\theta=\frac{\pi}{2}$ is
$\nabla_{\phi}\boldsymbol{V}=\frac{d\boldsymbol{V}}{d\phi}-(\hat{\boldsymbol{r}}\cdot\frac{d\boldsymbol{V}}{d\phi})\hat{\boldsymbol{r}}$.
Particularly, one can prove that $\nabla_{\phi}\hat{\boldsymbol{\phi}}=0$,
which implies that such great circle is a geodesic curve. In general, all
great circles of the $2$-sphere are geodesic curves.

Figure 1.1: Sphere embedded in the 3-dimensional Euclidean space. The vector
fields $\hat{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\phi}}$ are tangent
to the sphere. On the right hand side it is illustrated that the covariant
derivative of a vector field tangent to the sphere is the projection of the
ordinary derivative onto the plane tangent to the spherical surface.

####  1.4 Symmetries and Conserved Quantities

Suppose that a space-time is symmetric on the direction of the coordinate
vector $\boldsymbol{K}=\partial_{1}$, i.e., it looks the same irrespective of
the value of the coordinate $x^{1}$. This implies that in this coordinate
system we have $\partial_{1}\,g_{\mu\nu}=0$. Then, using the fact that
$K^{\mu}=\delta^{\mu}_{1}$ and the expression for the Christoffel symbol in
terms of the derivatives of the metric, we easily find that:

$\nabla_{\mu}\,K_{\nu}\,=\,\frac{1}{2}\,\left(\partial_{\mu}\,g_{\nu
1}\,-\,\partial_{\nu}\,g_{\mu
1}\right)\;\Rightarrow\;\;\nabla_{\mu}\,K_{\nu}\,+\,\nabla_{\nu}\,K_{\mu}\,=\,0\,.$
(1.10)

Conversely, if a vector field $\boldsymbol{K}$ satisfies
$\nabla_{(\mu}K_{\nu)}=0$ then it is simple matter to prove that on a
coordinate system in which $\boldsymbol{K}$ is a coordinate vector the
relation $K^{\mu}\partial_{\mu}g_{\rho\sigma}=0$ holds. The equation
$\nabla_{(\mu}K_{\nu)}=0$ is the so-called Killing equation and the vector
field $\boldsymbol{K}$ is called a Killing vector field. In general the
symmetries of a space-time are not obvious from the expression of the metric.
For example, the Minkowski space-time has 10 independent Killing vector
fields, although only 4 symmetries are obvious from the usual expression of
this metric. That is the reason why the Killing vectors are so important, they
characterize the symmetries of a manifold without explicitly using
coordinates.

From the Noether theorem it is known that continuous symmetries are associated
to conserved charges. So the Killing vector fields must be related to
conserved quantities. Indeed, if $\boldsymbol{K}$ is a Killing vector and
$\boldsymbol{T}$ is the affinely parameterized vector field tangent to a
geodesic curve then the scalar $T^{\mu}K_{\mu}$ is constant along such
geodesic,
$T^{\nu}\nabla_{\nu}(T^{\mu}K_{\mu})=T^{\nu}T^{\mu}\nabla_{(\nu}K_{\mu)}=0$.
Physically, this means that along free-falling orbits the component of the
momentum along the direction of a Killing vector is conserved. The use of
these conserved quantities are generally quite helpful to find the solutions
of the geodesic equation. For instance, since the Schwarzschild space-time has
4 independent Killing vectors it follows that the geodesic trajectories can be
found without solving the geodesic equation. But, in addition to the Killing
vectors, there are other tensors associated with the symmetries of a manifold.
For example, let $K_{\nu_{1}\nu_{2}\ldots\nu_{p}}$ be a completely symmetric
tensor obeying to the equation

$\nabla_{(\mu}\,K_{\nu_{1}\nu_{2}\ldots\nu_{p})}\,=\,0\,,$

then the scalar $K_{\nu_{1}\ldots\nu_{p}}T^{\nu_{1}}\ldots T^{\nu_{q}}$ is
conserved along the geodesic generated by $\boldsymbol{T}$. The tensor
$K_{\nu_{1}\nu_{2}\ldots\nu_{p}}$ is called a Killing tensor of order $p$.

Another important class of tensors associated to symmetries is formed by the
Killing-Yano (KY) tensors. These are skew-symmetric tensors,
$Y_{\nu_{1}\nu_{2}\ldots\nu_{p}}=Y_{[\nu_{1}\nu_{2}\ldots\nu_{p}]}$, that obey
to the equation
$\nabla_{\mu}Y_{\nu_{1}\ldots\nu_{p}}+\nabla_{\nu_{1}}Y_{\mu\ldots\nu_{p}}=0$.
If $T^{\mu}$ generates an affinely parameterized geodesic then
$Y_{\nu_{1}\nu_{2}\ldots\nu_{p}}T^{\nu_{p}}$ is covariantly constant along the
geodesic. Note also that if $Y_{\mu\nu}$ is a Killing-Yano tensor then
$K_{\mu\nu}=Y_{\mu}^{\phantom{\mu}\rho}Y_{\rho\nu}$ is a Killing tensor of
order two. Although we can always construct Killing tensors out of KY tensors,
not all Killing tensors are made from KY tensors [8]. For more details about
KY tensors see [5].

There are also tensors associated to scalars conserved only along null
geodesics. A totally symmetric tensor $\boldsymbol{L}$ is said to be a
conformal Killing tensor (CKT) when the equation
$\nabla_{(\nu}L_{\mu_{1}\ldots\mu_{p})}=g_{(\nu\mu_{1}}A_{\mu_{2}\ldots\mu_{p})}$
holds for some tensor $\boldsymbol{A}$. If $\boldsymbol{L}$ is a CKT of order
$p$ and $\boldsymbol{l}$ is tangent to an affinely parameterized null geodesic
then the scalar $L_{\mu_{1}\ldots\mu_{p}}l^{\mu_{1}}\ldots l^{\mu_{p}}$ is
constant along such geodesic. It is not so hard to prove that if
$\boldsymbol{K}$ is a Killing tensor on the manifold $(M,\boldsymbol{g})$ then
$L_{\mu_{1}\ldots\mu_{p}}=\Omega^{2p}\,K_{\mu_{1}\ldots\mu_{p}}$ is a CKT of
the manifold $(M,\tilde{\boldsymbol{g}})$ with
$\tilde{g}_{\mu\nu}=\Omega^{2}\,g_{\mu\nu}$. In the same vein, we say that a
completely skew-symmetric tensor $\boldsymbol{Z}$ is a conformal Killing-Yano
(CKY) tensor if it satisfies the equation
$\nabla_{(\nu}Z_{\mu_{1})\mu_{2}\ldots\mu_{p}}=g_{\nu[\mu_{1}}H_{\mu_{2}\ldots\mu_{p}]}+g_{\mu_{1}[\nu}H_{\mu_{2}\ldots\mu_{p}]}$
for some tensor $\boldsymbol{H}$ [5].

Generally it is highly non-trivial to guess whether a manifold possess a
Killing tensor, a KY tensor as well as its conformal versions. Therefore, such
tensors are said to represent hidden symmetries. Since the Kerr metric has
just 2 independent Killing vectors it is not possible to find the geodesic
trajectories using only these symmetries. But, in 1968, B. Carter was able to
discover another conserved quantity that enabled him to solve the geodesic
equation [9]. Two years later Walker and Penrose demonstrated that this “new”
conserved scalar is associated to a Killing tensor of order two [10].
Thereafter it has been proved that this Killing tensor is the “square” of a KY
tensor [8].

####  1.5 Einstein’s Equation

Hopefully we already convinced ourselves that the gravitational field is
represented by the metric, $g_{\mu\nu}$, of a curved manifold
$(M,\boldsymbol{g})$. But we do not know yet how to find this metric given the
distribution of masses throughout the space-time. For example, in the
Newtonian theory the gravitational field is represented by a scalar, the
gravitational potential $\phi$, whose equation of motion is
$\nabla^{2}\phi=4\pi G\varrho$, where $G$ is the gravitational constant and
$\varrho$ is the mass density. Analogously, we need to find the equation of
motion for the metric $g_{\mu\nu}$. It can already be expected that,
differently from the Newtonian theory, the source of gravity is not just the
mass density, but the energy content as a whole, since in relativity mass and
energy are equivalent.

A wise path to find the correct field equation satisfied by $g_{\mu\nu}$ is to
guess a reasonable action representing the gravitational field and its
interaction with the other fields. Let us start analyzing how the metric
couples to the matter fields. Well, this is simple: given the action of a
field in special relativity we just need to replace the Minkowski metric by
$\boldsymbol{g}$ and substitute the partial derivatives by covariant
derivatives. There is, however, an important detail missing. In order for the
action to look the same in any coordinate system we must impose for it to be a
scalar. It is simple matter to prove that the volume element of space-time
$d^{4}x=dx^{0}dx^{1}dx^{2}dx^{3}$ is not invariant under coordinate
transformations. This can be fixed by taking $\sqrt{|g|}d^{4}x$ as the volume
element, with $g$ being the determinant of $g_{\mu\nu}$. Regarding the action
of the gravitational field, the simplest non-trivial scalar that can be
constructed out of the metric is the Ricci scalar $R$, defined in section 1.2.
Therefore we find that a reasonable action is:

$S\,=\,\frac{1}{16\pi G}\int
R\,\sqrt{|g|}d^{4}x\,+\,\int\mathcal{L}_{m}(\varphi_{i},\nabla_{\mu}\varphi_{i},g_{\mu\nu})\,\sqrt{|g|}d^{4}x\,.$
(1.11)

Where $\mathcal{L}_{m}$ is the Lagrangian density of the matter fields
$\varphi_{i}$. Then, using the least action principle, we can prove that the
equation of motion for the field $g_{\mu\nu}$ is given by the so-called
Einstein’s equation [5]:

$R_{\mu\nu}\,-\,\frac{1}{2}R\,g_{\mu\nu}\,=\,8\pi G\,T_{\mu\nu}\;;\quad
T^{\mu\nu}\,\equiv\,\frac{2}{\sqrt{|g|}}\,\frac{\delta S_{m}}{\delta
g_{\mu\nu}}\,.$ (1.12)

The symmetric tensor $T_{\mu\nu}$ is the energy-momentum tensor of the matter
fields. Particularly, in vacuum we have $T_{\mu\nu}=0$. Einstein’s equation
matches the geometry of the space-time, on the left hand side, to the energy
content, on the right hand side. Note that this equation is highly non-linear,
since the Ricci tensor and the Ricci scalar depends on the square of the
metric as well as on the inverse of the metric. This non-linearity can be
easily grasped using physical intuition. Since the graviton carries energy it
produces gravity, which then interact with this graviton and so on. In other
words, the graviton interacts with itself. This differs from classical
electrodynamics, where the photon has zero electric charge and, therefore,
generates no electromagnetic field.

As a simple and important example let us work out the case where just the
electromagnetic field is present. In relativistic theory this field is
represented by a co-vector $A_{\mu}$, the vector potential. From this field
one can construct the skew-symmetric tensor
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$. The action of the
electromagnetic field is given by:

$S_{em}\,=\,-\frac{1}{16\pi}\,\int
g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\,\sqrt{|g|}d^{4}x\,.$ (1.13)

Taking the functional derivative of this action with respect to the metric
yields the following energy-momentum tensor for the electromagnetic field:

$\mathcal{T}_{\mu\nu}\,=\,\frac{1}{4\pi}\,\left(F_{\mu\sigma}F_{\nu}^{\phantom{\nu}\sigma}-\frac{1}{4}\,g_{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\right)\,.$
(1.14)

Furthermore, computing the functional derivative of the action (1.13) with
respect to $A_{\mu}$ and equating to zero yields $\nabla^{\nu}F_{\mu\nu}=0$,
which is equivalent to Maxwell’s equations without sources. The set of
equations $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G\mathcal{T}_{\mu\nu}$,
$\nabla^{\nu}F_{\mu\nu}=0$ and $F_{\mu\nu}=2\nabla_{[\mu}A_{\nu]}$ is called
Einstein-Maxwell’s equations.

In this section we have considered that the gravitational Lagrangian is given
by the Ricci scalar $R$, which yields Einstein’s theory. Although general
relativity has had several experimental confirmations it is expected that for
really intense gravitational fields this Lagrangian shall be corrected by
higher order terms, such as $R^{2}$,
$R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$, $\partial_{\mu}R\,\partial^{\mu}R$
and so on. Indeed, string theory predicts that the gravitational action
contains terms of all orders on the curvature. In this picture the Einstein-
Hilbert action, $S=\frac{1}{16\pi G}\int R\sqrt{|g|}d^{n}x$, is just a weak
field approximation for the complete action.

####  1.6 Differential Forms

Just as in section 1.2 it was valuable to say that the tangent space is
spanned by the differential operators $\partial_{\mu}$, it is also fruitful to
assume that the dual of this space, the space of linear functionals on
$T_{p}M$, is generated by the differentials $dx^{\mu}$. Thus if $A_{\mu}$ are
the components of a co-vector field in the coordinates $\\{x^{\mu}\\}$, then
we shall represent the abstract tensor $\boldsymbol{A}$ as follows:

$\boldsymbol{A}\,=\,A_{\mu}\,dx^{\mu}\,.$

With such definition it follows that $A_{\mu}$ will properly transform under
coordinate changes, see eq. (1.1). Therefore, an arbitrary tensor
$\boldsymbol{T}$ has the following abstract representation:

$\boldsymbol{T}\,=\,T^{\mu_{1}\ldots\mu_{p}}_{\phantom{\mu_{1}\ldots\mu_{p}}\nu_{1}\ldots\nu_{q}}\,\partial_{\mu_{1}}\otimes\ldots\otimes\partial_{\mu_{p}}\otimes
dx^{\nu_{1}}\otimes\ldots\otimes dx^{\nu_{q}}\,.$

Since formally $dx^{\mu}$ is a linear functional on the space of vector
fields, its action on a vector field gives a scalar. Such action is defined by
$dx^{\mu}(\partial_{\nu})=\delta^{\mu}_{\,\nu}$, so that if $\boldsymbol{A}$
is co-vector and $\boldsymbol{V}$ is a vector then
$\boldsymbol{A}(\boldsymbol{V})=A_{\mu}V^{\mu}$.

A particularly relevant class of tensors are the so-called differential forms,
which are tensors with all indices down and totally skew-symmetric. For
instance, $F_{\mu_{1}\ldots\mu_{p}}=F_{[\mu_{1}\ldots\mu_{p}]}$ is called a
$p$-form and the vectorial space generated by all $p$-forms at some point
$x\in M$ is denoted by $\wedge^{p}M|_{x}$. A fundamental operation when
dealing with forms is the exterior product, whose definition is:

$\boldsymbol{F}\wedge\boldsymbol{H}\,=\,\frac{(p+q)!}{p!\,q!}\,F_{[\mu_{1}\ldots\mu_{p}}\,H_{\nu_{1}\ldots\nu_{q}]}\,dx^{\mu_{1}}\otimes\ldots\otimes
dx^{\mu_{p}}\otimes dx^{\nu_{1}}\otimes\ldots\otimes dx^{\nu_{q}}\,.$

Where $\boldsymbol{F}$ is a $p$-form and $\boldsymbol{H}$ is a $q$-form, so
that their exterior product yields a $(p+q)$-form. As an example note that the
following relation holds:

$\displaystyle dx^{1}\wedge dx^{2}\wedge dx^{3}\,=\,$
$\displaystyle(dx^{1}\otimes dx^{2}\otimes dx^{3}+dx^{2}\otimes dx^{3}\otimes
dx^{1}+dx^{3}\otimes dx^{1}\otimes dx^{2}+$ $\displaystyle\,-$
$\displaystyle\phantom{(}dx^{2}\otimes dx^{1}\otimes dx^{3}-dx^{3}\otimes
dx^{2}\otimes dx^{1}-dx^{1}\otimes dx^{3}\otimes dx^{2}\,)\,.$

In $n$ dimensions the set $\\{1,dx^{\mu_{1}},dx^{\mu_{1}}\wedge
dx^{\mu_{2}},\ldots,dx^{1}\wedge\ldots\wedge dx^{n}\\}$, which contains
$2^{n}$ elements, forms a basis for the space of differential forms, called
exterior bundle. In particular, a general $p$-form $\boldsymbol{F}$ can be
written as:

$\boldsymbol{F}\,=\,\frac{1}{p!}\,F_{\mu_{1}\ldots\mu_{p}}\,dx^{\mu_{1}}\wedge
dx^{\mu_{2}}\wedge\ldots\wedge dx^{\mu_{p}}\,.$

A $p$-form is called simple when it can be expressed as the exterior product
of $p$ 1-forms. For instance, every $n$-form is simple.

Another important operation involving differential forms is the interior
product, which essentially is the contraction of a differential form
$\boldsymbol{F}$ with a vector field $\boldsymbol{V}$ yielding another form
$\boldsymbol{H}\equiv\boldsymbol{V}\lrcorner\boldsymbol{F}$. If
$\boldsymbol{F}$ is a $p$-form then the interior product of $\boldsymbol{V}$
and $\boldsymbol{F}$ is the $(p-1)$-form defined by
$H_{\mu_{2}\ldots\mu_{p}}\equiv V^{\mu_{1}}F_{\mu_{1}\mu_{2}\ldots\mu_{p}}$.
When $\boldsymbol{V}\lrcorner\boldsymbol{F}=0$ we say that the differential
form $\boldsymbol{F}$ annihilates $\boldsymbol{V}$.

Suppose that $(M,\boldsymbol{g})$ is an $n$-dimensional manifold. Then we can
introduce the so-called Levi-Civita symbol
$\varepsilon_{\mu_{1}\ldots\mu_{n}}$, defined as the unique object, up to a
sign, that is totally skew-symmetric and normalized as $\varepsilon_{12\ldots
n}=\pm 1$. Although this symbol is not a tensor we can use it to define the
important tensor $\boldsymbol{\epsilon}$ called the volume-form and defined by
[11]:

$\epsilon_{\mu_{1}\ldots\mu_{n}}\,\equiv\,\sqrt{|g|}\,\varepsilon_{\mu_{1}\ldots\mu_{n}}\;\Rightarrow\quad\boldsymbol{\epsilon}\,=\,\sqrt{|g|}\,dx^{1}\wedge\ldots\wedge
dx^{n}\,,$

where $g$ denotes the determinant of the matrix $g_{\mu\nu}$. After some
algebra it can be proved that this tensor obeys to the following useful
identity [11]:

$\epsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{p+1}\ldots\nu_{n}}\,\epsilon_{\mu_{1}\ldots\mu_{p}\,\sigma_{p+1}\ldots\sigma_{n}}\;=\;p!(n-p)!\,(-1)^{\frac{n-s}{2}}\delta_{\sigma_{p+1}}^{\;[\nu_{p+1}}\ldots\delta_{\sigma_{n}}^{\;\nu_{n}]}\,.$
(1.15)

Where $s$ is the signature of the metric. Moreover, the volume-form can be
used to define an important operation called Hodge dual. The Hodge dual of a
$p$-form $\boldsymbol{F}$ is a $(n-p)$-form denoted by $\star\boldsymbol{F}$
and defined by:

$\left(\star
F\right)_{\mu_{1}\ldots\mu_{n-p}}\;=\;\frac{1}{p!}\,\epsilon^{\nu_{1}\ldots\nu_{p}}_{\phantom{\nu_{1}\ldots\nu_{p}}\mu_{1}\ldots\mu_{n-p}}\,F_{\nu_{1}\ldots\nu_{p}}\,.$
(1.16)

Finally, the last relevant operation on the space of forms is the exterior
differentiation, $d$. This differential operation maps $p$-forms into
$(p+1)$-forms as follows:

$d\boldsymbol{F}=\frac{1}{p!}\,\partial_{\nu}F_{\mu_{1}\ldots\mu_{p}}\,dx^{\nu}\wedge
dx^{\mu_{1}}\wedge\ldots\wedge dx^{\mu_{p}}\,.$

Although we have used the partial derivative, we could have used the covariant
derivative and the result would be the same, because of the symmetry
$\Gamma^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\nu\mu}$ of the Christoffel symbol.
Therefore, the term on the right hand side of the above equation is indeed a
tensor. A remarkable property of the exterior derivative is that its square is
zero, $d(d\boldsymbol{F})=0$, which stems from the commutativity of the
partial derivatives.

As an application of this formalism note that the source-free Maxwell’s
equations can be elegantly expressed in terms of differential forms. The
vector potential $A_{\mu}$ is a 1-form, $\boldsymbol{A}=A_{\mu}dx^{\mu}$. The
field strength, $F_{\mu\nu}\equiv\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$, is
nothing more than the exterior derivative of $\boldsymbol{A}$,
$\boldsymbol{F}=d\boldsymbol{A}$. In particular, this implies that
$d\boldsymbol{F}=0$. The missing equation is $\nabla^{\nu}F_{\mu\nu}=0$, which
can be proved to be equivalent to $d(\star\boldsymbol{F})=0$. Hence, in the
absence of sources, the electromagnetic field is represented by a 2-form,
$\boldsymbol{F}$, obeying the equations $d\boldsymbol{F}=0$ and
$d(\star\boldsymbol{F})=0$.

####  1.7 Cartan’s Structure Equations

Up to now we have adopted the coordinate frames $\\{\partial_{\mu}\\}$ and
$\\{dx^{\mu}\\}$ as bases for the tangent space and for its dual respectively.
Often it is convenient to use a non-coordinate frame
$\\{\boldsymbol{e}_{a}=e_{a}^{\phantom{a}\mu}\partial_{\mu}\\}$, where the
index $a$ is not a vectorial index, but rather a label for the $n$ vector
fields composing the frame. Associated to this non-coordinate vector frame is
the so-called dual frame
$\\{\boldsymbol{e}^{a}=e^{a}_{\phantom{a}\mu}dx^{\mu}\\}$, defined to be such
that $\boldsymbol{e}^{a}(\boldsymbol{e}_{b})=\delta^{a}_{\,b}$. Given a
tensor, say $T^{\mu}_{\phantom{\mu}\nu}$, its components in the frame
$\\{\boldsymbol{e}_{a}\\}$ are defined by $T^{a}_{\phantom{a}b}\equiv
T^{\mu}_{\phantom{\mu}\nu}e^{a}_{\phantom{a}\mu}e_{b}^{\phantom{b}\nu}$. In
particular, note that
$g_{ab}=\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})$. Once fixed the
frame $\\{\boldsymbol{e}_{a}\\}$, let us define the set of $n^{2}$ connection
1-forms $\boldsymbol{\omega}^{a}_{\phantom{a}b}$ by the following relation:

$V^{\mu}\nabla_{\mu}\boldsymbol{e}^{a}\,=\,-\,\boldsymbol{\omega}^{a}_{\phantom{a}b}(\boldsymbol{V})\,\boldsymbol{e}^{b}\,,\quad\forall\;\textrm{
vector field }\;\boldsymbol{V}\,.$ (1.17)

Then expanding $\boldsymbol{e}^{a}$ in a coordinate frame and using equation
(1.6) we can, after some algebra, prove the following identities [12]:

$d\boldsymbol{e}^{a}+\boldsymbol{\omega}^{a}_{\phantom{a}b}\wedge\boldsymbol{e}^{b}\,=\,0\quad;\quad\frac{1}{2}R^{a}_{\phantom{a}bcd}\,\boldsymbol{e}^{c}\wedge\boldsymbol{e}^{d}\,=\,d\boldsymbol{\omega}^{a}_{\phantom{a}b}+\boldsymbol{\omega}^{a}_{\phantom{a}c}\wedge\boldsymbol{\omega}^{c}_{\phantom{c}b}\,.$
(1.18)

Where $R^{a}_{\phantom{a}bcd}$ are the components of the Riemann tensor with
respect to the frame $\\{\boldsymbol{e}_{a}\\}$. These equations are known as
the Cartan structure equations. Moreover, defining the scalars
$\omega_{ab}^{\phantom{ab}c}\equiv\boldsymbol{\omega}^{c}_{\phantom{c}b}(\boldsymbol{e}_{a})$
we can easily prove that
$\nabla_{a}\boldsymbol{e}_{b}=\omega_{ab}^{\phantom{ab}c}\boldsymbol{e}_{c}$.

Sometimes it is of particular help to work with frames such that $g_{ab}$ is a
constant scalar. In this case the components of the connection 1-forms obey to
the constraint $\omega_{abc}=-\omega_{acb}$, where
$\omega_{abc}\equiv\omega_{ab}^{\phantom{ab}d}\,g_{dc}$. Indeed, using the
fact that the metric is covariantly constant along with the Leibniz rule
yield:

$0\,=\,\nabla_{c}\,\left[\,\boldsymbol{g}(\boldsymbol{e}_{a},\boldsymbol{e}_{b})\,\right]\,=\,\boldsymbol{g}(\nabla_{c}\boldsymbol{e}_{a},\boldsymbol{e}_{b})+\boldsymbol{g}(\boldsymbol{e}_{a},\nabla_{c}\boldsymbol{e}_{b})\,=\,\omega_{ca}^{\phantom{ca}d}\,g_{db}+\omega_{cb}^{\phantom{cb}d}\,g_{ad}\,.$

Just as the language of differential forms provides an elegant and fruitful
way to deal with Maxwell’s equations, Cartan’s structure equations do the same
in Riemannian geometry. Particularly, equation (1.18) gives, in general, the
quicker way to compute the Riemann tensor of a manifold. For applications and
geometrical insights on the meaning of these equations see [2].

From the physical point of view, the relevance of Cartan’s structure equations
stems from its relation with the formulation of general relativity as a gauge
theory. It is well-known that, except for gravity, the fundamental
interactions of nature are currently described by gauge theories, more
precisely Yang-Mills theories. Although not widely advertised, it turns out
that general relativity can also be cast in the language of gauge
theories333Actually, the most simple gauge formulation of gravity, called
Einstein-Cartan theory, is equivalent to general relativity just in the
absence of spin. In the presence of matter with spin the former theory allows
a non-zero torsion [13].. In this approach the gauge group of gravity is the
group of Lorentz transformations, $SO(3,1)$ [13]. Indeed, those acquainted
with the formalism of non-abelian gauge theory will recognize the second
identity of (1.18) as the equation defining the curvature associated to the
connection $\boldsymbol{\omega}^{a}_{\phantom{a}b}$.

####  1.8 Distributions and Integrability

Let $(M,\boldsymbol{g})$ be an $n$-dimensional manifold, then a
$q$-dimensional distribution in $M$ is a smooth map that associates to every
point $p\in M$ a vector subspace of dimension $q$, $\Delta_{p}\subset T_{p}M$.
We say that the set of vector fields $\\{\boldsymbol{V}_{i}\\}$ generates this
distribution when they span the vector subspace $\Delta_{p}$ for every point
$p\in M$. For instance, a non-vanishing vector field generates a 1-dimensional
distribution. We say that a distribution of dimension $q$ is integrable when
there exists a smooth family of submanifolds of $M$ such that the tangent
spaces of these submanifolds are $\Delta_{p}$. This means that locally $M$
admits coordinates $\\{x^{1},\ldots,x^{q},y^{1},\ldots,y^{n-q}\\}$ such that
the vector fields $\\{\partial_{x^{i}}\\}$ generate $\Delta_{p}$. In this case
the family of submanifolds is given by the hyper-surfaces of constant
$y^{\alpha}$.

Given a set of $q$ vector fields $\\{\boldsymbol{V}_{i}\\}$ that are linearly
independent at every point then it generates a $q$-dimensional distribution
denoted by $Span\\{\boldsymbol{V}_{i}\\}$. One might then wonder, how can we
know if such distribution is integrable? Before answering this question it is
important to introduce the Lie bracket. If $\boldsymbol{V}$ and
$\boldsymbol{Z}$ are vector fields then their Lie bracket is another vector
field defined by:

$[\boldsymbol{V},\boldsymbol{Z}]\,\equiv\,V^{\mu}\nabla_{\mu}\boldsymbol{Z}-Z^{\mu}\nabla_{\mu}\boldsymbol{V}\,=\,\left(V^{\mu}\partial_{\mu}\,Z^{\nu}-Z^{\mu}\partial_{\mu}\,V^{\nu}\right)\partial_{\nu}\,.$

As a warming exercise let us work out an example on the $n$-dimensional
Euclidian space, $(\mathbb{R}^{n},\delta_{\mu\nu})$. Let $f(\boldsymbol{r})$
be some function on this manifold, then generally the surfaces of constant $f$
foliate the space, with the leafs being orthogonal to $\boldsymbol{\nabla}f$
as depicted in figure 1.2. Therefore, if $\boldsymbol{V}$ is some vector field
tangent to the foliating surfaces then
$\boldsymbol{V}\cdot\boldsymbol{\nabla}f=0$. Differentiating this last
equation we get

$\partial_{\mu}(\boldsymbol{V}\cdot\boldsymbol{\nabla}f)\,=\,0\;\;\Rightarrow\;\;(\partial_{\mu}V^{\nu})\,\partial_{\nu}f\,+\,V^{\nu}\,\partial_{\mu}\partial_{\nu}f\,=\,0\,.$

Therefore, if $\boldsymbol{Z}$ is another vector field tangent to the leafs of
constant $f$ then

$[\boldsymbol{V},\boldsymbol{Z}]\cdot\boldsymbol{\nabla}f=\left(V^{\mu}\partial_{\mu}Z^{\nu}-Z^{\mu}\partial_{\mu}V^{\nu}\right)\,\partial_{\nu}f=-V^{\mu}Z^{\nu}\partial_{\mu}\partial_{\nu}f+Z^{\mu}V^{\nu}\partial_{\mu}\partial_{\nu}f=0\,.$

This means that the Lie bracket of two vector fields tangent to the foliating
surfaces yield another vector field tangent to these surfaces. Now let
$\boldsymbol{\theta}\neq 0$ be a 1-form proportional to $df$,
$\boldsymbol{\theta}=h\,df$. Then note that a vector field $\boldsymbol{V}$ is
tangent to the leafs of constant $f$ if, and only if,
$\boldsymbol{\theta}(\boldsymbol{V})=0$. In addition, note that
$d\boldsymbol{\theta}\wedge\boldsymbol{\theta}=0$ and that
$d(\frac{1}{h}\boldsymbol{\theta})=0$.

Figure 1.2: The space is foliated by the surfaces of constant $f$. The vector
field $\boldsymbol{\nabla}f$ is orthogonal to the leafs of the foliation,
while $\boldsymbol{V}$ and $\boldsymbol{Z}$ are tangent.

The results obtained in the preceding paragraph are just a special case of a
well-known theorem called the Frobenius theorem, which states that the
distribution generated by the vector fields $\\{\boldsymbol{V}_{i}\\}$ is
integrable if, and only if, there exists a set of functions $C_{ij}^{k}$ such
that $[\boldsymbol{V}_{i},\boldsymbol{V}_{j}]=C_{ij}^{k}\,\boldsymbol{V}_{k}$.
In other words, this distribution is integrable if, and only if, the vector
fields $\boldsymbol{V}_{i}$ form a closed algebra under the Lie brackets [14].

The Frobenius theorem can be presented in a “dual” version, in terms of
differential forms. Let $\\{\boldsymbol{V}_{i}\\}$ be a set of $q$ vector
fields generating a $q$-dimensional distribution. Then we can complete this
set with more $(n-q)$ vector fields, $\\{\boldsymbol{U}_{\alpha}\\}$, so that
$\\{\boldsymbol{V}_{i},\boldsymbol{U}_{\alpha}\\}$ spans the tangent space at
every point. Associated to this frame is a dual frame of 1-forms
$\\{\boldsymbol{\omega}^{i},\boldsymbol{\theta}^{\alpha}\\}$ such that
$\boldsymbol{\omega}^{i}(\boldsymbol{V}_{j})=\delta^{i}_{\,j}$,
$\boldsymbol{\omega}^{i}(\boldsymbol{U}_{\alpha})=0$,
$\boldsymbol{\theta}^{\alpha}(\boldsymbol{V}_{i})=0$ and
$\boldsymbol{\theta}^{\alpha}(\boldsymbol{U}_{\beta})=\delta^{\alpha}_{\,\beta}$.
Note that a vector field is tangent to the distribution if, and only if, it is
annihilated by all the $(n-q)$ 1-forms $\boldsymbol{\theta}^{\alpha}$. The
dual version of the Frobenius theorem then states that the distribution
generated by $\\{\boldsymbol{V}_{i}\\}$ is integrable if, and only if,

$d\boldsymbol{\theta}^{\alpha}\wedge\boldsymbol{\theta}^{1}\wedge\boldsymbol{\theta}^{2}\wedge\ldots\wedge\boldsymbol{\theta}^{(n-q)}\,=\,0\quad\;\forall\;\;\alpha\in\\{1,\ldots,(n-q)\\}\,.$
(1.19)

Defining
$\boldsymbol{\Theta}\equiv\boldsymbol{\theta}^{1}\wedge\ldots\wedge\boldsymbol{\theta}^{(n-q)}$,
then note that a vector field $\boldsymbol{V}$ is tangent to the distribution
generated by $\\{\boldsymbol{V}_{i}\\}$ if, and only if,
$\boldsymbol{V}\lrcorner\boldsymbol{\Theta}=0$. Now suppose that there exists
a non-zero function $h$ such that $d(h\boldsymbol{\Theta})=0$, then expanding
this equation and taking the wedge product with $\boldsymbol{\theta}^{\alpha}$
we arrive at the equation (1.19). Conversely, if the distribution generated by
$\\{\boldsymbol{V}_{i}\\}$ is integrable then, by definition, one can
introduce coordinates $\\{x^{1},\ldots,x^{q},y^{1},\ldots,y^{n-q}\\}$ such
that the vector fields $\\{\partial_{x^{i}}\\}$ generate this distribution.
Since $dy^{\alpha}(\partial_{x^{i}})=0$, it follows that
$\boldsymbol{\Theta}=\frac{1}{h}(dy^{1}\wedge\ldots\wedge dy^{n-q})$ for some
non-vanishing function $h$, which implies that $d(h\boldsymbol{\Theta})=0$. We
proved, therefore, that the distribution annihilated by $\boldsymbol{\Theta}$
is integrable if, and only if, there exists some non-zero function $h$ such
that $d(h\boldsymbol{\Theta})=0$. Equivalently, it can be stated that the
distribution annihilated by a simple form $\boldsymbol{\Theta}$ is integrable
if, and only if, there exists a 1-form $\boldsymbol{\varphi}$ such that
$d\boldsymbol{\Theta}=\boldsymbol{\varphi}\wedge\boldsymbol{\Theta}$.

The integrability of distributions plays an important role in Caratheodory’s
formulation of thermodynamics. In his formalism, the equilibrium states of a
thermodynamical system form a differentiable manifold $\mathcal{M}$. In such a
manifold it is defined a global function $U$, the internal energy, and two
$1$-forms, $\boldsymbol{W}$ and $\boldsymbol{Q}$, representing the work done
and the received heat, respectively. The first law of thermodynamics is then
written as $dU=\boldsymbol{Q}-\boldsymbol{W}$. A curve in this manifold is
called adiabatic if its tangent vector field is annihilated by
$\boldsymbol{Q}$. According to Caratheodory, the second law of thermodynamics
says that in the neighborhood of every point $x\in\mathcal{M}$ there are
points $y$ such that there is no adiabatic curve joining $x$ to $y$. He was
able to prove that this formulation of the second law guarantees that the
distribution annihilated by $\boldsymbol{Q}$ is integrable. Particularly, this
implies that there exist functions $T$ and $S$ such that $\boldsymbol{Q}=TdS$.
Physically, these functions are the temperature, $T$, and the entropy, $S$.
For more details see [14] and references therein.

####  1.9 Higher-Dimensional Spaces

Einstein’s general relativity postulates that we live in a 4-dimensional
Lorentzian manifold, which means that the space-time has 3 spatial dimensions
and one time dimension. There are, however, some theories claiming that our
space-time can have more spatial dimensions. Particularly, in order to provide
a consistent quantum theory, superstring theory requires the space-time
dimension to be 10 or 11 [15]. Which justifies the study of higher-dimensional
general relativity.

One might wonder: If these extra dimensions exist then why they have not been
perceived yet? A reasonable reason is that these dimensions can be highly
wrapped. For example, if we look at a long pipe that is far from us it will
appear that it is just a one-dimensional line. But as we get closer and closer
to the pipe we will note that it is actually a cylinder, which has two
dimensions. An instructive example for understanding the role played by a
curled dimension is to solve Schrödinger equation for a particle of mass $m$
inside an infinite well. Let the space be 2-dimensional with one of the
dimensions being a circle of radius R while the other dimension is open and
has an infinite well of size L, then the energy spectrum of this system is
easily proved to be [16]:

$E_{p,q}\,=\,\frac{\hbar^{2}\pi^{2}}{2m}\,\left(\frac{p}{\textsf{L}}\right)^{2}\,+\,\frac{\hbar^{2}}{2m}\,\left(\frac{q-1}{\textsf{R}}\right)^{2}\,,\quad
p,q\in\\{1,2,3,\ldots\\}\,.$

The first term on the right hand side of this equation is just the regular
spectrum of a 1-dimensional infinite well of size L, while the second term is
the contribution from the extra dimension. Then note that if R is very small,
$\textsf{R}\ll\textsf{L}$, then it will be necessary a lot of energy to excite
the modes with quantum number $q$. Thus in the limit $\textsf{R}\rightarrow 0$
the system will remain in a state with $q=1$, which implies that we retrieve
the spectrum of a 1-dimensional well. Thus if the extra dimensions are very
tiny the only hope to detect them is through very energetic experiments444In
closed string theory a new phenomenon emerges. Since strings can wrap around a
curled dimension there exist winding modes that need little energy to be
excited when R is much smaller than the Planck length. Furthermore, due to a
symmetry called $T$-duality, in closed string theory very small radius turns
out to be equivalent to very large radius.. Indeed, currently the LHC555LHC is
the abbreviation for Large Hadron Collider, the most energetic particle
accelerator in the world. is probing the existence of extra dimensions.

In addition to the possibility of our universe having extra dimensions and to
the obvious mathematical relevance, the study of higher-dimensional curved
spaces has other applications. For example, in classical mechanics the phase
space of a system is a $2p$-dimensional manifold endowed with a symplectic
structure, where $p$ is the number of degrees of freedom [17]. As a
consequence, higher-dimensional spaces are also of interest to thermodynamics
and statistical mechanics.

It is needless to explain the physical relevance of the Lorentzian signature.
But it is worth highlighting that other signatures are also important in
physics, let alone in mathematics. Spaces with split signature are of
relevance for the theory of integrable systems, Yang-Mills fields and for
twistor theory [19]. Moreover, the Euclidean signature emerges when we make a
Wick rotation on the time coordinate in order to make path integrals
convergent. The Euclidean curved spaces are sometimes called gravitational
instantons, although it is more appropriate to define a gravitational
instanton as a complete 4-dimensional Ricci-flat Euclidean manifold that is
asymptotically-flat and whose Weyl tensor is self-dual [18]. Analogously to
the instantons solutions of Yang-Mills theory, gravitational instantons
provide a dominant contribution to Feynman path integral, justifying its
physical interest [18]. Non-Lorentzian signatures are also of relevance for
string theory.

Given the importance of these topics, the present thesis will investigate some
properties of higher-dimensional curved spaces of arbitrary signature. The
path adopted here is to work with complexified manifolds so that the results
can be carried to any signature by judiciously choosing a reality condition
[20]. The technique of using complexified geometry with the aim of extracting
results for real spaces can be fruitful and enlightening, an approach that was
advocated by McIntosh and Hickman in a series of papers [21], where
4-dimensional general relativity was explored using complexified manifolds.

###  Chapter 2 Petrov Classification, Six Different Approaches

The Petrov classification is an algebraic classification for the curvature,
more precisely for the Weyl tensor, valid in 4-dimensional Lorentzian
manifolds. It has been of invaluable relevance for the development of general
relativity, in particular it played a prominent role on the discovery of Kerr
metric [22], which is probably the most important solution of general
relativity. Furthermore, guided by such classification and a theorem due to
Goldberg and Sachs [23], Kinnersley was able to find all type $D$ vacuum
solutions [24], a really impressive accomplishment since Einstein’s equation
is non-linear. Moreover, this classification contributed for the study of
gravitational radiation [25, 26], the peeling theorem being one remarkable
example [27].

Such classification was created by the Russian mathematician Alexei
Zinovievich Petrov in 1954111Petrov obtained this classification in a previous
article published in 1951 but, as himself acknowledges in [28], the proofs in
this first work were not precise. [28] with the intent of classifying Einstein
space-times. A. Z. Petrov has worked on differential geometry and general
relativity, and he has been one of the most important scientists responsible
for the spread of Einstein’s gravitational theory inside the Soviet Union222A
short biography of A. Z. Petrov can be found in Kazan University’s website
[29].. In particular, around 1960 he has written a really remarkable book on
general relativity that certainly has been of great relevance for the
dissemination of this theory on such an isolated nation [30].

In its original form, this classification consisted only of three types, $I$,
$II$ and $III$. Few years later, in 1960, Roger Penrose developed spinorial
techniques to general relativity and, as a consequence, has found out that
these types could be further refined, adding the types $D$ and $N$ to the
classification [31]. It is worth mentioning that by the same time Robert
Debever and Louis Bel arrived at such refinement by a different path [25, 32],
in particular they have developed an alternative approach to define the Petrov
types, the so-called Bel-Debever criteria.

The route adopted by A. Z. Petrov to arrive at his classification amounts to
reinterpreting the Weyl tensor as an operator acting on the space of
bivectors. As time passed by, several other methods to attack such
classification were developed. Since these approaches look very different from
each other, it comes as a surprise that all of them are equivalent. The intent
of the present chapter is to describe six different ways to attain this
classification. As one of the goals of this thesis is to describe an
appropriate generalization for the Petrov classification valid in dimensions
greater than four, the analysis of these different approaches proves to be
important because in higher dimensions many of these methods are not
equivalent anymore. Therefore, in order to find a suitable higher-dimensional
generalization for the Petrov classification it is helpful to investigate the
benefits and flaws of each method in 4 dimensions.

Throughout this chapter it will be assumed that the space-time is a
4-dimensional manifold endowed with a metric of Lorentzian signature,
$(M,\boldsymbol{g})$. Furthermore, the tangent bundle is assumed to be endowed
with the Levi-Civita connection, hence the curvature referred here is with
respect to this connection. All calculations are assumed to be local, in a
neighborhood of an arbitrary point $p\in M$.

####  2.1 Weyl Tensor as an Operator on the Bivector Space

In this section the so-called bivector approach will be used to define the
Petrov classification. To this end the results of appendices A and B will be
necessary, so that the reader is advised to take a look at these appendices
before proceeding.

The Weyl tensor is the trace-less part of the Riemann tensor, it has the
following symmetries (see section 1.2):

$C_{\mu\nu\rho\sigma}=C_{[\mu\nu][\rho\sigma]}=C_{\rho\sigma\mu\nu}\;;\;\;C^{\mu}_{\phantom{\mu}\nu\mu\sigma}=0\;;\;\;C_{\mu[\nu\rho\sigma]}=0\,.$
(2.1)

Skew-symmetric tensors of rank 2 are called a bivectors,
$B_{\mu\nu}=B_{[\mu\nu]}$. Since the Weyl tensor is anti-symmetric in the
first and second pairs of indices, it follows that this tensor can be
interpreted as a linear operator that maps bivectors into bivectors in the
following way:

$B_{\mu\nu}\mapsto
T_{\mu\nu}=C_{\mu\nu\rho\sigma}B^{\rho\sigma}\;,\;\textrm{where}\;\;B_{\mu\nu}=B_{[\mu\nu]}\;,T_{\mu\nu}=T_{[\mu\nu]}\,.$
(2.2)

Studying the possible eigenbivectors of this operator we arrive at the Petrov
classification, actually this was the original path taken by A. Z. Petrov
[28]. In order to enlighten the analysis it is important to review some
properties of bivectors in four dimensions. Let us denote the volume-form of
the 4-dimensional Lorentzian manifold $(M,g_{\mu\nu})$ by
$\epsilon_{\mu\nu\rho\sigma}$. This is a totally anti-symmetric tensor,
$\epsilon_{\mu\nu\rho\sigma}=\epsilon_{[\mu\nu\rho\sigma]}$, whose non-zero
components in an orthonormal frame are $\pm 1$. It is well-known that it
satisfies the following identity [11]:

$\epsilon^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\,\epsilon_{\mu_{1}\mu_{2}\sigma_{1}\sigma_{2}}=-2\,\left(\,\delta_{\sigma_{1}}^{\nu_{1}}\,\delta_{\sigma_{2}}^{\nu_{2}}\,-\,\delta_{\sigma_{2}}^{\nu_{1}}\,\delta_{\sigma_{1}}^{\nu_{2}}\,\right)\,.$
(2.3)

By means of the volume-form we can define the Hodge dual operation that maps
bivectors into bivectors. The dual of the bivector $\boldsymbol{B}$ is defined
by

$\left(\star
B\right)_{\mu\nu}\,\equiv\,\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}B^{\rho\sigma}\,.$
(2.4)

Let us denote by $\mathfrak{B}_{\mathbb{C}}$ the complexification of the
bivector bundle. Using equation (2.3) it is easy matter to see that the double
dual of a bivector is it negative, $[\star(\star B)]_{\mu\nu}=-B_{\mu\nu}$.
This implies that the 6-dimensional space $\mathfrak{B}_{\mathbb{C}}$ can be
split into the direct sum of the two 3-dimensional eigenspaces of the dual
operation.

$\mathfrak{B}_{\mathbb{C}}=\mathfrak{D}\oplus\mathfrak{\overline{D}}$ (2.5)
$\mathfrak{D}=\\{Z_{\mu\nu}\in\mathfrak{B}_{\mathbb{C}}\,|\,\left(\star
Z\right)_{\mu\nu}=iZ_{\mu\nu}\\}\;;\;\mathfrak{\overline{D}}=\\{Y_{\mu\nu}\in\mathfrak{B}_{\mathbb{C}}\,|\,\left(\star
Y\right)_{\mu\nu}=-iY_{\mu\nu}\\}$

The elements of $\mathfrak{D}$ are called self-dual bivectors, whereas a
bivector belonging to $\mathfrak{\overline{D}}$ is dubbed anti-self-dual. By
means of the volume-form it is also possible to split the Weyl tensor into a
sum of the dual part, $C^{+}$, and the anti-dual part, $C^{-}$:

$C_{\mu\nu\rho\sigma}=C^{+}_{\mu\nu\rho\sigma}+C^{-}_{\mu\nu\rho\sigma}\;\;;\;\;C^{\pm}_{\mu\nu\rho\sigma}\equiv\frac{1}{2}\left(C_{\mu\nu\rho\sigma}\mp\frac{i}{2}\,C_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\epsilon_{\alpha\beta\rho\sigma}\right)\,.$
(2.6)

It is then immediate to verify the following relations:

$C^{+}_{\mu\nu\rho\sigma}\,Y^{\rho\sigma}\,=\,0\quad\forall\;\boldsymbol{Y}\in\overline{\mathfrak{D}}\quad;\quad
C^{-}_{\mu\nu\rho\sigma}\,Z^{\rho\sigma}\,=\,0\quad\forall\;\boldsymbol{Z}\in\mathfrak{D}\,.$

This means that in order to analyse the action of Weyl tensor on
$\mathfrak{B}_{\mathbb{C}}$ it is sufficient to study the action of $C^{+}$ in
$\mathfrak{D}$ and the action of $C^{-}$ in $\overline{\mathfrak{D}}$.
However, by the definition on eq. (2.6), $C^{-}$ is the complex conjugate of
$C^{+}$, so that it is enough to study just the operator
$C^{+}:\,\mathfrak{D}\rightarrow\mathfrak{D}$. Since this operator is trace-
less and act on a 3-dimensional space it follows that it can have the
following algebraic types according to the refined Segre classification (see
appendix A):

$\left\\{\begin{array}[]{ll}\textbf{Type O}&\rightarrow\;C^{+}\,=\,0\\\
\textbf{Type I}&\rightarrow\;C^{+}\,\textrm{ is type }\;[1,1,1|\,]\,\textrm{
or }[1,1|1]\\\ \textbf{Type D}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[(1,1),1|\,]\\\ \textbf{Type II}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[2,1|\,]\\\ \textbf{Type N}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[\,|2,1]\\\ \textbf{Type III}&\rightarrow\;C^{+}\,\textrm{ is type
}\;[\,|\,3]\,.\end{array}\right.$ (2.7)

These are the so-called Petrov types. Therefore, in order to determine the
Petrov classification of the Weyl tensor using this approach we must follow
four steps: 1) Choose a basis for the space of self-dual bivectors
$\mathfrak{D}$; 2) Calculate the action of the operator defined by (2.2) in
this basis in order to find a $3\times 3$ matrix representation for $C^{+}$;
3) Find the eigenvalues and eigenvectors of this matrix; 4) Use this
eigenvalue structure to determine the algebraic type of such matrix according
to the refined Segre classification (appendix A) and after this use equation
(2.7).

With the aim of making connection with the forthcoming sections, let us follow
some of these steps explicitly. Once introduced a null tetrad frame
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$
(see appendix B), the ten independent components of the Weyl tensor can be
written in terms of five complex scalars:

$\Psi_{0}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}l^{\rho}m^{\sigma}\;;\;\Psi_{1}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}l^{\rho}m^{\sigma}\;;\;\Psi_{2}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}\overline{m}^{\rho}n^{\sigma}$
$\Psi_{3}\equiv
C_{\mu\nu\rho\sigma}l^{\mu}n^{\nu}\overline{m}^{\rho}n^{\sigma}\;;\;\Psi_{4}\equiv
C_{\mu\nu\rho\sigma}n^{\mu}\overline{m}^{\nu}n^{\rho}\overline{m}^{\sigma}\,.$
(2.8)

These are the so-called Weyl scalars. A basis to the space of self-dual
bivectors, $\mathfrak{D}$, is given by:

$Z^{1}_{\mu\nu}=2\,l_{[\mu}m_{\nu]}\;;\;Z^{2}_{\mu\nu}=2\,\overline{m}_{[\mu}n_{\nu]}\;;\;Z^{3}_{\mu\nu}=2\,n_{[\mu}l_{\nu]}+2\,m_{[\mu}\overline{m}_{\nu]}$
(2.9)

In this basis the representation of operator
$C^{+}:\,\mathfrak{D}\rightarrow\mathfrak{D}$ is

$[C^{+}]=2\left[\begin{array}[]{ccc}\Psi_{2}&\Psi_{4}&-2\Psi_{3}\\\
\Psi_{0}&\Psi_{2}&-2\Psi_{1}\\\ \Psi_{1}&\Psi_{3}&-2\Psi_{2}\\\
\end{array}\right]$ (2.10)

Note that this matrix has vanishing trace, as claimed above equation (2.7).
Thus, in order to get the Petrov type of the Weyl tensor we just have to
calculate the Weyl scalars, using eq. (2.1), plug them on the above matrix and
investigate the algebraic type of such matrix.

When the Weyl tensor is type $I$ it is said to be algebraically general,
otherwise it is called algebraically special. If the Weyl tensor is type O in
all points we say that the space-time is conformally flat, which means there
exists a coordinate system such that $g_{\mu\nu}=\Omega^{2}\eta_{\mu\nu}$.
Note that the Petrov classification is local, so that the type of the Weyl
tensor can vary from point to point on space-time. In spite of this it is
interesting that the majority of the exact solutions has the same Petrov type
in all points of the manifold. For instance, all known black holes are type
$D$ and the plane gravitational waves are type $N$.

As pointed out at the beginning of this chapter, when Petrov classification
first emerged only three types were defined, known as types I, II and III [26,
28]. With the contributions of Penrose, Debever and Bel these types were
refined as depicted below.

$\textrm{I}-\textrm{Refinement}-^{\nearrow}_{\searrow}\begin{array}[]{l}I\\\
D\end{array}\quad;\quad\textrm{II}-\textrm{Refinement}-^{\nearrow}_{\searrow}\begin{array}[]{c}II\\\
N\end{array}\quad;\quad\textrm{III}\longrightarrow III$

Indeed, from the definition of Petrov types presented on equation (2.7) it is
already clear that the type $D$ can be seen as special case of the type $I$,
while type $N$ is a specialization of type $II$333 It is worth mentioning that
in ref. [25] L. Bel has used a different convention, denoting the types $I$,
$D$, $II$, $III$ and $N$ by $I$, $II_{a}$, $II_{b}$, $III_{a}$ and $III_{b}$
respectively..

More details about the bivector method will be given in chapter 4, where this
approach will be used to classify the Weyl tensor in any signature, see also
[33]. In particular, chapter 4 advocates that the bivector approach is endowed
with an enlightening geometrical significance. A careful investigation of the
bivector method in higher dimensions was performed in [34].

####  2.2 Annihilating Weyl Scalars

In this section a different characterization of the Petrov types will be
presented. In this approach the different types are featured by the
possibility of annihilating some Weyl tensor components using a suitable
choice of basis. As a warming up example let us investigate the type $D$.
According to eq. (2.7), in this case the algebraic type of $C^{+}$ is
$[(1,1),1|\,]$, which means that such operator can be put on the diagonal form
$\operatorname{diag}(\lambda,\lambda,\lambda^{\prime})$. But since
$\operatorname{tr}(C^{+})=0$, we must have $\lambda^{\prime}=-2\lambda$, hence
$C^{+}=\operatorname{diag}(\lambda,\lambda,-2\lambda)$. Now, looking at eq.
(2.10) we see that this is compatible with the Weyl scalars
$\Psi_{0},\Psi_{1},\Psi_{3}$ and $\Psi_{4}$ being all zero. In general, each
Petrov type enables one to find a suitable basis where some Weyl scalars can
be made to vanish.

The Lorentz transformations at point $p\in M$ is the set of linear
transformations on tangent space, $T_{p}M$, which preserves the inner
products. These transformations can be obtained by a composition of the
following three simple operations in a null tetrad frame
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$:

(i) Lorentz Boost

$\boldsymbol{l}\rightarrow\lambda\boldsymbol{l}\;;\;\;\boldsymbol{n}\rightarrow\lambda^{-1}\boldsymbol{n}\;;\;\;\boldsymbol{m}\rightarrow
e^{i\theta}\boldsymbol{m}\;;\;\;\overline{\boldsymbol{m}}\rightarrow
e^{-i\theta}\overline{\boldsymbol{m}}$ (2.11)

(ii) Null Rotation Around $\boldsymbol{l}$

$\boldsymbol{l}\rightarrow\boldsymbol{l}\,;\;\,\boldsymbol{n}\rightarrow\boldsymbol{n}+w\boldsymbol{m}+\overline{w}\,\overline{\boldsymbol{m}}+|w|^{2}\boldsymbol{l}\,;\;\,\boldsymbol{m}\rightarrow\boldsymbol{m}+\overline{w}\boldsymbol{l}\,;\;\,\overline{\boldsymbol{m}}\rightarrow\overline{\boldsymbol{m}}+w\boldsymbol{l}$
(2.12)

(iii) Null Rotation Around $\boldsymbol{n}$

$\boldsymbol{l}\rightarrow\boldsymbol{l}+\overline{z}\boldsymbol{m}+z\overline{\boldsymbol{m}}+|z|^{2}\boldsymbol{n}\,;\;\,\boldsymbol{n}\rightarrow\boldsymbol{n}\,;\;\,\boldsymbol{m}\rightarrow\boldsymbol{m}+z\boldsymbol{n}\,;\;\,\overline{\boldsymbol{m}}\rightarrow\overline{\boldsymbol{m}}+\overline{z}\boldsymbol{n}.$
(2.13)

Where $\lambda$ and $\theta$ are real numbers while $z$ and $w$ are complex,
composing a total of six real parameters. This should be expected from the
fact that the Lorentz group, in a 4-dimensional space-time, has 6 dimensions.
In order to verify that these transformations do indeed preserve the inner
products, note that the metric
$g_{\mu\nu}=2l_{(\mu}n_{\nu)}-2m_{(\mu}\overline{m}_{\nu)}$ remains invariant
by them.

Now let us try to annihilate the maximum number of Weyl scalars by
transforming the null tetrad under the Lorentz group. After performing a null
rotation around $\boldsymbol{n}$ the Weyl scalars change as follows:

$\displaystyle\Psi_{0}\rightarrow\Psi^{\prime}_{0}(z)\,=\,\Psi_{0}\,+\,4\,z\,\Psi_{1}\,$
$\displaystyle+\,6\,z^{2}\,\Psi_{2}\,+\,4\,z^{3}\,\Psi_{3}\,+\,z^{4}\,\Psi_{4}\;;$
$\displaystyle\Psi_{1}\rightarrow\Psi^{\prime}_{1}(z)\,=\,\frac{1}{4}\frac{d}{dz}\Psi^{\prime}_{0}(z)\;\;\;;$
$\displaystyle\;\;\;\Psi_{2}\rightarrow\Psi^{\prime}_{2}(z)=\frac{1}{3}\frac{d}{dz}\Psi^{\prime}_{1}(z)\;;$
(2.14)
$\displaystyle\Psi_{3}\rightarrow\Psi^{\prime}_{3}(z)\,=\,\frac{1}{2}\frac{d}{dz}\Psi^{\prime}_{2}(z)\;\;;$
$\displaystyle\;\;\Psi_{4}\rightarrow\Psi^{\prime}_{4}(z)=\frac{d}{dz}\Psi^{\prime}_{3}(z)=\Psi_{4}\,,$

which can be proved using equations (2.1) and (2.13). Now if we set
$\Psi^{\prime}_{0}=0$ we will have a fourth order polynomial in $z$ equal to
zero444Here it is being assumed that $\Psi_{4}\neq 0$, which is always allowed
if the Weyl tensor does not vanish identically. Indeed, if the Weyl tensor is
non-zero and $\Psi_{4}=0$ then by means of a null rotation around
$\boldsymbol{l}$ we can easily make $\Psi_{4}\neq 0$.. Thus, in general we
have four distinct values of the parameter $z$ which accomplish this, call
these values $\\{z_{1},z_{2},z_{3},z_{4}\\}$. Then the Petrov types can be
defined as follows:

$\left\\{\begin{array}[]{ll}\textbf{Type O}&\rightarrow\;\textrm{Weyl tensor
is zero}\\\ \textbf{Type I}&\rightarrow\;\textrm{All roots are different}\\\
\textbf{Type D}&\rightarrow\;\textrm{Two pairs of roots
coincide},\,z_{1}=z_{2}\neq z_{3}=z_{4}\\\ \textbf{Type
II}&\rightarrow\;\textrm{Two roots coincide},\,z_{1}=z_{2}\neq z_{3}\neq
z_{4}\neq z_{1}\\\ \textbf{Type III}&\rightarrow\;\textrm{Three roots
coincide},\,z_{1}=z_{2}=z_{3}\neq z_{4}\\\ \textbf{Type
N}&\rightarrow\;\textrm{All roots
coincide},\,z_{1}=z_{2}=z_{3}=z_{4}\,.\end{array}\right.$ (2.15)

These four roots define four Lorentz transformations. By means of eq. (2.13)
such transformations lead to four privileged null vector fields
$\boldsymbol{l}^{\prime}_{i}$, which are the ones obtained by performing these
transformations on the vector field $\boldsymbol{l}$ of the original null
tetrad:

$\boldsymbol{l}\,\rightarrow\,\,\boldsymbol{l}^{\prime}_{i}\,=\,\boldsymbol{l}+\overline{z_{i}}\,\boldsymbol{m}+z_{i}\,\overline{\boldsymbol{m}}+|z_{i}|^{2}\,\boldsymbol{n}\;,\quad
i\in\\{1,2,3,4\\}\,.$ (2.16)

These real null directions are called the principal null directions (PNDs) of
the Weyl tensor. Moreover, when $z_{i}$ is a degenerated root the PND
$\boldsymbol{l}^{\prime}_{i}$ is said to be a repeated PND555The concept of
repeated PND can also be extracted from the bivector formalism of section 2.1,
as proved on reference [35].. When $z_{i}$ is a root of order $q$, we say that
the associated PND has degeneracy $q$. By the above definition of Petrov
classification it then follows that the Petrov type $I$ admits four distinct
PNDs; in type $D$ there are two pairs of repeated PNDs; in type $II$ there
exists three distinct PNDs, one being repeated; in type $III$ we have two
PNDs, one of which is repeated with triple degeneracy; in type $N$ there is
only one PND, this PND in repeated and has degree of degeneracy four.

In type $I$ once we set $\Psi^{\prime}_{0}=0$, by making $z=z_{i}$, the other
Weyl scalars are all different from zero, as can be seen from equations (2.14)
and (2.15). Then performing a null rotation around $\boldsymbol{l}$, which
makes $\Psi^{\prime}_{\alpha}\rightarrow\Psi^{\prime\prime}_{\alpha}$, it is
possible to make $\Psi^{\prime\prime}_{4}$ vanish while keeping
$\Psi^{\prime\prime}_{0}=0$, no other scalars can be made to vanish. Thus in
type $I$ the Weyl scalars $\Psi_{0}$ and $\Psi_{4}$ can always be made to
vanish by a judicious choice of null tetrad. As a further example let us treat
the type $D$. In the type $D$ setting $z=z_{1}$ it follows from equations
(2.14) and (2.15) that $\Psi^{\prime}_{0}=\Psi^{\prime}_{1}=0$. After this we
can perform a null rotation around $\boldsymbol{l}$ in order to set
$\Psi^{\prime\prime}_{3}=\Psi^{\prime\prime}_{4}=0$ while keeping
$\Psi^{\prime\prime}_{0}=\Psi^{\prime\prime}_{1}=0$. The table below sums up
what can be accomplished using this kind of procedure.

Type $O$ $-$ All | Type $II$ $-$ $\Psi_{0},\Psi_{1},\Psi_{4}$ | Type $D$ $-$ $\Psi_{0},\Psi_{1},\Psi_{3},\Psi_{4}$
---|---|---
Type $I$ $-$ $\Psi_{0},\Psi_{4}$ | Type $III$ $-$ $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{4}$ | Type $N$ $-$ $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{3}$
Table 2.1: Weyl scalars that can be made to vanish, by a suitable choice of
basis, on each Petrov type.

Although the definition of the Petrov types given in the present section looks
completely different from the one given in section (2.1) it is not hard to
prove that they are actually equivalent. As an example let us work out the
type $N$ case. According to the table 2.1, if the Weyl tensor is type $N$ it
follows that it is possible to find a null tetrad on which the only non-
vanishing Weyl scalar is $\Psi_{4}$. In this basis eq. (2.10) yield that
$C^{+}$ has the following matrix representation:

$C_{N}\,=\,2\left[\begin{array}[]{ccc}0&\Psi_{4}&0\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right]\,.$

Along with appendix A this means that the algebraic type of the operator
$C_{N}$ is $[\,|2,1]$, which perfectly matches the definition of eq. (2.7).
More details about the approach adopted in this section can be found in [12].

####  2.3 Boost Weight

In this section the boost transformations, eq. (2.11), will be used to provide
another form of expressing the Petrov types. In order to accomplish this we
first need to see how the Weyl scalars behave under Lorentz boosts. Inserting
eq. (2.11) into the definition of the Weyl scalars, eq. (2.1), we easily find
the following transformation:

$\Psi_{\alpha}\,\longrightarrow\,\gamma^{(2-\alpha)}\,\Psi_{\alpha}\
\;\;,\quad\,\gamma\equiv
e^{i\theta}\,\lambda\;,\,\;\alpha\in\\{0,1,2,3,4\\}\,.$ (2.17)

In jargon we say that the Weyl scalar $\Psi_{\alpha}$ has boost weight
$\mathfrak{b}=(2-\alpha)$. Note, particularly, that the maximum boost weight
(b.w.) that a component of the Weyl tensor can have is $\mathfrak{b}=2$, while
the minimum is $\mathfrak{b}=-2$.

Given the components of the Weyl tensor on a particular basis, we shall denote
by $\mathfrak{b}_{+}$ the b.w. of the non-vanishing Weyl scalar with maximum
boost weight. Analogously, $\mathfrak{b}_{-}$ denotes the b.w. of the non-
vanishing Weyl tensor component with minimum boost weight. For instance, using
eq. (2.17) and table (2.1) we see that if the Weyl tensor is type $III$ then
it is possible to find a null frame in which $\mathfrak{b}_{+}=-1$. In general
we can define the Petrov types using this kind of reasoning, the bottom line
is summarized below:

$\left\\{\begin{array}[]{ll}\textbf{Type I}&\rightarrow\;\textrm{There is a
frame in which}\;\mathfrak{b}_{+}=+1\\\ \textbf{Type
II}&\rightarrow\;\textrm{There is a frame in which}\;\mathfrak{b}_{+}=\,0\\\
\textbf{Type III}&\rightarrow\;\textrm{There is a frame in
which}\;\mathfrak{b}_{+}=-1\\\ \textbf{Type N}&\rightarrow\;\textrm{There is a
frame in which}\;\mathfrak{b}_{+}=-2\\\ \textbf{Type
D}&\rightarrow\;\textrm{There is a frame in
which}\;\mathfrak{b}_{+}=\;\mathfrak{b}_{-}=0\\\ \textbf{Type
O}&\rightarrow\;\textrm{Weyl tensor vanishes
identically}\,.\end{array}\right.$ (2.18)

On the boost weight approach the different Petrov types have a hierarchy: The
type $I$ is the most general, type $II$ is a special case of the type $I$,
type $III$ is a special case of type $II$ and the type $N$ is a special case
of type $III$. The type $D$ is also a special case of type $II$, in this type
all non-vanishing components of the Weyl tensor have zero boost weight.

A classification for the Weyl tensor using the boost weight method can be
naturally generalized to higher dimensions, which yields the so-called CMPP
classification [36]. The CMPP classification has been intensively investigated
in the last ten years, see, for example, [37, 38] and references therein.

####  2.4 Bel-Debever and Principal Null Directions

Few years after the release of Petrov’s original article defining his
classification, Bel and Debever have, independently, found an equivalent, but
quite different, way to define the Petrov types [25, 32]. On such approach the
Petrov types are defined in terms of algebraic conditions involving the Weyl
tensor and the principal null directions defined in section 2.2.

Since the null tetrad frame at a point $p\in M$ forms a local basis for the
tangent space $T_{p}M$, it follows that the Weyl tensor can be expanded in
terms of the tensorial product of this basis. Because of the symmetries of
this tensor, eq. (2.1), it follows that the expansion shall be expressed in
terms of the following kind of combination:

$\langle
e,v,u,t\rangle_{\mu\nu\rho\sigma}\,\equiv\,4\,e_{[\mu}v_{\nu]}\,u_{[\rho}t_{\sigma]}\,+\,4\,u_{[\mu}t_{\nu]}\,e_{[\rho}v_{\sigma]}\,.$

Once introduced a null tetrad
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\overline{\boldsymbol{m}}\\}$,
the Weyl tensor can be written as the following expansion:

$\displaystyle
C_{\mu\nu\rho\sigma}=\Big{\\{}\,\frac{1}{2}(\Psi_{2}+\overline{\Psi}_{2})\big{[}\langle
l,n,l,n\rangle+\langle
m,\overline{m},m,\overline{m}\rangle\big{]}+\Psi_{0}\langle
n,\overline{m},n,\overline{m}\rangle+$ $\displaystyle+\Psi_{4}\langle
l,m,l,m\rangle-\Psi_{2}\langle
l,m,n,\overline{m}\rangle-\frac{1}{2}(\Psi_{2}-\overline{\Psi}_{2})\langle
l,n,m,\overline{m}\rangle+$ (2.19) $\displaystyle+\Psi_{1}\big{[}\langle
l,n,n,\overline{m}\rangle+\langle
n,\overline{m},\overline{m},m\rangle\big{]}+\Psi_{3}\big{[}\langle
l,m,m,\overline{m}\rangle-\langle
l,n,l,m\rangle\big{]}+\;c.c.\,\Big{\\}}_{\mu\nu\rho\sigma}\,.$

Where $c.c.$ denotes the complex conjugate of all previous terms inside the
curly bracket. In particular, note that the right hand side of the above
equation is real and has the symmetries of the Weyl tensor. We can verify that
such expansion is indeed correct by contracting equation (2.19) with the null
frame and checking that equation (2.1) is satisfied. Now, contracting equation
(2.19) with $l^{\nu}l^{\rho}$ yield:

$C_{\mu\nu\rho\sigma}l^{\nu}l^{\rho}=\left[\Psi_{1}(l_{\mu}\overline{m}_{\sigma}+\overline{m}_{\mu}l_{\sigma})+c.c.\right]-2\left(\Psi_{0}\overline{m}_{\mu}\overline{m}_{\sigma}+c.c.\right)-2\left(\Psi_{2}+\overline{\Psi}_{2}\right)l_{\mu}l_{\sigma}\,.$

The above expression, in turn, immediately implies the following identities:

$l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}\,=\,\left(\Psi_{1}\,l_{[\alpha}\overline{m}_{\mu]}l_{\sigma}+c.c.\right)\,-\,2\,\left(\Psi_{0}\,l_{[\alpha}\overline{m}_{\mu]}\overline{m}_{\sigma}+c.c.\right)\,,$
(2.20)
$l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}\,=\,-2\,\left(\Psi_{0}\,l_{[\alpha}\overline{m}_{\mu]}\overline{m}_{[\sigma}l_{\beta]}+c.c.\right)\,.$
(2.21)

From which we conclude that the combination on the left hand side of eq.
(2.21) vanishes if, and only if, $\Psi_{0}=0$. Hence, by the definition given
in section 2.2, it follows that $\boldsymbol{l}$ is a principal null direction
if, and only if,
$l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}=0$. Analogously,
eq. (2.20) and the definition below eq. (2.16) imply that $\boldsymbol{l}$ is
a repeated PND if, and only if,
$l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0$. In the same vein, the
following relations can be proved:

$\displaystyle\Psi_{0}=\Psi_{1}=\Psi_{2}=0\;\;\Leftrightarrow\;\;$
$\displaystyle C_{\mu\nu\rho[\sigma}l_{\alpha]}l^{\rho}\,=\,0$
$\displaystyle\Psi_{0}=\Psi_{1}=\Psi_{2}=\Psi_{3}=0\;\;\Leftrightarrow\;\;$
$\displaystyle C_{\mu\nu\rho\sigma}l^{\rho}\,=\,0$

Using these results and table 2.1 it is then simple matter to arrive at the
following alternative definition for the Petrov types:

$\left\\{\begin{array}[]{ll}\textbf{Type I}&\rightarrow\;\textrm{ exists
$\boldsymbol{l}$ such that
}l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}=0\\\ \textbf{Type
II}&\rightarrow\;\textrm{ exists $\boldsymbol{l}$ such that
}l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0\\\ \textbf{Type
III}&\rightarrow\;\textrm{ exists $\boldsymbol{l}$ such that
}C_{\mu\nu\rho[\sigma}l_{\alpha]}l^{\rho}=0\\\ \textbf{Type
N}&\rightarrow\;\textrm{ exists $\boldsymbol{l}$ such that
}C_{\mu\nu\rho\sigma}l^{\rho}=0\\\ \textbf{Type D}&\rightarrow\;\textrm{ exist
$\boldsymbol{l},\boldsymbol{n}$ such that
}l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0=n_{[\alpha}C_{\mu]\nu\rho\sigma}n^{\nu}n^{\rho}\\\
\textbf{Type O}&\rightarrow\;\textrm{ exist $\boldsymbol{l},\boldsymbol{n}$
such that
}C_{\mu\nu\rho\sigma}l^{\rho}=0=C_{\mu\nu\rho\sigma}n^{\rho}\,.\end{array}\right.$

Where it was assumed that $\boldsymbol{l}$ and $\boldsymbol{n}$ are real null
vectors such that $l^{\mu}\,n_{\mu}=1$. On such definition it is assumed that
the Petrov types obey the same hierarchy of the preceding section:

$O\,\subset\,N\,\subset\,III\,\subset\,II\,\subset\,I\;\textrm{ and
}\;O\,\subset\,D\,\subset\,II\,.$

These algebraic constraints involving the Weyl tensor and null directions are
called Bel-Debever conditions. In reference [39] these conditions were
investigated in higher-dimensional space-times and connections with the CMPP
classification were made.

####  2.5 Spinors, Penrose’s Method

In this section we will take advantage of the spinorial formalism in order to
describe the Petrov classification, an approach introduced by R. Penrose [31].
Here it will be assumed that the reader is already familiar with the spinor
calculus in 4-dimensional general relativity. For those not acquainted with
this language, a short course is available in [40]. For a more thorough
treatment with diverse applications [41] is recommended. Appendix C of the
present thesis provides the general formalism of spinors in arbitrary
dimensions.

On the spinorial formalism of 4-dimensional Lorentzian manifolds we have two
types of indices, the ones associated with Weyl spinors of positive chirality,
$A,B,C,...\in\\{1,2\\}$, and the ones related to semi-spinors of negative
chirality, $\dot{A},\dot{B},\dot{C},...\in\\{1,2\\}$. It is also worth
mentioning that the complex conjugation changes the chirality of the spinorial
indices. In this language a vectorial index is equivalent to the “product” of
two spinorial indices, one of positive chirality and one of negative
chirality:

$V_{\mu}\,\sim\,V_{A\dot{A}}\,.$

The spaces of semi-spinors are endowed with skew-symmetric metrics
$\varepsilon_{AB}=\varepsilon_{[AB]}$ and
$\overline{\varepsilon}_{\dot{A}\dot{B}}=\overline{\varepsilon}_{[\dot{A}\dot{B}]}$.
This anti-symmetry implies, for instance, that
$\zeta^{A}\zeta_{A}=\zeta^{A}\varepsilon_{AB}\zeta^{B}=0$ for every spinor
$\boldsymbol{\zeta}$. These spinorial metrics are related to the space-time
metric by the relation
$g_{\mu\nu}\sim\varepsilon_{AB}\overline{\varepsilon}_{\dot{A}\dot{B}}$. In
this formalism the Weyl tensor is represented by

$C_{\mu\nu\rho\sigma}\,\sim\,\left(\,\Psi_{ABCD}\,\overline{\varepsilon}_{\dot{A}\dot{B}}\overline{\varepsilon}_{\dot{C}\dot{D}}\,+\,c.c.\,\right)\,.$
(2.22)

Where $\Psi$ is a completely symmetric object, $\Psi_{ABCD}=\Psi_{(ABCD)}$,
and $c.c.$ denotes the complex conjugate of the previous terms inside the
bracket. Since $\boldsymbol{\varepsilon}$ carry the degrees of freedom of the
space-time metric, it follows that the degrees of freedom of the Weyl tensor
are entirely contained on $\boldsymbol{\Psi}$. Therefore, classify the Weyl
tensor is then equivalent to classify $\boldsymbol{\Psi}$.

It is a well-known result in this formalism that every object with completely
symmetric chiral indices, $S_{A_{1}A_{2}\ldots A_{p}}=S_{(A_{1}A_{2}\ldots
A_{p})}$, can be decomposed as a symmetrized direct product of spinors,
$S_{A_{1}A_{2}\ldots
A_{p}}=\zeta^{1}_{(A_{1}}\zeta^{2}_{A_{2}}\ldots\zeta^{p}_{A_{p})}$ [40].
Particularly, we can always find spinors
$\boldsymbol{\zeta},\boldsymbol{\theta},\boldsymbol{\xi}$ and
$\boldsymbol{\chi}$ such that

$\Psi_{ABCD}\,=\,\zeta_{(A}\,\theta_{B}\,\xi_{C}\,\chi_{D)}\,.$ (2.23)

We can then easily classify the Weyl tensor according to the possibility of
the spinors $\boldsymbol{\zeta},\boldsymbol{\theta},\boldsymbol{\xi}$ and
$\boldsymbol{\chi}$ being proportional to each other. Denoting de
proportionality of the spinors by “$\leftrightarrow$” and the non-
proportionality by “$\nleftrightarrow$”, we shall define:

$\left\\{\begin{array}[]{ll}\textbf{Type
I}&\rightarrow\;\boldsymbol{\zeta},\boldsymbol{\theta},\boldsymbol{\xi}\textrm{
and }\boldsymbol{\chi}\textrm{ are non-propotional to each other}\\\
\textbf{Type II}&\rightarrow\;\textrm{One pair coincide,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\nleftrightarrow\xi\nleftrightarrow\boldsymbol{\chi}\nleftrightarrow\boldsymbol{\zeta}\\\
\textbf{Type III}&\rightarrow\;\textrm{Three spinors coincidence,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\leftrightarrow\boldsymbol{\xi}\nleftrightarrow\boldsymbol{\chi}\\\
\textbf{Type D}&\rightarrow\;\textrm{Two pairs coincide,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\nleftrightarrow\boldsymbol{\xi}\leftrightarrow\boldsymbol{\chi}\\\
\textbf{Type N}&\rightarrow\;\textrm{All spinors coincide,
}\boldsymbol{\zeta}\leftrightarrow\boldsymbol{\theta}\leftrightarrow\boldsymbol{\xi}\leftrightarrow\boldsymbol{\chi}\\\
\textbf{Type
O}&\rightarrow\;\boldsymbol{\zeta}\,=\,\boldsymbol{\theta}\,=\,\boldsymbol{\xi}\,=\,\boldsymbol{\chi}\,=\,0\,.\end{array}\right.$
(2.24)

The spinors that appear on the decomposition of $\boldsymbol{\Psi}$ are called
the principal spinors of the Weyl tensor, since they are intimately related to
the principal null directions. Indeed, the real null vectors generated by
these spinors,

$l_{1}^{\,\mu}\,\sim\,\zeta^{A}\overline{\zeta}^{\dot{A}}\;\;;\;\;l_{2}^{\,\mu}\,\sim\,\theta^{A}\overline{\theta}^{\dot{A}}\;\;;\;\;l_{3}^{\,\mu}\,\sim\,\xi^{A}\overline{\xi}^{\dot{A}}\;\;;\;\;l_{4}^{\,\mu}\,\sim\,\chi^{A}\overline{\chi}^{\dot{A}}\,,$

point in the principal null directions of the Weyl tensor. Hence, the
coincidence of the principal spinors is equivalent to coincidence of PNDs,
which makes a bridge between the spinorial approach to the Petrov
classification and the approach adopted in section 2.2.

The spinorial formalism allows us to see quite neatly which Weyl scalars can
be made to vanish by a suitable choice of null tetrad frame on each Petrov
type. If $\\{o_{{}_{A}},\iota_{{}_{A}}\\}$ forms a spin frame,
$o_{{}_{A}}\iota^{A}=1$, then we can use them to build a null tetrad frame, as
shown in appendix B. So using equations (2.1) and (B.1) we can prove that the
Weyl scalars are given by:

$\displaystyle\Psi_{0}=\Psi_{ABCD}o^{A}o^{B}o^{C}o^{D}\;;\;\Psi_{1}=\Psi_{ABCD}o^{A}o^{B}o^{C}\iota^{D}\;;\;\Psi_{2}=\Psi_{ABCD}o^{A}o^{B}\iota^{C}\iota^{D}$
$\displaystyle\Psi_{3}=\Psi_{ABCD}o^{A}\iota^{B}\iota^{C}\iota^{D}\;;\;\Psi_{4}=\Psi_{ABCD}\iota^{A}\iota^{B}\iota^{C}\iota^{D}\,.$
(2.25)

Thus, for example, if the Weyl tensor is type $D$ according to eq. (2.24) then
there exists non-zero spinors $\boldsymbol{\zeta}$ and $\boldsymbol{\xi}$ such
that $\Psi_{ABCD}=\zeta_{(A}\zeta_{B}\xi_{C}\xi_{D)}$. Since
$\boldsymbol{\zeta}\nleftrightarrow\boldsymbol{\xi}$ it follows that
$\zeta_{A}\xi^{A}=w\neq 0$. Therefore, setting $o_{{}_{A}}=\zeta_{{}_{A}}$ and
$\iota_{{}_{A}}=w^{-1}\xi_{{}_{A}}$ it follows that
$\\{\boldsymbol{o},\boldsymbol{\iota}\\}$ forms a spin frame. Then using
equation (2.25) we easily find that in this frame
$\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0$, which agrees with table 2.1. By means
of the same reasoning it is straightforward to work out the other types and
verify that the definitions of the Petrov types presented on (2.24) perfectly
matches the table 2.1.

In the same vein, the bivector method of section 2.1 can be easily understood
on the spinorial formalism. In the spinorial language a self-dual bivector is
represented by a symmetric spinor $\phi^{AB}=\phi^{(AB)}$, so that the map
$C^{+}$ is represented by
$\phi_{AB}\mapsto\phi^{\prime}_{AB}=\Psi_{ABCD}\phi^{CD}$. Thus, for example,
if the Weyl tensor is type $N$ then we can find a spin frame
$\\{\boldsymbol{o},\boldsymbol{\iota}\\}$ such that
$\Psi_{ABCD}=o_{{}_{A}}o_{{}_{B}}o_{{}_{C}}o_{{}_{D}}$. Then defining
$\phi_{1}^{AB}=o^{A}o^{B}$, $\phi_{2}^{AB}=o^{(A}\iota^{B)}$ and
$\phi_{3}^{AB}=\iota^{A}\iota^{B}$, it follows that the action of $C^{+}$ in
this basis of self-dual bivectors yields $\phi^{\prime}_{1}=0$,
$\phi^{\prime}_{2}=0$ and $\phi^{\prime}_{3}=\phi_{1}$, which agrees with
equation (2.7).

####  2.6 Clifford Algebra

In this section the formalism of Clifford algebra will be used to describe
another form to arrive at the Petrov classification. For those not acquainted
with the tools of geometric algebra, appendix C introduces the necessary
background. Let
$\\{\hat{\boldsymbol{e}}_{0},\hat{\boldsymbol{e}}_{1},\hat{\boldsymbol{e}}_{2},\hat{\boldsymbol{e}}_{3},\\}$
be a local orthonormal frame on a 4-dimensional Lorentzian manifold
$(M,\boldsymbol{g})$,

$\frac{1}{2}\left(\hat{\boldsymbol{e}}_{a}\hat{\boldsymbol{e}}_{b}\,+\,\hat{\boldsymbol{e}}_{b}\hat{\boldsymbol{e}}_{a}\right)\,=\,\boldsymbol{g}(\hat{\boldsymbol{e}}_{a},\hat{\boldsymbol{e}}_{b})\,=\,\eta_{ab}\,=\,\operatorname{diag}(1,-1,-1,-1)\,.$

Denoting by $\eta^{ab}$ the inverse matrix of $\eta_{ab}$, we shall define
$\hat{\boldsymbol{e}}^{a}=\eta^{ab}\hat{\boldsymbol{e}}_{b}$. Let us denote
the space spanned by the bivector fields by $\Gamma(\wedge^{2}M)$. Then, in
the formalism of geometric calculus [42, 43] the Weyl tensor is a linear
operator on the space of bivectors,
$\mathcal{C}:\Gamma(\wedge^{2}M)\rightarrow\Gamma(\wedge^{2}M)$, whose action
is666All results in this thesis are local, so that it is always being assumed
that we are in the neighborhood of some point. Thus, formally, instead of
$\Gamma(\wedge^{2}M)$ we should have written $\Gamma(\wedge^{2}M)|_{N_{x}}$,
which is the restriction of the space of sections of the bivector bundle to
some neighborhood $N_{x}$ of a point $x\in M$. So we are choosing a particular
local trivialization of the bivector bundle.

$\mathcal{C}(\boldsymbol{V}\wedge\boldsymbol{U})\,=\,V^{a}\,U^{b}\,C_{abcd}\,\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d}\,,$
(2.26)

where $C_{abcd}$ are the components of the Weyl tensor on the frame
$\\{\hat{\boldsymbol{e}}_{a}\\}$. In the above equation
$\boldsymbol{V}\wedge\boldsymbol{U}$ means the anti-symmetrized part of the
Clifford product of $\boldsymbol{V}$ and $\boldsymbol{U}$,
$\boldsymbol{V}\wedge\boldsymbol{U}=\frac{1}{2}(\boldsymbol{V}\boldsymbol{U}-\boldsymbol{U}\boldsymbol{V})$.
Then using (2.26) and equation (C.4) of appendix C we find:

$\displaystyle\hat{\boldsymbol{e}}^{a}\,\mathcal{C}(\hat{\boldsymbol{e}}_{a}\wedge\hat{\boldsymbol{e}}_{b})\,$
$\displaystyle=\,C_{abcd}\,\hat{\boldsymbol{e}}^{a}\,(\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d})=C_{abcd}\,\hat{\boldsymbol{e}}^{a}\,\frac{1}{2}(\hat{\boldsymbol{e}}^{c}\hat{\boldsymbol{e}}^{d}-\hat{\boldsymbol{e}}^{d}\hat{\boldsymbol{e}}^{c})$
$\displaystyle=\,C_{abcd}\,\frac{1}{2}(2\,\eta^{ac}\,\hat{\boldsymbol{e}}^{d}-2\,\eta^{ad}\,\hat{\boldsymbol{e}}^{c}+2\,\hat{\boldsymbol{e}}^{a}\wedge\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d})$
$\displaystyle=\,2\,C_{\phantom{c}bcd}^{c}\,\hat{\boldsymbol{e}}^{d}\,-\,C_{b[acd]}\,\hat{\boldsymbol{e}}^{a}\wedge\hat{\boldsymbol{e}}^{c}\wedge\hat{\boldsymbol{e}}^{d}$
(2.27)

Equation (2.27) makes clear that on the Clifford algebra formalism the single
equation
$\hat{\boldsymbol{e}}^{a}\mathcal{C}(\hat{\boldsymbol{e}}_{a}\wedge\hat{\boldsymbol{e}}_{b})=0$
is equivalent to the trace-less property and the Bianchi identity satisfied by
the Weyl tensor. There are two other symmetries satisfied by this tensor, see
(2.1), which are the anti-symmetry on the first and second pairs of indices,
$C_{abcd}=C_{[ab][cd]}$ and the symmetry by the exchange of these pairs,
$C_{abcd}=C_{cdab}$. But the latter symmetry can be derived from the Bianchi
identity, while the former is encapsulated in the present formalism by the
fact that the operator $\mathcal{C}$ maps bivectors into bivectors. Thus we
conclude that on the Clifford algebra approach all the symmetries of the Weyl
tensor are encoded in the following relations:

$\mathcal{C}:\,\Gamma(\wedge^{2}M)\rightarrow\Gamma(\wedge^{2}M)\quad;\quad\hat{\boldsymbol{e}}^{a}\,\mathcal{C}(\hat{\boldsymbol{e}}_{a}\wedge\hat{\boldsymbol{e}}_{b})\,=\,0\,.$
(2.28)

Before proceeding let us define the following bivectors:

$\boldsymbol{\sigma}_{i}\,=\,\hat{\boldsymbol{e}}_{0}\wedge\hat{\boldsymbol{e}}_{i}\;\,;\quad\boldsymbol{I}\boldsymbol{\sigma}_{i}\,=\,\frac{1}{2}\epsilon^{ijk}\,\hat{\boldsymbol{e}}_{j}\wedge\hat{\boldsymbol{e}}_{k}$

Where $i,j,k$ are indices that run from 1 to 3, $\epsilon^{ijk}$ is a totally
anti-symmetric object with $\epsilon^{123}=1$ and
$\boldsymbol{I}=\hat{\boldsymbol{e}}_{0}\hat{\boldsymbol{e}}_{1}\hat{\boldsymbol{e}}_{2}\hat{\boldsymbol{e}}_{3}$
is the pseudo-scalar defined on appendix C. In particular, using these
definitions and the Bianchi identity it is not difficult to prove that the
following equation holds:

$\mathcal{C}(\boldsymbol{\sigma}_{i})\,=\,-2\left[\,C_{0i0j}+\boldsymbol{I}\,C_{0kli}\epsilon^{klj}\,\right]\,\boldsymbol{\sigma}_{j}$
(2.29)

Also, expanding equation (2.28) we find the following explicit relations:

$\displaystyle\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{\sigma}_{1})\,+\,\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{\sigma}_{2})\,+\,\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{\sigma}_{3})\,=\,0$
$\displaystyle\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{\sigma}_{1})\,=\,\boldsymbol{I}\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{2})\,+\,\boldsymbol{I}\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{3})$
$\displaystyle\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{\sigma}_{2})\,=\,\boldsymbol{I}\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{1})\,+\,\boldsymbol{I}\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{3})$
(2.30)
$\displaystyle\boldsymbol{\sigma}_{3}\,\mathcal{C}(\boldsymbol{\sigma}_{3})\,=\,\boldsymbol{I}\boldsymbol{\sigma}_{1}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{1})\,+\,\boldsymbol{I}\boldsymbol{\sigma}_{2}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{2})$

Summing the last three relations above and then using the first one, we find
$\sum_{i}\boldsymbol{I}\boldsymbol{\sigma}_{i}\,\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{i})=0$.
Then using this identity on the last three relations of (2.30) we conclude
that
$\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{i})=\boldsymbol{I}\mathcal{C}(\boldsymbol{\sigma}_{i})$.
By means of this and the identity $\boldsymbol{I}^{2}=-1$ we also find that
$\mathcal{C}(\boldsymbol{I}\,\boldsymbol{I}\boldsymbol{\sigma}_{i})=\boldsymbol{I}\mathcal{C}(\boldsymbol{I}\boldsymbol{\sigma}_{i})$.
Since $\\{\boldsymbol{\sigma}_{i},\boldsymbol{I}\boldsymbol{\sigma}_{i}\\}$ is
a basis for the bivector space it follows that in general

$\mathcal{C}(\boldsymbol{I}\boldsymbol{B})\,=\,\boldsymbol{I}\,\mathcal{C}(\boldsymbol{B})\quad\;\forall\;\boldsymbol{B}\in\Gamma(\wedge^{2}M)\,.$
(2.31)

Now recall from appendix C that the pseudo-scalar $\boldsymbol{I}$ commutes
with the elements of even order, in particular it commutes with all bivectors.
Moreover, equation (2.31) guarantees that $\boldsymbol{I}$ commutes with the
Weyl operator. Therefore, when dealing with the Weyl operator acting on the
bivector space we can treat the $\boldsymbol{I}$ as if it were a scalar.
Furthermore, since $\boldsymbol{I}^{2}=-1$ we can pretend that
$\boldsymbol{I}$ is the imaginary unit, $\boldsymbol{I}\sim i=\sqrt{-1}$, so
that we can reinterpret the operator $\mathcal{C}$ as an operator on the
complexification of the real space generated by
$\\{\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2},\boldsymbol{\sigma}_{3}\\}$.
With these conventions the equation (2.29) can be written as777A similar
phenomenon happens on the Clifford algebra of the space $\mathbb{R}^{3}$. In
this case the pseudo-scalar commutes with all elements of the algebra and
obeys to the relation $\boldsymbol{I}^{2}=-1$, so that it can actually be
interpreted as the imaginary unit, $\boldsymbol{I}\sim i=\sqrt{-1}$. This is
the geometric explanation of why the complex numbers are so useful when
dealing with rotations in 3 dimensions.:

$\mathcal{C}(\boldsymbol{\sigma}_{i})\,=\,\mathcal{C}_{ij}\,\boldsymbol{\sigma}_{j}\quad;\;\mathcal{C}_{ij}\sim-2\left(C_{0i0j}+i\,C_{0kli}\epsilon^{klj}\right)$
(2.32)

Now we can easily define a classification for the Weyl tensor. Using equation
(2.32) and the symmetries of the Weyl tensor it is trivial to prove that this
matrix is trace-less, $\mathcal{C}_{ii}=0$. Therefore, the possible algebraic
types for the operator $\mathcal{C}$ are the same as the ones listed on eq.
(2.7).

Note that this classification is, in principle, different from the one shown
on subsection 2.1. While the latter uses the space of self-dual bivectors to
define a 3-dimensional operator, the operator introduced in the present
subsection acts on the space generated by
$\\{\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2},\boldsymbol{\sigma}_{3}\\}$,
which is not the space of self-dual bivectors. The remarkable thing is that
these two classifications turns out to be equivalent. This can be seen by
noting that to every eigen-bivector of $\mathcal{C}$ we can associate a self-
dual bivector that is eigen-bivector of $C^{+}$ with the “same” eigenvalue.
Indeed, if $\boldsymbol{B}$ is an eigen-bivector of the operator $\mathcal{C}$
on the Clifford algebra approach then
$\mathcal{C}(\boldsymbol{B})=(\lambda_{1}+\boldsymbol{I}\lambda_{2})\boldsymbol{B}$,
where $\lambda_{1}$ and $\lambda_{2}$ are real numbers. Then using equation
(C.6) of appendix C we see that
$\boldsymbol{B}_{+}=(1-i\boldsymbol{I})\boldsymbol{B}$ is a self-dual
bivector. Moreover, we can use equation (2.31) to prove that
$\mathcal{C}(\boldsymbol{B}_{+})=(\lambda_{1}+i\lambda_{2})\boldsymbol{B}_{+}$.
To finish the proof just note that the Weyl operator defined on (2.26) agrees
with the definition of the section 2.1, see equation (2.2). Hence we have that
$C^{+}(\boldsymbol{B}_{+})=(\lambda_{1}+i\lambda_{2})\boldsymbol{B}_{+}$. More
details about this method can be found in [43, 44]. In particular, reference
[43] has exploited the Clifford algebra formalism to find canonical forms for
the Weyl operator for each algebraic type. As an aside, it is worth mentioning
that the whole formalism of general relativity can be translated to the
Clifford algebra language with some advantages [45].

####  2.7 Interpreting the PNDs

In the previous sections it has been proved that every space-time with non-
vanishing Weyl tensor admits some privileged null directions, four at most,
called the principal null directions (PNDs). In the present section we will
investigate the role played by these directions both from the geometrical and
physical points of view.

According to [46, 39], in 1922 Élie Cartan has pointed out that the Weyl
tensor of a general 4-dimensional space-time defined four distinguished null
directions endowed with some invariance properties under the parallel
transport over infinitesimal closed loops. It turns out that these directions
were the principal null directions of the Weyl tensor, in spite of Petrov’s
article defining his classification have appeared three decades later. Suppose
that a vector $\boldsymbol{v}$ belonging to the tangent space at a point $p\in
M$ is parallel transported along an infinitesimal parallelogram with sides
generated by $\boldsymbol{t}_{1}$ and $\boldsymbol{t}_{2}$, as illustrated on
the figure below.

It is a well-known result of Riemannian geometry that the change on the vector
$\boldsymbol{v}$ caused by the parallel transport over the loop is given by

$\delta v^{\mu}\equiv
v^{\prime\mu}-v^{\mu}=-\epsilon\,R^{\mu}_{\phantom{\mu}\nu\rho\sigma}\,v^{\nu}\,t_{1}^{\rho}\,t_{2}^{\sigma}\,.$
(2.33)

Where $\boldsymbol{v}^{\prime}$ is the vector after the parallel transport and
$\epsilon$ is proportional to the area of the parallelogram. In vacuum, as
henceforth assumed in this section, Einstein’s equation implies that the
Riemann tensor is equal to the Weyl tensor. So that in this case one can
substitute $R^{\mu}_{\phantom{\mu}\nu\rho\sigma}$ by
$C^{\mu}_{\phantom{\mu}\nu\rho\sigma}$ in equation (2.33). Now let us search
for null directions that are preserved by this kind of parallel transport.

Let $\boldsymbol{v}=\boldsymbol{l}$ be a PND and $\boldsymbol{n}$ a null
vector such that $l^{\mu}n_{\mu}=1$. Then, from section 2.4, we have that
$l_{[\alpha}C_{\mu]\nu\rho[\sigma}l_{\beta]}l^{\nu}l^{\rho}=0$. Contracting
this equation with $t_{2}^{\mu}n^{\beta}$ we easily find that
$C^{\sigma}_{\phantom{\sigma}\nu\rho\mu}l^{\nu}l^{\rho}t_{2}^{\mu}\propto
l^{\sigma}$ for any $\boldsymbol{t}_{2}$ orthogonal to $\boldsymbol{l}$. Thus
PNDs are the null directions with the property of being invariant by the
parallel transport around infinitesimal parallelograms generated by the PND
itself and any direction orthogonal to it. In the same vein, if
$\boldsymbol{l}$ is a repeated principal null direction then
$l_{[\alpha}C_{\mu]\nu\rho\sigma}l^{\nu}l^{\rho}=0$. Contracting this last
equation with $t_{2}^{\sigma}n^{\alpha}$ we find that $\delta l^{\mu}\propto
l^{\mu}$ for any parallelogram such that one of the sides is generated by
$\boldsymbol{l}$. If $\boldsymbol{l}$ is a triply degenerated PND then
$C_{\mu\nu\rho[\sigma}l_{\alpha]}l^{\rho}=0$, which by contraction with
$t_{1}^{\mu}t_{2}^{\nu}n^{\alpha}$ yield that $\delta l^{\mu}\propto l^{\mu}$
for any parallelogram. Finally, if $\boldsymbol{l}$ is a PND with degree of
degeneracy four then $C_{\mu\nu\rho\sigma}l^{\sigma}=0$, so that $\delta
l^{\mu}=0$ for any parallelogram. Table 2.2 summarizes these geometric
properties of the PNDs.

$q\,=\,1$ | $q\,=\,2$ | $q\,=\,3$ | $q\,=\,4$
---|---|---|---
$\boldsymbol{t}_{1}\,=\,\boldsymbol{l}$ | $\boldsymbol{t}_{1}\,=\,\boldsymbol{l}$ |  $\boldsymbol{t}_{1}$ arbitrary |  $\boldsymbol{t}_{1}$ arbitrary
$\;t_{2}^{\mu}\,l_{\mu}=0$ |  $\boldsymbol{t}_{2}$ arbitrary |  $\boldsymbol{t}_{2}$ arbitrary |  $\boldsymbol{t}_{2}$ arbitrary
$\delta l^{\mu}\propto l^{\mu}$ | $\delta l^{\mu}\propto l^{\mu}$ | $\delta l^{\mu}\propto l^{\mu}$ | $\delta l^{\mu}=0$
Table 2.2: Invariance of the PNDs under parallel transport over an
infinitesimal parallelogram with sides generated by $\boldsymbol{t}_{1}$ and
$\boldsymbol{t}_{2}$. In the first row $q$ denotes the degeneracy of the PND
$\boldsymbol{l}$.

In ref. [47] it was shown another geometric interpretation for the principal
null directions. Glossing over the subtleties, it was proved there that a null
direction is a PND when the Riemannian curvature of a 2-space generated by
this null direction and a space-like vector field $\boldsymbol{t}$ is
independent of $\boldsymbol{t}$.

One of the first physicists to investigate the physical meaning of the Petrov
types was F. Pirani. In ref. [26] he has tried to find a plausible definition
of gravitational radiation by comparing with the electromagnetic case. In this
article it has been shown that the energy-momentum tensor associated with
electromagnetic radiation admits no time-like eigenvector and one null
eigenvector at most, this null vector turned out to point in the direction of
the radiation propagation. Searching for an analogous condition in general
relativity Pirani investigated the eigenbivectors of Riemann tensor. The
intersection of the planes generated by such eigenbivectors defined what he
called Riemann principal directions (RPDs), which are not the PNDs, as they
are not necessarily null. But it turns out that the null Riemann principal
directions are repeated PNDs. Thus, mimicking the electromagnetic case, Pirani
arrived at the conclusion that if a space-time admits a time-like RPD then no
gravitational radiation should be present. Along with the results of Bel [25],
this means that no gravitational radiation is allowed on Petrov types $I$ and
$D$, which is reasonable since all static space-times are either type $I$ or
$D$. Pirani and Bel interpreted the repeated PNDs of types $II$, $III$ and $N$
as the direction of the gravitational radiation propagation [25, 26].

In order to understand the physical meaning of the PNDs, the analogy between
the electromagnetic theory and general relativity was also exploited by other
physicists. In [48, 25] L. Bel has introduced a tensor of rank four that is
quadratic on the Riemann tensor and that in vacuum has properties that
perfectly mimics the electromagnetic energy-momentum tensor. Such tensor is
now called the Bel-Robinson tensor [41]. Then Debever proved that in vacuum
this tensor is completely determined by the principal null directions of the
Weyl tensor [32], a result that can be easily verified using the spinorial
formalism. In ref. [31], Penrose has argued that the PNDs are related to the
gravitational energy density, enforcing and complementing Debever’s results.
Penrose also concluded that pure gravitational radiation should be present
only in type $N$ space-times, since only in this case the Weyl tensor
satisfies the massless wave-equation.

Finally, according to the Goldberg-Sachs theorem, the repeated PNDs in vacuum
are tangent to a congruence of null geodesics that is shear-free. This
celebrated theorem is behind the integrability of Einstein’s equation for
space-times of type $D$ [24]. This important result will be deeply exploited
on the forthcoming chapters. One of the goals of this thesis is to prove a
suitable generalization of this theorem valid in higher dimensions, which will
be accomplished in chapters 5 and 6.

####  2.8 Examples

1) Schwarzschild space-time
Schwarzschild space-time is the unique spherically-symmetric solution of
Einstein’s equation in vacuum. In a static and spherically symmetric
coordinate system its metric is given by

$ds^{2}=f^{2}\,dt^{2}-f^{-2}\,dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\,,\;\;f^{2}=1-\frac{2M}{r}\,.$

A suitable orthonormal frame and a suitable null tetrad are then,

$\displaystyle\hat{\boldsymbol{e}}_{0}=f^{-1}\,\partial_{t}\;;\;\;\hat{\boldsymbol{e}}_{1}=f\,\partial_{r}\;;\;\;\hat{\boldsymbol{e}}_{2}=\frac{1}{r}\partial_{\theta}\;;\;\;\hat{\boldsymbol{e}}_{3}=\frac{1}{r\sin\theta}\partial_{\varphi}\,;\,\textrm{
and }$
$\displaystyle\boldsymbol{l}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}_{1})\,;\;\boldsymbol{n}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{0}-\hat{\boldsymbol{e}}_{1})\,;\;\boldsymbol{m}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}+i\hat{\boldsymbol{e}}_{3})\,;\;\overline{\boldsymbol{m}}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{e}}_{2}-i\hat{\boldsymbol{e}}_{3})\,.$

Since the vector field $\partial_{t}=f\hat{\boldsymbol{e}}_{0}$ is a time-like
hyper-surface orthogonal Killing vector field, the space-time is called
static. In other words this means that the above metric is invariant by the
transformations $t\rightarrow-t$ and $t\rightarrow t+\epsilon$, where
$\epsilon$ is a constant. Such symmetries imply that the Weyl tensor cannot be
of Petrov types $II$, $III$ or $N$. For instance, if some static space-time
were type $N$ it would have just one PND,
$\boldsymbol{l}=\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}$ where
$\hat{\boldsymbol{e}}$ is some space-like vector of unit norm. But using the
symmetry $t\rightarrow-t$ we conclude that the null vector
$\boldsymbol{l}^{\prime}=-\hat{\boldsymbol{e}}_{0}+\hat{\boldsymbol{e}}$
should also be a PND, which contradicts the type $N$ hypothesis. Thus the
Schwarzschild solution must be either type $I$ or $D$. Indeed, calculating the
Weyl scalars, by means of (2.1), on the above null frame we get:

$\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\;\;;\;\;\Psi_{2}=\frac{M}{r^{3}}\,.$

Then, thanks to table 2.1, we conclude that the Schwarzschild space-time has
Petrov type $D$, with $\boldsymbol{l}$ and $\boldsymbol{n}$ being repeated
PNDs. Actually, it can be proved that the whole family of Kerr-Newman
solutions is type $D$.

2) Plane Gravitational Waves
Physically, plane waves are characterized by the existence of plane wave-
fronts (equipotentials) orthogonal to the direction of propagation. Since the
graviton is a massless particle, it follows that the gravitational field
propagates along a null direction $\boldsymbol{l}$. In order for all the
points on a wave-front remain on the same phase as propagation occurs, the
null vector field $\boldsymbol{l}$ must be covariantly constant throughout the
space-time. In particular, this implies that $\boldsymbol{l}$ remains
unchanged by parallel transport, which according to table 2.2 implies that the
space-time must be type $N$ if vacuum is assumed. Therefore, a manifold that
represents the propagation of plane gravitational waves might be type $N$.
Indeed, if a space-time admits a covariantly constant null vector