$\displaystyle\frac{\pi^{2}}{30}N^{*}(T_{R})(kT_{R}^{4})\sim
V(\phi\sim\phi_{T}).$ (122)

In this case, if we take for instance $N^{*}\sim 10^{2}$ in our quartic model,
this would entail

$\displaystyle kT_{R}\sim\lambda^{1/4}M_{p}\sim 10^{15}GeV,$ (123)

which is still around the scale of GUT phase transitions. However, the
temperature of reheating generally turns out to be orders of manitude lower to
that of thermal equlibrium, due to the inneficiency of reheating if we require
the interactions of $\phi$ to other fields in nature to be sufficiently weak
[4]. In this case, one can transition from an inflationary phase to a usual
hot universe described in our standart model, liberated from the fundamental
issues of its extrapolation to arbitrarily early times.

##  ?chaptername? 5 Conclusion

Vacuum is complex. If there is a single sentence that captures the message of
this dissertation, we believe this should be it. As surprising as it may seem
to our intuition this concept turns out to spawn an incredibly rich structure,
which may be intimately related to some of the most profound questions that we
have about our own universe.

When we switch our fundamental perspective from the notion of particles to
that of fields, such that the former comes to be conceived just as an emergent
manifestation of the latter, we find that a corresponding notion of vacuum as
“a state devoid of particles” cannot make unambiguous sense. Particularly, in
quantum theory, different observers can measure very different particle
contents in the same field state, depending on their state of motion (even
when they are at the same spacetime region). Moreover, even a fixed inertial
observer probing the field in a fixed state may observe very different
particle contents at different times, giving rise to the phenomenon of
particle creation. Nonetheless, all observers converge on their notions of
particle occupations at arbitrarily high-energies, corresponding to
increasingly localized short wavelength, which allows for a meaningful,
although approximate and nonunique, extension of the concept of vacuum in
curved spacetimes, which plays a key role in the renormalization of localized
quantities, such as field amplitudes and energy densities.

The vacuum energy density, in particular, is found to play significant roles
in many contexts. Even classically it may behave in a nontrivial manner,
allowing for the description of a finite inflationary phase for the universe.
At a quantum level, however, it reveals an even richer scope of possibilities.
As long as one can systematically eliminate the divergencies that appear in
our description of quantum field theory, quite surprising physical predictions
emerge. Even in a description in flat space, where gravity plays no role
whatsoever, it was possible to predict and experimentally verify that the
electromagnetic vacuum will present negative pressures and induce an
attractive force between two conducting plates. In curved spaces, it is found
to give rise to a cosmological-constant kind of term, with constant positive
energy and negative pressure of the same magnitude, which could conceivably
account for the puzzling cosmic component that we now call Dark Energy.
Further still, primordial vacuum fluctuations seem like a promising candidate
to explain the tiny fluctuations that we observe in the far background sky,
and that were ultimately responsible for the formation of the many structures
that we observe in the universe today, including the sun, the earth, and the
all life that emerged on it.

Of course, there is still much research to be done in the subject before we
can extrapolate from appealing theoretical pictures to making strong claims
about the workings of the real world. Particularly, on the observational side,
increasingly precise and varied measurements of relics from the early universe
should allow for significant, and possibly quite surprising, improvements of
our understanding of it. Nonetheless, we hope that the present work may serve
as a comprehensible introduction to the theoretical window of such a
fascinating subject.

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##  ?chaptername? 6 Distributions

The subject of distributions is one hard to ignore in physics, and yet it is
seldom given a proper treatment in the exposition of topics for which it is
relevant (ranging from point-charges in electrostatics, going through bras and
kets in ordinary quantum mechanics, and up to its ubiquitous presence in field
theory). The present exposition of the topic, far from exhaustive or fully
rigorous, aims at laying a few basic definitions and providing a clear picture
for its applications in the scope of this dissertation. For a longer but
straightforward and physically-oriented exposition, we recommend [18]; or, for
a more general and rigorous covering of the topic, see [19].

$\bullet$ Distributions as linear functionals.

The subject of distributions is one of linear algebra. When one is handling an
ordinary finite-dimensional vector space $\mathbb{V}$, one fundamental concept
is that of linear operators acting on $\mathbb{V}$ to produce scalars. Those
operators, in their turn, form another linear space, the _dual space_
$\mathbb{V}^{*}$. $\mathbb{V}^{*}$ can be easily shown to be isomorphic to
$\mathbb{V}$, although there is no natural identification between them.111On
the other hand, the dual of $\mathbb{V}^{*}$, $\mathbb{V}^{**}$, can be
naturally identified with $\mathbb{V}$: given $V\in\mathbb{V}^{**}$ one
associates it with the unique vector $v\in\mathbb{V}$ that satisfies
$V(\sigma)=\sigma(v),\,\forall\sigma\in\mathbb{V}^{*}$.

In this case, let $n=\dim(\mathbb{V})=\dim(\mathbb{V}^{*})$ and consider an
arbitrary pair of a vector $v\in\mathbb{V}$ and a dual vector
$\sigma\in\mathbb{V}^{*}$. Given a basis $\\{\mathbb{e}_{i}\\}$ of
$\mathbb{V}$ and its dual basis $\\{\mathbb{e}^{*}_{i}\\}$ in $\mathbb{V}^{*}$
($\mathbb{e}^{*}_{i}(\mathbb{e}_{j})=\delta_{ij}$), we can write $\sigma(v)$
in terms of their components:

$\displaystyle\sigma(v)=\sum_{i=1}^{n}\sigma_{i}v^{i}.$ (1)

However, in the case of infinite-dimensional vector spaces, such as many
function spaces, it is no longer generally true that $\mathbb{V}$ and
$\mathbb{V}^{*}$ are isomorphic. In fact $\mathbb{V}^{*}$ is generally bigger
than $\mathbb{V}$: that means one can generally associate to each vector
$v\in\mathbb{V}$ a dual vector $u\in\mathbb{V}^{*}$, but the converse is not
always true. Specifically, for function spaces (usually, for “well-behaved”
functions defined on some open set of a sufficiently smooth manifold
$\mathcal{O}\subset\mathcal{M}$ ), these dual vectors are linear functionals,
and they are called distributions.

We shall generally denote the function space under consideration
$\mathcal{F}$, whose domain $\mathcal{O}$ is assumed to have a measure
$\mu$222 For a metric manifold $(\mathcal{M},g_{ab})$, this just amounts to
familiar integrals in $\mathbb{R}^{d}$:
$\int_{\mathcal{M}}d\mu_{g}(x)f(x)=\int_{\mathbb{R}^{d}}d^{d}x|g(x)|^{\frac{1}{2}}f(x)$.,
and the space of distributions that act on it $\mathcal{D}$
($=\mathcal{F}^{*}$); then, the action of a distribution on a function will
generally take the form of a Lebesgue integral. A case of particular interest
is when $\mathcal{F}$ is a space of smooth functions on open intervals
$U\subset\mathbb{R}^{d}$, $\mathcal{C}^{\infty}(U)$ (the case of smooth
manifolds can be mapped in (a countable sum of) this one333See appendix B of
[5] for details of integration on manifolds.);0 for a brute-force guarantee
that one will not have to worry with boundary terms, one often resorts to the
subset of these functions with compact support $\mathcal{C}^{\infty}_{0}(U)$.
Throughout this Appendix, to avoid being cumbersome with technical remarks, we
shall always assume our functions to be sufficiently well-behaved so that our
assertions make sense (for example, that one may take derivatives to a desired
order, that certain integrals converge and that there will be no contributions
from boundary terms).

Analogously to (1), when we have a distribution
$\mathcal{\sigma}\in\mathcal{F}^{*}$ _that can be identified with a locally
integrable function_ $\tilde{\sigma}\in\mathcal{F}$, we may write its action
on any function $f\in\mathcal{F}$ as

$\displaystyle\sigma[f]=\int d\mu(x)\tilde{\sigma}(x)f(x).$ (2)

However, that is not always the case. Consider the linear functional
$\delta_{x}$, $x\in\mathcal{O}$, whose action upon a function
$f\in\mathcal{F}$ produces the value of $f$ in $x$, that is:
$\delta_{x}[f]\equiv f(x)$. Clearly, there is no locally integrable function
$\tilde{\delta}_{x}$ that satisfies this property in the continuum444For a
slightly longer discussion of this point, see chapter 1 of [18]. However, we
would still like to represent this functional in that form:

$\displaystyle\delta_{x}[f]=f(x)=\int_{\mathcal{O}}d\mu(y)f(y)\delta(x,y)=\int_{U}d^{d}yf(y)\delta(x-y)=\int_{U}d^{d}yf(y-x)\delta(y),$
(3)

so that one often speaks of the “Dirac delta fuction” $\delta(x)$, as though
one was actually integrating a function in (3) – or, extrapolating further,
one speaks of the delta function outside an integration sign, where it makes
much less formal sense. Our aim here is not to be pedantic about definitions;
on the contrary, _we are interested in potentializing their practical use
without incurring in any operational pitfalls_. It is usually harmless, and
often quite useful, to make informal manipulations with distributions as
though they were functions (as it is to “pass $dx$ multiplying” in a
differential equation, or use “the wave-function of a particle of momentum
$k$”: $\psi_{k}(x)=(2\pi)^{-\frac{1}{2}}e^{ikx}$). However, there are
situations where such approaches break and produce implausible or paradoxical
results, and that often happens because one failed to appreciate that he/she
is not dealing with a function, but rather with a distribution, and that more
care to a particular operation was needed (just like one may be led to the
absurd conclusion that $1=2$ by carelessly performing algebraic manipulations
involving a division by $0$). A few sensible/troublesome manipulations with
distributions include trying to evaluate objects like $\braket{x}{x}$,
$\int\delta^{2}(x)f(x)dx$, which would be perfectly natural if one was
operating ordinary functions or vectors.

In the context of ordinary nonrelativistic quantum mechanics, one finds that
plane waves, the eigenfunctions of the momentum operator, are not square-
integrable in $\mathbb{R}^{n}$, and, as such, they cannot properly represent
state vectors in the Hilbert space
$\mathcal{H}=\mathcal{E}\subset\mathcal{L}^{2}(\mathbb{R}^{n})$ (we take a
subset $\mathcal{E}$, not the entire $\mathcal{L}^{2}$, to ensure we are
restricted to sufficiently well-behaved functions). The case is even worse for
the eigenvectors of the position operator555Note that these are not vectors
belonging to $\mathcal{H}$, but rather to a larger space: $\mathcal{H}^{**}$
(the dual of the dual of $\mathcal{H}$)., which are not even functions to
start with. Nevertheless, all actual wave-functions $\psi\in\mathcal{E}$ have
well-defined Fourier transforms, as well as (obviously) well-defined values at
each point. Thus, although position and momentum eigenstates
$\ket{x}:X\ket{x}=x\ket{x}$ and $\ket{p}:P\ket{p}=p\ket{p}$ do not make
rigorous sense as state vectors belonging to the Hilbert space $\mathcal{H}$,
it is perfectly sensible to evaluate them as distributions:
$\forall\ket{\psi}\in\mathcal{H}$,

$\displaystyle\psi(x)$
$\displaystyle=\braket{x}{\psi},\quad\braket{x}{X}{\psi}=x\psi(x),\quad\ket{\psi}=\int
dx\ket{x}\braket{x}{\psi}=\int dx\ket{x}\psi(x),$ (4)
$\displaystyle\bar{\psi}(p)$
$\displaystyle=\braket{p}{\psi},\quad\braket{p}{P}{\psi}=p\bar{\psi}(p),\quad\ket{\psi}=\int
dp\ket{p}\braket{p}{\psi}=\int dp\ket{p}\bar{\psi}(p).$ (5)

We also have the famous orthornormality relations, which can be rigorously
stated in the distributional sense:
$\braket{x}{x^{\prime}}=\delta(x-x^{\prime})$,
$\braket{p}{p^{\prime}}=\delta(p-p^{\prime})$,
$\braket{x}{p}=(2\pi)^{-1/2}e^{\pm ipx}$, and which allow one to evaluate any
scalar products $\braket{\phi}{\psi}$ or operator transformations
$\braket{\phi}{A}{\psi}$ in terms of the $\\{\ket{x}\\}$ or $\\{\ket{p}\\}$
bases.

$\bullet$ Field operators as distributions and applications to 2-point
functions:

Similarly, in the context of QFT, it occurs that the fundamental observables
(namely, field configurations or canonical momenta configurations) do not have
proper eigenstates in their respective Hilbert Spaces. Nonetheless, one often
just writes

$\displaystyle\ket{\phi^{\prime}}:$
$\displaystyle\;\phi(x)\ket{\phi^{\prime}}=\phi^{\prime}(x)\ket{\phi^{\prime}},\quad\forall
x\in\Sigma,$ (6) $\displaystyle\ket{\pi^{\prime}}:$
$\displaystyle\;\pi(x)\ket{\pi^{\prime}}=\pi^{\prime}(x)\ket{\pi^{\prime}},\quad\forall
x\in\Sigma,$ (7)

where we used primes ′ to distinguish between the field (momentum) operator
$\phi(x)$ ($\pi(x)$) – evaluated at any event $x$ belonging to a certain
Cauchy surface666Note that we cannot have a single eingenstate of the field
operator $\phi(x)$ in all spacetime, just like we cannot have a wave-function
perfectly localized at all times: this would amount to having a sharp
(classical) trajectory for $\phi$. Determining it (or its momentum) at an
entire simultaneity surface $\Sigma$ corresponds to obtaining its maximal
information in a quantum description. $\Sigma$ – and its eigenvalue
$\phi^{\prime}(x)$ ($\pi^{\prime}(x)$).

To be more precise, the quantized field observable $\phi(x)$ is actually an
_operator-valued distribution_ (such that
$\braket{\Psi_{1}}{\phi(x)}{\Psi_{2}}$ is an ordinary (number-valued)
distribution, $\forall\ket{\Psi_{1}},\ket{\Psi_{2}}\in\mathcal{H}$). (Further
considerations an implications of that point can be found in chapter 3 of [3])

There are some basic operations that one may perform with distributions. Sum
and mulplication by scalars are the most elementary ones, stemming directly
from the vector space structure of $\mathcal{D}$. Further uselful operations
that one may perform with distributions are _tensor products_ , _convolutions_
and _taking derivatives_. All of them are well-defined and quite intuitive to
handle operationally by recurring to their function correspondents; we refer
the reader to [18] for a more detailed definition and examples of each
operation (see, respectively, chapters 2, 4 and 5).

Therefore, we have that bilinear objects such as $\phi(x)\phi(x^{\prime})$, or
any combination of them that involves derivatives (such as the terms that
appear in the two-point stress tensor $T_{\mu\nu}(x,x^{\prime})$), are
generally well-defined as distributions. They will often show a singular
behaviour when one attempts to evaluate them as $x\rightarrow x^{\prime}$,
whereas they are usually regular for $x\neq x^{\prime}$. Such a behaviour is
not at all surprising when we think of the paradigmatic example of Dirac
deltas: If we define a test function $F\in\mathcal{F}\otimes\mathcal{F}$, it
is perfectly sensible to evaluate the ‘double delta’ distribution
$\Delta(x,x^{\prime})\equiv\delta_{x}\otimes\delta_{x^{\prime}}\in\mathcal{D}\otimes\mathcal{D}$:

$\displaystyle\Delta(x,x^{\prime})[F]$ $\displaystyle\equiv\iint
d\mu(y)d\mu(y^{\prime})F(y,y^{\prime})\delta(x,y)\delta(x^{\prime},y^{\prime})$
(8) $\displaystyle=F(x,x^{\prime}).$ (9)

It is also perfectly sensible to evaluate the convolution of two Dirac deltas,
$(\delta*\delta)_{x}$, in a test function $f\in\mathcal{F}$:

$\displaystyle(\delta*\delta)_{x}[f]$ $\displaystyle\equiv\int
dyf(x-y)(\delta*\delta)(y)$ $\displaystyle\equiv\iint
dydzf(x-y)\delta(y-z)\delta(z)$ $\displaystyle=\int dyf(x-y)\delta(y)$
$\displaystyle=f(x).$ (10)

So one immediately finds that $(\delta*\delta)_{x}=\delta_{x}$ (of course,
this last equality is not true for _any_ distribution $\sigma$, i.e.
$\sigma_{x}\not\equiv(\sigma*\sigma)_{x}$). Now, _the product of 2
distributions is not generally defined_. Although for any two functions, $g$,
and $u$, one could evaluate their action in a test function $f$ like

$\displaystyle(gu)[f]=\int dx\,g(x)u(x)f(x),$ (11)

the same is _not true_ for two distributions $\sigma$ and $\theta$, for which
it generally makes no sense to evaluate

$\displaystyle(\sigma\theta)[f]=\int\sigma(x)\theta(x)f(x),$ (12)

obvious exceptions being the case where one of these distributions can be
identified with a function (or when
$\operatorname{supp}(\sigma)\cap\operatorname{supp}(\theta)=\emptyset$, for
which we could trivially define $\sigma\theta=0$). Particularly, there is no
direct way to make sense of an object like $\delta^{2}(x)$.

Then, it should not be surprising that the attempt to directly evaluate
expected values such as $\braket{\Psi}{\phi^{2}(x)}{\Psi}$ does not make
direct sense, and generally yields divergent results. Much more appaling is
the fact that, for a quite large variety of field theories, these divergencies
can actually be systematically handled and subtracted to yield meaningful
finite physical results (although these procedures often require very
sophisticated techniques and cumbersome calculations, and they are not
generally free from ambiguities).

$\bullet$ Discontinuities and singularities; principal value of distributions:

In field theory, it is also not unusual that one must handle distributions
involving integrals that go directly through singularities in their
integrands. In such cases, there is a variety of ways through which one may
obtain a meaningful value of the integration, giving rise to ambiguities to
such singular distributions. Among the many ways to define the action of
singular distributions, one conventional one is their so-called principal
value. Ultimately, evaluating the principal value is one convenient way to
cancel out infinities and obtain meaningful finite results; here, we shall
give just a superficial glimpse in the subject, applying it to simple
distributions that appear in this dissertation (again, we refer the reader to
chapter 2 of [18] for a more rigurous and thorough exposition of the subject)

A case of particular interest to us will be calculating the principal value of
integrals around order 1 poles. Thus, for a start, let us consider a
functional $\sigma\in\mathcal{D}(\mathcal{C}^{\infty}_{0})$ defined through
the function $1/x$, which has a singularity at $x=0$. How, then, should we
interpret its action on a test function $f\in\mathcal{C}_{0}^{\infty}$,
$\sigma[f]$? Well, since $\frac{1}{x}$ can be written as $\frac{d}{dx}\ln|x|$
for $x\neq 0$, _one particular_ way to evaluate it is:

$\displaystyle\sigma[f]$
$\displaystyle=\int_{-\infty}^{\infty}\\!\\!dx\frac{f(x)}{x}=-\int_{-\infty}^{\infty}\\!\\!dxf^{\prime}(x)\ln|x|$
$\displaystyle\equiv-\lim_{\epsilon\rightarrow
0^{+}}\biggl{[}\int_{-\infty}^{-\epsilon}\\!\\!dxf^{\prime}(x)\ln(-x)+\int_{\epsilon}^{\infty}\\!\\!dxf^{\prime}(x)\ln(x)\biggl{]}$
$\displaystyle=\lim_{\epsilon\rightarrow
0^{+}}\biggl{[}\cancelto{\mathcal{O}(\epsilon\ln\epsilon)\rightarrow
0}{\bigl{(}f(\epsilon)-f(-\epsilon)\bigl{)}\ln(\epsilon)}+\int_{-\infty}^{-\epsilon}\\!\\!dx\frac{f(x)}{x}+\int_{\epsilon}^{\infty}\\!\\!dx\frac{f(x)}{x}\biggl{]}$
$\displaystyle=\lim_{\epsilon\rightarrow
0^{+}}\biggl{[}\int_{-\infty}^{-\epsilon}\\!\\!dx\frac{f(x)}{x}+\int_{\epsilon}^{\infty}\\!\\!dx\frac{f(x)}{x}\biggl{]}$
$\displaystyle\equiv\mathcal{P}\left(\int_{-\infty}^{\infty}dx\frac{f(x)}{x}\right),$
(13)

which is how we _define_ the principal value of $\frac{1}{x}$,
$\mathcal{P}\bigl{(}\frac{1}{x}\bigl{)}$.

Note that (i) the second equality is a particular (arbitrary) way to take the
limit in the domain around $x=0$ (the results could be different if we
approached $0$ at different rates from the left and the right) and (ii) since
the logarithm is also singular at $x=0$ (although it is the derivative of a
piecewise continuous function) this integration by parts is also not free of
ambiguities. For instance, we could have defined:

$\displaystyle\frac{1}{x}$ $\displaystyle=\frac{d}{dx}\ln(-x),\;$
$\displaystyle x<0$ $\displaystyle\frac{1}{x}$
$\displaystyle=\frac{d}{dx}\ln(x)+a,$ $\displaystyle x>0$

so that the same procedure would yield

$\displaystyle\sigma[f]$
$\displaystyle=\int_{-\infty}^{\infty}\\!\\!dx\frac{f(x)}{x}=-\int_{-\infty}^{0}\\!\\!dxf^{\prime}(x)\ln(-x)+\int_{0}^{\infty}dxf^{\prime}(x)\bigl{(}\ln
x+a\bigl{)}$ $\displaystyle\equiv-\lim_{\epsilon\rightarrow
0^{+}}\biggl{[}\int_{-\infty}^{-\epsilon}\\!\\!dxf^{\prime}(x)\ln(-x)+\int_{\epsilon}^{\infty}\\!\\!dxf^{\prime}(x)\ln(x)+a\int_{\epsilon}^{\infty}\\!\\!dxf^{\prime}(x)\biggl{]}$
$\displaystyle=\lim_{\epsilon\rightarrow
0^{+}}\biggl{[}\int_{-\infty}^{-\epsilon}\\!\\!dx\frac{f(x)}{x}+\int_{\epsilon}^{\infty}\\!\\!dx\frac{f(x)}{x}\biggl{]}+af(0)$
$\displaystyle\equiv\mathcal{P}\left(\int_{-\infty}^{\infty}dx\frac{f(x)}{x}\right)+af(0).$
(14)

This gives us the distribution
$\sigma(x)\equiv\mathcal{P}(\frac{1}{x})+a\delta(x)$. Since taking the value
$a=0$ is just an arbitrary choice as any other value, so will be the result of
the distribution that we try to associate to $1/x$, in regards to its singular
region. The point here is that the behaviour of the function in its nonsigular
region does not uniquely determine a distribution, and additional information
regarding its singular region may be required to define it. For a order 1
pole, the particular way of defining its principal value is adding up the
divergent contributions aroud 0 symmetrically, so that they cancel out.

Another interesting application of considering a $1/x$ distribution as the
derivative of $ln(x)$ emerges when we consider functions in the complex plane.
Let $z$ be a complex number, $z=x+iy=|z|e^{i\arg(z)}$, so that its logarithm
is defined as

$\displaystyle\ln(z)=\ln|z|+i\arg(z).$ (15)

Then, if we take the limit $y\rightarrow 0^{+}$, we have for all finite $x$
that

$\displaystyle\lim_{\;y\rightarrow
0^{+}}\ln(z)=\ln|x|+i\pi(1-\Theta(x))\qquad\Rightarrow\qquad\frac{d}{dx}\Bigl{(}\lim_{\;y\rightarrow
0^{+}}\ln(z)\Bigl{)}=\frac{1}{x}-i\pi\delta(x).$ (16)

Similarly, for negative values of $y$:

$\displaystyle\lim_{\;y\rightarrow
0^{-}}\ln(z)=\ln|x|-i\pi(1-\Theta(x))\qquad\Rightarrow\qquad\frac{d}{dx}\Bigl{(}\lim_{\;y\rightarrow
0^{-}}\ln(z)\Bigl{)}=\frac{1}{x}+i\pi\delta(x).$ (17)

We can then use the complex identity $z^{-1}=\frac{d}{dz}\ln(z)$ to obtain the
following distribution in the reals:

$\displaystyle\sigma(x)=\lim_{\epsilon\rightarrow 0^{+}}\frac{1}{x\pm
i\epsilon}\equiv\mathcal{P}\left(\frac{1}{x}\right)\mp i\pi\delta(x).$ (18)

With this identity, one can analyze (particular values of) integrals along the
real axis, by displacing their poles infinitesimally in the complex plane
(either above or below the axis, depending on the application at hand).

##  ?chaptername? 7 Some geometrical derivations

Most texts in QFTCS already assume the reader to be familiar, to a fair
extent, with both QFT in Minkowski spacetime and General Relativity. In this
work, while we do provide a full introductory chapter to QFT, we shall not
present a thorough and comprehensive introduction to GR (for that, we refer
the reader to the excellent textbook of R. Wald [5], where this author
personally learned the subject; alternatively, see [25]). Still, the need was
felt to provide an appendix discussing some fundamentals and covering more
specific geometric derivations. It should serve both to lay the basic
definitions and notations, and to explicitly develop some useful tools for our
discussion in the main text, avoiding gaps in our derivations. Besides
defining the fundamental geometrical objects used in the formulation of the
theory, such as curvature and covariant derivatives, the topics in this
appendix include an introduction to the computations of variations in respect
to the metric, as well as a few useful geometrical structures, like Lie
derivatives, Killing fields, and conformal transformations.

###  1 Fundamental building blocks of GR

The theory of General Relavity, whose original formulation was culminated in
Einstein’s work, succeeded to incorporate two very simple founding physical
principles111Namely, the local invariance of the speed of light and the
equivalence principle. in a geometrical formalism for spacetime. In this
formulation, spacetime came to be conceived as a curved, pseudo-Riemannian
manifold, whose dynamics are governed by the matter propagating in it. Of
course, one can trace the roots of this theory back to the simpler geometrical
formulation of special relativity, built in Minkowski spacetime
$(\mathbb{R}^{4},\eta_{ab})$, from which one can find a generalization in
curved spacetimes $(\mathcal{M},g_{ab})$, suitable to general relativity.

However, unlike Minkowski spacetime (or even prerelativistic Galilean
spacetime), whose affine structure allows for fairly simple geometrical
constructions and manipulations with little more than linear algebra and
calculus tools, in curved spaces one requires a quite more sophisticated
paraphernalia from differential geometry to carry various relevant
calculations.

To start with, one can no longer use a single vector space structure to define
vectors $v^{a}$, dual vectors (also called covectors) $\omega_{a}$ and general
tensors $T^{abc...}_{\;def...}$ _in the entire spacetime_. Instead, one must
build _tangent spaces to each event $x\in\mathcal{M}$_, $\mathbb{V}_{x}$, and
work with tangent vectors $v^{a}(x)$, dual vectors (also called _cotangent
vectors_) $\omega_{a}(x)$ and general tensors $T^{abc...}_{\;def...}(x)$, with
no natural identification between $\mathbb{V}_{x}$ and
$\mathbb{V}_{x^{\prime}}$ for two distinct events $x\neq x^{\prime}$.222Here,
we are using the _abstract index notation_ (see List of Symbols). Throughout
this work, we often switch between concrete (greek) indices and abstract ones
(latin, from a to h), using the former more often in the context of quantum
field theory and the latter in ‘purely geometrical’ contexts, maintaining
similarities with the literature. This, on its turn, prevents one from having
a _unique_ geometrical notion of derivatives for tensor fields, which must be
specified through further physical postulates. First, one usually requires
that it acts symmetrically on scalars, that is:

$\displaystyle\nabla_{\\!a}\nabla_{\\!b}\phi-\nabla_{\\!b}\nabla_{\\!a}\phi=0.$
(1)

Since generally one could have that

$\displaystyle\nabla_{\\!a}\nabla_{\\!b}\phi-\nabla_{\\!b}\nabla_{\\!a}\phi=T_{ab}\phi,$
(2)

being $T_{ab}=-T_{ba}$ the so-called the _torsion tensor_ , this is referred
to as the null torsion condition (or, equivalently, one says that
$(\mathcal{M},g_{ab})$ is a torsion-free space).

Generally, two derivative operators $\nabla_{\\!a}$ and
$\tilde{\nabla}_{\\!a}$ may differ in the following way:

$\displaystyle\tilde{\nabla_{\\!b}}v^{a}=\nabla_{\\!b}v^{a}+C^{a}_{\;bc}v^{c},$
(3)

where $C^{a}_{\;bc}$ is called a connection. For torsion free spaces, it will
be symmetrical in the lower indices: $C^{a}_{\;bc}=C^{a}_{\;cb}$.

Thus, the fundamental objects in GR are the metric field $g_{ab}$ and a
preferred derivative operator $\nabla_{a}$ (or, equivalently, a preferred
connection). In the standard formulation of GR, motivated by the equivalence
principle, one requires the physical derivative operator – the so-called
covariant derivative $\nabla_{\\!a}$ – to be that with respect to which local
variations of the metric vanish:

$\displaystyle\nabla_{\\!c}\,g_{ab}=0.$ (4)

That is not to say the metric is spacetime homogeneous, but rather that a
fundamental notion of a locally nonvarying quantity is defined in closed
proximity with it333In section 3, when we define Lie derivatives and construct
the notion of continuous isometry groups, we shall ascribe a clearer meaning
to the notions of variations of the metric along spacetime from a purely
geometrical perspective. .

Although the formulation so far has been carried in a coordinate-free way, one
must often adopt a coordinate system to carry calculations. In practice, one
must be able to compute covariant derivatives in terms of the metric
components $g_{\mu\nu}$ and ordinary coordinate derivatives $\partial_{\mu}$.
These can be calculated as a particular instance of (3), using the Christoffel
Symbols $\Gamma^{a}_{\;bc}$:

$\displaystyle\nabla_{\\!\mu}v^{\nu}=\partial_{\mu}v^{\nu}+\Gamma^{\nu}_{\mu\alpha}v^{\alpha},$
(5)

where $\Gamma^{\alpha}_{\;\mu\nu}$ can be calculated as a function of metric
components as

$\displaystyle\Gamma^{\alpha}_{\;\mu\nu}=\tfrac{1}{2}g^{\alpha\beta}\bigl{[}\partial_{\mu}g_{\nu\beta}+\partial_{\nu}g_{\mu\beta}-\partial_{\beta}g_{\mu\nu}\bigl{]}.$
(6)

For practical computations, it is very useful to write a contraction of these
symbols in termos of _the determinant_ of the matrix of metric components,
$g$, which reads

$\displaystyle\Gamma^{\alpha}_{\;\,\alpha\mu}=\tfrac{1}{2}g^{\alpha\beta}\partial_{\mu}g_{\alpha\beta}=\tfrac{1}{2}g^{-1}\partial_{\mu}g=\partial_{\mu}\ln|g|^{\frac{1}{2}}=|g|^{-\frac{1}{2}}\partial_{\mu}|g|^{\frac{1}{2}}.$
(7)

The most useful application of this formula for us will be in computing the
D’Alembertian for a scalar field. Note that, for a scalar field, while first
derivatives will be simply given by partial derivatives, second derivatives
will involve Christoffel symbols. We can then write

$\displaystyle\Box\phi\equiv g^{\mu\nu}\nabla_{\\!\mu}\nabla_{\\!\nu}\phi$
$\displaystyle=\nabla_{\\!\mu}\bigl{(}g^{\mu\nu}\partial_{\nu}\phi\bigl{)}$
$\displaystyle=\partial_{\mu}\bigl{(}g^{\mu\nu}\partial_{\nu}\phi\bigl{)}+\Gamma^{\mu}_{\;\,\mu\alpha}g^{\alpha\nu}\partial_{\nu}\phi$
$\displaystyle=\partial_{\mu}\bigl{(}g^{\mu\nu}\partial_{\nu}\phi\bigl{)}+g^{\alpha\nu}\partial_{\nu}\phi\bigl{(}|g|^{-\frac{1}{2}}\partial_{\alpha}|g|^{\frac{1}{2}}\bigl{)}$
$\displaystyle=|g|^{-\frac{1}{2}}\partial_{\mu}\bigl{(}|g|^{\frac{1}{2}}g^{\mu\nu}\partial_{\nu}\phi\bigl{)}$
(8)

Now, we would like to construct an intrinsic notion of curvature for our
spacetime, defined uniquely by the metric $g_{ab}$. Ultimately, this curvature
will refer to _derivatives_ of the metric (up to second order). This may sound
very strange at this point, since we postulated the covariant derivative of
the metric to be identically null in (4). It happens that this equation
precisely defines how the physical notion of derivatives depend on the metric.
Thus, we define the notion of curvature indirectly, through covariant
derivatives. To start with, we define the Riemann curvature
tensor444Conventions may somewhat vary in the literature, due both to the
choice of metric signature and conventions on different indices. by

$\displaystyle\nabla_{\\!a}\nabla_{\\!b}\,\omega_{c}-\nabla_{\\!b}\nabla_{\\!a}\,\omega_{c}=-R_{abc}^{\;\;\;\;\;d}\omega_{d},$
(9)

so that, indirectly, one can uniquely associate a curvature tensor to a given
metric, by equations (4) and (9).

One can also cumpute the curvature components in terms of Christoffel Symbols
(that is, in terms of partial derivatives of the metric components) as

$\displaystyle
R_{\mu\nu\alpha}^{\;\;\;\;\;\;\beta}=2\partial_{[\mu}\,\Gamma^{\beta}_{\;\,\nu]\alpha}+2\Gamma^{\lambda}_{\;\,\alpha[\nu}\,\Gamma^{\beta}_{\;\,\mu]\lambda}.$
(10)

It is also worth listing here a few of the symmetries and identities obeyed by
the Riemann tensor. They can each be worked with some algebraic effort, and we
shall state them without proofs. First we have three independent antissymmetry
properties:

$\displaystyle 1.$ $\displaystyle\;R_{abcd}=-R_{bacd},$ (11a) $\displaystyle
2.$ $\displaystyle\;R_{abcd}=-R_{abdc},$ (11b) $\displaystyle 3.$
$\displaystyle\;R_{[abc]d}=0$ (11c) (where $2$ makes explicit use that
$\nabla_{\\!a}$ is the covariant derivative in $(\mathcal{M},g_{ab})$).
Together, they imply the following symmetry:

$\displaystyle R_{abcd}=R_{cdab}.$ (11d)

Finally, we state the Bianchi identity:

$\displaystyle\nabla_{\\![a}R_{bc]de}=0.$ (11e)

Contractions of the Riemann tensor play a key role in GR. Because of all its
antissymetry properties, there is only one independent rank $(0,2)$
contraction, which will be the Ricci tensor. We then define the Ricci
curvature $R_{ab}$ and the curvature scalar as $R$ as

$\displaystyle R_{ac}$ $\displaystyle\equiv R_{abc}^{\;\;\;\;\;b},$ (12)
$\displaystyle R$ $\displaystyle\equiv g^{ac}R_{ac}.$ (13)

Note that in virtue of eq (11d), $R_{ab}=R_{ba}$. An important combination of
$R_{ab}$ and $R$ is the so-called _Einstein tensor_ , defined as

$\displaystyle G_{ab}=R_{ab}-\tfrac{1}{2}Rg_{ab}.$ (14)

It is this tensor that will appear at the left side of Einstein equations.
Finally, we note that the Bianchi identity will imply that $G_{ab}$ is
covariantly conserved:

$\displaystyle\nabla_{\\!a}G^{ab}=0.$ (15)

###  2 Variations with respect to the metric

As we have seen thus far, the construction of geometrical quantities from the
metric is often very indirect, being determined by the particular way that it
fixates the covariant derivative in our spacetime. Thus, it is useful to
compile a few results for the systematic computation of variations of these
quantities as we vary the metric. Our main interest in computing these
variations will be to derive functional derivatives of curvature tensors in
the context of field theory. However, another very important and immediate
application of such results lies in obtaining perturbative solutions for
Einsteins’ equations near some known solution, so we make our derivations
directly in this latter context. Once we have the desired results at hand, we
directly interpret them in terms of infinitesimal variations.

We start by considering a dynamic equation for a generic field variable $g$
(concretely, in our context of interest, this will be Einstein’s Equations for
the metric), which can be put in the form:

$\displaystyle\mathcal{E}(g)=0,$ (16)

where $\mathcal{E}$ is some local differential functional of $g$.

Let us supose now that we know an exact solution $\prescript{0}{}{g}$ of (16),
and that we are considering a situation in which the deviation from a certain
solution of interest, $g$, with respect to $\prescript{0}{}{g}$ is small. In
fact, to express this assertion in a mathematically meaningful way, we assume
the existence of a (1-parameter) family of exact solutions of (16),
$g(\lambda)$, i.e.

$\displaystyle\mathcal{E}\bigl{(}g(\lambda)\bigl{)}=0,\qquad\forall\lambda\in
I$ (17)

(where the domain $I\subset\mathbb{R}$ contains an open interval around $0$),
such that:

$\displaystyle(i)$ $\displaystyle\,g(\lambda)\text{ is a differentiable
function of }\lambda$ (18) $\displaystyle(ii)$
$\displaystyle\,g(0)=\prescript{0}{}{g}$ (19)

One may then think of $\lambda$ as a parameter that quantifies the deviation
of some (unknown) exact solution $g(\lambda)$ to our known solution
$\prescript{0}{}{g}$. This way, for arbitrarily small values of $\lambda$, we
obtain solutions that will be arbitrarily close to $\prescript{0}{}{g}$.
However, since (16) may be too difficult (or impossible) to solve exactly, we
may then obtain an approximate solution by noting that

$\displaystyle\frac{d}{d\lambda}\Bigl{[}\mathcal{E}\bigl{(}g(\lambda)\bigl{)}\Bigl{]}=0$
(20)

(since (17) is valid for all $\lambda\in I$). Particularly, this equality must
hold for $\lambda=0$. If we then define a perturbation in our field as $\delta
g\equiv\lambda\gamma$, that is:

$\displaystyle\gamma\equiv\frac{dg(\lambda)}{d\lambda}\biggl{|}_{\lambda=0},$
(21)

then, by Leibniz’s rule, it is easy to see that (20) will give us a linear
equation for $\gamma$, in the form

$\displaystyle\mathscr{L}(\gamma)=0,$ (22)

where $\mathscr{L}$ is a linear local differentiable operator. One then calls
(22) the ‘linearization of (16) around $\prescript{0}{}{g}$’.

Then, our case of interest will be when $g$ represents the metric field
$g_{ab}$ and $\mathcal{E}(g_{ab})=G_{ab}$, such that (16) will represent the
exact Einstein Equations in the vacuum555 For simplicity, our presentation
will be focused on the vacuum case _for the unperturbed equations_ , but it is
straightforward to generalize it to account for matter sources.  (which can
actually be written simply as $R_{ab}=0$). In this case, let us explicitly
derive the _linearized_ equations (22) for the perturbation in the metric.

To achieve that, we must calculate the Ricci tensor $R_{ab}(\lambda)$
associated with the metric $g_{ab}(\lambda)$ in a useful expression. More
specifically, we want an expression for it in terms of the background metric
$\prescript{0}{}{g}_{ab}$ and with explicit algebraic functions of $\lambda$,
so that we may clearly take derivatives with respect to it. The challenge in
doing that lies in the fact that the curvature is only indirectly defined in
respect to the metric, through covariant derivatives (9), which are bound to
obey eq (4). We then begin by noting that the covariant derivatives
$\prescript{\lambda}{}{\nabla}_{\\!a}$ and $\prescript{0}{}{\nabla}_{\\!a}$
($\prescript{\lambda}{}{\nabla}_{\\!a}g_{bc}(\lambda)=0=\prescript{0}{}{\nabla}_{\\!a}\prescript{0}{}{g}_{bc}$)
differ when acting on cotangent vectors by a connection in the form:

$\displaystyle\prescript{\lambda}{}{\nabla}_{\\!a}\omega_{b}=\prescript{0}{}{\nabla}_{\\!a}\omega_{b}-C^{c}_{\;ab}(\lambda)\omega_{c}.$
(23)

If we then write $C^{c}_{\;ab}(\lambda)$ in terms of
$\prescript{0}{}{\nabla}_{\\!a}$ and $g_{ab}(\lambda)$, we obtain

$\displaystyle
C^{c}_{\;ab}(\lambda)=\tfrac{1}{2}g^{cd}\bigl{[}\prescript{0}{}{\nabla}_{\\!a}g_{bd}(\lambda)+\prescript{0}{}{\nabla}_{\\!b}g_{ad}(\lambda)-\prescript{0}{}{\nabla}_{\\!d}g_{ab}(\lambda)\bigl{]}.$
(24)

Then, with a little algebraic effort, we can compute the Riemann curvature
tensor:

$\displaystyle
R_{abc}^{\;\;\;\;\;d}(\lambda)=\prescript{0}{}{R}_{abc}^{\;\;\;\;\;d}+2\prescript{0}{}{\nabla}_{\\![a}C^{d}_{\;b]c}-2C^{e}_{\;c[a}C^{d}_{\;b]e},$
(25)

from which we obtain the Ricci:

$\displaystyle
R_{ac}(\lambda)=\prescript{0}{}{R}_{ac}+2\prescript{0}{}{\nabla}_{\\![a}C^{b}_{\;b]c}-2C^{e}_{\;c[a}C^{b}_{\;b]e}.$
(26)

Then, differentiating this entire expression with respect to $\lambda$ and
evaluating it at $\lambda=0$, we obtain

$\displaystyle\dot{R}_{ac}=2\prescript{0}{}{\nabla}_{\\![a}\dot{C}^{b}_{\;b]c},$
(27)

where we are using a dot to denote a derivative with respect to $\lambda$ at
$\lambda=0$. We note that the term quadratic in $C^{c}_{\;\,ab}$ will not
yield any contribution, since $C^{c}_{\;\,ab}(\lambda\\!=\\!0)=0$.

Thus, denoting the metric derivatives by $\gamma$,
$\gamma_{ab}\equiv\dot{g}_{ab}$, and by noting that
$\prescript{0}{}{\nabla}_{\\!a}\prescript{0}{}{g}_{bc}=0$, we can easily
compute $\dot{C}^{c}_{\;ab}$, yielding

$\displaystyle\dot{C}^{c}_{\;ab}=\frac{1}{2}\prescript{0}{}{g}^{cd}\bigl{[}\prescript{0}{}{\nabla}_{\\!a}\gamma_{bd}+\prescript{0}{}{\nabla}_{\\!b}\gamma_{ad}-\prescript{0}{}{\nabla}_{\\!d}\gamma_{ab}\bigl{]}.$
(28)

Then, subtistituting on (27), we obtain

$\displaystyle\dot{R}_{ac}=\frac{1}{2}\prescript{0}{}{g}^{bd}\bigl{[}\prescript{0}{}{\nabla}_{\\!a}\prescript{0}{}{\nabla}_{\\!c}\gamma_{bd}+\prescript{0}{}{\nabla}_{\\!b}\prescript{0}{}{\nabla}_{\\!d}\gamma_{ac}-2\prescript{0}{}{\nabla}_{\\!b}\prescript{0}{}{\nabla}_{\\!(c}\gamma_{a)d}\bigl{]}.$
(29)

At this point, we simplify our notation, dropping the prescript 0 for
background quantities and using the background metric to raise and lower
indices. In this notation, we obtain:

$\displaystyle\dot{R}_{ac}=\tfrac{1}{2}\nabla_{\\!a}\nabla_{\\!c}\gamma+\tfrac{1}{2}\nabla^{b}\nabla_{\\!b}\gamma_{ac}-\nabla^{b}\nabla_{\\!(c}\gamma_{a)b},$
(30)

where we have defined $\gamma\equiv g^{ab}\gamma_{ab}$.

Now, if we multiply this entire equation by an infinitesimal variation
$\lambda$, we obtain the form for infinitesimal variations of the curvature,
in the familiar notation of chapter 2:

$\displaystyle\delta{R}_{ac}=\tfrac{1}{2}g^{bd}\nabla_{\\!a}\nabla_{\\!c}\delta
g_{bd}+\tfrac{1}{2}\nabla^{b}\nabla_{\\!b}\delta
g_{ac}-\nabla^{b}\nabla_{\\!(c}\delta g_{a)b}.$ (31)

As a final remark, we note that variations of the inverse metric $g^{ab}$ are
_not_ simply given by raising the indexes of $\delta g_{ab}$ with the
unperturbed metric. We can calculate these by using that
$g^{ab}g_{bc}=\delta^{a}_{\;c}$, and thus

$\displaystyle 0=\delta(g^{ab}g_{bc})=g_{bc}\delta g^{ab}+g^{ab}\delta
g_{bc}\qquad\Rightarrow\qquad\delta g_{ab}=-g_{ad}g_{bc}\delta g^{dc}.$ (32)

###  3 Lie derivatives, Killing fields, and conformal transformations

We have already seen that a crucial operational toolbox to handle curved
spaces (in the form of smooth manifolds $(\mathcal{M},g_{ab})$) is their
differential structure. We have throughout been using tangent spaces at each
point to define tensor fields and compute many local quantities in our theory.
Correspondingly, when one must handle _extensive_ geometrical quantities and
operations (such as finite arclengths or displacements, the parallel transport
of vectors and tensors, etc.) one must develop a suitable integral structure
to extend differential structures throughout spacetime.

In doing so, a central geometrical structure are _vector fields_ , which we
can used to build integral orbits (curves) and meaningfully transport local
quantities throughout spacetime. Let us consider a differentiable vector field
$\xi^{a}$ defined on the spaces tangent to $\mathcal{M}$ at each event; we can
use this field to find integral curves $C\subset\mathcal{M}$ (such that
$\xi^{a}$ will be tangent to them at each event), and we can define a
1-parameter family of diffeomorphisms
$\phi:\mathbb{R}\times\mathcal{M}\longrightarrow\mathcal{M}$ which act
translating all points along these curves by a variable amount. More
precisely, for any parameters $t,s\in\mathbb{R}$, we will have diffeomorphisms
obeying

$\displaystyle\phi_{t+s}=\phi_{t}\circ\phi_{s},\qquad\Rightarrow\qquad\phi_{0}=\mathbb{1}_{\mathcal{M}}.$
(33)

These diffeomorphisms will induce natural maps between tangent vectors (or,
more generally, tensors) at point $p$ and tangent vectors (tensors) at points
$\phi_{t}(p)$ along its orbits. These maps are called pushforwards (as they
“push tensors forward” from $p$ to $\phi_{t}(p)$) and they are denoted
$\phi_{t}^{*}$:

$\displaystyle
T^{a_{1}...a_{l}}_{b_{1}...b_{m}}\in\mathcal{T}_{p}(l,m)\;\longrightarrow\;\phi^{*}_{t}T^{a_{1}...a_{l}}_{b_{1}...b_{m}}\in\mathcal{T}_{\phi_{t}(p)}(l,m).$
(34)

With these maps, one can define a particular notion of a derivative to a
tensor field along the integral orbits of $\xi^{a}$, the so-called Lie
Derivative. Since we cannot in general subtract tensors defined at different
points (i.e., at different tangent spaces), a way to evaluate their difference
at the points $p$ and $\phi_{t}(p)$ is to “pull the latter back to $p$” by the
induced map $\phi^{*}_{-t}$. The Lie derivative is thus defined as

$\displaystyle\pounds_{\xi}(T^{a_{1}...a_{l}}_{b_{1}...b_{m}})_{p}=\lim_{t\rightarrow
0}\biggl{[}\frac{(\phi^{*}_{-t}T^{a_{1}...a_{l}}_{b_{1}...b_{m}})_{p}-(T^{a_{1}...a_{l}}_{b_{1}...b_{m}})_{p}}{t}\biggl{]}.$
(35)

On the first sight, it may seem that this would just yield a directional
derivative along $\xi^{a}$,
$\xi^{c}\nabla_{\\!c}T^{a_{1}...a_{l}}_{b_{1}...b_{n}}$. Note, however, that
we have not imposed any restrictions in the magnitudes or orientations of
$\xi^{a}$ throughout $\mathcal{M}$ (except that they should vary smoothly),
such that fixed parameter displacement $t$ may produce displacements of varied
magnitudes and directions throughout spacetime666 This is particularly clear
when we consider, for instance, a rotation: the magnitudes of the displacement
for a fixed angular variation $\theta$ will produce larger displacements at
larger radii, and go towards different directions for each position.. Thus, a
Lie derivative will generally carry information about the variations of
$\xi^{a}$ as well, which will manifest in the form of terms proportional to
its covariant derivative.

We shall not derive here how to obtain an expression for the Lie derivative in
terms of purely geometrical operations (see appendix C of [5] for a complete
and pedagogical derivation), but we quote here the result, which we shall use
throughout this thesis:

$\displaystyle\pounds_{\xi}T^{a_{1}...a_{l}}_{b_{1}...b_{m}}=\xi^{c}\nabla_{\\!c}T^{a_{1}...a_{l}}_{b_{1}...b_{m}}-\sum_{i=1}^{l}T^{a_{1}...c...a_{l}}_{b_{1}...b_{m}}\nabla_{\\!c}\xi^{a_{i}}+\sum_{j=1}^{m}T^{a_{1}...a_{l}}_{b_{1}...c...b_{m}}\nabla_{\\!b_{j}}\xi^{c}.$
(36)

A particular important class of diffeomorphisms in $(\mathcal{M},g_{ab})$ are
isometries, that is, transformations that leave the metric field $g_{ab}$
invariant. A vector field that generates a 1-parameter family of isometries
$\phi_{t}^{*}$, $\phi^{*}_{t}g_{ab}=g_{ab}$ is called a _Killing Field_. We
immediately see from equation (35) that the Lie derivative of the metric along
any Killing field vanishes. Then, eq (36) yields

$\displaystyle\pounds_{\xi}g_{ab}=\nabla_{\\!a}\xi_{b}+\nabla_{\\!b}\xi_{a}=2\nabla_{\\!(a}\xi_{b)}=0,$
(37)

since $\nabla_{c}g_{ab}=0$.

In fact, in terms of the Lie derivative in respect to a Killing field, we may
then define a precise notion of what it means for the metric to remain
constant or vary throughout spacetime, since its covariant derivative is
trivially null. We can say that Minkowski spacetime, for instance, has a
constant metric _everywhere_ , since it is maximally symmetric, and one can
get from any event $p$ into any distinct event $q$ by following the integral
orbits of a Killing field (i.e. by following an isometry). In fact, the same
is true for any homogeneous spacetimes, such as Einstein’s Static Universe, or
de Sitter spaces.

Equation (37) is called the Killing equation, and, by solving it, one can find
the generators of various isometries in one spacetime, if it has any. Another
very important family of vector fields in a spacetime are the so-called
_conformal_ Killing fields. They obey a relation similar to (37), called the
conformal Killing equation:

$\displaystyle\pounds_{\xi}g_{ab}=2\nabla_{\\!(a}\xi_{b)}=\lambda g_{ab}.$
(38)

That is, the metric can only vary along a conformal Killing field parallel to
itself. Thus, the associated integral transformations, called _conformal
transformations_ 777In the literature, they are sometimes referred to as
conformal isometries, but we avoid this term since it may be misleading. will
at most stretch or contract the metric, but will always preserve angles. We
note, however, that not all conformal transformations (and not all isometries)
must belong to a continuous group generated by a (conformal) Killing field.
Generally, one can write a conformal transformation $\psi$ as a spacetime
diffeomorphism, whose induced map in the metric ($\psi^{*}g_{ab}$) will act in
the form

$\displaystyle
g_{ab}(x)\longrightarrow\tilde{g}_{ab}(x)=\Omega^{2}(x)g_{ab}(x),$ (39)

where $\Omega^{2}(x)>0$ is a positive function of spacetime.

Conformal transformations are particularly useful in GR because they allow one
to distort spacetime distances while preserving all angles. Particularly, it
is obvious from (39) that a vector will be spacelike, timelike or null with
respect to $\tilde{g}_{ab}$ if, and only if, it is respectively spacelike,
timelike or null with respect to $g_{ab}$. This means that any two conformally
related spacetimes will have the same causal structure, even though they may
have widely different geometries.

For the aforementioned reasons, it can be often more convenient to carry a
geometrical or dynamical analysis originally defined in one spacetime in
another, conformally related one. For such, it is useful to write geometric
tensors from one spacetime in terms of the other. If $\nabla_{a}$ and
$\tilde{\nabla}_{a}$ are the covariant derivatives related to $g_{ab}$ and
$\tilde{g}_{ab}$, respectively, we may express their difference through a
connection $C^{c}_{\;ab}$, defined by

$\displaystyle\tilde{\nabla}_{\\!a}\omega_{b}=\nabla_{\\!a}\omega_{b}-C^{c}_{\;ab}\omega_{c},$
(40)

such that it can be written as

$\displaystyle C^{c}_{\;ab}$
$\displaystyle=\tfrac{1}{2}\tilde{g}^{cd}\bigl{[}{\nabla}_{\\!a}\tilde{g}_{bd}+{\nabla}_{\\!b}\tilde{g}_{ad}-{\nabla}_{\\!d}\tilde{g}_{ab}\bigl{]}$
$\displaystyle=\Omega^{-1}\bigl{[}2\delta^{c}_{(a}\nabla_{\\!b)}\Omega-
g_{ab}g^{cd}\nabla_{d}\Omega\bigl{]}.$ (41)

From this connection, one can derive with some algebraic effort the values of
the tensor curvatures $\tilde{R}_{abc}^{\;\;\;\;d}$, $\tilde{R}_{ab}$ and
$\tilde{R}$ in terms of their conformally related counterparts and $\Omega$.
Particularly, we will be interested in the Ricci curvature and the curvature
scale, which read

$\displaystyle\tilde{R}^{b}_{\;a}$
$\displaystyle=\;\Omega^{-2}\bigl{\\{}R^{b}_{\;a}-(n-2)g^{bc}\nabla_{\\!c}\nabla_{\\!a}(\ln\Omega)-\delta^{b}_{\;a}g^{cd}\nabla_{\\!c}\nabla_{\\!d}(\ln\Omega)$
$\displaystyle\quad+(n-2)g^{bc}(\nabla_{\\!c}\ln\Omega)(\nabla_{\\!a}\ln\Omega)-(n-2)\delta^{b}_{\;a}g^{cd}(\nabla_{\\!c}\ln\Omega)(\nabla_{\\!d}\ln\Omega)\bigl{\\}},$
(42) $\displaystyle\tilde{R}\;\;$
$\displaystyle=\;\Omega^{-2}\bigl{\\{}R-2(n-1)g^{cd}\nabla_{\\!c}\nabla_{\\!d}\ln\Omega-(n-1)(n-2)g^{cd}(\nabla_{\\!c}\ln\Omega)(\nabla_{\\!d}\ln\Omega)\bigl{\\}}.$
(43)

Finally, we would like to analyze how our field equations transform upon a
conformal transformation. Since we are often interested in working in simpler,
conformally related spacetimes than that of direct interest to our problem, it
would be particularly convenient to know if there is some conformal scaling
that we may perform in our field, $\phi\rightarrow\tilde{\phi}=\Omega^{s}\phi$
(where $s$ is called a conformal weight), so that the _form_ of the field
equations remains invariant. More precisely, if our field equations are
defined by spacetime differential operator $L_{x}$, we would like to find an
invariance in the form

$\displaystyle
L_{x}\phi(x)=0\quad\Leftrightarrow\quad\tilde{L}_{x}\tilde{\phi}(x)=0,$ (44)

where $\tilde{L}_{x}$ is the conformally transformed operator.

A particularly simple case is that of massless scalar field for which
$L=\Box=g^{ab}\nabla_{\\!a}\nabla_{\\!b}$. We see that a conformally
transformed equation would read

$\displaystyle
0=\tilde{g}^{ab}\tilde{\nabla}_{\\!a}\tilde{\nabla}_{\\!b}\tilde{\phi}$
$\displaystyle=\Omega^{-2}g^{ab}\bigl{[}\nabla_{\\!a}\nabla_{\\!b}(\Omega^{s}\phi)-C^{c}_{\;ab}\nabla_{\\!c}(\Omega^{s}\phi)\bigl{]}$
$\displaystyle=\Omega^{s-2}g^{ab}\nabla_{\\!a}\nabla_{\\!b}\phi+(2s+n-2)\Omega^{s-3}g^{ab}\nabla_{\\!a}\Omega\nabla_{\\!b}\phi$
(45) $\displaystyle\quad+s\Omega^{s-3}\phi
g^{ab}\nabla_{\\!a}\nabla_{\\!b}\Omega+s(n+s-3)\Omega^{s-4}\phi
g^{ab}\nabla_{\\!a}\Omega\nabla_{\\!b}\Omega.$

For this equation to be made equivalent with
$g^{ab}\nabla_{\\!a}\nabla_{\\!b}\phi=0$, we must require that all terms
except the first identically vanish. Well, we can immediately see that, for
$n\neq 2$, there will be no choice of $s$ that allows for such cancelling.
However, a very convenient covariant way of modifying this field equation
(which furthermore recovers the same field equation $\Box\phi=0$ in flat
spaces) is by adding a coupling with scalar curvature, making

$\displaystyle L=g^{ab}\nabla_{\\!a}\nabla_{\\!b}+\xi R,\qquad\xi=cte.$ (46)

In this case, one can verify that there is indeed a special value of $\xi$
(which will depend on $n$) that will produce a convenient cancellation of
terms in the modified equation, namely:

$\displaystyle\xi(n)=\frac{n-2}{4(n-1)}.$ (47)

With this value, we can put our equations in a conformally invariant form by
choosing the appropriate conformal weight $s=\frac{2-n}{2}$:

$\displaystyle\Bigl{[}\Box+\frac{n-2}{4(n-1)}R(x)\Bigl{]}\phi(x)\rightarrow\Bigl{[}\tilde{\Box}+\frac{n-2}{4(n-1)}\tilde{R}(x)\Bigl{]}\tilde{\phi}(x)=\Omega^{s-2}(x)\Bigl{[}\Box+\frac{n-2}{4(n-1)}R(x)\Bigl{]}\phi(x).$
(48)

Particularly, for $n=4$, we have $\xi(4)=1/6$ and $s=-1$.

Finally, we note that we cannot maintain conformal invariance if we add a mass
term to our equation (unlike the other two terms it will scale as
$\Omega^{s}$, not $\Omega^{s-2}$). One can interpret this fact by noting that
$m$ will introduce a natural (inverse) length scale to the theory; since we
keep this scale unchanged when we operate a conformal transformation (which
distorts distances), this will necessarily provoke a nontrivial distortion in
relative scales in our theory.