\chapter{A new window on the Universe}
\label{GW}
The approach given in this thesis to introduce gravitational waves starts from the concept of gravity: gravity is not a force, but it is a property of the spacetime. This statement has been proved by General Relativity.\\
In this chapter we will briefly go through the reason why gravitational waves are an astrophysical phenomenon and how it is possible to detect them.\\
The purpose of the writer is to offer a contribution to the improvement of techniques of gravitational waves detection, extremely sophisticated in practice as well as elegant and straightforward in theory.

\section{Space, time and gravitation}
%RELATIVITA' GENERALE E EEs
%In the first years of XVIII century, the assumption that the fundamental laws of physics were invariant for all inertial observers was undisputed; however, they find hard applications in the electromagnetic theory: this led Einstein to develop the theory of relativity. He found that the Maxwell's equations imply that the electromagnetic fiends propagate in space as waves at the speed of light and that they are not invariant for all inertial observers in the Newtonian dynamics, but they are invariant in a frame were the space and time coordinates are linked together to be a unified system. Thus, for inertial observers moving with different relative velocities, space and time form a four-dimensional structure, called \textit{spacetime}.\\


%\noindent
The replacement of the Newtonian concepts of absolute space and absolute time with the merged concept of \textit{spacetime} introduced the Special Relativity\footnote{It is not in the aims of this work to demostrate the Theory of Relativity: a suggested detailed resource about the important transition between classical to relativistic physics is \cite{wei}.}. In this theory the light speed is assumed as preserved everywhere and every time for all interactions. However, gravitation seems not following this rule: the gravitational attraction seems to be instantaneous, or to propagate with \textit{infinite} speed.\\
%As well as the Newtonian space and time concepts have been modified to adapt to electromagnetic propagation
The Special Relativity then needs to be modified to adapt to gravity, taking into account its features: wherever there is matter, there is a gravitational effect, which is always present and acts as a radial inverse-square law. When trying to apply Newtonian gravitation to Special Relativity, the problem to solve becomes non-linear \cite{nar}; the key feature used by Einstein to adapt Special relativity to the presence of gravitation, is to consider that the permanence of gravitational effects makes gravity to be an intrinsic feature of space and time. Einstein's intuition marks the transition to the General Relativity: gravity is not anymore a force but an effect on the \textit{geometry} of the spacetime.\\

\noindent
General Relativity is then built on a non-Euclidean geometry able to explain the presence of gravity without considering it as a force: in such a geometry, matter under no force should move in straight like paths with uniform speeds.\\
This is possible if the spacetime is \textit{curved}.

\paragraph{Physics in a curved spacetime}
In General Relativity, the way to describe the physics in the curved spacetime modifies the usual Cartesian coordinate system used to measure the distance $s$ in space and time for two observers in an inertial frame:

\begin{equation}
\centering
ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2
\end{equation}

\noindent
where $c = 3 \times 10^8$ m/s is the speed of light, with the following:

\begin{equation}
\centering
ds^2 = \Sigma_{i,k = 0}^{3}  g_{ik}dx^{i}dx^{k}
\label{a}
\end{equation}

\noindent
where the coordinates are now called $x^i$, with $i$=1,2,3 representing the three space coordinates and $i$=0 is the time coordinate\footnote{From now on we will omit the $\Sigma$ symbol, thanks to the summation convention of indeces: \textit{whenever an index appears as subscribt and superscript in the same expression, it is summed over all values}.} . The new entry is the coefficient $g_{ik}$, which is a series of functions of $x^i$. Eq. \ref{a} describes the geometry of the spacetime: its properties depend on $g_{ik}$, which transforms as a covariant tensor\footnote{A covariant vector is a quantity that transforms as $A'_{k} = \frac{\partial x^i}{\partial x^{'k}} B_i $. A tensor T is defined as the 4 $\times$ 4 product of two vectors and a covariant tensor transforms as $T'_{ik} = \frac{\partial x^m}{\partial x^{'i}}\frac{\partial  x^n}{\partial x^{'k}} T_{mn}$.} and it is called \textit{metric tensor} (a simple demonstration of the tensorial property of $g_{ik}$ can be found in \cite{nar}).\\

\noindent
%All physical interactions and how they behave in presence of gravitation can be described in this geometry in terms of vectors and tensors and the effect of gravitation enters in a condition of \textit{curved spacetime}:
The defined metric explains how to measure distances, while in order to define parallel vectors in a curved spacetime (and build equations of motion in such conditions), we need functions $\Gamma$ of space and time that can describe variations of components of a vector $B_i$ during a displacement $\delta x^k$ along the curve:

\begin{equation}
\centering
\delta B_i = \Gamma^{l}_{ik} B_l \delta x^k
\end{equation}

\noindent
that are determined once the metric tensor is known \cite{nar}:

\begin{equation}
\centering
\Gamma^{l}_{ik} = \frac{1}{2} g^{im} \left(\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{lm}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m}\right).
\end{equation}
\\
\noindent
The spacetime curvature is instead described by the Riemann tensor:

\begin{equation}
\centering
R_{ikn}^{m} \equiv \frac{\partial \Gamma^{m}_{ik}}{\partial x^n} - \frac{\partial \Gamma^{m}_{in}}{\partial x^n} + \Gamma^{l}_{ik}\Gamma^{m}_{ln} - \Gamma^{l}_{in}\Gamma^{m}_{lk} = 0.
\end{equation}
\\
\noindent
Spacetime is \textit{flat} if the Riemann tensor is null everywhere, otherwise it is \textit{curved}.\\
In this frame, the energy-momentum tensor of matter $T_{ik}$ is defined from the action principle and a variation of the metric over a volume V:

\begin{equation}
\centering
\delta \Sigma cm_a \int ds_a = \frac{1}{2c} \int_{V} \delta g_{ik} T_{ik} d^{4}x
\end{equation}
\\
\noindent
where m is the mass of a particle a.\\
Now that we have the elements describing geometry, displacements, the curvature of the spacetime and the energy distribution, physical interactions can be built on them. In particular, we are interested in the gravitational fields.

\subsection{The Einstein's equations}
%NON DERIVARLE NEL DETTAGLIO, SPIEGA IL METODO EURISTICO
%SPIEGA LE APPROSSIMAZIONI PER MANIPOLARLE
The dynamical equations derivated by Einstein consider the energy-momentum tensor $T_{ik}$ acting as a source of gravity  propagating in the spacetime as a wave, having the form of series of wave equations for the metric $g_{ik}$ in presence of curvature\footnote{A heuristic derivation of the Eintein's equations can be found in \cite{nar}, while a derivation based on the non-linear effect of the gravitational fields can be found in \cite{wei}.}:

\begin{equation}
\centering
G_{ik} = R_{ik} - \frac{1}{2}g_{ik}R = \frac{8\pi G}{c^4}T_{ik}
\label{EE}
\end{equation}
\\
\noindent
with $G = 6,67 \time 10^{-11}$ Nm$^2$/Kg$^2$.
The Einstein's equations expressed in the form of eq. \ref{EE} are valid in the weak field (or Newtonian) approximation:
\begin{enumerate}
\item The motion of particles is non-relativistic.
\item The gravitational fields are weak: $g_{ik} = \eta_{ik} + h_{ik}$, where the term $\eta_{ik}$ = (-,+,+,+) expresses the flat-space metric and the perturbation is $\mid h_{ik}\mid$ $\ll$ 1.
\item The fields change slowly with time, so time derivatives are negligible with respect to space derivatives.
\end{enumerate}

\noindent
This approximation allows to handle the field equations in a linearized condition.

\paragraph{Gravitational-wave radiation}
%RISOLUZIONE DELLE EEs PER TROVARE L'EQ D'ONDA DELLE GWs\\
%SOLUZIONI DELL'EQ D'ONDA: COME SI PROPAGANO LE GW\\
The wave equations in the weak field approximation and under linearized conditions describe the gravitational radiation:

\begin{equation}
\centering
\square \bar{h}_{ik} = -\frac{16\pi G}{c^4}T_{ik},
\label{GW}
\end{equation}
\\
\noindent
where $\square = \eta_{ik}\partial ^{i} \partial ^{k}$ and $\bar{h}_{ik} = h_{ik} - \frac{1}{2}\eta_{ik}h$.\\
Physically, eqs. \ref{GW} say that bodies acting as sources of gravitational radiation move in a flat spacetime along a trajectory defined by their mutual influence. The gravitational field propagating in waves is known as \textit{Gravitational Waves}.\\
Solutions of the wave equations will tell how the gravitational waves propagate, how they interact with other bodies, the energy carried by the waves and the physical objects that could be gravitational waves sources.\\
If we consider what happens far from the source, i.e. when gravitational waves propagate, we need $T_{ik}$ = 0: in this way we just need to solve $\square \bar{h}_{ik}$ = 0. This gives a matrix-type form of the tensor when the wave propagates along the direction z\footnote{Full computations to derive the solutions of the wave equations for inspiriling masses can be found in \cite{mag}}:

\begin{equation}
\centering
h_{ab}^{TT}(t,z) =
\begin{bmatrix}
h_+ & h_\times & 0\\
h_\times & -h_+ & 0\\
0 & 0 & 0
\end{bmatrix}_{ab}\cos \left[\omega \left(t-\frac{z}{c}\right)\right].
\end{equation}
\\
\noindent
The TT index indicates a Transverse-Traceless condition of the solution. The two amplitudes of the gravitational wave depend on the mass of two objects at distance $r$ from each other \cite{mag}:

\begin{equation}
\centering
h_+ = \frac{4}{r} \left(\frac{GM_c}{c^2}\right)^{5/3} \left(\frac{\pi f_{gw}}{c}\right)^{2/3} \frac{1+\cos ^2 \theta}{2} \cos (2\pi f_{gw}t + 2\phi);
\end{equation}

\begin{equation}
\centering
h_\times = \frac{4}{r} \left(\frac{GM_c}{c^2}\right)^{5/3} \left(\frac{\pi f_{gw}}{c}\right)^{2/3} \cos \theta \sin (2\pi f_{gw}t + 2\phi)
\end{equation}

\noindent
where 0 $< \theta <$ $\pi/2$ and $f_{gw} = 2 \omega/(2\pi)$ is the frequency of the gravitational wave emitted. The mass of the two-system source of $m_1$ and $m_2$ each is defined as \textit{chirp mass}:

\begin{equation}
\centering
M_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}.
\end{equation}\\
\noindent
Typically, the amplitude of a gravitational wave is of the order of $h \sim 10^{-21} $ $1/\sqrt{Hz}$, very small: masses able to deform the fabric of the spacetime and generate gravitational waves are of the order of more than the solar mass $M_{\odot}$, so they need to be looked for in the Universe.
%servono masse molto grandi, quindi vanno cercate nell'Universo: possibili candidati

