~josealberto4444/phd-thesis

706ff9835a4a372171cb066148b32822fe53bd5d — José Alberto Orejuela García 2 months ago 97db7bb
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In this thesis we will study Lovelock Theories, that is, some extensions to General Relativity with particularly good properties, for example, giving second-order differential equations and having Levi-Civita connection as a solution of first-order formalism. Despite their advantages, these theories had never been studied so deeply and in this thesis we will present several new results.

First of all, we explain basic concepts and set the mathematical base. In second chapter, we study the Einstein-Hilbert action. We will see that the solution to the metric-affine formalism is not only the Levi-Civita connection, but a set of connection that we will call Palatini connections. In third chapter, we talk about general properties of every Lovelock Theory, especially about projective invariance, which explains why Palatini connections are solutions of these theories. Finally, we study the Gauss-Bonnet action and we give a non-trivial solution of metric-affine formalism that is physically distinguishable of Levi-Civita, hence demonstrating the non-equivalence between metric and metric-affine formalisms.
First of all, we explain basic concepts and set the mathematical base. In second chapter, we study the Einstein-Hilbert action. We will see that the solution to the metric-affine formalism is not only the Levi-Civita connection, but a set of connections that we will call Palatini connections. In third chapter, we talk about general properties of every Lovelock Theory, especially about projective invariance, which explains why Palatini connections are solutions of these theories. Finally, we study the Gauss-Bonnet action and we give a non-trivial solution of metric-affine formalism that is physically distinguishable of Levi-Civita, hence demonstrating the non-equivalence between metric and metric-affine formalisms.

\begin{center}
  \rule{0.618\textwidth}{0.4pt}

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\vspace{0.3cm}
\noindent
\foreignlanguage{spanish}{Garantizamos, al firmar esta tesis doctoral, que el trabajo ha sido realizado por la doctoranda bajo la dirección de los directores de la tesis y hasta donde nuestro conocimiento alcanza, en la realización del trabajo, se han respetado los derechos de otros autores a ser citados cuando se han utilizado sus resultados o publicaciones.}
\foreignlanguage{spanish}{Garantizamos, al firmar esta tesis doctoral, que el trabajo ha sido realizado por el doctorando bajo la dirección de los directores de la tesis y hasta donde nuestro conocimiento alcanza, en la realización del trabajo, se han respetado los derechos de otros autores a ser citados cuando se han utilizado sus resultados o publicaciones.}

\vspace{0.3cm}
\noindent

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  \label{eq:recurrence-same-Riemann}
  \nabla_\fii B_\fiii = B_\fii B_\fiii.
\end{equation}
Apart from this highly exceptional case, the symmetric part of the Ricci tensor does not coincide con Levi-Civita any more, so we cannot recover the Einstein equation from it and then we cannot recover the dynamics.
Apart from this highly exceptional case, the symmetric part of the Ricci tensor does not coincide with Levi-Civita any more, so we cannot recover the Einstein equation from it and then we cannot recover the dynamics.

\subsection{Geodesic deviation}
As geodesics are the same, we are going to take a look at the geodesic deviation. We look at this because in geodesic deviation we can see contributions of the Riemann tensor itself, not through its contraction, the Ricci tensor. As the Riemann tensor of the Palatini connections has a greater discrepancy of its Levi-Civita equivalent than the Ricci tensor, this is an interesting case.