ETD

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Tesi etd-05172018-101035


Tipo di tesi
Tesi di laurea magistrale
Autore
LADU, ROBERTO
URN
etd-05172018-101035
Titolo
On some positive mass theorems for Chern-Gauss-Bonnet masses.
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Dott. Mari, Luciano
correlatore Prof. Malchiodi, Andrea
Parole chiave
  • positive mass
  • mass
  • Lovelock tensors
  • geometric analysis
  • general relativity
  • Gauss-Bonnet
  • differential geometry
  • conformally flat
  • Chern
  • ADM mass
  • riemannian geometry
  • spin manifolds
  • Witten
Data inizio appello
08/06/2018
Consultabilità
Completa
Riassunto
An asymptotically flat (AF) manifold is a Riemannian manifold that models a spacelike hypersurface in a Lorentzian spacetime obeying the Einstein field equations
\begin{equation}\label{EinsteinFieldEquation}
\mathcal{E}^g_2 = T
\end{equation}
(where $\mathcal{E}^g_2 = \mathrm{Ric}_g - \frac 1 2 S_g g$ is the Einstein tensor of $g$ and $T$ is the stress-energy tensor)
and representing an \emph{isolated} gravitational system.
A prototype example is the Schwarzschild space that models
a spherically symmetric static mass distribution, like an idealized sun in an otherwise empty universe.
For AF manifolds there is a well established notion of total mass of the system, the ADM mass (named after Arnowitt,Deser and Misner \cite{ADM1, ADM2, ADM3}),
that is given as a kind of flux integral.
More precisely, if $ (g_{ij} )_{i j} $ are the components of the metric in a suitable coordinate system, the ADM mass is given by (up to a multiplicative constant)
\begin{equation}
m(g) = \lim_{r\to \infty} \int_{S_r} (g_{ji,i} - g_{ii,j}) \partial_j\ \lrcorner \ dV,
\end{equation}
where $S_r$ is a coordinate sphere of radius $r$ and $dV$ is the Euclidean volume element.
In the case of a Schwarzschildean space, $m(g)$ coincides with the mass of the system $m$.
From the definition though, it is not clear whether the ADM mass is positive.
The positive mass conjecture (PMC) states that an AF manifold of non negative scalar curvature and suitable decaying condition at infinity
has non negative ADM mass.
The hypothesis of non negative scalar curvature is not restrictive as it is inherited from the dominant energy condition of the spacetime when the spacelike hypersurface is totally geodesic.
Two important results in this direction are due to Schoen and Yau and Witten.
The former two authors proved \cite{SchoenYau} the PMC in dimension less than seven, and the latter \cite{Witten} for spin manifolds in any dimensions.
Recently \cite{SchoenYau2017} Schoen and Yau announced they have found a proof valid for any dimension, currently their paper is under review.
The PMC has played an important role also in Riemannian Geometry, since \cite{SchoenYau} allowed Schoen to deal with the cases that were left open in the famous Yamabe problem, a challenging question arising from conformal geometry.
\begin{equation*}
\quad
\end{equation*}

As a matter of fact, Einstein gravity is the simplest example of gravitational theories based on tensor identities of type
\begin{equation}
\mathcal{L} = T,
\end{equation}
where $\mathcal{L}$ is a symmetric, divergence-free and natural tensor of the second order (see \cite{Garraffo}).
These tensors arise from higher order corrections to the Einstein-Hilbert action in any sensible theory of quantum gravity and are regarded by physicists as some of the most natural generalizations of the Einstein-Hilber action to dimension larger than four (cf. \cite{Camanho}).
The tensors $\mathcal{L}$, constitute a vector space that has been exhaustively studied by Lovelock in the 70's, providing
a set of generators now known as Lovelock tensors $\{\mathcal{E}_{2k}^g\}_{k\in \mathbb{N}}$.
In particular, in $k$-pure Gauss-Bonnet gravities the Einstein field equation \eqref{EinsteinFieldEquation} is replaced by
\begin{equation}
\mathcal{E}^g_{2k} = T
\end{equation}
where $\mathcal{E}_{2k}^g$ is the $2k$-Lovelock tensor, given in a local orthonormal frame $\{e_i\}_{i=1,\dots, n}$ by
\begin{equation}
\mathcal{E}_{2k}^g (e_i, e_j) = -\frac{1}{2^{k+1}} \delta_{j,j_1,\dots, j_{2q}}^{i,i_1,\dots, i_{2q}}R^{i_1,i_2}_{j_1,j_2}\cdots R^{i_{2k-1},i_{2q}}_{j_{2q-1},j_{2q}},
\end{equation} here $R$ is the Riemann curvature tensor of $g$
and $\delta^{M_1,\dots, M_p}_{L_1,\dots, L_p} = \det (\delta_{L_i}^{M_j})_{i,j}$ is the generalized Kronecker delta. The Einstein tensor is then obtained as a particular case when $k=1$.

The aim of this thesis was to investigate the definitions of mass $m_k$ for pure Lovelock gravities, focusing on their geometrical interpretation,
together with some related positive mass theorems.
The analogous of the ADM mass for $k$-pure Gauss-Bonnet gravity is the $k$-Chern-Gauss-Bonnet mass (CBG mass) $m_k$, that includes as a special case for $k=1$ the ADM mass. CGB masses have been introduced only recently, firstly in 2014 for the case $k=2$ by Ge,Wang and Wu \cite{WangWuNewMass} and then in 2017 for general $k$ by the latter two authors \cite{WangWuCGB}.
As for the case of the ADM mass, the positivity of the CBG masses has been conjectured ($k$-PMC) but, to the authors knowledge, it has been proved only in two cases: for conformally flat manifolds \cite{GeWangWuConformallyFlat} and for Euclidean graphs \cite{WeiXiong, WangWuNewMass,GiraoSousa}.
We aim to give an unified treatment of these masses, discussing several equivalent definitions of them, in particular we worked on a variational characterization of $m_k$ that seems to be new.
We also provide proofs, of the $k$-PMC, recently appeared in the literature
together with Witten's proof for spin manifold \cite{Witten}. In doing so, we make use of the double forms formalism
that allows to efficiently compute curvature invariants and to highlight the symmetry of the relevant geometric quantities.
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