Tesi etd-04092021-195317 |
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Tipo di tesi
Tesi di laurea magistrale
Autore
ZUCCOTTI, ALESSANDRO
URN
etd-04092021-195317
Titolo
Wormhole solutions in Einstein-Weyl gravity
Dipartimento
FISICA
Corso di studi
FISICA
Relatori
relatore Prof. Bonanno, Alfio
relatore Prof. Anselmi, Damiano
relatore Prof. Anselmi, Damiano
Parole chiave
- relatività
- gravitazione
- simmetria
- quantistica
- quantum
- wormhole
- fakeons
- asymptotic
- Riemann
- Kretschmann
- gravitazionale
- gravitational
- safety
- Bonanno
- Anselmi
- weyl
- einstein
- gravità
- gravity
- quadratic
- gravitation
- buchi neri
- curvatura
- force
- forza
- black hole
- generale
- Ricci
- spazio tempo
- relativity
- spacetime
- curvature
- general
- vuoto
- vacuum
- statiche
- static
- solutions
- wormholes
- spherical
- metric
- symmetry
- sferica
Data inizio appello
26/04/2021
Consultabilità
Tesi non consultabile
Riassunto
In this work we are going to study wormhole solutions in Einstein-Weyl gravity.
Such solutions emerge when looking for a static spherically symmetric metric in the
more general context of classical quadratic gravity.
Classical quadratic gravity is the theory of gravitation that comes out when including
quadratic terms in the curvature in the Einstein-Hilbert action of general relativity.
The study of such theory is motivated by the presence of quadratic corrections in
almost all the attempts to find a consistent description of quantum gravity. Indeed,
it is well known that general relativity can be consistent as a quantum field theory
only as a low-energy eective theory. We are not going to discuss the quantum
aspects of the quadratic action: instead, we consider what happens to the classical
description of the space-time when quadratic corrections are taken into account. In
order to do that, we restrict to the simplest non-trivial case, that is a static spherically symmetric vacuum space-time. Given these restrictions in general relativity,
we have the well known Schwarzschild solution, i.e. black hole solution. In classical
quadratic gravity the Schwarzschild solution is still present, but we can also find
many different classes of solutions: the aim of this thesis is to classify the various
solutions families, as well as to characterize a specific family that covers a large part
of the solutions space, i.e. the wormhole solutions.
We solved the geodesic equation in such solutions which shows the reason why we
call them traversable wormholes: an observer that passes through the radius r0 of
the wormhole does not reach the region r < r0, but ends up in a new copy of r > r0.
We reported all the solutions families found in the previous work and we added a
new subfamily of the generic wormholes solutions that we have discovered in our
analysis. In contrast to what happens in general relativity, the various solutions
found in quadratic gravity are not always asymptotically flat.
When studying the different classes of solutions we are assisted by a Lichnerowicz type theorem which removes the contributions of the R^2
term from the equations
of motion under some assumptions, in particular when an horizon is present. When
such contribution is absent, the quadratic theory reduces to Einstein-Weyl gravity.
By numerically solving the equations of motion in the Einstein-Weyl theory, we
have classified the various solutions families in a phase diagram of the theory, restricting to the asymptotically flat solutions. By using the shooting method for the
boundary value problem between spatial infinity and the radius r0, we found for
the rst time the geometric properties of the wormhole solutions, and in particular
we characterized the behavior of these solutions in function of their position on the
phase diagram. Then we used these results to explore both the interior r < r0 of the
wormholes and the new copy of r > r0 that emerges in such solutions. We proposed
a way to extend the solutions for the interior of the wormholes r < r0, and we made
an analysis of the metric in this region. However, a precise method to connect the
interior and the exterior of such solutions is still lacking.
The last part is dedicated to explore the new region r > r0, which is completely
determined by the solutions around r0. We have done a qualitative analysis of the
behavior of the metric in this new copy of r > r0, discovering that this new region
has a finite volume for r → ∞ for almost all the solutions, due to a strongly non-flat
behavior of the metric.
Finally we concluded by showing what happens to an observer that falls into these
solutions, finding that the surface r → ∞ in the new region of the wormholes acts
like a strongly attractive surface.
Such solutions emerge when looking for a static spherically symmetric metric in the
more general context of classical quadratic gravity.
Classical quadratic gravity is the theory of gravitation that comes out when including
quadratic terms in the curvature in the Einstein-Hilbert action of general relativity.
The study of such theory is motivated by the presence of quadratic corrections in
almost all the attempts to find a consistent description of quantum gravity. Indeed,
it is well known that general relativity can be consistent as a quantum field theory
only as a low-energy eective theory. We are not going to discuss the quantum
aspects of the quadratic action: instead, we consider what happens to the classical
description of the space-time when quadratic corrections are taken into account. In
order to do that, we restrict to the simplest non-trivial case, that is a static spherically symmetric vacuum space-time. Given these restrictions in general relativity,
we have the well known Schwarzschild solution, i.e. black hole solution. In classical
quadratic gravity the Schwarzschild solution is still present, but we can also find
many different classes of solutions: the aim of this thesis is to classify the various
solutions families, as well as to characterize a specific family that covers a large part
of the solutions space, i.e. the wormhole solutions.
We solved the geodesic equation in such solutions which shows the reason why we
call them traversable wormholes: an observer that passes through the radius r0 of
the wormhole does not reach the region r < r0, but ends up in a new copy of r > r0.
We reported all the solutions families found in the previous work and we added a
new subfamily of the generic wormholes solutions that we have discovered in our
analysis. In contrast to what happens in general relativity, the various solutions
found in quadratic gravity are not always asymptotically flat.
When studying the different classes of solutions we are assisted by a Lichnerowicz type theorem which removes the contributions of the R^2
term from the equations
of motion under some assumptions, in particular when an horizon is present. When
such contribution is absent, the quadratic theory reduces to Einstein-Weyl gravity.
By numerically solving the equations of motion in the Einstein-Weyl theory, we
have classified the various solutions families in a phase diagram of the theory, restricting to the asymptotically flat solutions. By using the shooting method for the
boundary value problem between spatial infinity and the radius r0, we found for
the rst time the geometric properties of the wormhole solutions, and in particular
we characterized the behavior of these solutions in function of their position on the
phase diagram. Then we used these results to explore both the interior r < r0 of the
wormholes and the new copy of r > r0 that emerges in such solutions. We proposed
a way to extend the solutions for the interior of the wormholes r < r0, and we made
an analysis of the metric in this region. However, a precise method to connect the
interior and the exterior of such solutions is still lacking.
The last part is dedicated to explore the new region r > r0, which is completely
determined by the solutions around r0. We have done a qualitative analysis of the
behavior of the metric in this new copy of r > r0, discovering that this new region
has a finite volume for r → ∞ for almost all the solutions, due to a strongly non-flat
behavior of the metric.
Finally we concluded by showing what happens to an observer that falls into these
solutions, finding that the surface r → ∞ in the new region of the wormholes acts
like a strongly attractive surface.
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