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With a focus on modified gravity this book presents a review of the
recent developments in the fields of gravity and cosmology,
presenting the state of the art, high-lighting the open problems,
and outlining the directions of future research. General Relativity
and the CDM framework are currently the standard lore and
constitute the concordance paradigm of cosmology. Nevertheless,
long-standing open theoretical issues, as well as possible new
observational ones arising from the explosive development of
cosmology in the last two decades, offer the motivation and lead a
large amount of research to be devoted in constructing various
extensions and modifications. In this review all extended theories
and scenarios are first examined under the light of theoretical
consistency, and are then applied in various geometrical
backgrounds, such as the cosmological and the spherical symmetric
ones. Their predictions at both the background and perturbation
levels, and concerning cosmology at early, intermediate and late
times, are then confronted with the huge amount of observational
data that astrophysics and cosmology has been able to offer in the
last two decades. Theories, scenarios and models that successfully
and efficiently pass the above steps are classified as viable and
are candidates for the description of Nature, allowing readers to
get a clear overview of the state of the art and where the field of
modified gravity is likely to go. This work was performed in the
framework of the COST European Action "Cosmology and Astrophysics
Network for Theoretical Advances and Training Actions" - CANTATA.
This book provides an itinerary to quantum mechanics taking into
account the basic mathematics to formulate it. Specifically, it
features the main experiments and postulates of quantum mechanics
pointing out their mathematical prominent aspects showing how
physical concepts and mathematical tools are deeply intertwined.
The material covers topics such as analytic mechanics in Newtonian,
Lagrangian, and Hamiltonian formulations, theory of light as
formulated in special relativity, and then why quantum mechanics is
necessary to explain experiments like the double-split, atomic
spectra, and photoelectric effect. The Schroedinger equation and
its solutions are developed in detail. It is pointed out that,
starting from the concept of the harmonic oscillator, it is
possible to develop advanced quantum mechanics. Furthermore, the
mathematics behind the Heisenberg uncertainty principle is
constructed towards advanced quantum mechanical principles.
Relativistic quantum mechanics is finally considered.The book is
devoted to undergraduate students from University courses of
Physics, Mathematics, Chemistry, and Engineering. It consists of 50
self-contained lectures, and any statement and theorem are
demonstrated in detail. It is the companion book of "A Mathematical
Journey to Relativity", by the same Authors, published by Springer
in 2020.
Beyond Einstein's Gravity is a graduate level introduction to
extended theories of gravity and cosmology, including variational
principles, the weak-field limit, gravitational waves, mathematical
tools, exact solutions, as well as cosmological and astrophysical
applications. The book provides a critical overview of the research
in this area and unifies the existing literature using a consistent
notation. Although the results apply in principle to all
alternative gravities, a special emphasis is on scalar-tensor and
f(R) theories. They were studied by theoretical physicists from
early on, and in the 1980s they appeared in attempts to renormalize
General Relativity and in models of the early universe. Recently,
these theories have seen a new lease of life, in both their metric
and metric-affine versions, as models of the present acceleration
of the universe without introducing the mysterious and exotic dark
energy. The dark matter problem can also be addressed in extended
gravity. These applications are contributing to a deeper
understanding of the gravitational interaction from both the
theoretical and the experimental point of view. An extensive
bibliography guides the reader into more detailed literature on
particular topics.
Beyond Einstein's Gravity is a graduate level introduction to
extended theories of gravity and cosmology, including variational
principles, the weak-field limit, gravitational waves, mathematical
tools, exact solutions, as well as cosmological and astrophysical
applications. The book provides a critical overview of the research
in this area and unifies the existing literature using a consistent
notation. Although the results apply in principle to all
alternative gravities, a special emphasis is on scalar-tensor and
f(R) theories. They were studied by theoretical physicists from
early on, and in the 1980s they appeared in attempts to renormalize
General Relativity and in models of the early universe. Recently,
these theories have seen a new lease of life, in both their metric
and metric-affine versions, as models of the present acceleration
of the universe without introducing the mysterious and exotic dark
energy. The dark matter problem can also be addressed in extended
gravity. These applications are contributing to a deeper
understanding of the gravitational interaction from both the
theoretical and the experimental point of view. An extensive
bibliography guides the reader into more detailed literature on
particular topics.
This book provides an itinerary to quantum mechanics taking into
account the basic mathematics to formulate it. Specifically, it
features the main experiments and postulates of quantum mechanics
pointing out their mathematical prominent aspects showing how
physical concepts and mathematical tools are deeply intertwined.
The material covers topics such as analytic mechanics in Newtonian,
Lagrangian, and Hamiltonian formulations, theory of light as
formulated in special relativity, and then why quantum mechanics is
necessary to explain experiments like the double-split, atomic
spectra, and photoelectric effect. The Schroedinger equation and
its solutions are developed in detail. It is pointed out that,
starting from the concept of the harmonic oscillator, it is
possible to develop advanced quantum mechanics. Furthermore, the
mathematics behind the Heisenberg uncertainty principle is
constructed towards advanced quantum mechanical principles.
Relativistic quantum mechanics is finally considered.The book is
devoted to undergraduate students from University courses of
Physics, Mathematics, Chemistry, and Engineering. It consists of 50
self-contained lectures, and any statement and theorem are
demonstrated in detail. It is the companion book of "A Mathematical
Journey to Relativity", by the same Authors, published by Springer
in 2020.
With a focus on modified gravity this book presents a review of the
recent developments in the fields of gravity and cosmology,
presenting the state of the art, high-lighting the open problems,
and outlining the directions of future research. General Relativity
and the CDM framework are currently the standard lore and
constitute the concordance paradigm of cosmology. Nevertheless,
long-standing open theoretical issues, as well as possible new
observational ones arising from the explosive development of
cosmology in the last two decades, offer the motivation and lead a
large amount of research to be devoted in constructing various
extensions and modifications. In this review all extended theories
and scenarios are first examined under the light of theoretical
consistency, and are then applied in various geometrical
backgrounds, such as the cosmological and the spherical symmetric
ones. Their predictions at both the background and perturbation
levels, and concerning cosmology at early, intermediate and late
times, are then confronted with the huge amount of observational
data that astrophysics and cosmology has been able to offer in the
last two decades. Theories, scenarios and models that successfully
and efficiently pass the above steps are classified as viable and
are candidates for the description of Nature, allowing readers to
get a clear overview of the state of the art and where the field of
modified gravity is likely to go. This work was performed in the
framework of the COST European Action "Cosmology and Astrophysics
Network for Theoretical Advances and Training Actions" - CANTATA.
This book opens with an axiomatic description of Euclidean and
non-Euclidean geometries. Euclidean geometry is the starting point
to understand all other geometries and it is the cornerstone for
our basic intuition of vector spaces. The generalization to
non-Euclidean geometry is the following step to develop the
language of Special and General Relativity. These theories are
discussed starting from a full geometric point of view.
Differential geometry is presented in the simplest way and it is
applied to describe the physical world. The final result of this
construction is deriving the Einstein field equations for
gravitation and spacetime dynamics. Possible solutions, and their
physical implications are also discussed: the Schwarzschild metric,
the relativistic trajectory of planets, the deflection of light,
the black holes, the cosmological solutions like de Sitter,
Friedmann-Lemaitre-Robertson-Walker, and Goedel ones. Some current
problems like dark energy are also scketched. The book is
self-contained and includes details of all proofs. It provides
solutions or tips to solve problems and exercises. It is designed
for undergraduate students and for all readers who want a first
geometric approach to Special and General Relativity.
This volume summarizes the many alternatives and extensions to
Einstein's General Theory of Relativity, and shows how symmetry
principles can be applied to identify physically viable models. The
first part of the book establishes the foundations of classical
field theory, providing an introduction to symmetry groups and the
Noether theorems. A quick overview of general relativity is
provided, including discussion of its successes and shortcomings,
then several theories of gravity are presented and their main
features are summarized. In the second part, the 'Noether Symmetry
Approach' is applied to theories of gravity to identify those which
contain symmetries. In the third part of the book these selected
models are tested through comparison with the latest experiments
and observations. This constrains the free parameters in the
selected models to fit the current data, demonstrating a useful
approach that will allow researchers to construct and constrain
modified gravity models for further applications.
This book presents information on the study of gravitational
theories which can be seen as modifications or extensions of
General Relativity. The basic principles that a gravitational
theory should follow, and their geometrical interpretation, are
analysed in a broad perspective which highlights the basic
assumptions of General Relativity and suggests possible
modifications. In particular, the invariance principles that are
related to the gravitational field are presented, showing how
gravity can emerge as a gauge theory. Also, the boundle approach to
gravity and derive geometrical quantities which intervene in the
construction of gravitational field are discussed.
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