Chapter 1
Einstein Gravity and the Need for Its
Modification
This review presents itself as a collection of the lecture notes on modified gravity
based on lectures given at UFPB (Joao Pessoa, Brazil), CBPF (Rio de Janeiro, Brazil),
UFC (Fortaleza, Brazil), and Universidad del Bio-Bio (Concepcion, Chile).
General relativity (GR) is clearly one of the most successful physical theories.
Being formulated as a natural development of the special relativity, it has made a
number of fundamental physical predictions which have been confirmed experimentally with a very high degree of precision. Among these predictions, the special role
is played by expansion of the Universe and precession of Mercure perihelion, which
have been proved many years ago, while other important claims of GR such as gravitational waves and black holes, have been confirmed through direct observations
only recently.
By its concept, the general relativity is an essentially geometric theory. Its key
idea consists in the fact that the gravitational field manifests itself through modifications of the space-time geometry. Thus, one can develop a theory where the fields
characterizing geometry, that is, metric and connection, become dynamical variables
so that a non-trivial space can be described in terms of curvature and/or torsion. It has
been argued in [1] that there are eight types of geometry characterized by possibilities of zero or non-zero curvature tensor, torsion and so-called homothetic curvature
tensor, with all these objects are constructed on the base of metric and connection.
Nevertheless, the most used formulation of the gravity is based on the Riemannian
approach where the connection is symmetric and completely characterized by the
metric. Within these lecture notes, we present namely Riemannian description of
gravity where the action is described by functions of geometric invariants described
exclusively in terms of metric (i.e. various contractions of Riemann curvature tensor, its covariant derivatives and a metric), and possibly some extra fields, scalar or
vector ones. So, let us introduce some basic definitions of quantities used within
Riemannian approach.
By definition, the infinitesimal interval in a curved space-time is given as
ds
2
= g μν (x)dx
μ dx
ν . The metric tensor g μν (x) is considered as the only independent
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_1
1
Einstein Gravity and the Need for Its
Modification
This review presents itself as a collection of the lecture notes on modified gravity
based on lectures given at UFPB (Joao Pessoa, Brazil), CBPF (Rio de Janeiro, Brazil),
UFC (Fortaleza, Brazil), and Universidad del Bio-Bio (Concepcion, Chile).
General relativity (GR) is clearly one of the most successful physical theories.
Being formulated as a natural development of the special relativity, it has made a
number of fundamental physical predictions which have been confirmed experimentally with a very high degree of precision. Among these predictions, the special role
is played by expansion of the Universe and precession of Mercure perihelion, which
have been proved many years ago, while other important claims of GR such as gravitational waves and black holes, have been confirmed through direct observations
only recently.
By its concept, the general relativity is an essentially geometric theory. Its key
idea consists in the fact that the gravitational field manifests itself through modifications of the space-time geometry. Thus, one can develop a theory where the fields
characterizing geometry, that is, metric and connection, become dynamical variables
so that a non-trivial space can be described in terms of curvature and/or torsion. It has
been argued in [1] that there are eight types of geometry characterized by possibilities of zero or non-zero curvature tensor, torsion and so-called homothetic curvature
tensor, with all these objects are constructed on the base of metric and connection.
Nevertheless, the most used formulation of the gravity is based on the Riemannian
approach where the connection is symmetric and completely characterized by the
metric. Within these lecture notes, we present namely Riemannian description of
gravity where the action is described by functions of geometric invariants described
exclusively in terms of metric (i.e. various contractions of Riemann curvature tensor, its covariant derivatives and a metric), and possibly some extra fields, scalar or
vector ones. So, let us introduce some basic definitions of quantities used within
Riemannian approach.
By definition, the infinitesimal interval in a curved space-time is given as
ds
2
= g μν (x)dx
μ dx
ν . The metric tensor g μν (x) is considered as the only independent
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_1
1