Chapter 5
(Non-)Uniqueness of Einstein–Palatini
Gravity
Bert Janssen, Alejandro Jiménez-Cano, José Alberto Orejuela, and
Pablo Sánchez-Moreno
Abstract We analyse the most general connection allowed by Einstein–Hilbert
theory in Palatini formalism. We also consider a matter lagrangian independent of
the affine connection. We show that any solution of the equation of the connection is
essentially Levi-Civita up to a term that contains an undetermined 1-form. Finally, it
is proved that these connections and Levi-Civita describe a completely equivalent
physics.
Talk given by A. J. C. and based on the publication (Bernal et al., Phys Lett B
768:280–287, 2017).
5.1 Introduction and Mathematical Notions
Since the publication of the Einstein’s theory of General Relativity in 1915, we
understand gravitation as a geometrical effect. Many extensions of this theory have
been formulated in order to solve various problems in theoretical physics, such as
dark matter or the first corrections to General Relativity that could come from the
quantum gravity regime.
In the geometrical framework introduced by Einstein, the spacetime is defined as
a differentiable manifold M. Omitting some mathematical details, a D-dimensional
manifold is essentially a topological space that looks, locally, as the Euclidean space
ℝ
D . For example, spheres, planes and hyperboloids are 2-dimensional manifolds.
B. Janssen · A. Jiménez-Cano (*) · J. A. Orejuela
Departamento de Física Teórica y del Cosmos, Universidad de Granada, Granada, Spain
Centro Andaluz de Física de Partículas Elementales, Universidad de Granada, Granada, Spain
e-mail: bjanssen@ugr.es; alejandrojc@ugr.es; josealberto@ugr.es
P. Sánchez-Moreno
Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada, Spain
e-mail: pablos@ugr.es
© Springer Nature Switzerland AG 2021
B. G. Sidharth et al. (eds.), Fundamental Physics and Physics Education Research,
https://doi.org/10.1007/978-3-030-52923-9_5
49
(Non-)Uniqueness of Einstein–Palatini
Gravity
Bert Janssen, Alejandro Jiménez-Cano, José Alberto Orejuela, and
Pablo Sánchez-Moreno
Abstract We analyse the most general connection allowed by Einstein–Hilbert
theory in Palatini formalism. We also consider a matter lagrangian independent of
the affine connection. We show that any solution of the equation of the connection is
essentially Levi-Civita up to a term that contains an undetermined 1-form. Finally, it
is proved that these connections and Levi-Civita describe a completely equivalent
physics.
Talk given by A. J. C. and based on the publication (Bernal et al., Phys Lett B
768:280–287, 2017).
5.1 Introduction and Mathematical Notions
Since the publication of the Einstein’s theory of General Relativity in 1915, we
understand gravitation as a geometrical effect. Many extensions of this theory have
been formulated in order to solve various problems in theoretical physics, such as
dark matter or the first corrections to General Relativity that could come from the
quantum gravity regime.
In the geometrical framework introduced by Einstein, the spacetime is defined as
a differentiable manifold M. Omitting some mathematical details, a D-dimensional
manifold is essentially a topological space that looks, locally, as the Euclidean space
ℝ
D . For example, spheres, planes and hyperboloids are 2-dimensional manifolds.
B. Janssen · A. Jiménez-Cano (*) · J. A. Orejuela
Departamento de Física Teórica y del Cosmos, Universidad de Granada, Granada, Spain
Centro Andaluz de Física de Partículas Elementales, Universidad de Granada, Granada, Spain
e-mail: bjanssen@ugr.es; alejandrojc@ugr.es; josealberto@ugr.es
P. Sánchez-Moreno
Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada, Spain
e-mail: pablos@ugr.es
© Springer Nature Switzerland AG 2021
B. G. Sidharth et al. (eds.), Fundamental Physics and Physics Education Research,
https://doi.org/10.1007/978-3-030-52923-9_5
49