Related papers: Non-coordinates basis in General Relativity and Ca…
We extend the non-commutative coordinates relationship into other than the Minkowski space-time. We clarify the non-commutativity dependency to the geometrical structure. As well as, we find an inverse map between Riemann's normal and…
Strong field (exact) solutions of the gravitational field equations of General Relativity in the presence of a Cosmological Constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres-Szafron…
In complex general relativity, Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold, with holomorphic connection and holomorphic curvature tensor. A multisymplectic analysis shows that the Hamiltonian…
This paper reframes Riemannian geometry as a generalized Lie algebra allowing the equations of both RG and then General Relativity to be expressed as commutation relations among fundamental operators. We begin with an Abelian Lie algebra of…
We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that in…
An explicit, geometric description of the first-class constraints and their Poisson brackets for gravity in the Palatini-Cartan formalism (in space-time dimension greater than three) is given. The corresponding Batalin- Fradkin-Vilkovisky…
The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the…
Spacetime geometry is described by two -- {\em a priori} independent -- geometric structures: the symmetric connection $\Gamma$ and the metric tensor $g$. Metricity condition of $\Gamma$ (i.e. $\nabla g = 0$) is implied by the Palatini…
Non-geometric frames in string theory are related to the geometric ones by certain local O(D,D) transformations, the so-called $\beta$-transforms. For each such transformation, we show that there exists both a natural field redefinition of…
As true as it is that a bricklayer needs a plumb line and a T-square, so it is that a physicist using general relativity needs how to draw geodesics and use fields of congruent vector frames of reference. While the first part of the…
We use the theory of Cartan connections to analyze the geometrical structures underpinning the gauge-theoretical descriptions of the gravitational interaction. According to the theory of Cartan connections, the spin connection $\omega$ and…
Admitting non-Riemannian geometries, Double Field Theory extends the notion of spacetime beyond the Riemannian paradigm. We identify a class of singular spacetimes known in General Relativity with regular non-Riemannian geometries. The…
In this paper we study the possibility of assigning a geometric structure to the Lie groups. It is shown the Poincar\'{e} and Weyl groups have geometrical structure of the Riemann-Cartan and Weyl space-time respectively. The geometric…
Nonrelativistic string theory is described by a sigma model with a relativistic worldsheet and a nonrelativistic target spacetime geometry, that is called string Newton-Cartan geometry. In this paper we obtain string Newton-Cartan geometry…
A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played…
Two classic field theories of metric gravitation are given as constant-coefficient Exterior Differential Systems (EDS) on the flat orthonormal frame bundle over ten dimensional space. They are derivable by variation of Cartan 4-forms, and…
The gravitational instability of inhomogeneities in the expanding universe is studied by the relativistic second-order approximation. Using the tetrad formalism we consider irrotational dust universes and get equations very similar to those…
We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction…
We analyze the properties of foliations in presence of non-metricity, deriving the generalized Gauss-Codazzi relations in full generality. These results are employed to study the teleparallel framework of non-metric geometry, obtaining…
I consider an extension of General Relativity by an auxiliary non-dynamical dimension that enables our space-time to acquire an extrinsic curvature. Obtained gravitational equations, without or with a cosmological constant, have a…