A modified gravity model coupled to a Dirac field in 2D space–times with quadratic nonmetricity and curvature

Abstract

After summarizing the basic concepts for the exterior algebra, we first discuss the gauge structure of the bundle over base manifold for deciding the form of the gravitational sector of the total Lagrangian in any dimensions. Then we couple minimally a Dirac spinor field to our gravitational Lagrangian 2-form which is quadratic in the nonmetricity and both linear and quadratic in the curvature in two dimensions. Subsequently, we obtain field equations by varying the total Lagrangian with respect to the independent variables. Finally, we find some classes of solutions of the vacuum theory and then a solution of the Dirac equation in a specific background and analyze them.

Weyl covariance, second clock effect and proper time in theories of symmetric teleparallel gravity

“…b where T denotes the transpose matrix. This shows that the transformation elements, L a a , obtained from a local GCT generate the Lorentz group SO (1,3) in O F * (M) [16,35,36]. That is the reason for {e a } to be called the Lorentzian co-frame.…”

“…This is called the coincident gauge. 1 Of course, one can pass to a different gauge (or coordinate system) via a local GCT (16) such that ω μ ν = 0, but still Q μ ν = 0, T μ = 0 and R μ ν = 0. In the case of coincident gauge the standard rule of parallel transport of a tangent vector (34) yields…”

“…However, there is a plenty of work containing nonmetricity which is performed in the context of modified theories of gravity in literature, e.g. see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein. Thus since it is to be worthy to strive for clarifying if there is definitely a SCE or a way for avoiding that in spacetimes with non-metricity, there have been researches [24][25][26][27][28][29][30][31][32][33][34].…”

Weyl covariance, second clock effect and proper time in theories of symmetric teleparallel gravity

“…This choice is called as the coincident gauge. Of course, one can pass to a different gauge (or coordinate system) via a local GCT (16). In the case of coincident gauge the standard rule of parallel transport of a vector (34) yields…”

“…We prefer to work in the othonormal co-frame because of user friendly and coordinate independent behavior of the Hodge dual star, * 1 = e 0 ∧ e 1 ∧ e 2 ∧ e 3 . Besides, one can consult for [15], [16] to see how the gauge spirit leads to that lagrangian. It is invariant under the Lorentz transformation meaning that gauge group is SO(1, 3).…”