Conserved Quantities And Symmetries For The F(T) And F(R)Gravity Theories

In the modified gravity theories, the explicit expressions of Lagrangian decide the exact solutions of the field equations and are important to gain insight into the physical content of the theories. Noether’s theory offers us a useful tool to research the conserved quantities and symmetries of the modified gravity theories. f(T) theory, a generally modified teleparallel gravity, and f(R) theory have been proposed to account for the dark energy phenomena. By the Noether symmetry approach, we investigate the condi-tions of the power law form, exponential form and polynomial form with corresponding conserved quantities and symmetries. All forms concerned in this work possess the time translational symmetry, which is related with energy condition or Hamilton constraint. Constructing the viable cosmological model for the scale in the solar system, the form of theories need to approach the teleparallel gravity or General Relativity. So it is more suited to add a linear term to the form as T+Tn or R+Rn. The corresponding Noether symmetry only exists the time translational symmetry. Through numerically calculating and observation data fitting, we get the optimal solution αT+βT-1for f(T) theory, which is close to the ACDM. Due to the corresponding Hubble quantity is a litter larger than the standard model, the cosmic age is lightly reduced. Proper constraints on the ratio β/α are also given, which could be converted into constraints on the ratio of the magnitude of torsion between large and small scales. And we find that αT+β√T is equivalent to γT for the same solution of H(z).In the last part, we prove that Birkhoff s theorem holds not only in the diagonal tetrad case, but also the off diagonal tetrad case. Moreover, we discuss respectively the results of the external vacuum and internal gravitational field in f(T) gravity frame-work, as well as the extended meaning of this theorem. We also investigate the validity of Birkhoff s theorem in the frame of f(T) gravity via conformal transformation by regarding the Brans-Dicke-like scalar as effective matter, and study the equivalence be-tween both Einstein frame and Jordan frame. Then we also investigate specifically the range of validity of Birkhoff’s theorem with the general tetrad field via perturbative ap- proach. At zero order, Birkhoff s theorem is valid and the solution is the well known Schwarzschild-(A)dS metric. Then considering the special case of the diagonal tetrad field, we present a new spherically symmetric solution in the frame of f(T) gravity up to the perturbative order. The results with the diagonal tetrad field satisfy the physical equivalence between the Jordan and the so-called Einstein frames, which are realized via conformal transformation, at least up to the first perturbative order.