On Three Special Topics Of Extended Theories Of Gravity

This thesis mainly studies three kind of models of extended gravity:critical gravity, pseudo-supergravity and f(R) gravity. In critical gravity, we study the parameter space of D-dimensional cosmological Einstein gravity together with quadratic curvature terms. We show that for appropriate choice of the parameters, there exists a critical point for one of the vacua, for which there are only massless tensor, but neither massive tensor nor scalar. At criticality, the linearized excitations have vanishing energy, as do asymptotically Kerr-AdS and Schwarzschild-AdS black holes. A further restriction of the parameters gives a one-parameter cosmological Einstein plus Weyl square model with a unique vacuum. We further perform linearized analysis around the black holes and show that in general the spec-trum consists of the usual spin-2 massless and ghost massive modes. At the critical point, the massless spin-2 modes have zero energy whilst the massive spin-2 modes are replaced by the log modes. There always exist certain linear combination of massless and log modes that has negative energy. Thus the stability of the black holes requires that the log modes to be truncated out by the boundary condition. In pseudo-supergravity, we find that there only exists two kinds of gravity-dilaton-form field systems which admit consistant Killing spinor equations, one is the low-energy effective action of the bosonic string and the other is Kaluza-Klein theory. When scalar potential is present, both theories can still admit Killing spinor equations. We show that the conformal anomaly term is the only scalar potential that one can add into the bosonic string action, and for Kaluza-Klein theory, one can add a special scalar potential which can generate an AdS spacetime. We also construct solutions of these theories. When pseudo-gravitino and dilatino fields are added to the Kaluza-Klein theory, we find that the full Lagrangian is invariant under the pseudo-supersymmetric trans-formation, up to quadratic order in fermion fields. We find that the theory possesses a U(1) global symmetry that can be gauged. The gauging process generates a scalar potential that has a maximum, leading to the AdS vacuum. Whist the highest dimension for gauged AdS supergravity is seven, our gauged AdS pseudo-supergravities can exist in arbitrary dimen-sions. In f(R) gravity, we derive the condition on f(R) gravities that admit Killing spinor equations and construct explicit such examples. The Killing spinor equations can be used to reduce the fourth-order differential equations of motion to the first order for both the domain wall and FLRW cosmological solutions. We obtain exact "BPS"domain walls that describe the smooth Randall-Sundrumâ…¡, AdS wormholes and the RG flow from IR to UV. We also obtain exact smooth cosmological solutions. An important feature is that the scalar curvature R of all these metrics is varying rather than a constant. Another intriguing feature is that there are two different f(R) gravities that give rise to the same "BPS" solution.