Massive Gravity Theory And Its Applications In Gauge/gravity Duality

General relativity proposed by Einstein is the most successful theory of gravity until today,the equations of motion of which are the Einstein equations.The Einstein equations can be viewed as the equations of motion of a massless spin-2 graviton,and a natural idea of modified theories of gravity is to consider a massive graviton,and this is massive gravity theory.Unlike Einstein's gravity theory,massive gravity theory does not enjoy diffeomorphism invariance.The original massive gravity theory,i.e.Fierz-Pauli(FP)theory has the van Dam-Veltman-Zakharov(vDVZ)discontinuity.Though this can be solved by the Vainshtein mechanism of the nonlinear theory,the nonlinear theory suffers from the Boulware-Deser(BD)ghost problem.The de Rham-Gabadadze-Tolley(dRGT)massive gravity theory built later is a general nonlinear massive gravity theory which can evade the BD ghost.This theory contains two fundamental tensor fields,the spacetime metric which is dynamical,and the reference metric which is a fixed background.The reference metric can take different forms,so the diffeomor-phism invariance is broken in different ways.Gauge/gravity duality renders a classi-cal gravity system in asymptotically anti-de Sitter(AdS)spacetime correspondent to a quantum field theory system on the spacetime boundary.This is a strong-weak duality,and a weakly coupled gravity system is dual to a strongly coupled field theory system.By studying this weakly coupled gravity system,one can obtain some properties of the dual strongly coupled field theory system.In recent years,a model of dRGT theory which contains a degenerate reference metric has been applied to gauge/gravity dual-ity to describe the condensed matter system with the translational symmetries broken.This has drawn wide attention and has caused wide discussions.We derive the boundary counterterms of the gravity action by analyzing the asymptotic behavior of the action near the boundary.These counterterms cancels the divergent part of the action and the boundary energy-momentum tensor.We obtain finite thermodynamic quantities and the conserved charge,and the results are the same as those in the literature except some finite constant terms.We also introduce the generalized pseudo inverse of a degenerate tensor on a Lorentz manifold and rigorously derive the equations of motion for a degen-erate reference metric.By using the equations of motion we study the uniquess of the solutions of the model and prove a Birkhoff type theorem.