Complexity Of Black Holes In Nonlinear Electrodynamics And Chern-simons Modified Gravity

AdS/CFT duality has provided a brand new perspective to learn theoretical physics.It reveals that the conformal field theory on the boundary is somehow equivalent to the bulk of the anti-de Sitter space-time,which brings us great convenience to study the complicated conformal field on the boundary.Recently,Susskind et.al.proposed a conjecture of complexity/action duality,which says that the quantum computational complexity of the boundary state of a black hole is equivalent to the action in the Wheeler-DeWitt patch,or more simply,the complexity growth rate of a black hole is equivalent to its action growth rate.Based on this hypothesis,we calculate the complexity growth rate of two types of black holes.The first type is those in the framework of the coupled theory of non-linear electrodynamics and Einstein gravity where the double horizon black holes exist.In this thesis,we start from the action of the static spherically symmetric metric and calculate both the electric potential and the action on the Wheeler-DeWitt patch.We prove that the action growth rate is equal to the difference of the product of charge and electric potential evaluated on the inner and outer horizons,which finally gives us the complexity growth rate agreeing with the special cases of Reissner-Nordstr?m black hole and Born-Infeld black hole.Another example is the black hole in Chern-Simons modified gravity.Again,we start from its action and calculate the complexity of spherically symmetric black holes,which leads no modification compared with situations in Einstein gravity.The black hole which gives non-trivial modification is the slowly rotating black hole.After careful calculations,we get its action growth rate,which successfully goes back to the result of Schwarzschild black holes and Kerr black holes.In light of the conjecture of complexity/action duality,these results may have implications on the quantum complexity of a state in the dual field theory.