Generally, The Time Evolution Of High-order Derivative Gravitational Complexity

Maldacena found the first concrete example of the holographic principle in 1997-Ad S/CFT correspondence.The Ad S/CFT correspondence indicates that the theory of gravity in the Anti-de Sitter spacetime of(d+1)dimension is equivalent to the conformal field theory of the d-dimensional boundary.As always,it is very difficult to calculate the complexity of black holes directly,except for a few simple models.The Ad S/CFT correspondence opens a new window on the complexity study of black holes.Based on Ad S/CFT correspondence,Susskind team successively proposed Complexity/Length duality and Complexity/Volume duality,and finally developed Complexity/Action duality after gradual improvement.The Complexity/ Action duality indicates the quantum computational complexity of d-dimensional boundary holographic states dual to the classical action of(d+1)dimension Wheeler-Dewitt patch.The Complexity/Action duality reduces the problem of complexity of a black hole to the calculation of the gravitational action.After several years of development,people have obtained the results of the full time evolution of the complexity of some black holes by using the Complexity/Action duality.The main work and arrangement of this thesis are as follows: In the first chapter,the introduction briefly introduces the background of the research,including the Complexity/Action duality and the higher derivative gravity;In chapter 2 and chapter 3,the complexity time evolution formula of neutral black hole in general higher derivative gravity theory is derived by using the Complexity/Action duality,and the complexity time evolution of several simple neutral black holes is discussed in detail by using the numerical method,including the planar Gauss-Bonnet black holes and the planar third order Lovelock black holes.In chapter 4,the complexity of charged black holes in general higher derivative gravity theory is studied,the complexity time evolution formula of charged black holes in general higher derivative gravity theory is derived,and the complexity time evolution of several simple charged black holes is discussed in detail by numerical method.In chapter 5,conclusion and discussion,summarizes the results of this thesis and prospects the future work.By research,We observe that at early times,the critical time of the Wheeler-Dewitt patch of neutral Lovelock black holes is a decreasing function of the higher order coupling constants,which implies that the complexity evolves of neutral Lovelock black holes faster than that of Schwarzschild black holes.At late times,the rate of change of complexity of neutral Lovelock black holes is essentially determined by the Gibbons-Hawking-York boundary term evaluated at the future singularity,and its ratio to black hole mass is a characteristic constant,independent of the higher order couplings.Thus,in the vanishing coupling limit,the result of the time evolution of complexity in general does not reduce to that of Schwarzschild black holes,in spite of that the metric reduces to the latter as well as the gravitational action.In fact,the two differ by a constant during the the time evolution of complexity.Including the next-to-leading order term around late times,we find that as the Einstein case,the late time limit is always approached from above,thus violating any conjectured upper bound given by the late time result.For charged Lovelock black holes,we find that with sufficient charge,the time evolution of complexity roughly behaves the same as the Einstein case.However,for smaller charges,the two have some significant differences.In particular,unlike the Einstein case,in the uncharged limit the complexity growth rate does not match with the neutral case,differing by a constant in the whole time evolution.