我們從黑洞熱力學出發，並將宇宙常數與壓力做連結，在黑洞熱力學第一定律中加入壓力項，來討論黑洞相變相關的問題。而所使用的方法為Ruppeiner geometry，其概念是用黎曼幾何，將熱力學中的變量如溫度、體積等等作為參數，把一個狀態演化為另一個狀態的機率看作是流形中兩點的距離，距離越長代表機率越小，越短則代表機率越高。並有相關研究指出Ruppeiner geometry 所推導出的純量曲率發散點與熱力學中的spinodal point 一致，並且純量曲率之正負也與系統中微觀結構上的交互作用是排斥力還是吸引力有關。本文會回顧前人以溫度及體積作為參數所看到的現象，之後在RN-AdS black holes 的案例中加入電荷項，考慮允許黑洞與外界交換電荷之情況，推導出其度規張量並計算純量曲率，觀察在允許黑洞與外界交換電荷的情況下與其他情況之差異。;We start from black hole thermodynamics, and associate cosmology constant with pressure. Then we add the pressure term into the first law of black hole thermodynamics, to discuss some properties about black hole phase transition. We use the Ruppeiner geometry, which is analogous to the Riemannian geometry to study black hole thermodynamics by treating temperature, volume and charge as parameters. We regard the probability of one state evolving to another state as the distance between two points. Longer distance represents smaller probability, and shorter distance represents bigger probability of transition between two states. It was pointed out that the scalar curvature divergent points are the same as the spinodal points of thermodynamics. Moreover, the sign of scalar curvature is also related to whether the interaction on the microstructure is a repulsive force or attractive force. This thesis will review the phenomena that previous researchers had discussed with temperature and volume as thermodynamic variables. Later, we study the charged RN-AdS black holes, considering the situation that black holes can exchange charge with outside. We derive the metric tensor, calculate the scalar curvature, and finally discuss the difference between the case in which black holes exchange charge or not with outside.
