Thermodynamics and phase transition of Black hole in high dimensional massive gravity Ruihong Yue Center for Gravity and Cosmology, Yangzhou University 5th Dark Matter Dark Energy and Matter-Antimatter Asymmetry， 2018.12.29 at FoGuangshan Collaborater: Decheng Zou, Yunqi Liu, Zhanying Yang Dr.s Ming Zhang, Bo Liu

Outline Part A: Introduction Part B : Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity Part C: Water-like Phase Transitions and Triple Points of higher-dimensional Ads-BH in dRGT Part D: Behavior of quasinormal modes and Van derWaals-like phase transition of charged AdS black holes in massive gravity Part E：Reentrant Phase Transitions and Triple Points of Topological AdS Black Holes in Born-Infeld-Massive Gravity Part F: Disscussion

2. Accelerated expansion of universe: cosmological constant Part A: Introduction Einstein: spin-2, massless Possible extension 2. Accelerated expansion of universe: cosmological constant dark matter dark energy 3. Massive gravity possibly provides an explanation for late-time acceleration

1. Fierz and Pauli：linear theory Dam，Veltman，Zakharov discontinuity Nonlinear theory：Boulware-Deser ghost 2. De Rham-Gabadadze-Tolley. spherically symmetric BH solution, charged BH, black string solution, BTZ-like BH 3. Thermodynamics Ads BH has been generalized into the extended phase space by treating cosmologic constant as the pressure of BH.

Outline Part A: Introduction Part B : Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity Part C: Water-like Phase Transitions and Triple Points of higher-dimensional Ads-BH in dRGT Part D: Behavior of quasinormal modes and Van derWaals-like phase transition of charged AdS black holes in massive gravity Part E：Reentrant Phase Transitions and Triple Points of Topological AdS Black Holes in Born-Infeld-Massive Gravity Part F: Disscussion

Λ: cosmological constant, m the mass Part B. Reentrant phase trans. AdS bh in dRGT massive gravity 1. Model and solution: Λ: cosmological constant, m the mass Consider a spherical symmetric solution: EPJC77:256, DOI 10.1140/epjc/s10052-017-4822-9

Final sol. Using the definition the horizon of BH, f(r+)=0, one can find the mass, entropy and temperature of BH to be

2. Critical Behavior of d-dim. Ads BH equation of state: Determine the critical point

d>5 Gibbs free energy：

P-r figure with w2=-1,w3=1：

G-T figure with w2=-1,w3=1,w4=-0.7

P-T figure with w2=-1,w3=1,w4=-0.7

Outline Part A: Introduction Part B : Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity Part C: Water-like Phase Transitions and Triple Points of higher-dimensional Ads-BH in dRGT Part D: Behavior of quasinormal modes and Van derWaals-like phase transition of charged AdS black holes in massive gravity Part E：Reentrant Phase Transitions and Triple Points of Topological AdS Black Holes in Born-Infeld-Massive Gravity PartF: Disscussion

1. Model and solution: BH sol. arxiv1810.07885 Part C. Water-like Phase Transitions and Triple Points in dRGT 1. Model and solution: N BH sol. arxiv1810.07885

Determining the critical point equation of state: Determining the critical point

P-r picture with different temperatures

G-T picture with different pressures

P-T phase transition picture

Case 2

Outline Part A: Introduction Part B : Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity Part C: Water-like Phase Transitions and Triple Points of higher-dimensional Ads-BH in dRGT Part D: Behavior of quasi-normal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity Part E：Reentrant Phase Transitions and Triple Points of Topological AdS Black Holes in Born-Infeld-Massive Gravity Part F: Disscussion

Part D: Behavior of quasinormal modes of charged AdS BH EPJC（2017）77:365

1.Perturbation of charged Ads-BH in massive gravity by massless scalar field satisfying Decomposing Psi in terms of radial function phi and harmonic function Ylm and an time oscillating factor

A:Isobaric phase trans. ( P fixed <Pc) q=2,m=1, c1=0.05,c2=2 Arrow indicates the increase of BH horizon radius

A: Isobar phase trans. ( P fixed=pc) B: Isothermal phase trans. ( T fixed<Tc)

B: Isothermal phase trans. ( T fixed<Tc)

B:Isothermal phase trans. ( T fixed, T=Tc)

Outline Part A: Introduction Part B : Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity Part C: Water-like Phase Transitions and Triple Points of higher-dimensional Ads-BH in dRGT Part D: Behavior of quasinormal modes and Van derWaals-like phase transition of charged AdS black holes in massive gravity Part E：Reentrant Phase Transitions and Triple Points of Topological AdS Black Holes in Born-Infeld-Massive Gravity Part F: Disscussion

The action of BIMG di=d-i Advance in High energy Physics (2017) 3819246 di=d-i

The thermodynamic quantities

3. There are Van der Waals SBH/LBH and RPT SBH/IBH in BI-dRGT. Discussion: 1.There are Van der Waals SBH/LBH and RPT SBH/IBH in dRGT. 2.The quasi-normal modes behave differently in SBH and LBH regions, indicating the PT of charged AdsBH in dRGT. 3. There are Van der Waals SBH/LBH and RPT SBH/IBH in BI-dRGT. 4. There are Water-like SBH/LBH , SBH/IBH and IBH/LBH in dRGT. 5. Expecting the rich structure of PT in N-dimension gravity with multi-parameters.

Thank you for your attention!

The spherical BH in 5D GB and perturbation of scaler field

A: Isobar phase trans. ( P fixed<pc)