Gravity with higher curvature terms BUM-HOON LEE SOGANG UNIVERSITY - - PowerPoint PPT Presentation
Gravity with higher curvature terms BUM-HOON LEE SOGANG UNIVERSITY Sogang University APRIL 12, 2017 1970 1975 : ,
BUM-HOON LEE SOGANG UNIVERSITY
Sogang University
APRIL 12, 2017
고 송희성 교수님 추모 심포지엄
1970년대 1975 : 관악 캠퍼스로 이전, 학부 4학년 시절 문리대 물리학과 + 공대 응용물리학과 + 사범대 물리교육학과 1976~ 1978 대학원 시절, 양자역학 수강 1980년대 1985년 경, 뉴욕 방문시, 1990 년대 및 2000년대 이론물리센터 (SRC) 물리학회 활동, 학회장 출마,
17 May 2014 THE 48TH WORKSHOP ON GRAVITATION AND NUMERICAL RELATIVITY FOR APCTP TOPICAL RESEARCH PROGRAM
고 송희성 교수님에 대한 기억 들
3
Werner Israel(1967), Brandon Carter(1971,1977), David Robinson (1975)
Stationary black holes (in 4-dim Einstein Gravity) are completely described by 3 parameters of the Kerr-Newman metric : mass, charge, and angular momentum (M, Q, J) Exists the minimum mass of BH Affects the stability, etc.
1) Effects to the Black Holes.
Hairy black hole solution ? In the dilaton-Gauss-Bonnnet theory → Yes!
2) Effects in the Early Universe. Low energy effective theory from string theory → Einstein Gravity + higher curvature terms Gauss-Bonnet term is the simplest leading term. Q : What is the physical effects of Gauss-Bonnet terms?
Werner Israel(1967), Brandon Carter(1971,1977), David Robinson (1975)
No-Hair Theorem of Black Holes
Stationary black holes (in 4-dim Einstein Gravity) are completely described by 3 parameters of the Kerr-Newman metric : mass, charge, and angular momentum (M, Q, J)
with angular momentum) has an ergoregion around the outside of the event horizon
time themselves are dragged along with the rotation of the black hole Hairy black hole solution is possible in the dilaton-Gauss- Bonnnet theory.
Colliding Galaxies: A Black Hole Merger
NASA / CXC / MPE / S. Komossa, et al.
Actual observations provide evidence and data for computer simulations. What does it look like when black holes collide?
Colliding Black Holes : A Black Hole Merger + Gravitational Wave Q: A Black Hole unstable ? splitting into two Black Holes ?
GW150914
(asymptotic) AdS Black Hole in d+1 dim ↔ Quantum System in d dim. Instability of Black Holes ↔ instability of Quantum System Hence, instability of AdS BH ↔ phase transitions in Quantum System
* * Black ho hole les i in hi higher r dim imensio ions are re quite te div ivers rse !
Action
where and
Horizon Note :
𝑒𝑡2= - (1-
2𝑁 𝑠 ) 𝑒𝑢2 + 𝑒𝑠2
(1−2𝑁
𝑠 ) +𝑠2𝑒Ω2
𝑠𝐼= 2𝑁
Black Hole solution 𝑇 = න 𝑒4𝑦 − 1 2𝜆2 𝑆 − 1 2 𝜈𝜉𝜖𝜈𝜚𝜖𝜉𝜚 𝜚 No hair
𝑠
𝐼= 2𝑁 2𝑁
𝑠𝐼 W.Ahn, B. Gwak, BHL, W.Lee, Eur.Phys.J.C (2015)
Action
where and The Gauss-Bonnet term :
Guo,N.Ohta & T.Torii, Prog.Theor.Phys. 120,581(2008);121 ,253 (2009); N.Ohta &Torii,Prog.Theor.Phys.121,959; 122,1477(2009);124,207 (2010); K.i.Maeda,N.Ohta Y.Sasagawa, PRD80, 104032(2009); 83,044051 (2011)
,064002 (2013).
1) The symmetry under
allows choosing γ positive values without loss of generality.
Note :
2) The coupling α dependency could be absorbed by the r → r/ α transformation. with non-zero α coupling cases being generated by α scaling. However, the behaviors for the α = 0 case cannot be generated in this way. Hence, we keep the parameter α, to show a continuous change to α = 0. Boundary term if 𝛿 = 0
I.e., there does not exist black hole solutions without a hair in DEGB theory. (If we have Φ = 0, dilaton e.o.m. reduces to 𝑆𝐻𝐶
2
= 0. so it cannot satisfy the dilaton e.o.m..) Hair Charge Q is not zero, and is not independent charge either.
Note :
The EGB black hole solution is the same as that of the Schwarzschild one. However, the GB term contributes to the black hole entropy and influence stability.
the limit of γ going to zero. These solutions depend on the coupling γ. Coupling γ dependency of the minimum mass for fixed α 1/16. Singular pt S & the min. mass C exist for γ = √2. Singular pt S coincides w/ pt C btwn γ=1.29(blue) & 1.30(cyan). No lower branch below γ=1.29 As γ→0, the solution →Schw BH.
𝑁) exist. Note : γ=√2(green), γ=1.3(cyan), γ=1. 29(blue ), γ=1/2(red), γ=1/6(black), γ=0(purple)
I.e., there are two black holes for a given mass in which the smaller one is unstable under perturbations.
Q: How about the properties, such as Stability Implication to the cosmology etc ?
𝑇 = න 𝑒4𝑦 − 1 2𝜆2 𝑆 − 1 2 𝜈𝜉𝜖𝜈𝜚𝜖𝜉𝜚 − 𝑊 𝜚 − 1 2 𝜊 𝜚 𝑆𝐻𝐶
2
𝑆𝐻𝐶
2
= 𝑆𝜈𝜉𝜍𝜏𝑆𝜈𝜉𝜍𝜏 − 4𝑆𝜈𝜉𝑆𝜈𝜉 + 𝑆2 Gauss-Bonnet term
Einstein and Field equations yield:
ሶ 𝐼 = − 𝜆2 2 ሶ 𝜚2 − 2𝐿 𝜆2𝑏2 − 4 ሷ 𝜊 𝐼2 + 𝐿 𝑏2 − 4 ሶ 𝜊𝐼 2 ሶ 𝐼 − 𝐼2 − 3𝐿 𝑏2 ሷ 𝜚 + 3𝐼 ሶ 𝜚 + 𝑊
,𝜚 + 12𝜊,𝜚 𝐼2 + 𝐿
𝑏2 ሶ 𝐼 + 𝐼2 = 0 𝐼2= 𝜆2 3 1 2 ሶ 𝜚2 + 𝑊 − 3𝐿 𝜆2𝑏2 + 12 ሶ 𝜊𝐼 𝐼2 + 𝐿 𝑏2
𝑒𝑡2 = - 𝑒𝑢2 + 𝑏2 𝑢
𝑒𝑠2 1−𝐿𝑠2 + 𝑠2(𝑒θ2 + 𝑡𝑗𝑜2θ 𝑒φ2)
𝐻μν=𝜆2 𝑈
μν + 𝑈 μν 𝐻𝐶
𝜆2= 8π𝐻 𝐻μν ≡ 𝑆μν −
1 2 μν 𝑆
□𝜚 − 𝑊
,𝜚 𝜚 − 1
2 𝑈𝐻𝐶 = 0 𝑈
μν 𝐻𝐶 = 4 𝜖𝜍𝜖𝜏𝜊𝑆𝜈𝜍𝜉𝜏 − □𝜊𝑆𝜈𝜉 + 2𝜖𝜍𝜖(𝜈𝜊𝑆𝜉) 𝜍 − 1
2 𝜖𝜈𝜖𝜉𝜊𝑆 − 2 2𝜖𝜍𝜖𝜏𝜊𝑆𝜍𝜏 − □𝜊𝑆 μν 𝑈
μν = 𝜖𝜈𝜚𝜖𝜉𝜚 + 𝑊 𝜚 − 1
2 μν 𝜍𝜏𝜖𝜍𝜚𝜖𝜏𝜚 + 2𝑊
𝑈𝐻𝐶 = 𝜊,𝜚 𝜚 𝑆𝐻𝐶
2
Guo,N.Ohta &Torii, Pr.Th.P.120,581(2008);121 ,253 (2009); N.Ohta &Torii,Pr.Th.P.121,959;122,1477(2009);124,207 (2010); Maeda,N.Ohta Sasagawa, PRD80,104032(2009); 83,044051 (2011)
Einstein and Field equations yield:
ሶ 𝐼 = − 𝜆2 2 ሶ 𝜚2 − 2𝐿 𝜆2𝑏2 − 4 ሷ 𝜊 𝐼2 + 𝐿 𝑏2 − 4 ሶ 𝜊𝐼 2 ሶ 𝐼 − 𝐼2 − 3𝐿 𝑏2 ሷ 𝜚 + 3𝐼 ሶ 𝜚 + 𝑊
,𝜚 + 12𝜊,𝜚 𝐼2 + 𝐿
𝑏2 ሶ 𝐼 + 𝐼2 = 0 𝐼2= 𝜆2 3 1 2 ሶ 𝜚2 + 𝑊 − 3𝐿 𝜆2𝑏2 + 12 ሶ 𝜊𝐼 𝐼2 + 𝐿 𝑏2
𝑊
0 = 0.5 × 10−12
𝜊0 = 0(black), 𝜊0 = 3 × 106 (red), and 𝜊0= 3 × 107 (blue).
The duration of inflation gets shorter as the Gauss- Bonnet coupling constant increases. Increasing
the Gauss-Bonnet coupling constant makes the effective potential steeper such that the scalar field rolls faster than it does in models without Gauss-Bonnet term Hence inflation ends earlier in models with a large Gauss-Bonnet coupling.
𝑇 = න 𝑒4𝑦 − 1 2𝜆2 𝑆 − 1 2 𝜈𝜉𝜖𝜈𝜚𝜖𝜉𝜚 − 𝑊 𝜚 − 1 2 𝜊 𝜚 𝑆𝐻𝐶
2
PRD90 (2014) ) no. no.6, 06 063527
arX rXiv:1610.04360
Let us consider a mode with comoving wavenumber 𝑙∗ which crosses the horizon during inflation when the scale factor is 𝑏∗. The comoving Hubble scale 𝑏∗𝐼∗ = 𝑙∗ at the horizon crossing time can be related to that of the present time as where 𝑏0, 𝑏∗, 𝑏end, and 𝑏th denote the scale factor at present, the horizon crossing, the end of inflation, and the end of reheating, respectively. By taking logarithm from both sides, we rewrite where 𝑂∗ ≡ ln(𝑏end/𝑏∗) is the number of e-foldings between the time of mode exits the horizon and the end
reheating. (6) (7)
where We obtain the number of e-folding 𝑂th , as well as the temperature of reheating 𝑈th.
Figure 3: Plots of 𝑂th and 𝑈th using Eq. (23) with 𝑊
0 = 0.5 × 10−12 and n = 2. The black color indicates the result without the Gauss-Bonnet term, ξ0 = 0
while the result with a Gauss-Bonnet term is presented in red color, 𝜊0= 2.9537 × 106. The light blue shaded region shows the current 1σ bounds on 𝑜𝑡 from Planck while the gray shaded one shows the 1σ bounds of a further CMB experiment with sensitivity ±103 [9,10], using the same central 𝑜𝑡 value as Planck. The region below the blue horizontal line at 100GeV indicates below the electro-weak scale.
Figure 5: Plots of Eq. (18), the duration of reheating, for the model Eq. (23) with 𝑊
0 = 0.5 × 10−12 and n = 2.
Numerical inputs and the background shaded areas are same as Figure 3. However, for the red lines, we use central ns = 0.9682 value. The blue dot indicates the instantaneous reheating that occurs at 𝜊0= 2.9537 × 106.
Figure 5: Plots of Eq. (18), the duration of reheating, for the model Eq. (23) with 𝑊
0 = 0.5 × 10−12 and n = 2.
Numerical inputs and the background shaded areas are same as Figure 3. However, for the red lines, we use central ns = 0.9682 value. The blue dot indicates the instantaneous reheating that occurs at 𝜊0= 2.9537 × 106.
asymptotically flat spacetime There exists minimum mass, etc.
We have studied the Black Hole with Gauss-Bonnet term
When the scalar field on the horizon is the maximum, the DGB black hole solution has the minimum horizon size. The amount of black hole hair decreases as the DGB black hole mass increases. DGB black hole configurations go to EGB black hole cases for small and The DGB black hole phase is unstable under fragmentation, even if these phases are stable under perturbation. We have found the phase diagram
the fragmentation instability for a black hole mass and two couplings.
case, r is enhanced for 𝛽 > 0 while it is suppressed for 𝛽 < 0.
would be interesting to search for the alternatives to reconcile Planck data with BICEP2 besides consideration of the running spectral index.
Other Effects such as the reheating under investigation
exponential potential and exponential coupling. In this case, we could find the valid model parameter range for inflation to happen, unfortunately, these parameter ranges are not favored by the data sets.
this case, r is enhanced for 𝛽 > 0 while it is suppressed for 𝛽 < 0.
would be interesting to search for the alternatives to reconcile Planck data with BICEP2 besides consideration of the running spectral index.
Other Effects such as the reheating under investigation