backreaction effects of matter coupled higher derivative ...€¦ · introduction and motivation...
TRANSCRIPT
![Page 1: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/1.jpg)
Backreaction effects of matter coupled higherderivative gravity
Lata Kh Joshi(Based on arXiv:1409.8019, work done with Ramadevi)
Indian Institute of Technology, Bombay
DAE-HEP, Dec 09, 2014
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 1 / 14
![Page 2: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/2.jpg)
Introduction and Motivation
Introduction
Strongly coupled field theories
AdS − CFT Correspondence
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 2 / 14
![Page 3: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/3.jpg)
Introduction and Motivation
Introduction
Strongly coupled field theories
AdS − CFT Correspondence
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 2 / 14
![Page 4: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/4.jpg)
Introduction and Motivation
Introduction
Strongly coupled field theories
AdS − CFT Correspondence
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 2 / 14
![Page 5: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/5.jpg)
Introduction and Motivation
AdS − CFT correspondenceI AdS5/CFT4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]
Type IIB string theory on AdS5×S5 ←→ N = 4 SYM theory.
A correspondence to relate theories of two very different kind.One in gravity and other in field theory.
Strong coupling ⇐⇒ Weak coupling
AdS CFT
Specific limit:
Fluid-Gravity Correspondence
Hydrodynamics: Large length scales in field theories.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 3 / 14
![Page 6: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/6.jpg)
Introduction and Motivation
AdS − CFT correspondenceI AdS5/CFT4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]
Type IIB string theory on AdS5×S5 ←→ N = 4 SYM theory.
A correspondence to relate theories of two very different kind.One in gravity and other in field theory.
Strong coupling ⇐⇒ Weak coupling
AdS CFT
Specific limit:
Fluid-Gravity Correspondence
Hydrodynamics: Large length scales in field theories.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 3 / 14
![Page 7: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/7.jpg)
Introduction and Motivation
AdS − CFT correspondenceI AdS5/CFT4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]
Type IIB string theory on AdS5×S5 ←→ N = 4 SYM theory.
A correspondence to relate theories of two very different kind.One in gravity and other in field theory.
Strong coupling ⇐⇒ Weak coupling
AdS CFT
Specific limit:
Fluid-Gravity Correspondence
Hydrodynamics: Large length scales in field theories.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 3 / 14
![Page 8: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/8.jpg)
Introduction and Motivation
To locate transition temperature precise behavior of η/s withT is needed.
Figure: arXiv:1206.3581
The figure shows expectation taking analogy with H2O and Heplots.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 4 / 14
![Page 9: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/9.jpg)
Introduction and Motivation
To locate transition temperature precise behavior of η/s withT is needed.
Figure: arXiv:1206.3581
The figure shows expectation taking analogy with H2O and Heplots.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 4 / 14
![Page 10: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/10.jpg)
Using AdS/CFT
Calculation of the correlator?
Use AdS-CFT correspondence
〈e i∫∂M φ0O〉 = e iScl [φ]
O is an operator in the boundary; φ field in the bulk, with φ0 value atboundary.
Giving,
〈O(x1)O(x2)〉 = i2 δ2
δφ0(x1)δφ0(x2)eiScl [φ0] at φ0 → 0
Note: We need to know the background geometry for the abovecalculations.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 5 / 14
![Page 11: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/11.jpg)
Using AdS/CFT
Calculation of the correlator?
Use AdS-CFT correspondence
〈e i∫∂M φ0O〉 = e iScl [φ]
O is an operator in the boundary; φ field in the bulk, with φ0 value atboundary.
Giving,
〈O(x1)O(x2)〉 = i2 δ2
δφ0(x1)δφ0(x2)eiScl [φ0] at φ0 → 0
Note: We need to know the background geometry for the abovecalculations.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 5 / 14
![Page 12: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/12.jpg)
Using AdS/CFT
Calculation of the correlator?
Use AdS-CFT correspondence
〈e i∫∂M φ0O〉 = e iScl [φ]
O is an operator in the boundary; φ field in the bulk, with φ0 value atboundary.
Giving,
〈O(x1)O(x2)〉 = i2 δ2
δφ0(x1)δφ0(x2)eiScl [φ0] at φ0 → 0
Note: We need to know the background geometry for the abovecalculations.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 5 / 14
![Page 13: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/13.jpg)
Background Solutions
Background solutions in gravity
Einstein gravity with higher derivatives
S =
∫dDx√−g(R − 2Λ + c1R
2 + c2RµνRµν + c3RµνρσR
µνρσ)
(1)
with Λ = −(D − 1)(D − 2)
2(2)
ds2 =1
r 2
((−f (r)dt2 +
(d~x2)) +
dr 2
f (r)
)(3)
where, f (r) = 1− rD−1 + δ + κr 2(D−1)
δ and κ being dependent on D, c1, c2 and c3(arXiv: 0712.0743)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 6 / 14
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Background Solutions
Background solutions in gravity
Einstein gravity with higher derivatives
S =
∫dDx√−g(R − 2Λ + c1R
2 + c2RµνRµν + c3RµνρσR
µνρσ)
(1)
with Λ = −(D − 1)(D − 2)
2(2)
ds2 =1
r 2
((−f (r)dt2 +
(d~x2)) +
dr 2
f (r)
)(3)
where, f (r) = 1− rD−1 + δ + κr 2(D−1)
δ and κ being dependent on D, c1, c2 and c3(arXiv: 0712.0743)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 6 / 14
![Page 15: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/15.jpg)
Background Solutions
Matter coupling
To know the properties of QGP, it becomes necessary to considerdilaton field in the gravity
Consider the action
S =1
2κ2D
∫dDx [R − 4
D − 2(∂Φ(r))2 − V (Φ(r))] ; V (Φ(r)) = 2ΛeαΦ(r)
(4)The already known solution to this action shows effects of matteron the background metric
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 7 / 14
![Page 16: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/16.jpg)
Background Solutions
Matter coupling
To know the properties of QGP, it becomes necessary to considerdilaton field in the gravity
Consider the action
S =1
2κ2D
∫dDx [R − 4
D − 2(∂Φ(r))2 − V (Φ(r))] ; V (Φ(r)) = 2ΛeαΦ(r)
(4)The already known solution to this action shows effects of matteron the background metric
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 7 / 14
![Page 17: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/17.jpg)
Background Solutions
It is interesting to look upon the case of matter coupled higherderivative gravity
S =1
2κ2D
∫dDx(R − 4
D − 2(∂Φ(r))2 − V (Φ(r)) + βG (Φ(r))RµνσρR
µνσρ)
(5)
with V (Φ) = 2ΛeαΦ ; G (Φ) = eγΦ.
Varying the action with respect to metric field and scalar field gives,
Rµν −1
2gµνR = Tmatter
µν + T hdµν (6)
8
D − 2�Φ(r) = ∂Φ(βG (Φ(r))RµνσρR
µνσρ − V (Φ(r))) (7)
where,Tµν = − 1√−g
∂√−gLM∂gµν
(8)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 8 / 14
![Page 18: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/18.jpg)
Background Solutions
It is interesting to look upon the case of matter coupled higherderivative gravity
S =1
2κ2D
∫dDx(R − 4
D − 2(∂Φ(r))2 − V (Φ(r)) + βG (Φ(r))RµνσρR
µνσρ)
(5)
with V (Φ) = 2ΛeαΦ ; G (Φ) = eγΦ.
Varying the action with respect to metric field and scalar field gives,
Rµν −1
2gµνR = Tmatter
µν + T hdµν (6)
8
D − 2�Φ(r) = ∂Φ(βG (Φ(r))RµνσρR
µνσρ − V (Φ(r))) (7)
where,Tµν = − 1√−g
∂√−gLM∂gµν
(8)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 8 / 14
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Background Solutions
Choosing an ansatz is the most crucial part in solving the aboveequation.
To find the solution up to O(β), we choose the ansatz as:
ds2 = −r−2a(1− r c(β,r))dt2 +dr2
r−2a(1− r c(β,r))r4+ r−2ad~x2
Φ(r) = mLog(r) + βΦ1(r) (9)
where c[β, r ] is chosen as:
c[r ] = c +Log(1− βκ(r))
Log(r)(10)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 9 / 14
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Background Solutions
Choosing an ansatz is the most crucial part in solving the aboveequation.
To find the solution up to O(β), we choose the ansatz as:
ds2 = −r−2a(1− r c(β,r))dt2 +dr2
r−2a(1− r c(β,r))r4+ r−2ad~x2
Φ(r) = mLog(r) + βΦ1(r) (9)
where c[β, r ] is chosen as:
c[r ] = c +Log(1− βκ(r))
Log(r)(10)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 9 / 14
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Background Solutions
The first order correction κ(r), is obtained as:
−(D−4)
4 +(D − 4)(D − 1)16D−
6((−1)D+1)mD−2 (g × r)−aD+1
(D − 2) ((D − 2)2α2 − 16(D − 1))
+2Λr
m2
(α+γ)
(D − 2)
(rmγ+a(D−1)
((D − 2)2α (3(D − 2)α + 4(D − 3)γ)− 16(D − 4)(D − 3)
)((D − 2)2αγ + 16(D − 1))
−rmα(4(D − 1)(D − 2)2α2 + 8(D − 3)(D − 2)2αγ + 64(D − 4)
)((D − 2)2α(α + 2γ) + 16(D − 1))
),
Horizon gets a linear order correction as: rh = 1 + βr1.
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 10 / 14
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Shear viscosity to entropy ratio
Shear viscosityKubo formula for viscosity Low energy and low momentum limit ofretarded Green’s function of stress tensor in CFT.
η = − limω→0
1
ωImGR
xy ;xy (ω, k = 0) (11)
Translate the calculation of the correlator to a holographic one
perturbation in the xy components of the metric.
gxy = g0xy + εhxy (r , x) = g0
xy + εgiiφ(r , x),[x = (t,−→x )
]with
φ(r , x) =1
(2π)4
∫d4ke−ik.xφ(r , k),[k = (ω,
−→k ), k .x = kµx
µ]
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 11 / 14
![Page 23: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/23.jpg)
Shear viscosity to entropy ratio
Shear viscosityKubo formula for viscosity Low energy and low momentum limit ofretarded Green’s function of stress tensor in CFT.
η = − limω→0
1
ωImGR
xy ;xy (ω, k = 0) (11)
Translate the calculation of the correlator to a holographic one
perturbation in the xy components of the metric.
gxy = g0xy + εhxy (r , x) = g0
xy + εgiiφ(r , x),[x = (t,−→x )
]with
φ(r , x) =1
(2π)4
∫d4ke−ik.xφ(r , k),[k = (ω,
−→k ), k .x = kµx
µ]
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 11 / 14
![Page 24: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/24.jpg)
Shear viscosity to entropy ratio
Entropy density
Wald formula
S = −2π
∫ΣdD−2x
√−h δL
δRµνρσεµνερσ (12)
1 εµνεµν = −2
2 εµν = ξµην − ξνηµThe L above is chosen by writing the action as
Saction ∼∫
d5x√−gL
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 12 / 14
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Shear viscosity to entropy ratio
Backreaction effects on η and s
For the matter coupled higher derivative action the shear viscosityand entropy density is
η =1
2κ2D
[rPh −
4β(D − 2)2αγ(α2(D − 2)2 − 16(D − 1)
)rM0
(16 + α2(D − 2)2)2
](13)
s =1
2κ2D
[4πrPh −
128πβ(D − 4)(α2(D − 2)2 − 16(D − 1)
)rN0
(16 + α2(D − 2)2)2
](14)
Giving,
η
s=
1
4π
[1−
4β(−16(D − 1) + (D − 2)2α2
) (−8(D − 4) + (D − 2)2αγ
)(16 + (D − 2)2α2)2
].
(15)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 13 / 14
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Shear viscosity to entropy ratio
Backreaction effects on η and s
For the matter coupled higher derivative action the shear viscosityand entropy density is
η =1
2κ2D
[rPh −
4β(D − 2)2αγ(α2(D − 2)2 − 16(D − 1)
)rM0
(16 + α2(D − 2)2)2
](13)
s =1
2κ2D
[4πrPh −
128πβ(D − 4)(α2(D − 2)2 − 16(D − 1)
)rN0
(16 + α2(D − 2)2)2
](14)
Giving,
η
s=
1
4π
[1−
4β(−16(D − 1) + (D − 2)2α2
) (−8(D − 4) + (D − 2)2αγ
)(16 + (D − 2)2α2)2
].
(15)
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 13 / 14
![Page 27: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/27.jpg)
Shear viscosity to entropy ratio
Summary
Found the first order correction to the background metric formatter coupled higher derivative AdS gravity.
Horizon gets a linear order correction due to matter presence.
Shear viscosity and Entropy density get linear order correctionvia the correction in horizon
However, the ratio η/s remains unaffected due to thebackreaction in the metric.
Puzzle: Backreaction to the matter solution still remains unknown
Thank You
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 14 / 14
![Page 28: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/28.jpg)
Shear viscosity to entropy ratio
Summary
Found the first order correction to the background metric formatter coupled higher derivative AdS gravity.
Horizon gets a linear order correction due to matter presence.
Shear viscosity and Entropy density get linear order correctionvia the correction in horizon
However, the ratio η/s remains unaffected due to thebackreaction in the metric.
Puzzle: Backreaction to the matter solution still remains unknown
Thank You
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 14 / 14
![Page 29: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/29.jpg)
Shear viscosity to entropy ratio
Summary
Found the first order correction to the background metric formatter coupled higher derivative AdS gravity.
Horizon gets a linear order correction due to matter presence.
Shear viscosity and Entropy density get linear order correctionvia the correction in horizon
However, the ratio η/s remains unaffected due to thebackreaction in the metric.
Puzzle: Backreaction to the matter solution still remains unknown
Thank You
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 14 / 14
![Page 30: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/30.jpg)
Shear viscosity to entropy ratio
Summary
Found the first order correction to the background metric formatter coupled higher derivative AdS gravity.
Horizon gets a linear order correction due to matter presence.
Shear viscosity and Entropy density get linear order correctionvia the correction in horizon
However, the ratio η/s remains unaffected due to thebackreaction in the metric.
Puzzle: Backreaction to the matter solution still remains unknown
Thank You
Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP 2014 14 / 14
![Page 31: Backreaction effects of matter coupled higher derivative ...€¦ · Introduction and Motivation AdS CFT correspondence I AdS 5=CFT 4 [Maldacena, Adv.Theor.Math.Phys.2:231-252,1998]](https://reader036.vdocuments.us/reader036/viewer/2022070914/5fb5608d61f10013ce187927/html5/thumbnails/31.jpg)
Shear viscosity to entropy ratio
Anti de Sitter space
Solution to Einstein equation with negative cosmological constant.
Space of Lorentzian signature (−,+,+..+), but of constant negativecurvature
On boundary
−x20 +
d−1∑i=1
x2i − x2
d+1 = R2
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Shear viscosity to entropy ratio
Rµν −1
2gµνR +
1
2gµνΛ = 0 (16)
=⇒ R = Λ(d + 1)
d − 1(17)
Thus constant negative curvature
Rµν =1
d − 1Λgµν (18)
Parametrize,
Λ = −d(d − 1)
L2
=⇒ R = −d(d + 1)
L2(19)
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Shear viscosity to entropy ratio
Conformal Field Theory
Guage field theory with enhanced symmetries.
Poincare symmetry + dilatations + special conformal symmetry
Dilatations: xµ → λxµ
special conformal: Translation+Inversion+Translation
Under special conformal: x ′µ = xµ + 2xµb.x − x2bµ
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Shear viscosity to entropy ratio
Conformal Field Theory
Guage field theory with enhanced symmetries.
Poincare symmetry + dilatations + special conformal symmetry
Dilatations: xµ → λxµ
special conformal: Translation+Inversion+Translation
Under special conformal: x ′µ = xµ + 2xµb.x − x2bµ
Back
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Shear viscosity to entropy ratio
Translation:Pµ = −i∂µ (20)
Dilatations:D = −ixµ∂µ (21)
Rotations:Lµν = i(xµ∂ν − xν∂µ) (22)
SCT:Gµ = −2ixµ(x .∂) + ix2∂µ (23)
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