On The Throttling Process Of AdS BlackHole With A Global Monopole

Naveena Kumara ANational Institute of Technology Karnataka

Outline for Section 1

1. Joule Thomson Effect in van der Waals’ fluid1.1 Throttling Process1.2 Joule Thomson Effect1.3 Isenthalpic and Inversion curves1.4 van der Waals’ fluid

2. The charged AdS black hole with global monopole2.1 Thermodynamics of Black Hole2.2 Joule Thomson expansion of Black Hole2.3 Isenthalpic and Inversion curves2.4 Conclusion

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Throttling ProcessEnthalpy remains constant

Expansion of a fluid from a regionof high pressure to a region oflow pressure through a porous plug.

ExampleWater passing through a faucet.

Ideal gas - no temp. change.

Real gas - temp. may increase,decrease or remain same.

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Joule Thomson EffectEnthalpy remains constant

Joule Thomson Coeffiecent,

μJ =(dTdP

)H

Pure right : heating

Pure left : cooling

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Isenthalpic and Inversion curvesfor van der Walls’ fluid

0 5 10 15 20 25

1.0

1.5

2.0

2.5

3.0

Pr

Tr

0 2 4 6 80

2

4

6

8

Pr

Tr

Cooling

Heating

Isenthalpic curve:the locus of all pointswith the same enthalpy

Inversion curve:locus of inversion temperatures

Region of cooling : μJ positiveRegion of heating : μJ negative

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van der Waals’ fluidshows critical behavior

2 4 6 8 10 12V

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

P

T > Tc

T Tc

T < Tc

T < Tc

Equation of state:(P+ a

V2m

)(Vm − b) = RT.

Below critical temperature : Stable - Unstable - Stable states.Indication of Phase transition

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Outline for Section 2

1. Joule Thomson Effect in van der Waals’ fluid1.1 Throttling Process1.2 Joule Thomson Effect1.3 Isenthalpic and Inversion curves1.4 van der Waals’ fluid

2. The charged AdS black hole with global monopole2.1 Thermodynamics of Black Hole2.2 Joule Thomson expansion of Black Hole2.3 Isenthalpic and Inversion curves2.4 Conclusion

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The charged AdS black hole with global monopoleThis is our metric

The Lagrangian :

Lgm =1

2∂μa∂μ∗a −

γ

4

(a∗a − η20

)2, (1)

where a is a multiplet of scalar field, γ is a constant and η0 is theenergy scale of symmetry breaking.Solving Einstein equation, we have the line element

ds2 = −f(r)dt2 + f(r)−1dr2 + (1− η2)r2dΩ2, (2)

with

f(r) = 1−2m

r+

q2

r2+

r2

l2. (3)

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The charged AdS black hole with global monopoleADMMass and Pressure, Extended Phase space

The ADMmass:

M =(1− η2)

2r+ +

Q2

2r+(1− η2)+

r3+(1− η2)

2l2. (4)

In the extended phase space the cosmological constant correspondsto the thermodynamic variable pressure (P), and its conjugatequantity corresponds to the thermodynamic volume (V),

P = −Λ

8π=

3

8πl2, V =

4

3π(1− η2)r3+ . (5)

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Thermodynamics of Black HolePhase Transition of Black Hole!

0 5 10 15 20 25 30v

0.001

0.002

0.003

0.004

0.005

P

T > Tc

T > Tc

T Tc

T < Tc

T < Tc

The Hawking temperature:

T = 14πr+

(1+

3r2+l2 −

Q2

(1−η2)2r2+

).

Equation of state :

P = Tv −

12πv2 +

2Q2

π(1−η2)2v4 .

The Pv plot is similar to that of van der Waals’ fluid.

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Joule Thomson expansion of Black Holecharged AdS blackhole with monopole term

The expression for Joule Thomson coefficient,

μJ =(∂T

∂P

)H=

1

CP

[T

(∂V

∂T

)P− V

]. (6)

As μJ = 0 defines the inversion temperature we have

Ti = V(∂T

∂V

)P. (7)

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Joule Thomson expansion of Black Holecharged AdS blackhole with monopole term

Expression for inversion temperature (Ti) in terms of inversionpressure, charge and monopole parameter ,

Ti =

pPi

(1+ 16πPiQ2

(1−η2)2 −p

24PiπQ2+(1−η2)2(1−η2)

)p2π

(−1+

p24PiπQ2+(1−η2)2(1−η2)

)3/2 . (8)

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Inversion curvesInversion curves showing the dependence of charge Q fordifferent η values.

1

2

10

20

0.0 0.2 0.4 0.6 0.8 1.0P

1

2

3

4

T

Q Valuesη 0.1

1

2

10

20

0.0 0.2 0.4 0.6 0.8 1.0P

1

2

3

4

T

Q Valuesη 0.3

1

2

10

20

0.0 0.2 0.4 0.6 0.8 1.0P

1

2

3

4

T

Q Valuesη 0.5

1

2

10

20

0.0 0.2 0.4 0.6 0.8 1.0P

1

2

3

4

T

Q Valuesη 0.7

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Inversion and isenthalpic curvesCrossing diagrams between inversion and isenthalpic curvesfor different values of η and Q = 1

1.5

2

2.5

3

0 2 4 6 8 10 12P

1

2

3

4

T

M Values

η 0.1

1.5

2

2.5

3

0 2 4 6 8 10P

1

2

3

4

T

M Values

η 0.3

1.5

2

2.5

3

0 2 4 6 8P

1

2

3

4

T

M Values

η 0.5

1.5

2

2.5

3

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5P

0.5

1.0

1.5

2.0

2.5

T

M Values

η 0.7

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ConclusionAnother outcome of AdS-CFT Correspondence

Joule Thomson effect, similar to van der Waals gas isobserved in AdS black hole.

Monopole term plays an important role in the thermodynamicsand Joule Thomson expansion of AdS black hole.

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Bibliography[1] Ahmed Rizwan C. L., Naveena Kumara A., Deepak Vaid, and K. M. Ajith.

Joule-Thomson expansion in AdS black hole with a global monopole.arXiv:1805.11053, 2018.

[2] Özgür Ökcü and Ekrem Aydıner.Joule–Thomson expansion of the charged AdS black holes.Eur. Phys. J. C, 77, 2017.

[3] David Kubizňák and Robert B. Mann.P - V criticality of charged AdS black holes.Journal of High Energy Physics, 2012(7), 2012.

[4] Gao-Ming Deng, Jinbo Fan, Xinfei Li, and Yong-Chang Huang.Thermodynamics and phase transition of charged AdS black holes with a globalmonopole.International Journal of Modern Physics A, 33(03):1850022, jan 2018.

[5] Manuel Barriola and Alexander Vilenkin.Gravitational field of a global monopole.Phys. Rev. Lett., 63:341–343, Jul 1989.

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Team NITKMembers

Ahmed Rizwan C.L. Dr. Deepak Vaid Dr. Ajith K. M.

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