Thermodynamics of McVittie Metric - National Centre for Physics

Einstein’s Equations (1915):Ricci TensorGravitational Constant1R − g R = 8 GT2πμν μν μν{Geometry matter (energy-momentum)}C = 1

a) Brief Introduction to Black Hole Thermodynamics1. Black Hole Solutions to the Einstein field equation2. Cosmological solutions to the Einstein Field EquationBlack Hole (By Wheeler in 1967):A spacetime region in which gravitational field is so strong thatit precludes even light from escaping to infinityNote: Velocity required to leave the boundary of the blackhole is equal to the speed of light ( Classical Picture)Penrose (1967):proposed that the black hole must contain non-zero entropy

Bekenstein, Bardeen, Carter, Hawking (1970-1974):Jacob Bekenstein et al proposed that a black holeshould have an entropy, and that it should beproportional to its horizon area.Since black holes do not classically emit radiation,the thermodynamic viewpoint seemed simply ananalogy.Four Laws Black hole ThermodynamicsFour Laws of ThermodynamicsHowever, in 1974, Hawking applied quantum fieldtheory to the curved spacetime around the eventhorizon and discovered that black holes emitHawking radiation, a form of thermal radiation,which implied they had a positive temperature. Hedetermined the constant of proportionality.

Schwarzschild Black Hole: Units G = c =1M = mass of the black HoleEvent HorizonSurface Gravity

TemperatureSurface gravityEntropySimple ExampleA = 4π r 2Schwarzschild Black Hole (1916): Mass MhorizonT dS = dMMore general:Kerr - Newmann Black Holes (M, J, Q)dM = T dS + Ωd J+Φd QReference: Black Hole Physics, ByValeri P. Frolov and Igor D. Novikov

Einstein’s Field Equation | (At horizon)First Law of ThermodynamicsFirst by T. JacobsonHe found that it is indeed possible to derive the Einstein equations from theproportionality of entropy to the horizon area together with the fundamentalrelation δQ = T dST. Jacobson, Phys. Rev. Lett. 75 (1995) 1260Thermodynamics of Spacetime: The Einstein Equation of StateHorizon Thermodynamics: T. Padmanabhan, Class. Quant. Grav.( 2002)[

c) From the First Law to the Friedmann EquationsFriedman-Robertson-Walker Universe:22 2 2 dr 2 2 2 2 2ds =− dt + a ()( t + r dθ+ r sin θdϕ)21−kr1) k = -1open2) k = 0flat3) k =1closed

Friedmann Equations:Where:

Our goal :Some related works:(1) A. Frolov and L. Kofman, JCAP 0305 (2003) 009(2) Ulf H. Daniesson, PRD 71 (2005) 023516(3) R. Bousso, PRD 71 (2005) 064024

22 2 2 dr 2 2 2 2 2ds =− dt + a ()( t + r dθ+ r sin θdϕ)21−kr

Apparent Horizon in FRW Universe:Apparent Horizon:A marginally trapped surface with vanishing expansion

Apply the first law to the apparent horizon:Make two ansatzes:The only problem is to get dE

Suppose that the perfect fluid is the source, thenThe energy-supply vector is: The work density is:S. A. Hayward, 1997,1998)Then, the amount of energy crossing the apparent horizon withinthe time interval dt

By using the continuity equation:1. Cai and Kim, JHEP 0502 (2005) 0502. Akbar and Cai, Phys. Rev. D75:084003,2007

McVittie Thermodynamics:McVittie 1933:McVittie MetricM 0 = Mass of the black holea (t) = scale factor of the universeNotice That: M 0 = 0Flat FRW universea (t) = 1Isotropic form of Schwarzschild solution

Properties:(i) The metric seems to represent a Schwarzschild black hole embedded in spatiallyflat FRW universe [D. J. Shaw et al Phys. Rev. D73 (2006) and others](ii) As r → ∞, The McVittie metric reduces to a Robertson-Walker (RW) universe.(iii) It is singular on the 2-sphere r = M 0 / 2. Also Pressure and Ricci scalar divergeat r = M0 / 2McVittie can be rewritten as:ds2AB22 2 4 2 2 2= − dt + a( t)B dr + R dΩ2WhereA =M−2a(t)r1 0B=M+102a(t)r2R=a( t)rB

Apparent Horizon: abh ∂ R∂R = 0abh ab=diag( −AB22,a ( t )2B4)H2R3A−RA+2M0= 0Hubble parameterApparent HorizonCubic equation in apparent horizon radius

The different cases for the roots of the above equation are obtained bycomparing it withx 3 +p x =q,having discriminant D = (p/3) 3 + (q/2) 2D = -1 ⁄ 27 H 6 + M 02⁄ 4 H 4D > 0 → One real root which is negativeD = 0 → all real roots and two are equal (one positive and one negative)D < 0 → all roots are real and un-equal (two positive and one negative)

Entropy:S = 4π R A2Surface Gravityκ=•H2 M0+ H RA−22 HRAT=κ / 2π

Field Equation at apparent Horizon:Misner - Sharp EnergyE = 4π ⁄ 3 R A2ρ(t) + M 0dS= 2 π RA•dtRT dS = dE +PdWWhere W= (1 ⁄ 2) (ρ –P) is the work density

Conclusion:‣ We worked out the explicit expressions forthe apparent horizons of the McVittie universe.‣ The temperatures associated with thesehorizon are obtained.‣ It is also shown that these thermal quantitiessatisfy first law of thermodynamics.

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