Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Black holes with positive specific heat S ( E , V ) -thermodynamics T.S. Biró 1 , P . Ván 1 , V.G. Czinner 1 , 2 1 Heavy Ion Research Group MTA Research Centre for Physics, Budapest 2 Department for Physics University of Lisbon, Lisbon September 18, 2017 Biró Ván Czinner BH positive 1 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes What is it About? Hawking radiation, temperature, entropy Negative specific heat With pressure positive specific heat and 4 times the original entropy (at the same temperature) Biró Ván Czinner BH positive 2 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Outline Traditional horizon entropy and temperature (Hawking) 1 Homogenity class assumption about volume dependence 2 Proof of positive specific heat 3 Charged black holes 4 Biró Ván Czinner BH positive 3 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Outline Traditional horizon entropy and temperature (Hawking) 1 Homogenity class assumption about volume dependence 2 Proof of positive specific heat 3 Charged black holes 4 Biró Ván Czinner BH positive 4 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Unruh effect on monochromatic source Accelerating source → smeared Doppler Biró Ván Czinner BH positive 4 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Bekenstein-Hawking entropy Simple estimate Entropy = 1/4 Horizon Area At Schwarzschild radius R = 2 GM / c 2 the gravitational acceleration is g = GM / R 2 = c 2 / 2 R . The Unruh temperature in proper units becomes � g k B T U = 2 π c . (1) Clausius’ entropy if E = Mc 2 becomes � d ( Mc 2 ) c 3 S � G π R 2 = 1 A = = . (2) k B k B T 4 L 2 P Biró Ván Czinner BH positive 5 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Temperature Unruh, Hawking, spectral Planck scale units: c = 1, L P = GM P , L P = � / M P . Entropy units: k B = 1. (constant) acceleration occurs as (constant) temperature T U ( g ) = g 2 π . (3) Radial space time metric ds 2 = f ( r ) dt 2 − dr 2 f ( r ) , (4) � � Maupertuis action S = m ds = Lds in terms of proper time. Being a static t = f ( r )˙ metric, E = ∂ L t is constant. Therefore ∂ ˙ r 2 = K 2 − f ( r ) . ˙ (5) Its total derivative delivers the comoving aceleration r = − 1 2 f ′ ( r ) ¨ (6) Biró Ván Czinner BH positive 6 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Entropy to Unruh and Hawking temperatures Size of horizon Horizon at r = R with f ( R ) = 0. Here g = | f ′ ( R ) | / 2. � 1 � 1 � f ′ ( R ) � f ′ ( R ) � . � � � � T H = T U = (7) � 2 4 π For the Schwarzschild metric f ( r ) = 1 − 2 M / r , R = 2 M and f ′ ( R ) = 1 / 2 M 1 T H = 8 π M . (8) Clausius’ entropy: � dE � dM � � S = = 4 π | f ′ ( R ) | = 4 π δ ( f ( r , M )) drdM . (9) T As a volume integral it defines entropy density: � �� � � � � s d 3 r = r 2 f ( r , M ) d 3 r . S = δ dM (10) for Schwarzschild S = π R 2 = A / 4. Biró Ván Czinner BH positive 7 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Horizon entropy thermodynamics negative specific heat Traditional analysis considers E = M and the EoS S = π R 2 = 4 π E 2 . (11) Derivatives � T = ∂ S 1 1 ∂ E = 8 π M = . (12) T H Instability problem: � Vc V T 2 = ∂ 2 S 1 − ∂ E 2 = 8 π > 0 (13) leads to c V < 0 and instable entropy maximum. Biró Ván Czinner BH positive 8 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Outline Traditional horizon entropy and temperature (Hawking) 1 Homogenity class assumption about volume dependence 2 Proof of positive specific heat 3 Charged black holes 4 Biró Ván Czinner BH positive 9 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes S ( E , V ) thermodynamics E = M , R = 2 M put at the end only Scaling anstaz: S ( E , V ) = V s ( E / V ) . Energy density: ε = E / V in the above ansatz. Homogeneous EoS, s ( ε ) , based thermodynamics. 1 ∂ S ∂ E = s ′ ( ε ) , = T ∂ S p ∂ V = s ( ε ) − ε s ′ ( ε ) . = (14) T This satisfies T s = ε + p Biró Ván Czinner BH positive 9 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes EoS from radial integrals 2 M � s ( r ) 4 π r 2 dr = aS H = 4 π aM 2 , S = 0 2 M � ε ( r ) 4 π r 2 dr = M , E = (15) 0 results in ε ( r ) = 1 / 8 π r 2 , s = a / 2 r , (16) to conclude at the equation of state Scaling Horizon Equation of State √ s ( ε ) = a 2 πε. (17) Biró Ván Czinner BH positive 10 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Consequences of scaling EoS Hawking radiation pressure From the s ( ε ) = κ √ ε type EoS it follows 1 S p s T = s T = 2 ε = 2 E , 2 . (18) and Scaling Horizon Equation of State p ( ε ) = ε. (19) 1-dim ideal radiation, causal Biró Ván Czinner BH positive 11 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Outline Traditional horizon entropy and temperature (Hawking) 1 Homogenity class assumption about volume dependence 2 Proof of positive specific heat 3 Charged black holes 4 Biró Ván Czinner BH positive 12 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Consequences of scaling EoS Thermal stability ∂ 2 S ∂ 2 S 1 − ε ∂ E 2 = s ′′ ( ε ) ∂ E ∂ V = 0 . det det (20) V ε 2 ∂ 2 S ∂ 2 S − ε ∂ V ∂ E ∂ V 2 Heat capacity and specific heat ∂ 2 S ∂ E s ′ ( ε ) = 1 ∂ V s ′′ ( ε ) . ∂ E 2 = (21) ∂ 2 S ∂ T = − 1 1 ∂ T 1 ∂ E 2 = ∂ E = − C V T 2 . (22) ∂ E T 2 Stable if ∂ 2 S ∂ E 2 < 0, i.e. with positive C V . Biró Ván Czinner BH positive 12 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Positivity of specific heat c V = C V / V (heat capacity per volume) T 2 s ′′ ( ε ) = − s ′ ( ε ) 2 1 c V = − (23) s ′′ ( ε ) In our case with horizons s ′′ = − s / 4 ε 2 . It delivers c V = − ( s / 2 ǫ ) 2 √ ( − s / 4 ε 2 ) = s = a 2 πε > 0 . (24) Biró Ván Czinner BH positive 13 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Comparison with Hawking Hawking Present 4 π M 2 4 π aM 2 S H = S = 1 / T H = 8 π M 1 / T = 2 π aM ε = 3 / 32 π M 2 = 3 / 2 A p H = 0 p = c ( H ) = − 3 / 4 M < 0 c V = S / V = 3 a / 8 M > 0 V Biró Ván Czinner BH positive 14 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Which choice to make? a = 4 Relation to Hawking temperature (another view): 1 dS H = 1 � 1 · ∂ S ∂ E + dV dM · ∂ S � = T H dM a ∂ V � 1 � 1 T + 2 p dV = 1 + 2 pA 4 = = aT . (25) a T dR aT By the particular choice a = 4 we have p = ε = 3 c V = 3 / 2 M = − 2 c ( H ) 2 A , T = T H , S = 4 S H = A , V . (26) Proposal: black hole entropy is the total horizon area (in Planck units). Biró Ván Czinner BH positive 15 / 26
Traditional horizon entropy and temperature (Hawking) Homogenity class assumption about volume dependence Proof of positive specific heat Charged black holes Outline Traditional horizon entropy and temperature (Hawking) 1 Homogenity class assumption about volume dependence 2 Proof of positive specific heat 3 Charged black holes 4 Biró Ván Czinner BH positive 16 / 26
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