Quasilocal Smarr Relations in collaboration with Yein Lee(Kyung Hee U.), Mattew Richards(U. of McMaster), Sean Stotyn(U. of Calgary) 1 Miok Park Korea Institute for Advanced Study (KIAS), Seoul, S. Korea Strings and Fields 2019, at YITP August 19, 2019 1 arXiv:1809.07259 for AF (ver2 will be updated), arXiv:190X.XXXXX for AAdS
Content Introduction 1 a conserved charge in curved spacetime? what is Smarr Relation? why we need Quasilocal Formalism? Quasilocal Formalism 2 Quasilocal Quantities : how they are defined Quasilocal Smarr Relation in Asymptotically Flat spacetimes 3 QL Smarr Relation by Euler Theorem Example Quasilocal Smarr Relation in Asymptotically AdS spacetimes 4 QL Smarr Relation for AAdS by Euler Theorem Example Conclusion and Future Works 5 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 1 / 19
Introduction a conserved charge in curved spacetime? Conserved charge in curved spacetime There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR ∇ a ( T ab ξ b ) = 0 , where n a is the unit normal to Σ and t a a time-like Killing field. Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19
Introduction a conserved charge in curved spacetime? Conserved charge in curved spacetime There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR ∇ a ( T ab ξ b ) = 0 , where n a is the unit normal to Σ and t a a time-like Killing field. This guarantees that total energy is conserved. Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19
Introduction a conserved charge in curved spacetime? Conserved charge in curved spacetime There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR � ∇ a ( T ab ξ b ) = 0 , T ab n a t b → E = Σ where n a is the unit normal to Σ and t a a time-like Killing field. This guarantees that total energy is conserved. In GR R µν − 1 2 Rg µν = 8 π T µν the energy properties of matter are represented by T µν , but a gravitational field energy is not included in T µν . Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19
Introduction a conserved charge in curved spacetime? Conserved charge in curved spacetime There are difficulties to calculate a gravitational conserved energy in curved spacetime. In SR � ∇ a ( T ab ξ b ) = 0 , T ab n a t b → E = Σ where n a is the unit normal to Σ and t a a time-like Killing field. This guarantees that total energy is conserved. In GR R µν − 1 2 Rg µν = 8 π T µν the energy properties of matter are represented by T µν , but a gravitational field energy is not included in T µν . But some methods have been developed such as ADM method, Komar method, AD(T) method, and etc. Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 2 / 19
Introduction what is Smarr Relation? Smarr relation by Larry Smarr in 1973 2 du 2 + 2 dudr + 2 a sin 2 θ 1 − 2 Mr − Q 2 � � ( 2 Mr − Q 2 ) dud φ − 2 a sin 2 θ drd φ ds 2 RN = R 2 R 2 − R 2 d θ 2 + sin 2 θ � � ∆ a 2 sin 2 θ − ( a 2 + r 2 ) 2 d φ 2 , R 2 a ≡ J R 2 ≡ r 2 + a 2 cos 2 θ, ∆ ≡ r 2 + a 2 − 2 Mr + Q 2 , M The black hole’s area is written as � � 2 M 2 + 2 ( M 4 − J 2 − M 2 Q 2 ) 1 2 − Q 2 S = 4 π , 2 Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19
Introduction what is Smarr Relation? Smarr relation by Larry Smarr in 1973 2 du 2 + 2 dudr + 2 a sin 2 θ 1 − 2 Mr − Q 2 � � ( 2 Mr − Q 2 ) dud φ − 2 a sin 2 θ drd φ ds 2 RN = R 2 R 2 − R 2 d θ 2 + sin 2 θ � � ∆ a 2 sin 2 θ − ( a 2 + r 2 ) 2 d φ 2 , R 2 a ≡ J R 2 ≡ r 2 + a 2 cos 2 θ, ∆ ≡ r 2 + a 2 − 2 Mr + Q 2 , M The black hole’s area is written as � � 2 M 2 + 2 ( M 4 − J 2 − M 2 Q 2 ) 1 2 − Q 2 S = 4 π , � S � 1 16 π + 4 π J 2 + Q 2 2 + π Q 4 2 M = = M [ S , J , Q ] S S 2 Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19
Introduction what is Smarr Relation? Smarr relation by Larry Smarr in 1973 2 du 2 + 2 dudr + 2 a sin 2 θ 1 − 2 Mr − Q 2 � � ( 2 Mr − Q 2 ) dud φ − 2 a sin 2 θ drd φ ds 2 RN = R 2 R 2 − R 2 d θ 2 + sin 2 θ � � ∆ a 2 sin 2 θ − ( a 2 + r 2 ) 2 d φ 2 , R 2 a ≡ J R 2 ≡ r 2 + a 2 cos 2 θ, ∆ ≡ r 2 + a 2 − 2 Mr + Q 2 , M The black hole’s area is written as � � 2 M 2 + 2 ( M 4 − J 2 − M 2 Q 2 ) 1 2 − Q 2 S = 4 π , � S � 1 16 π + 4 π J 2 + Q 2 2 + π Q 4 2 M = = M [ S , J , Q ] S S dM = TdS + Ω dJ + Φ dQ 2 Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19
Introduction what is Smarr Relation? Smarr relation by Larry Smarr in 1973 2 du 2 + 2 dudr + 2 a sin 2 θ 1 − 2 Mr − Q 2 � � ( 2 Mr − Q 2 ) dud φ − 2 a sin 2 θ drd φ ds 2 RN = R 2 R 2 − R 2 d θ 2 + sin 2 θ � � ∆ a 2 sin 2 θ − ( a 2 + r 2 ) 2 d φ 2 , R 2 a ≡ J R 2 ≡ r 2 + a 2 cos 2 θ, ∆ ≡ r 2 + a 2 − 2 Mr + Q 2 , M The black hole’s area is written as � � 2 M 2 + 2 ( M 4 − J 2 − M 2 Q 2 ) 1 2 − Q 2 S = 4 π , � S � 1 16 π + 4 π J 2 + Q 2 2 + π Q 4 2 M = = M [ S , J , Q ] S S dM = TdS + Ω dJ + Φ dQ = ⇒ M = 2 TS + 2 Ω J + Φ Q 2 Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 3 / 19
Introduction what is Smarr Relation? Smarr relation by Larry Smarr in 1973 3 Euler theorem states that if a function f ( x , y , z ) obeys the scaling relation f ( α p x , α q y , α k z ) = α r f ( x , y , z ) , (1) then it satisfies � ∂ f � ∂ f � ∂ f � � � rf ( x , y , z ) = p x + q y + k z . (2) ∂ x ∂ y ∂ z M in terms of S , L and Q satisfies this relation having a following scaling S ∝ [ L ] 2 , J ∝ [ L ] 2 , M ∝ [ L ] , Q ∝ [ L ] , (3) then the Euler’s theorem yields � ∂ M � � ∂ M � � ∂ M � M = S + J + Q . (4) ∂ S ∂ J ∂ Q 3 Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 4 / 19
Introduction what is Smarr Relation? Smarr relation by Larry Smarr in 1973 3 Euler theorem states that if a function f ( x , y , z ) obeys the scaling relation f ( α p x , α q y , α k z ) = α r f ( x , y , z ) , (1) then it satisfies � ∂ f � ∂ f � ∂ f � � � rf ( x , y , z ) = p x + q y + k z . (2) ∂ x ∂ y ∂ z M in terms of S , L and Q satisfies this relation having a following scaling S ∝ [ L ] 2 , J ∝ [ L ] 2 , M ∝ [ L ] , Q ∝ [ L ] , (3) then the Euler’s theorem yields M = 2 TS + 2 Ω J + Φ Q . (4) 3 Larry Smarr, “Mass Formula for Kerr Black Holes”,Phys. Rev. Lett. 30, 521 Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 4 / 19
Introduction what is Smarr Relation? Smarr relation by Bardeen, Carter, and Hawking In 1973, this work resulted in “The Four laws of black hole mechanics 4 ” by Bardeen, Carter, and Hawking 4 J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973) Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19
Introduction what is Smarr Relation? Smarr relation by Bardeen, Carter, and Hawking In 1973, this work resulted in “The Four laws of black hole mechanics 4 ” by Bardeen, Carter, and Hawking M = − 1 � 1 � dS µν D µ k ν = − dS µ R µ ν ξ ν 8 π 8 π G ∂ V V 1 � � ν ξ ν − dS µ T µ dS µν D µ ξ ν = − 2 8 π G Σ H 4 J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973) Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19
Introduction what is Smarr Relation? Smarr relation by Bardeen, Carter, and Hawking In 1973, this work resulted in “The Four laws of black hole mechanics 4 ” by Bardeen, Carter, and Hawking M = − 1 � 1 � dS µν D µ k ν = − dS µ R µ ν ξ ν 8 π 8 π G ∂ V V 1 dS µν D µ ξ ν = Φ H Q + κ A � � ν ξ ν − dS µ T µ = − 2 8 π G 8 π Σ H 4 J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973) Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19
Introduction what is Smarr Relation? Smarr relation by Bardeen, Carter, and Hawking In 1973, this work resulted in “The Four laws of black hole mechanics 4 ” by Bardeen, Carter, and Hawking M = − 1 � 1 � dS µν D µ k ν = − dS µ R µ ν ξ ν 8 π 8 π G ∂ V V 1 dS µν D µ ξ ν = Φ H Q + κ A � � ν ξ ν − dS µ T µ = − 2 8 π G 8 π Σ H (Same as Smarr Relation!!) 4 J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973) Miok Park (KIAS) Quasilocal Smarr Relations August 19, 2019 5 / 19
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