A Model of Black Hole Evaporation and 4D Weyl Anomaly RIKEN-iTHES - PowerPoint PPT Presentation

A Model of Black Hole Evaporation and 4D Weyl Anomaly RIKEN-iTHES Yuki Yokokura with H. Kawai (kyoto university) [H. Kawai and Y. Y, Universe 3, 51, (2017)] 2017 August 7 @ Strings and Fields

Motivation What is the “black hole” in quantum mechanics? ⇒ I will give a possibility that in spherically- symmetric systems • No horizon appears in QM. • The quantum BH is a dense star that looks like the classical BH from the outside. ⇒ I will solve time evolution of the whole spacetime in a self-consistent manner: � 𝐻 𝜈𝜈 = 8 𝜌𝐻 𝑈 𝜈𝜈

Basic idea: step1 Consider a spherically-symmetric evaporating BH. Add a spherical thin shell (or a particle). Hawking radiation particle spherical thin shell BH What happens? ⇒ The shell will never reach “the horizon”.

The shell never crosses “the horizon”. 𝑠 Outside ≈ time-dependent “Schwarzschild” metric Hawking radiation 𝑠 𝑡 ( 𝑢 ) 𝑒𝑡 2 = − 𝑠 − 𝑏 𝑢 𝑠 𝑒𝑢 2 + 𝑠 − 𝑏 ( 𝑢 ) 𝑒𝑠 2 + 𝑠 2 𝑒Ω 2 . 𝑠 𝑒𝑢 𝑏 𝑢 = − 2𝜏 𝑒 𝑏 𝑢 2 𝑏 ( 𝑢 ) 𝑏 ≡ 2𝐻𝐻 intensity: 𝜏 = 𝑃 1 ~ ℏ𝐻𝐻 𝑢 ( 𝐻 =d.o.f. of fields) Δ𝑢 𝑚𝑚𝑚𝑚 ~ 𝑏 3 / 𝜏 e.o.m for 𝑠 𝑡 ~ 𝑏 : 𝑠 𝑡 ( 𝑢 ) 𝑒𝑠 𝑡 ( 𝑢 ) = − 𝑠 𝑡 𝑢 − 𝑏 𝑢 . 𝑒𝑢 𝑠 𝑡 𝑢 −𝑏 𝑢 𝑒𝑏 ( 𝑢 ) = 2𝜏 𝑏 ( 𝑢 ) − 𝑢 𝑡 𝑢 ≃ 𝑏 𝑢 − 𝑏 𝑢 𝑒𝑏 𝑢 𝑏 ( 𝑢 ) 𝑒𝑢 ⇒ 𝑠 + 𝐷𝑏 𝑢 𝑓 𝑏 ( 𝑢 ) 𝑒𝑢 2𝜏 → 𝑏 𝑢 + 𝑏 𝑢 Δ𝑢 ~ 𝑏 𝑢 2𝜏 ⇒ The shell will approach 𝑏 𝑢 + 𝑏 𝑢 .

The shell never crosses “the horizon”. 𝑠 Outside ≈ time-dependent “Schwarzschild” metric Hawking radiation 𝑠 𝑡 ( 𝑢 ) 𝑒𝑡 2 = − 𝑠 − 𝑏 𝑢 𝑠 𝑒𝑢 2 + 𝑠 − 𝑏 ( 𝑢 ) 𝑒𝑠 2 + 𝑠 2 𝑒Ω 2 . 𝑠 𝑒𝑢 𝑏 𝑢 = − 2𝜏 𝑒 𝑏 𝑢 2 𝑏 ( 𝑢 ) 𝑏 ≡ 2𝐻𝐻 intensity: 𝜏 = 𝑃 1 ~ ℏ𝐻𝐻 𝑢 ( 𝐻 =d.o.f. of fields) Δ𝑢 𝑚𝑚𝑚𝑚 ~ 𝑏 3 / 𝜏 e.o.m for 𝑠 𝑡 ~ 𝑏 : ・ The proper length is non-trivial 2 𝜏 𝑠 𝑡 ( 𝑢 ) 𝑒𝑠 𝑡 ( 𝑢 ) = − 𝑠 𝑡 𝑢 − 𝑏 𝑢 Δ𝑚 = 𝑕 𝑠𝑠 𝑏 ≈ 𝜏 ∼ 𝐻𝑚 𝑞 ≫ 𝑚 𝑞 . 𝑒𝑢 𝑠 𝑡 𝑢 −𝑏 𝑢 𝑒𝑏 ( 𝑢 ) = 2𝜏 𝑏 ( 𝑢 ) − 𝑢 if there are many matter fields: 𝑡 𝑢 ≃ 𝑏 𝑢 − 𝑏 𝑢 𝑒𝑏 𝑢 𝑏 ( 𝑢 ) 𝑒𝑢 ⇒ 𝑠 + 𝐷𝑏 𝑢 𝑓 𝑏 ( 𝑢 ) 𝐻 ≫ 1. 𝑒𝑢 (Note: The ( 𝑢 , 𝑠 ) -coordinates are complete to 2𝜏 → 𝑏 𝑢 + 𝑏 𝑢 describe this motion.) Δ𝑢 ~ 𝑏 𝑢 2𝜏 ⇒ The shell will approach 𝑏 𝑢 + 𝑏 𝑢 .

Basic idea: step2 After the shell comes sufficiently close to 𝑠 = 𝑏 + 2𝜏 𝑏 , the total system composed of the BH and the shell behaves like an ordinary BH with mass 𝐻 + Δ𝐻 . BH = shell BH 𝐻 + Δ𝐻 𝐻 Δ𝐻 radiation Indeed, we can check that, (radiation from the shell with Δ𝐻 ) (radiation from the shell with Δ𝐻 ) +(small redshift factor) (radiation from the core BH with 𝐻 ) +(redshift factor) (radiation from the core BH with 𝐻 ) = (radiation from a BH with 𝐻 + Δ𝐻 ) = (radiation from a BH with 𝐻 + Δ𝐻 )

Basic idea: step3 continuous collapsing matter (not BH) 𝑠 𝑠 ′ ( 𝑢𝑢 ) −𝑏 ′ 𝑒𝑏 ′ 𝑒𝑢𝑢 = 2𝜏 𝑏𝑢 𝑏𝑢 ( 𝑢𝑢 ) 𝑢𝑢 Regard this Apply the previous result as many shells. to each shell recursively. = BH 𝑠 = 𝑏 + 2𝜏 𝑏 Any object we recognize as a BH There is not a horizon but a surface . should be such an object.

Strong angular pressure is induced by 4D Weyl anomaly. Balance eq for spherical star (TOV eq) 𝑒𝑞 𝑠 𝑒𝑠 + 2 gravitational force 𝑠 𝑞 𝑠 − 𝑞 𝜄 + 𝑕 𝑠 𝜍 + 𝑞 𝑠 = 0 𝑞 𝑠 ≪ 𝑞 𝜄 balance ⇒ not a fluid ⇒ This object is stable!

Our strategy: self-consistent eq. � = quantum matter fields 𝜚 the self-consistent eq. � 𝑚 = 0 2 𝜚 𝛼 𝑕 � 𝐻 𝜈𝜈 = 8 𝜌𝐻 𝑈 𝜈𝜈 =collapsing matter 𝑕 𝜈𝜈 = a classical field + Hawking radiation

How to obtain the solution? We assume spherical symmetry. Step2: Evaluate 𝑈 𝜈𝜈 on 𝑕 𝜈𝜈 by considering conformal matter. the self-consistent eq. See [H.Kawai and Y.Y, PRD 2016] � for a more general case. 𝐻 𝜈𝜈 = 8 𝜌𝐻 𝑈 𝜈𝜈 Step1: Construct a candidate Step3: Put this and metric 𝑕 𝜈𝜈 by a simple model. determine the self-consistent 𝑕 𝜈𝜈 .

Step1: Construction of a candidate 𝑕 𝜈𝜈 (1/4) Preliminary: A single-shell model Consider time evolution of a spherical null shell. flat 𝑏 flat Assumption Scattering of radiations Radiation goes to infinity 𝑠 without reflection. scattering a collapsing (or mass) null shell 𝑣 𝑠 𝑡 ( 𝑣 ) 𝑠 𝑣 𝑏 ( 𝑣 ) 𝑉 inside = flat: 𝑒𝑡 2 = −𝑒𝑉 2 − 2𝑒𝑉𝑒𝑠 + 𝑠 2 𝑒Ω 2 𝑣

Step1: Construction of a candidate 𝑕 𝜈𝜈 (1/4) Preliminary: A single-shell model Consider time evolution of a spherical null shell. flat 𝑏 flat Assumption Radiation goes to infinity outside = Vaidya metric: without reflection. 𝑒𝑡 2 = − 1 − 𝑏 𝑣 𝑒𝑣 2 − 2𝑒𝑣𝑒𝑠 + 𝑠 2 𝑒Ω 2 a collapsing 𝑠 𝐻 𝑣𝑣 = − 𝑏̇ 𝑣 null shell 𝑠 2 , 𝐻 𝑝𝑢𝑝𝑚𝑠𝑡 = 0 𝑏 𝑣 is not fixed yet. 𝑠 𝑡 ( 𝑣 ) 𝑠 𝑣 𝑏 ( 𝑣 ) 𝑉 inside = flat: 𝑒𝑡 2 = −𝑒𝑉 2 − 2𝑒𝑉𝑒𝑠 + 𝑠 2 𝑒Ω 2 𝑣 Note: At this stage we don’t claim that this is the solution.

Step1: Construction of a candidate 𝑕 𝜈𝜈 (2/4) A multi-shell model Consider a continuous spherical matter. ⇒ Model this as many shells. 𝑠 𝑠 = − 𝑠 − 𝑏 ( 𝑣 ) 𝑜 𝑒𝑣 2 − 2𝑒𝑣𝑒𝑠 + 𝑠 2 𝑒Ω 2 2 𝑒𝑡 𝑝𝑣𝑢 𝑠 ⋯ 2 = − 𝑠 − 𝑏 𝑚 ( 𝑣 𝑚 ) 𝑠 𝑚 2 − 2𝑒𝑣 𝑚 𝑒𝑠 + 𝑠 2 𝑒Ω 2 𝑒𝑡 𝑚 𝑒𝑣 𝑚 𝑠 𝑠 𝑚−1 = − 𝑠 − 𝑏 𝑚−1 ( 𝑣 𝑚−1 ) 2 2 − 2𝑒𝑣 𝑚−1 𝑒𝑠 + 𝑠 2 𝑒Ω 2 𝑒𝑡 𝑚−1 𝑒𝑣 𝑚−1 ⋯ 𝑠 𝑏 𝑚 𝑏 𝑚−1 𝑠 1 2 = −𝑒𝑉 2 − 2𝑒𝑉𝑒𝑠 + 𝑠 2 𝑒Ω 2 𝑒𝑡 0 flat 𝑣

Step1: Construction of a candidate 𝑕 𝜈𝜈 (3/4): Self-consistent ansatz Ansatz: Each shell behaves like the ordinary evaporating BH: 𝑒𝑏 𝑚 = − 𝜏 2 , 𝑒𝑣 𝑚 𝑏 𝑚 and that each shell has already come close to 𝑠 𝑚 = 𝑏 𝑚 + 2 𝜏 𝑏 𝑚 𝑠 𝑠 𝑚 ( 𝑣 𝑚 ) 𝑒𝑏 𝑚 = 2𝜏 − 2 𝑏 𝑚 𝑒𝑣 𝑚 𝑏 𝑚 𝑏 𝑚 ( 𝑣 𝑚 ) 𝑣 𝑚 ⇒ After taking continuum limit ( Δ𝑏 𝑚 ≡ 𝑏 𝑚 − 𝑏 𝑚−1 → 0) , we obtain….

Step1: Construction of a candidate 𝑕 𝜈𝜈 (4/4): the metric − 1 − 𝑏 𝑣 𝑒𝑣 2 − 2𝑒𝑣𝑒𝑠 + 𝑠 2 𝑒𝛻 2 𝑠 𝑒𝑡 2 = − 2 𝜏 2 −𝑠 2 𝑒𝑣 2 − 2 𝑓 − 1 2 −𝑠 2 𝑒𝑣𝑒𝑠 + 𝑠 2 𝑒𝛻 2 𝑠 2 𝑓 − 1 2𝜏 𝑆 𝑏 𝑣 4𝜏 𝑆 𝑏 𝑣 Here 𝑒𝑏 𝑣 𝜏 𝑆 𝑏 ≡ 𝑏 + 2𝜏 = − 𝑏 𝑣 2 , 𝑏 𝑒𝑣 Note: At this stage, 𝜏 is not determined. 𝑠 = 𝑆 ( 𝑏 ( 𝑣 )) 𝑠 𝑣 Δ𝑣 𝑚𝑚𝑚𝑚 ~ 𝑏 3 / 𝜏 Note: At this stage we don’t claim yet that this is the solution.

Step2: Evaluation of 𝑈 𝜈𝜈 (1/3): Setup Consider the interior region. The background metric is static: 𝑠 2 𝑠 2 2 𝜏 𝑒𝑡 2 = − 2𝜏 𝑒𝑉 2 − 2 𝑓 4𝜏 𝑒𝑉𝑒𝑠 + 𝑠 2 𝑒𝛻 2 𝑠 2 𝑓 = −𝑓 𝜒 𝑠 ( 𝑉 , 𝑊 ) 𝑒𝑉𝑒𝑒 + 𝑠 𝑉 , 𝑒 2 𝑒Ω 2 , ⇒ 𝑈 𝜈𝜈 also should be static: 𝑈 𝜈𝜈 = 𝑈 𝜈𝜈 ( 𝑠 ) , 𝑈 𝑉𝑉 = 𝑈 𝑊𝑊 Physical origin of 𝑈 𝑉𝑊 ≠ 0 𝑠 ⇒ The use of Vaidya metric means 𝑈 𝑉𝑉 scattering 𝑈 𝑉𝑊 = 0, (or mass) 𝑈 𝑉𝑊 =0 we have to determine only 𝜄 . 𝑈 𝑉𝑉 , 𝑈 𝑉 𝜄 ( ⇒ We can remove this artificial assumption and generalize it to 〈𝑈 𝑉𝑊 〉 ≠ 0 .)

Step2: Evaluation of 𝑈 𝜈𝜈 (2/3): The relations of 𝑈 𝜈𝜈 ・ 1st eq. 𝜈 = 2 𝑕 𝑉𝑊 𝑈 𝜄 leads to 𝑈 𝑉𝑊 + 2 𝑈 𝜈 𝜄 𝜄 = 1 𝜈 𝑈 2 𝑈 𝜄 𝜈 All components are ・ 2nd eq. 𝜈 . expressed in terms of 𝑈 𝛼 𝜈 𝑈 𝜈 𝜈𝑉 = 0 provides 𝑠 𝑉𝑉 = 1 𝑠 2 𝑈 2 � 𝑒𝑠 ′ 𝑠 ′ 𝑓 𝜒 ( 𝑠 ′ ) 𝑈 + 𝑠 2 𝑈 𝜄 ( 𝑠 ′ ) 𝑠=0 , 𝑉𝑉 𝜄 0 𝑠 surface at 𝑠 = 𝑆 ( 𝑏 ( 𝑣 )) Boundary condition: 𝑈 𝜈𝜈 ( 𝑠 = 0) = 0 𝑒 = const. 𝑉 = const. flat 𝑉

Step2: Evaluation of 𝑈 𝜈𝜈 (3/3): 4D Weyl anomaly For simplicity, consider conformal matters. 𝜈 is determined by the 4D Weyl anomaly: ⇒ 𝑈 𝜈 𝜈 = ℏ𝑑 𝑥 ℱ − ℏ𝑏 𝑥 ℊ 𝑈 ← state-independent 𝜈 where ℊ ≡ 𝑆 𝜈𝜈𝜈𝜈 𝑆 𝜈𝜈𝜈𝜈 − 4 𝑆 𝜈𝜈 𝑆 𝜈𝜈 + 𝑆 2 ℱ ≡ 𝐷 𝜈𝜈𝜈𝜈 𝐷 𝜈𝜈𝜈𝜈 , ⇒ The metric determines 𝜈 = ℏ𝑑 𝑥 𝑈 3 𝜏 2 𝜈