1. Yoshinori Matsuo (KEK) in collaboration with Hikaru Kawai (Kyoto U.) Yuki Yokokura (Kyoto U.)

2. Black hole evaporation and information loss String theory implies information never be lost We construct a model for this process A solution including a backreaction from Hawking radiation ≇ ⊹⊺ ∽ ⊷ ∲ ≨ ≔ ⊹⊺ ≩ How about ordinary BH (i.e. Schwarzschild)? Basic idea  BH evaporates in finite time  Infinite time to get inside BH } Matter never get Inside the BH No information lost (For observer staying outside the BH)

3. ≴ ≲ ≲ ≳ ∨ ≴ ∩ ≲ ≨ ∽∲ ≍ Flat space Schwarzschild We consider a shock wave (shell) of collapsing matters The locus can be approximated by null geodesic Flat space and Schwarzschild are connected at the locus Shock wave approaches to the horizon The horizon of this geometry differs from Schwarzshild

4. Correction from Hawking radiation Schwarzschild is modified ≤≳ ∲ ∽ ⊡ ⊵ ∱ ⊡ ≡ ∨ ≵ ∩ ≲ ⊶ ≤≵ ∲ ⊡ ∲ ≤≵≤≲ ∫ ≲ ∲ ≤ ⊭ ∲ Vaidya metric One-way (incoming or outgoing) massless and conformal No angular momentum satisfies the following Einstein eq. ≇ ⊹⊺ ∽∸ ⊼≔ ⊹⊺ with Matters (Hawking rad.) are the following: a(u) : decreasing Other : 0 a(u) : decreasing Other : 0 ≔ ≵≵ ∽ ⊡ ≟ ≡ ∨ ≵ ∩ ∸ ⊼≲ ∲

5. Here we consider the scalar field for example. Evaporation is slowWe consider Schwarzschild ∱ ≲ ∲ ≀ ≲ ≲ ∨ ≲ ⊡ ≲ ≈ ∩ ≀ ≲ ⋁ ∨ ≲ ∩∫ ≲∡ ∲ ≲ ⊡ ≲ ≈ ⋁ ∨ ≲ ∩ ⊡ ≌ ∲ ≲ ∲ ⋁ ∨ ≲ ∩ ⊡ ≭ ∲ ⋁ ∨ ≲ ∩∽∰ ⋁ ∨ ≲ ∩ ⊻ ∨ ≲ ⊡ ≲ ≈ ∩ ⊧ ≩∡ Near the horizon, the field behaves as Angular momentum and mass have no contribution. Near the horizon, angular momentum and mass are effectively zero. We assume L 2 is not very large in NH: ≌ ∲ ⊿ ≲ ⊡ ≲ ≈ ≲ ≈

6. Weyl anomaly breaks conformal symmetry Incompatible to the Vaidya metric Anomaly for the conformal mode of the metric cancels that for the matter fields. We consider the following action: Schwinger-Dyson eq. ≨ ≇ ⊹⊺ ≩⊡ ⊷ ∲ ≨ ≔ ⊹⊺ ≩ ∽∰ ≓ ≛ ≧ ⊹⊺ ∻≘ ≝∽ ∱ ∲ ⊷ ∲ ≚ ≤ ∴ ≸ ≰ ⊡ ≧≒ ∫ ≓ ≭≡≴≴≥≲ ≛ ≧ ⊹⊺ ∻≘ ≝ does not have anomaly.

7. We separate the metric to background and fluctuation. ≧ ⊹⊺ ∽≞ ≧ ⊹⊺ ∫ ⊷≨ ⊹⊺ ≢ ≇ ⊹⊺ ⊡ ⊷ ∲ ≨ ≔ ≴≯≴≡≬ ⊹⊺ ≩ ⋃ ∽∰ Einstein-Hilbert action of background Matter action which contains the fluctuation. ≓ ≛≞ ≧ ⊹⊺ ∻ ≨ ⊹⊺ ∻⋁ ≝∽ ∱ ∲ ⊷ ∲ ≚ ≤ ∴ ≸ ≰ ⊡ ≞ ≧ ≢ ≒ ∫ ≓ ≴≯≴≡≬ ≭≡≴≴≥≲ ≛≞ ≧ ⊹⊺ ∻ ≨ ⊹⊺ ∻⋁ ≝ and treat the fluctuation as matter fields. ≞ ≧ ⊹⊺ ∡ ≞ ≧ ⊹⊺ ∫ ⊱ ≞ ≧ ⊹⊺ ≨ ⊹⊺ ∡ ≨ ⊹⊺ ⊡ ∱ ⊷ ⊱ ≞ ≧ ⊹⊺ We consider the following variation Then, the partition function is invariant: ∰∽ ⊱ ≚ ⋃ ≄ ≨ ⊹⊺ ≄ ⋁≥ ⊡ ≓≛≞≧∻≨∻⋁≝

8. EM tensor depends on the state ≞ ≧ ⊹⊺ ≨ ≔ ≴≯≴≡≬ ⊹⊺ ≩ ∽ ≣ ∱ ≒ ∲ ∫ ≣ ∲ ≒ ⊹⊺ ≒ ⊹⊺ ∫ ≣ ∳ ≒ ⊹⊺⊽⊾ ≒ ⊹⊺⊽⊾ However, the trace part comes only from the anomaly. can be balanced with b.g. and cannot be canceled with the background Weyl anomaly must be canceled We take the trace of the Einstein equation: ∰∽ ⊡ ∱ ⊷ ∲ ≢ ≒ ⊡ ≞ ≧ ⊹⊺ ⊭ ≔ ≴≯≴≡≬ ⊹⊺ ⊮ ⋃ ≞ ≧ ⊹⊺ ∽ ≨ ≧ ⊹⊺ ≩ ⋃ Here we take the background as does not depend on the state. No κ- dependence

9. ≵ ≲ ≲ ≳ ∨ ≵ ∩ ≲ ≨ ∨ ≵ ∩ ≡ ∨ ≵ ∩ Vaidya metric and flat space are connected. Time(null) in flat spaceTime(null) in Vaidya ⊡ ∲ ≤≲ ∽ ≤ ≾ ≵ ∽ ⊵ ∱ ⊡ ≡ ∨ ≵ ∩ ≲ ∨ ≵ ∩ ⊶ ≤≵ Inside of the shock wave is replaced by the flat space. Then, the singularity and horizon no longer exist. Inside of the shock wave is replaced by the flat space. Then, the singularity and horizon no longer exist. Junction conditionShock wave is null in both side.

10. Hawking radiation occurs even without the horizon. Method of Eikonal approximation Change of the wave profile is calculated by In the flat space side,, which is connected to Vaidya metric at the locus of the shock wave. So, we need to calculate relation r and u in Vaidya. ≾ ≵ ∽ ⊡ ∲ ≲ In the Vaidya side, locus is given by ≲ ≳ ∨ ≵ ∩∽ ≲ ≨ ∨ ≵ ∩∫ ≃ ≥≸≰ ⊷ ⊡ ≚ ≵ ≡ ∨ ≵ ∰ ∩ ≲ ∲ ≨ ∨ ≵ ∰ ∩ ≤≵ ∰ ⊸ exp で “past horizon” に近づく Hawking radiation ≁ ∡∻∡ ∰ ∽ ≚ ≤≵≥ ⊡ ≩∡ ∰ ≵ ≥ ≩∡≾≵∨≵∩

11. Hawking temperature is ≔ ∽ ∱ ∴ ⊼≡ ∨ ≵ ∩ Mass of BH changes very slowly. ≟ ≡ ∨ ≵ ∩⊢ ≵ ⊿ ≡ ∨ ≵ ∩ u dependence can be neglected: ≲ ≳ ∨ ≵ ∩∽ ≡ ∫ ≃≥ ⊡ ≵ ∲≡ same to ordinary BH Energy of Hawking radiation ⊻ ≡ ∲ ≔ ∴ ⊻ ∱ ≡ ∲ u dependence can be neglected up to. ⊢ ≵ ⊻ ≡ ∳ Next order correction is ∯ ≟ ≡ ∨ ≵ ∩ converges. ≡ ∨ ≵ ∩ ⊻ ≡

12. ≵ ≲ ≲ ≳ ∨ ≵ ∩ ≡ ∨ ≵ ∩ } Scattered ingoing radiation Flat space Potential barrier We have assumed that there are only outgoing radiation However, Hawking radiation is partially reflected by the potential barrier We need to generalize the Vaidya metric.

13. We consider geometry with matters which obey ≴≲ ≔ ⊹⊺ ∽∰ ≔ ⊵⊵ ∽ ≔ ∧∧ ∽∰ conformal Angular momentum and mass can be neglected Most general spherically symmetric geometry is ≤≳ ∲ ∽ ⊡ ≨ ∨ ≵∻≲ ∩ ⊡ ≦ ∨ ≵∻≲ ∩ ≤≵ ∲ ∫∲ ≤≵≤≲ ⊢ ∫ ≲ ∲ ≤ ⊭ ∲ ∲ ≇ ≵≲ ⊡ ≦ ∨ ≵∻≲ ∩ ≇ ≲≲ ∽∰ From the traceless condition for u-r surface, ≨ ∨ ≵∻≲ ∩∽ ≀ ≲ ∨ ≲≦ ∨ ≵∻≲ ∩∩ ≤≳ ∲ ∽ ⊡ ≀ ≲ ≰ ∨ ≵∻≲ ∩ ⊷ ≰ ∨ ≵∻≲ ∩ ≲ ≤≵ ∲ ∫∲ ≤≵≤≲ ⊸ ∫ ≲ ∲ ≤ ⊭ ∲ Then, the metric becomes ( p(u,r) = r f (u,r) )

14. ≤≳ ∲ ∽ ⊡ ≀ ≲ ≰ ∨ ≵∻≲ ∩ ⊷ ≰ ∨ ≵∻≲ ∩ ≲ ≤≵ ∲ ∫∲ ≤≵≤≲ ⊸ ∫ ≲ ∲ ≤ ⊭ ∲ p(u,r) satisfies the following EOM: ∲ ≲≀ ≵ ⊵ ≀ ∲ ≲ ≰ ∨ ≵∻≲ ∩ ≲≀ ≲ ≰ ∨ ≵∻≲ ∩ ⊶ ⊡ ∱ ≰ ∨ ≵∻≲ ∩ ≀ ≲ ⊵ ≰ ∲ ∨ ≵∻≲ ∩ ≀ ∲ ≲ ≰ ∨ ≵∻≲ ∩ ≲≀ ≲ ≰ ∨ ≵∻≲ ∩ ⊶ ∽∰ The Vaidya metric is a special case of ≰ ∨ ≵∻≲ ∩∽ ≲ ⊡ ≡ ∨ ≵ ∩ ≰ ∨ ≵∻≲ ∩ ∡ ≦ ∨ ≵ ∩ ≰ ∨ ≵∻≲ ∩ ≤≵ ∰ ∽ ≦ ∨ ≵ ∩ ≤≵ is absorbed by The ingoing energy flow can be expressed as ≓ ∨ ≵∻≲ ∩∽ ≰ ∲ ∨ ≵∻≲ ∩ ≔ ≲≲ ∨ ≵∻≲ ∩∽ ≰ ∲ ∨ ≵∻≲ ∩ ≀ ∲ ≲ ≰ ∨ ≵∻≲ ∩ ≲≀ ≲ ≰ ∨ ≵∻≲ ∩ The other condition determimes p(u,r).

15. The second equation is equivalent to EOM for p(u,r). ∰∽ ≀ ≵ ≓ ∨ ≵∻≲ ∩ ⊡ ≰ ∨ ≵∻≲ ∩ ∲ ≲ ≀ ≲ ≓ ∨ ≵∻≲ ∩ ⊡ ∲ ≀ ≵ ≰ ∨ ≵∻≲ ∩ ≰ ∨ ≵∻≲ ∩ ≓ ∨ ≵∻≲ ∩ Evaporation is very slowWe neglect u dependence. ≰ ∲ ∨ ≵∻≲ ∩ ≀ ∲ ≲ ≰ ∨ ≵∻≲ ∩ ≲≀ ≲ ≰ ∨ ≵∻≲ ∩ ∽ ≃ ∨ ≵ ∩ Where C can be estimated as ≔ ≲≲ ⊻ ≲ ∲ ≰ ∲ ≔ ≵≵ ≃ ∽ ≓ ∽ ≰ ∲ ≔ ≲≲ ⊻ ≟ ≡ ⊻ ⊵ ≬ ≰ ≡ ⊶ ∲ Conservation law is given by ∰∽ ≀ ≲ ≔ ≵≵ ∨ ≵∻≲ ∩∫ ∲ ≲ ≔ ≵≵ ∨ ≵∻≲ ∩ ⊡ ∱ ≲≰ ∨ ≵∻≲ ∩ ⊵ ⊡ ≀ ≵ ≰ ∨ ≵∻≲ ∩ ≰ ∨ ≵∻≲ ∩ ≓ ∨ ≵∻≲ ∩∫ ≀ ≵ ≓ ∨ ≵∻≲ ∩ ⊶

16. The solution can be approximated as ≰ ∫ ≡≃ ≬≯≧ ⊵ ≰ ⊡ ≡≃ ≡ ⊶ ∽ ≲ ⊡ ≡ ≥⊮ Numerical solution is plotted as For ≰ ⊻ ≲ ⊡ ≡ ≥⊮ ≲∾≡ ≲∼≡ ≰ ⊻ ≃≡ ∽≣≯≮≳≴∮ This implies that shock wave is smeared around. ≲ ⊻ ≡ Outside is approximated by Vaidya.

17. We put BH in a heat bath.In and Out of rad. are balanced finally goes some stationary state. ≤≳ ∲ ∽ ≰ ∰ ∨ ≲ ∩ ⊵ ⊡ ≰ ∨ ≲ ∩ ≲ ≤≴ ∲ ∫ ≲ ≰ ∨ ≲ ∩ ≤≲ ∲ ⊶ ∫ ≲ ∲ ≤ ⊭ ∲ Stationary geometry with ≇ ⊹⊺ ≧ ⊹⊺ ∽∰ and ≇ ⊵⊵ ∽ ≇ ∧∧ ∽∰ S(r) = C becomes exact. We assume that this metric can be used for r < a. ≰ ∨ ≲ ∩ ∧ ≃≡ This geometry cannot be used around r = 0, since ≔ ⊵ ⊵ ⊻ ∱ ≲ ∲

18. ⊽ ∽ ⊡ ≔ ≴ ≴ ∽ ∱ ∸ ⊼ ≃ ≲≰ ∨ ≲ ∩ ≰ ∰ ∨ ≲ ∩ ∸ ⊼≔ ≴≴ ∽ ≇ ≴≴ ∽ ≰ ∲ ∨ ≲ ∩ ≰ ∰∰ ∨ ≲ ∩ ≲ ∳ ≰ ∰ ∨ ≲ ∩ ∽ ≃ ≲ ∲ Energy-momentum tensor inside the BH is Then, the energy density of matter is given by Local temperature inside the BH is ≔ ∽ ≔ ≈ ≰ ⊡ ≧ ≴≴ ∽ ≔ ≈ ≲ ≲ ≰ ∨ ≲ ∩ ≰ ∰ ∨ ≲ ∩ where T H is the Hawking temperature ⊻ ∱ ∴ ⊼≡

19. Pressure is not isotropic We assume the following thermodynamic relation ⊽ ∽ ≔≳ Mass is reproduced by integrating the energy density ≍ ∽∲ ≚ ⊧ ⊵ ≔ ⊹⊺ ⊡ ∱ ∲ ≔≧ ⊹⊺ ⊶ ≮ ⊹ ⊻ ⊺ ≰ ≧ ⊧ ≤ ∳ ≸ ∽∲ ≚ ⊧ ⊽ ≰ ⊡ ≧≤ ∳ ≸ ∽ ≡ ∲ Then, the total entropy inside the BH is ≓ ∽ ≚ ⊧ ≤ ∳ ≸ ≰ ≧ ⊧ ≳ ∽ ≚ ≡ ∰ ≤≲ ∲ ⊼≡≃≲ ≰ ∨ ≲ ∩ ∽ ⊼≡ ∲ which reproduces the area law: ≓ ∽ ∱ ∴ ≁

20.  We have constructed a model of BH evaporation.  Collapsing matters are described by shock wave.  Geometry is obtained by connecting flat space and Vaidya.  This geometry has neither singularity nor horizon  Hawking radiation occurs without horizon.  No information is lost.  BH is ordinary thermodynamic object.  BH Entropy is reproduced from that of matters inside.