Spacetime Structure of Black Hole Evaporation

1. Spacetime Structure of Black Hole Evaporation Yoshinori Matsuo (KEK) in collaboration with Hikaru Kawai (Kyoto U.) Yuki Yokokura (Kyoto U.)

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

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

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

5. Matter fields near the horizon Here we consider the scalar field for example. We consider Schwarzschild Evaporation is slow ⊡ ≲ ≲ ∲ ∲ ∱ ≲ ∡ ≌ ≈ ⊧ ∲ ≩ ∡ ⊻ ⊿ ⊡ ⋁ ∨ ≲ ≌ ∩ ∨ ≲ ≲ ∩ ∲ ⊡ ⊡ ⊡ ≀ ≲ ∨ ≲ ≲ ∩ ≀ ⋁ ∨ ≲ ∩ ∫ ⋁ ∨ ≲ ∩ ⋁ ∨ ≲ ∩ ≭ ⋁ ∨ ≲ ∩ ∽ ∰ ≈ ≲ ≈ ≲ ≲ ∲ ∲ ⊡ ≲ ≲ ≲ ≲ ≈ ≈ 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 L2is not very large in NH:

6. Weyl anomaly Weyl anomalybreaksconformal 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-Dysoneq. does not have anomaly.

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

8. Weyl anomaly cancelation We take the trace of the Einstein equation: ⊭ ⊮ ∱ ≢ ⊹ ⊺ ≴ ≯ ≴ ≡ ≬ ∲ ⊹ ⊺ ⊹ ⊺ ≴ ≯ ≴ ≡ ≬ ⊹ ⊺ ⊽ ⊾ ≨ ≩ ⊡ ⊡ ≨ ≩ ≧ ≞ ≔ ∰ ∽ ∽ ≣ ≒ ≧ ≞ ∫ ≒ ∽ ≣ ≒ ≧ ≞ ≧ ≒ ≔ ∫ ≣ ≒ ≒ Here we take the background as ∱ ⊹ ⊺ ∲ ⊹ ⊹ ⊺ ⊺ ∳ ⊹ ⊺ ⊽ ⊾ ⊹ ⊺ ⊹ ⊺ ⋃ ∲ ⋃ ⊷ EM tensor depends on the state can be balanced with b.g. However, the trace part comes only from the anomaly. does not depend on the state. Noκ-dependence and cannot be canceled with the background Weyl anomaly must be canceled

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

10. Hawking radiation 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 radiation (2) Mass of BH changes very slowly. ≒ ≚ ∰ ∰ ≡ ∨ ≵ ∩ ≵ ∰ ∰ ⊡ ≤ ≵ ≵ ⊣ ⊡ ⊢ ⊡ ⊢ ⊤ ∰ ∰ ≡ ≟ ∨ ≵ ∩ ∲ ≵ ∨ ∰ ∩ ∨ ∩ ∰ ∰ ∰ ∰ ∰ ∰ ∰ ∰ ∨ ∱ ∩ ∨ ∰ ∩ ∨ ∰ ∩ ∲ ≲ ∨ ≵ ∩ ⊡ ⊡ ⊡ ≲ ∨ ≵ ∩ ∽ ≤ ≵ ≡ ∨ ≵ ∩ ≲ ∨ ≵ ∩ ≡ ∨ ≵ ∩ ≬ ≯ ≧ ≲ ∨ ≵ ∩ ≡ ∨ ≵ ∩ ≥ ≳ u dependence can be neglected: ≳ ≳ ≳ ∨ ∰ ∩ ∰ ≲ ∨ ≵ ∩ ≳ ∱ ∱ ≵ ⊡ ∲ ∴ ∳ ⊻ ∯ ⊻ ⊿ ⊻ ⊻ ≲ ≡ ≟ ∨ ∨ ≵ ≔ ≵ ∩ ∩ ≡ ⊢ ≡ ∽ ∽ ⊢ ∨ ≵ ≵ ≔ ≵ ≡ ∩ ≡ ≟ ∨ ∫ ≵ ≡ ∩ ≃ ≡ ≡ ∨ ≥ ≵ ∩ same to ordinary BH ∲ ≡ ≳ ∲ ∴ ⊼ ≡ ∨ ≵ ≡ ∩ Hawking temperature is converges. Energy of Hawking radiation Next order correction is u dependence can be neglected up to .

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

13. Introducing the ingoing energy flow 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) = rf (u,r) )

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

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

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

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

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

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

20. Conclusion • 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.