Spacetime Structure of

Black Hole EvaporationYoshinori Matsuo (KEK)

in collaboration with Hikaru Kawai (Kyoto U.) Yuki Yokokura (Kyoto U.)

IntroductionBlack hole evaporation and information loss

String theory implies information never be lost

We construct a model for this processA 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)

BH formation with shock wave of matter

≴

≲≲≳∨≴∩

≲≨∽∲≍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

Vaidya metricCorrection 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) : decreasingOther : 0≔≵≵∽⊡≟≡∨≵∩∸⊼≲∲

Matter fields near the horizonHere 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 L2 is not very large in NH: ≌∲⊿≲⊡≲≈≲≈

Weyl anomalyWeyl anomaly breaks conformal symmetryIncompatible 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.

Treating fluctuations as mattersWe 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:

∰∽⊱≚⋃≄≨⊹⊺≄⋁≥⊡≓≛≞≧∻≨∻⋁≝

Weyl anomaly cancelation

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

A model for black hole evaporation

≵

≲≲≳∨≵∩

≲≨∨≵∩≡∨≵∩

Vaidya metric and flat space are connected.

Time(null) in flat space Time(null) in Vaidya⊡∲≤≲∽≤≾≵∽⊵∱⊡≡∨≵∩≲∨≵∩

⊶≤≵

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.

Hawking radiationHawking 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

≁∡∻∡∰∽≚≤≵≥⊡≩∡∰≵≥≩∡≾≵∨≵∩

Hawking radiation (2)

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.

≡∨≵∩⊻≡

≲∨∱∩≳∨≵∩∽≚≵≤≵∰≟≡∨≵∰∩≲∨∰∩≳∨≵∰∩⊣≡∨≵∰∩⊡⊡≲∨∰∩≳∨≵∰∩⊡≡∨≵∰∩⊢≬≯≧⊡≲∨∰∩≳∨≵∰∩⊡≡∨≵∰∩⊢⊤≥⊡≒≵≵∰≤≵∰∰ ≡∨≵∰∰∩

∲∨≲∨∰∩≳∨≵∰∰∩∩∲

Reflection by gravitational potential

≵

≲≲≳∨≵∩≡∨≵∩

}Scattered ingoing radiation

Flat space

Potential barrier

We have assumed that there are only outgoing radiationHowever, Hawking radiation is partially

reflected by the potential barrier

We need to generalize the Vaidya metric.

Introducing the ingoing energy flowWe consider geometry with matters which obey ≴≲≔⊹⊺∽∰

≔⊵⊵∽≔∧∧∽∰conformalAngular 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) )

Generalized Vaidya metric

≤≳∲∽⊡≀≲≰∨≵∻≲∩⊷≰∨≵∻≲∩≲≤≵∲∫∲≤≵≤≲⊸∫≲∲≤⊭∲

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).

Conservation law for ingoing energy

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

≔≲≲⊻≲∲≰∲≔≵≵ ≃∽≓∽≰∲≔≲≲⊻≟≡⊻⊵≬≰≡

⊶∲

Conservation law is given by

∰∽≀≲≔≵≵∨≵∻≲∩∫∲≲≔≵≵∨≵∻≲∩⊡ ∱≲≰∨≵∻≲∩⊵⊡≀≵≰∨≵∻≲∩≰∨≵∻≲∩≓∨≵∻≲∩∫≀≵≓∨≵∻≲∩⊶

Approximated solutionThe solution can be approximated as

≰∫≡≃≬≯≧⊵≰⊡≡≃≡⊶∽≲⊡≡≥⊮

Numerical solution is plotted asFor

≰⊻≲⊡≡≥⊮≲∾≡

For ≲∼≡≰⊻≃≡∽≣≯≮≳≴∮

This implies that shock wave is smeared around . ≲⊻≡

Outside is approximated by Vaidya.

Static GeometryWe put BH in a heat bath.

In and Out of rad. are balancedfinally 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

≔⊵⊵⊻∱≲∲

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⊻ ∱∴⊼≡

Entropy

Pressure is not isotropicWe 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:≓∽∱∴≁

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.

Conclusion