Kerr/CFT対応におけるNon-extremal補正について

松尾 善典Based on

YM-Tsukioka-Yoo [arXiv:0907.0303]YM-Nishioka [arXiv:1010.4549]

Kerr/CFT 対応において Left mover は Extremal でのエントロピーを、 Right mover は Non-extremal の補正を与える。 Hidden conformal symmetry の解析から、 Central charge は

cL = cR = 12J となると予想される。 しかし、 Near horizon limit における Asymptotic symmetry を用いた解析では cL = 12J, cR = 0 となる。 そこで、新しい Near horizon limit を導入する。 この新しい Near horizon limit のもとで Asymptotic symmetryを用いて Central charge を計算すると cL = cR = 12J となる。 このとき Left mover と Right mover それぞれの Leading orderでのエントロピーへの寄与は、 Bekenstein-Hawking エントロピーの示唆する値と一致する。

Kerr/CFT対応におけるNon-extremal補正について

Kerr black hole

≤≳∲∽⊡⊢⊽∲∨≤≴⊡≡≳≩≮∲⊵≤⋁∩∲∫≳≩≮∲⊵⊽∲⊣∨≲∲∫≡∲∩≤⋁⊡≡≤≴⊤∲∫⊽∲⊢≤≲∲∫⊽∲≤⊵∲

⊢∽≲∲⊡∲≍≲∫≡∲∻ ⊽∲∽≲∲∫≡∲≣≯≳∲⊵

≍≁≄≍∽≍≇≎∻ ≊∽≡≍≇≎

≔≈∽≲∫⊡≲⊡∴⊼≍≲∫∻ ≓∽∲⊼≍≲∫≇≎

Metric of Kerr black hole is given by

where

2 parameters are related to the ADM mass and angular momentum as

Temperature and entropy are given by

Near horizon limit of Kerr black hole

⊲∡∰

≴∽∲⊲⊡∱≡≞≴∻ ≲∽≡∨∱∫⊲≞≲∩∻ ⋁∽≞⋁∫≴∲≡

≤≳∲∽⊡∨≞≲∲⊡≞≲∲≈∩≦∰∨⊵∩≤≞≴∲∫≦⋁∨⊵∩∨≤≞⋁∫≞≲≤≞≴∩∲∫≦∰∨⊵∩≤≞≲∲≞≲∲⊡≞≲∲≈∫≦∰∨⊵∩≤⊵∲

≍∽≡⊵∱∫⊲∲≞≲∲≈∲⊶

≦∰∨⊵∩∽≡∲∨∱∫≣≯≳∲⊵∩∻ ≦⋁∨⊵∩∽∴≡∲≳≩≮∲⊵∱∫≣≯≳∲⊵

We define near horizon coordinates as

We consider near-extremal case

And take the limit ofThen, the metric becomes

where

Asymptotic SymmetryWe first introduce a boundary condition

In this case⋂⊹⊺∽≲⊡≮⊹⊺

≨⊹⊺∽≏∨⋂⊹⊺∩Next, we introduce perturbations of same order

Then, if the metric satisfies the following condition:

⊱⊻≧⊹⊺∽∤⊻≧⊹⊺∽≏∨⋂⊹⊺∩≧⊹⊺∽⊹≧⊹⊺∫≨⊹⊺where

The geometry is asymptotically symmetric, namely,

⊹≧⊹⊺∫≏∨⋂⊹⊺∩∡⊹≧⊹⊺∫≏∨⋂⊹⊺∩

≏∨⋂⊹⊺∩

Asymptotic symmetry for left movers

≏∨⋂⊹⊺∩∽∰≂≂≂≂≂≀

≞≴ ≞≲ ≞⋁ ⊵≞≴≏∨≞≲∲∩≏∨≞≲⊡∲∩≏∨≞≲∰∩≏∨≞≲⊡∱∩≞≲ ≏∨≞≲⊡∳∩≏∨≞≲⊡∱∩≏∨≞≲⊡∲∩≞⋁ ≏∨≞≲∰∩≏∨≞≲⊡∱∩⊵ ≏∨≞≲⊡∱∩

∱≃≃≃≃≃≁

≛⊻≮∻⊻≭≝∽⊡≩∨≮⊡≭∩⊻≮∫≭

For left movers, the boundary condition is given by [Guica-Hartman-Song-Strominger,’08]

Then, the asymptotic symmetry is given by

This symmetry forms the Virasoro algebra

where ⊲⊻≮∨≞⋁∩∽≥≩≮≞⋁

⊻∽⊳⊲⊻∨≞⋁∩∫≏∨≲⊡∲∩⊴≀≞⋁∫⊳⊡≞≲⊲∰⊻∨≞⋁∩∫≏∨≲∰∩⊴≀≞≲∫⊳≃∫≏∨≲⊡∳∩⊴≀≞≴∫≏∨≞≲⊡∳∩≀⊵

We impose the following boundary conditionAsymptotic symmetry for right movers

≏∨⋂⊹⊺∩∽∰≂≂≂≂≂≀

≞≴ ≞≲ ≞⋁ ⊵≞≴≏∨≞≲∰∩≏∨≞≲⊡∳∩≏∨≞≲⊡∲∩≏∨≞≲⊡∳∩≞≲ ≏∨≞≲⊡∴∩≏∨≞≲⊡∳∩≏∨≞≲⊡∴∩≞⋁ ≏∨≞≲⊡∲∩≏∨≞≲⊡∳∩⊵ ≏∨≞≲⊡∳∩

∱≃≃≃≃≃≁

≛⊻≮∻⊻≭≝∽∨≮⊡≭∩⊻≮∫≭

Then, the asymptotic symmetry is given by

⊻∽⊳⊲⊻∨≞≴∩∫⊲∰∰⊻∨≞≴∩∲≞≲∲⊴≀≞≴∫⊳⊡≞≲⊲∰⊻∨≞≴∩∫⊲∰∰∰⊻∨≞≴∩∲≞≲⊴≀≞≲

∫⊳≃⊡⊲∰∰⊻∨≞≴∩≞≲⊴≀≞⋁∫≏∨≞≲⊡∳∩

This symmetry forms the Virasoro algebra

where⊲⊻≮∨≞≴∩∽≞≴≮∫∱

Asymptotic charge

≑⊻≛≨≝∽ ∱∸⊼≇≎≚

≀⊧≫⊻≛≨∻⊹≧≝≾≫⊹⊺⊻≛≨∻⊹≧≝∽∱∲

≨⊻⊹≄⊺≨⊡⊻⊹≄⊸≨⊸⊺∫⊡≄⊹≨⊺⊸⊢⊻⊸∫∱∲≨≄⊹⊻⊺

⊡≨⊹⊸≄⊸⊻⊺∫∱∲≨⊹⊸∨≄⊺⊻⊸∫≄⊸⊻⊺∩⊡∨⊹∤⊺∩≩≫⊻≛≨∻⊹≧≝∽≾≫⊹⊺⊻≛≨∻⊹≧≝⊡≤∲≸⊢⊹⊺

⊱⊳≑⊻∽ ∱∸⊼≇≎≚

≀⊧≫⊻≛∤⊳⊹≧∻⊹≧≝∫ ∱∸⊼≇≎≚

≀⊧≫⊻≛∤⊳≨∻⊹≧≝∱∸⊼≇≎

≚≀⊧≫⊻≮≛∤⊻≭⊹≧∻⊹≧≝∽⊡≩⊱≮∫≭∻∰≮∳≣∱∲∫≏∨≮∩

Asymptotic Charge is defined as [Barnich-Brandt-Compere]

where

We consider transform of the charge itself

The central charge can be read off from the first term

Central charges

∱∸⊼≇≎≚

≀⊧≫⊻≮≛∤⊻≭⊹≧∻⊹≧≝∽⊡≩⊱≮∫≭∻∰∨≮∳∫∲≮∩≡∲≇≎

∱∸⊼≇≎≚

≀⊧≫⊻≮≛∤⊻≭⊹≧∻⊹≧≝∽∰

≣≌∽∱∲≡∲≇≎⊻∱∲≊∻ ≣≒∽∰

For left movers, we obtain

And for right movers,

Then, the central charges become

Cardy formula and entropy

≔≌∽∱∲⊼∻ ≔≒∽≞≲≈∲⊼

Frolov-Thorne temperature is defined as

≥≸≰⊷⊡∡≔≈∫⊭≈≔≈≭⊸∽≥≸≰⊷⊡≮≌≔≌⊡≮≒≔≒

⊸In this case, we obtain

By using the Cardy formula,

≓∽⊼∲∳≣≌≔≌∫⊼∲

∳≣≒≔≒∽∲⊼≡∲≇≎

For left mover, the Cardy formula reproduce the entropy of the extremal Kerr black hole. For right mover, we obtain cR = 0, and does not contribute to the entropy.

Quasi-local chargeQuasi-local charge is defined in a similar fashion to the GKPW

⊰⊹⊺ : Induced metric≔⊹⊺∽ ∲≰⊡⊰⊱≓≧≲≡≶⊱⊰⊹⊺

The quasi-local charge is defined by

We first define the surface energy-momentum tensor

For Einstein gravity, it can be written as

≔⊹⊺∽ ∱∸⊼≇≎∨≋⊹⊺⊡⊰⊹⊺≋∩ ≋⊹⊺: extrinsic curvature

⊿⊹⊺∽≔⊹⊺⊡≔≣≴⊹⊺

We regularize the surface energy-momentum tensor as

≵⊹⊻⊹⊾⊹⊺≑⊻∽≚≤∲≸≰⊡⊾≵⊹⊿⊹⊺⊻⊺

: timelike unit normal: Killing vector: Induced metric on timeslice at boundary

Cardy formula for right mover

⊱≑⊻∽≡∲≇≎⊤⊲∰∰∰⊻∨≴∩

The central charge can be read off from the anomaly

where we put the boundary at ≲ ∽

⊤.Then, the central charge is

≣≒∽∱∲≡∲≇≎⊤

⊹≌∰∽≍∽≡∲∲≇≎⊤∨∲⊼≔∩∲

For finite temperature, we obtain

Then, the Cardy formula gives

≓∽∲⊼≲≣≒⊹≌∰∶ ∽∨∲⊼∩∲≡∲≔≇≎⊤

The Cardy formula gives the non-extremal correction of the entropy, if we identify .

If is kept finite, the geometry is approximated by near horizon geometry in near horizon region .

The boundary of the near horizon geometry should be taken around . Therefore, we identify .

Non-extremal correction

For near-extremal case, the entropy is

≓∽∲⊼≡∲≇≎∨∱∫⊲≞≲≈∫⊢⊢⊢∩

By using the Frolov-Thorne temperature,

≓∽∲⊼≡∲≞≲≈≇≎⊤

⊤∽∱∽⊲⊲ ≲⊡≲∫⊿≡≞≲∮⊲⊡∱ ⊤∽∱∽⊲

Hidden Conformal Symmetry

EOM for radial part of Scalar in Kerr background

For small ∡, this equation can be approximated as

[Castro-Maloney-Strominger ’10]We consider the scalar field in Kerr background.

≀⊹⊡≰⊡≧≧⊹⊺≀⊺⊩∨≴∻≲∻⋁∻⊵∩⊢∽∰∺Then, the scalar field can be factorized as

⊩∨≴∻≲∻⋁∻⊵∩∽≥⊡≩∡≴∫≩≭⋁≒∨≲∩≓∨⊵∩

⊷≀≲⊢≀≲∫∨∲≍≲∫∡⊡≡≭∩∲∨≲⊡≲∫∩∨≲∫⊡≲⊡∩⊡∨∲≍≲⊡∡⊡≡≭∩∲

∨≲⊡≲⊡∩∨≲∫⊡≲⊡∩⊸≒∨≲∩∽≋≒∨≲∩

⊷≀≲⊢≀≲∫∨∲≍≲∫∡⊡≡≭∩∲∨≲⊡≲∫∩∨≲∫⊡≲⊡∩⊡∨∲≍≲⊡∡⊡≡≭∩∲

∨≲⊡≲⊡∩∨≲∫⊡≲⊡∩∫∨≲∲∫∲≍≲∫∴≍∲∩∡∲⊸≒∨≲∩∽≋≒∨≲∩

We define conformal coordinates as

≷∫∽≲≲⊡≲∫≲⊡≲⊡≥∲⊼≔≒⋁∻ ≷⊡∽≲≲⊡≲∫≲⊡≲⊡≥∲⊼≔≌⋁⊡≴∲≍∻≹∽≲≲∫⊡≲⊡≲⊡≲⊡≥⊼∨≔≒∫≔≌∩⋁⊡≴∴≍∻

where

≔≌∽≲∫∫≲⊡∴⊼≡∻ ≔≒∽≲∫⊡≲⊡∴⊼≡Then, the laplacian becomes that on AdS3 .

≀≲⊢≀≲∫≩∨∲≍≲∫≀≴∫≡≀⋁∩∲∨≲⊡≲∫∩∨≲∫⊡≲⊡∩⊡≩∨∲≍≲⊡≀≴∫≡≀⋁∩∲

∨≲⊡≲⊡∩∨≲∫⊡≲⊡∩∽∱∴≹∳≀≹∱≹≀≹∫≹∲≀∫≀⊡

We define the “light-cone” coordinates as　 Then, the “laplacian” for radial part becomes

≸∫∽⋁∻ ≸⊡∽⋁⊡≡∲≍∲≴

HCS and BTZ black holeIn the Kerr background, ⋁ has a periodicity

⋁⊻⋁∫∲⊼The approximated background is not equivalent to the AdS3, but its quotient.

BTZ black hole

≀≲⊢≀≲∫∱⊢⊷∴≡∲≲∫∫≲⊡

⊵≲⊡≲∫∫≲⊡∲⊶≀∫≀⊡⊡≡∲≀∲∫⊡≡∲∨≲∫⊡≲⊡∩∲

∨≲∫∫≲⊡∩∲≀∲⊡⊸

≤≳∲∽⊡∨⊽∲⊡≲∲∫∩∨⊽∲⊡≲∲⊡∩⊽∲ ≤⊿∲∫ ≬∲⊽∲≤⊽∲∨⊽∲⊡≲∲∫∩∨⊽∲⊡≲∲⊡∩∫⊽∲⊵≤∧⊡≲∫≲⊡⊽∲≤⊿⊶∲

≸⊧∽∧⊧⊿∻ ⊽∲∽∨≲∫∫≲⊡∩≲⊡≲∫≲⊡

≔≌∽≲∫∫≲⊡∲⊼≬∻ ≔≒∽≲∫⊡≲⊡∲⊼≬

The metric of the BTZ black hole can be written as

The Frolov-Thorne temperatures are given by

By introducing the following coordinates

The laplacian in the BTZ background is expressed as

Therefore, the approximated laplacian on Kerr geometry equals to that in BTZ if we identify

≬∽∲≡

∴≬∲≀≲⊢≀≲∫∱⊢⊷ ∴≲∫∫≲⊡

⊵≲⊡≲∫∫≲⊡∲⊶≀∫≀⊡⊡≀∲∫⊡∨≲∫⊡≲⊡∩∲

∨≲∫∫≲⊡∩∲≀∲⊡⊸

New near horizon limit

≤≳∲∽⊡∨≞≲∲⊡≞≲∲≈∩≦∰∨⊵∩∨≤≸∫∩∲∫≦⋁∨⊵∩∨≤≸⊡∫≞≲≤≸∫∩∲∫≦∰∨⊵∩≤≞≲∲≞≲∲⊡≞≲∲≈∫≦∰∨⊵∩≤⊵∲

⊻∽⊳⊲⊻∨≸∫∩∫⊲∰∰⊻∨≸∫∩∲≞≲∲⊴≀∫∫⊳⊡≞≲⊲∰⊻∨≸∫∩∫⊲∰∰∰⊻∨≸∫∩∲≞≲

⊴≀≞≲∫⊳≃⊡⊲∰∰⊻∨≸∫∩≞≲⊴≀⊡∫≏∨≞≲⊡∳∩

In the near horizon limit, the metric becomes

This geometry has the following periodicity

The asymptotic symmetry for right mover is

which should be expanded as

≸∫⊻≸∫∫∲⊼≮⊲∻ ≸⊡⊻≸⊡∫∲⊼≮

⊲⊻≮∨≸∫∩∽⊲≥≩≮≸∫∽⊲

We define new near horizon coordinates as

≸∫∽⊲⋁∻ ≸⊡∽⋁⊡≡≴∲≍∲∻ ≲∽≡∨∱∫⊲≞≲∩

Entropy

≔∫∽≲∫⊡≲⊡∴⊼≡∡≞≲≈∲⊼⊲∻ ≔⊡∽≲∫∫≲⊡∴⊼≡∡∱∲⊼

≣≒∽∱∲≡∲≇≎∽≣≌

Integrating on a time-slice, the following component contributes to the central charge:

Then, the central charge becomes

The Frolov-Thorne temperatures are given by

The entropy can be reproduced by Cardy formula

∱∸⊼≇≎≚

≀⊧≾≫⊡≲⊻≭≛∤⊻≮⊹≧∻⊹≧≝≤≸∫≤⊵∽⊱≮∫≭∻∰≮∳≡∲≇≎

≓∽⊼∲∳≣≌≔≌∫⊼∲

∳≣≒≔≒∽∲⊼≡∲≇≎∨∱∫⊲≞≲≈∩⊻∲⊼≍≲∫≇≎

We define a new near horizon limit. By using this limit, we obtain the central charge cL = cR = 12J.

This new definition corresponds to a modification of the asymptotic symmetry.

There are higher order corrections from metric and Killing vectors of the asymptotic symmetry.

Left movers gives O(ε0) contributions but right movers gives O(ε ).

To be exact, we have calculated only the leading term for left and right movers, respectively.

They agree with the expected result. However, the next-to-leading term from the left movers is

at the same order to the leading term from right movers. It is left to be checked that this term vanishes.

Conclusion and outlook