motivation 柳 哲文 (APCTP) 石原秀樹, 木村匡志 (大阪市大) 丹澤優 (基研) 高次元時空の構造と BHの形成条件 柳　哲文 (APCTP) 石原秀樹, 木村匡志 (大阪市大) 丹澤優 (基研) My name is Chulmoon Yoo. I’m postdoc at APCTP. The title of my talk is hoop conjecture and horizon formation cross section in KK spacetimes. This work is collaboration with Hideki Ishihara, Masashi Kimura in Osaka City University and Sugure Tanzawa in YITP.

僕が知りたかったこと Kaluza-Klein的な時空においてブラックホールの形成条件として知られるフープ予想はどうなるのか？ BH最前線 motivation 僕が知りたかったこと Kaluza-Klein的な時空においてブラックホールの形成条件として知られるフープ予想はどうなるのか？ Let’ start. These are contents of my talk. After the brief introduction, I will explain hoop conjecture. One of the purpose in this work is to test the HHC. So, I will talk about our setup and how to test the HHC in our situations. After that I will give an extension of HHC and suggestion about cross section of black hole production. Finally I will give conclusions as a summary of my talk. BH最前線 柳　哲文

目次 導入 （ハイパー）フープ予想とは? Kaluza-Klein時空中の２質点系の初期データ 初期データを用いた形成条件のテスト motivation 目次 導入 （ハイパー）フープ予想とは? Kaluza-Klein時空中の２質点系の初期データ 初期データを用いた形成条件のテスト ハイパーフープ予想の拡張 ブラックホール生成断面積について 結論 Let’ start. These are contents of my talk. After the brief introduction, I will explain hoop conjecture. One of the purpose in this work is to test the HHC. So, I will talk about our setup and how to test the HHC in our situations. After that I will give an extension of HHC and suggestion about cross section of black hole production. Finally I will give conclusions as a summary of my talk. BH最前線 柳　哲文

motivation 導入 BH最前線 柳　哲文

TeV 高次元の空間構造を漸近的に ユークリッド空間に限る理由はない． • 超ひも理論 → ブレーンワールド → TeV 重力 motivation TeV • 超ひも理論 → ブレーンワールド → TeV 重力 • LHCや超高エネルギー宇宙線におけるBH生成 • 大きな余剰次元についての証拠が得られる？ • 漸近平坦（ミンコフスキー）BHに注目 :ミニブラックホールを考えると妥当 • しかし， About ten years ago motivated by the string theory, the brane world scenario which has large extra-dimension has been proposed. In this scenario, the fundamental scale of the gravity can be TeV scale. This is one of the solution for the hierarchy problem. If the fundamental scale is TeV, there is a possibility that mini black holes are produced in LHC or future linear collider. Then we might obtain evidences for existence of Large extra-dimension. If it is real, it’s very exciting. So far, asymptotically flat BHs are mainly considered as idealized model for small BH limit. However, there is no reason to restrict the asymptotic structures of higher dimensions to the flat space. 高次元の空間構造を漸近的に ユークリッド空間に限る理由はない． BH最前線 柳　哲文

KKバックグラウンドでのBH (4+n)次元空間 • もし (D-n-4) 個の方向のうち，一つでも rh と同程度の大きさを motivation KKバックグラウンドでのBH D次元 = 我々の４次元 + n 個の長さ l　にコンパクト化された方向 + (D-n-4) 個のプランク長 l* にコンパクト化された方向 (4+n)次元空間 • G4~Gn+4 / l n → プランク質量 m* ~ TeV • もし MBH~ k m* とすると For example, let us consider the D-dimensional spacetime with n compactified dimensions with scale l Which correspond to the large exra-dimensions, and D-n-4 compactified dimensions with scale l_*, where l*is the fundamental length scale. Since N+4 dimensional gravitational constant is given by this, The fundamental mass scale m* can be TeV scale. Suppose that a BH can be regarded as (n+4)-dimensional asymptotically flat and The mass of black hole is given by the k times fundamental mass scale m*, Then the horizon radius is given by this form. If, one of the remaining D-n-4 direction has the same scale as rh, this BH cannot be n+4 dimensional asymptotically flat But 4+n+1dimensional BH on KK background. • もし (D-n-4) 個の方向のうち，一つでも rh と同程度の大きさを 持つならば，このBHは (n+4)次元漸近ミンコフスキーというよりは， (4+n+1)次元Kaluza-Klein BH とみなさられるべきである． BH最前線 柳　哲文

rh と同程度の長さにコンパクト化された時空構造が， motivation なにが知りたいか rh と同程度の長さにコンパクト化された時空構造が， ブラックホール形成に与える影響 So, What we are interested in is the effect of the size of compactified directions Which can be comparable to rh On black hole formation process. The black hole production usually estimated by using the notion of the hoop conjecture. 「フープ予想」 BH最前線 柳　哲文

motivation フープ予想？ What is hoop conjecture? BH最前線 柳　哲文

motivation Thorneのフープ予想 (4次元時空) “Black holes with horizons form when and only when a mass M gets compacted into a region whose circumference C in every direction satisfies ざっくり言って エネルギーが十分小さい領域に 押し込められると ブラックホールが出来る This is thorne’s original hoop conjecture in 4-dimensional spacetimes. Black holes with horizons form when and only when a mass M gets compacted into a region Whose circumference in every direction satisfies this inequality. Roughly speaking, If energy is concentrated in a sufficiently small region, Black holes form. And if black holes form, the energy is concentrated in a sufficiently small region. There is no serious counter example so far. 深刻な反例はない． BH最前線 柳　哲文

明らかにブラックストリングの周囲の長さは無限 motivation 高次元でのフープ予想 ５次元ブラックストリング解の存在 明らかにブラックストリングの周囲の長さは無限 BHができる しかし, BHができる How about higher-dimensional spacetimes? For example, in 5-dimensional Einstein gravity, There is black string solution. Obviously, those circumference can be infinitely large. So, if this inequality is satisfied, black holes form. It is OK, but it is not a necessary condition. Therefore, we need modification in higher-dimensions. The problem is how the region should be small. 修正が必要 BH最前線 柳　哲文

motivation ハイパーフープ予想 (高次元時空) [Ida, Nakao(2002)] “Black holes with horizons form when and only when a mass M gets compacted into a region whose (D-3)-dimensional circumferential surface volume VD-3 in every direction satisfies ざっくり言って エネルギーが十分小さい領域に 押し込められると ブラックホールが出来る The extended version of the original hoop conjecture was proposed by Ida and Nakao. They proposed not to use circumference but to use D-3 dimensional volume. There is no serious counter example. 深刻な反例はない． BH最前線 柳　哲文

KK時空では? BH最前線 柳 哲文 motivation But how about in KK background? This is not so trivial. BH最前線 柳　哲文

検証に使う初期データ BH最前線 柳 哲文 motivation To consider that, we use a sequence of momentarily static initial data sets. BH最前線 柳　哲文

２質点系 時間反転対称初期データの構築 BH最前線 柳 哲文 motivation Let us consider an initial data set hab and kab on a 4-dim spacelike surface. Hab is the induced metric on 4-dim initial surface and kab is the extrincic curvature. Imposing these three assumptions, momentarily static vacuum and Conformally Ricci flat, we find the momentum constraint is trivial and Hamiltonian constraint is given by this form. This triangle is the Laplacian of this Ricci flat metric. 時間反転対称初期データの構築 BH最前線 柳　哲文

初期データの構成 初期データセット (hab, Kab) 時間 ◎時間反転対称 : Kab=0, ◎真空 : Tab =0, motivation 初期データの構成 初期データセット (hab, Kab) 時間 ◎時間反転対称　　　 : Kab=0, ◎真空　　 : Tab =0, ◎共形リッチ平坦　　 : hab, Kab Let us consider an initial data set hab and kab on a 4-dim spacelike surface. Hab is the induced metric on 4-dim initial surface and kab is the extrincic curvature. Imposing these three assumptions, momentarily static vacuum and Conformally Ricci flat, we find the momentum constraint is trivial and Hamiltonian constraint is given by this form. This triangle is the Laplacian of this Ricci flat metric. Momentum拘束条件 自明 Hamiltonian拘束条件 BH最前線 柳　哲文

４次元ユークリッド上の２質点 2a 2a a小 a大 acr • ハミルトニアン拘束条件 • ２質点系の解 二つを覆うホライズン AH motivation ４次元ユークリッド上の２質点 • ハミルトニアン拘束条件 • ２質点系の解 二つを覆うホライズン 2a 2a AH One of the most simple case is using 4-dimensional Euclid space as the Ricci flat metric. Then Hamiltonian constraint become like this. In this setup, we can put two point sources and the conformal factor can be given by this form. The a is the separation of two point sources. If a is smaller than some critical value, a coverall AH exists. If a is larger than acr, a coverall horizon does not exists. This case is asymptotically flat case. How can we compactify the space? a小 a大 acr BH最前線 柳　哲文

コンパクト化 S2 コンパクト化 歪む 一様 AHは常微分方程式で求まる 同一視 偏微分方程式が必要 BH最前線 柳 哲文 motivation コンパクト化 S2 コンパクト化 一様 AHは常微分方程式で求まる 歪む In the case of asymptotically flat, this section of coverall horizon is homogeneous S2. So in this case we can find AH by solving ODE. However, once we compactify the one direction, This S2 must be deformed because of the difference of b boundary conditions. Then we have to solve PDE. But, I didn’t want to solve PDE. 同一視 偏微分方程式が必要 BH最前線 柳　哲文

別の可能性 常微分方程式でOK • ２つの質点を覆うAHのトポロジーは S3/Zn 3次元空間 同一視 (Hopf ファイバー) motivation 別の可能性 3次元空間 同一視 (Hopf ファイバー) 一様な２次元面 There is another possibility. This is 3-dimensional space, Inside this space, let us consider this kind of 2-dimensional surface and and identification. In this figure, the topology of the coverall horizon looks different from S3 but S2 cross S1, but because of this fibration, The topology of the coverall horizon can be S^3 / Zn. This n is determined by the way of fibration. 常微分方程式でOK • ２つの質点を覆うAHのトポロジーは S3/Zn BH最前線 柳　哲文

Gibbons-Hawking計量 • Hamiltonian 拘束条件 : • F がψに依存しないとする, • ２質点系の解 motivation Gibbons-Hawking計量 3dim Euclid • Hamiltonian 拘束条件 : • F がψに依存しないとする, • ２質点系の解 How to do this is shown here. The point is to use Gibbons-Hawking metric which is known as self-dual metric. This metric is equivalent to the two center Taub-NUT. By using this metric, we have Hamiltonian constraint as this form. Assuming independence of F on psi, This equation reduces to this form. A solution with two point sources are given by this form. In this space, Abbott Deser mass can be calculated as 6 pi m. • Abbott-Deser 質量 (基準は m=0 の場合 ) BH最前線 柳　哲文

座標 x φ r z θ ψ y 0 ≤θ≤ π, 0 ≤φ≤ 2π, 0 ≤ψ≤ 4π →２質点を覆うAHのトポロジーは S3/Z2 motivation 座標 x φ r z θ ψ The relation between these coordinates and the previous figures are shown here. R, theta, phi coordinate are in this space and psi corresponds to this direction. If we put the range of coordinates like this, The topology of the coverall horizon is given by S3/Z2. y 0 ≤θ≤ π, 0 ≤φ≤ 2π, 0 ≤ψ≤ 4π →２質点を覆うAHのトポロジーは S3/Z2 BH最前線 柳　哲文

漸近計量 r →∞ 余剰次元のサイズ ３次元ユークリッド空間 コンパクト化された１次元 BH最前線 柳 哲文 motivation This is asymptotic behavior of the metric. You can see that the l corresponds to the size of extra-dimension in the asymptotic region. BH最前線 柳　哲文

空間の対称性 AHは常微分方程式を解けば得られる． • ∂Φ, ∂ψ : Killingベクトル • AHは r=rh(θ) BH最前線 motivation 空間の対称性 • ∂Φ, ∂ψ : Killingベクトル • AHは r=rh(θ) You can easily see these two vector fields are killing vector fields in this space. So it is clear that we can find apparent horizon by solving ODE. It’s easier than naïve compactification. AHは常微分方程式を解けば得られる． BH最前線 柳　哲文

フープ予想の検証 BH最前線 柳 哲文 motivation What can we do using this metric as a test of the hoop conjecture. BH最前線 柳　哲文

Schwarzschild BHで等式になるよう motivation ハイパーフープ予想の再解釈 ハイパーフープ予想はとてもあいまい Black holes with horizons form when and only when a mass M gets compacted into a region whose (D-3)-dimensional circumferential surface volume in every direction satisfies • 適当に再解釈する必要がある Total mass (Abbott-Deser mass) Mass M First, we have to fix the notion of HHC in reasonable way. Because HHC has several ambiguities. What is this mass? What is this D-3 dimensional circumferential surface volume in every direction? What does mean this lesssim and numerical factor alpha. We use Total mass as the Mass M, And we use maximum value of the volume of geodetic surfaces on coverall AH as VD-3. We fix alpha so that this becomes equality in the case of Schwarzschild BH. (D-3)-dimensional circumferential surface volume in every direction 二つの質点を覆うAH上の測地面 の面積の内最大の物 α Schwarzschild BHで等式になるよう BH最前線 柳　哲文

motivation 測地面 Coverall horizon Vn = Max{ , , } BH最前線 柳　哲文

満たされるべき性質 • V2 はAH上で測られる，aに依存 • AH は a > acr で消える motivation 満たされるべき性質 • V2 はAH上で測られる，aに依存 • AH は a > acr　で消える • 不等式は a > acrでやぶれなければならない 2a Now the question is what kind os properties should Vn have. Since the inequality should be violated in a larger than acr, We can expect tht Vn is a monotonic increasing function of a in the vicinity of acr. And Vn acr is the same order as r.h.s.. We will check these two properties. (1) V2(a) はaの単調増加関数（少なくとも acr近傍で） (2) BH最前線 柳　哲文

AH上の測地面 • 2-dim surface • 定義 BH最前線 柳 哲文 motivation To calculate Vn we have used following geodetic surfaces on coverall AH. For 1-dimensional object, these four geodetic sufaces and For 2-dimensional surface, these three kinds of surfaces and We define V1 and V2 as the maximum value of these. • 定義 BH最前線 柳　哲文

V2(a) m/l2=0.0025 m/l2=1 m/l2=4 m/l2=100 BH最前線 柳 哲文 motivation We have calculated area of three surfaces solving ordinary differential equations to Find AH. in the large l this phi=0 surface has largest value and It is monotonic increasing function of a. And the value at acr is same order as the value of The Schwarzschild case. But in smaller l psi=0 surface becomes larger and this is monotonic decreasing function of A in addition, acr can be much larger than that of the Schwarzschild black hole. m/l2=4 m/l2=100 BH最前線 柳　哲文

結果 V2 (1) (2) l2 << m No l2 >> m Yes motivation 結果 (1) V2(a) は a の単調増加関数 (2) Variable V2 Property (1) (2) l2 << m No l2 >> m Yes This is the summary of results. This no is because of the existence of black string solution. You can see that there is no appropriate variable. In large l, we can use V2. But in small l we cannot use both of them. ハイパーフープ予想はl2<<mでは使えない． BH最前線 柳　哲文

motivation 拡張してみる Now, we try to extend the HHC. BH最前線 柳　哲文

l2 << m の場合はどう扱われるべきか motivation l2 << m の場合はどう扱われるべきか • 素朴には KKコンパクト化の後4次元フープ予想が適用できる (1) V1(a) は a の単調増加関数 The problem is how to treat the l^2 much smaller than m case. Naively thinking, 4-dimensional hoop conjecture should be satisfied with KK reduction. The original hoop conjecture gives this inequality. Using this relation, we have this inequality. Now we want to find the appropriate variable W which gives the criterion As this form. Comparing this inequality with this inequality, We find W should be alpha2V1l for l^2 much smaller than m. (2) BH最前線 柳　哲文

AH上の測地線 • 1-dim circumference • 定義 BH最前線 柳 哲文 motivation To calculate Vn we have used following geodetic surfaces on coverall AH. For 1-dimensional object, these four geodetic sufaces and For 2-dimensional surface, these three kinds of surfaces and We define V1 and V2 as the maximum value of these. • 定義 BH最前線 柳　哲文

V1(a) m/l2=1 m/l2=0.0025 Large value of V1(acr) m/l2=4 m/l2=100 BH最前線 motivation V1(a) m/l2=1 m/l2=0.0025 Large value of V1(acr) We have calculated area of three surfaces solving ordinary differential equations to Find AH. in the large l this phi=0 surface has largest value and It is monotonic increasing function of a. And the value at acr is same order as the value of The Schwarzschild case. But in smaller l psi=0 surface becomes larger and this is monotonic decreasing function of A in addition, acr can be much larger than that of the Schwarzschild black hole. m/l2=4 m/l2=100 BH最前線 柳　哲文

５次元的なら面積，４次元的なら長さを使うのがよろしい motivation 結果のまとめ (1) V(a) は a の単調増加関数 (2) Variable V2 V1 Property (1) (2) l2 << m No Yes l2 >> m This is the summary of results. This no is because of the existence of black string solution. You can see that there is no appropriate variable. In large l, we can use V2. But in small l we cannot use both of them. ５次元的なら面積，４次元的なら長さを使うのがよろしい BH最前線 柳　哲文

An alternative variable motivation An alternative variable Then one of the possibility for the W is given by this equation. Actually, this definition of W satisfies two conditions. W は (1)， (2) をどちらも満たす BH最前線 柳　哲文

motivation 一般化 “Black holes with horizons form when and only when a mass M gets compacted into a region whose n-dimensional circumferential surface volumes Vn (n=1…D-3) in every direction satisfy • Note 1- 左辺は系のコンパクトさを表す量だけでなく，余剰次元のサイズも含む Generalizing this discussion, we get at an extension of HHC as this. Note 1 left hand side contains not only the variables which characterize the size of The system but also the size of extra-dimensions. 2 This conjecture includes the HHC by Ida-Nakao by large extra-dimension limit. 2- ハイパーフープ予想も含む BH最前線 柳　哲文

BH生成断面積 BH最前線 柳 哲文 motivation Let me move to the cross section estimation. BH最前線 柳　哲文

断面積の評価 • 厳密に計算する方法を知らない． • 素朴にはものすごくひしゃげたBHは出来ない（？） motivation 断面積の評価 • 厳密に計算する方法を知らない． • 素朴にはものすごくひしゃげたBHは出来ない（？） • W の中のV1 ，V2をσpを使って置き換える Now, no exact way to estimate the cross section. So, from the notion of hoop conjecture, we can expect cross section Can be estimated using V1in small l and V2 in large l. To express in unified way, we define sigma as like this referring to the definition of W. And adopt sigma a cr as an estimate of sigmap. BH最前線 柳　哲文

σp ∝m2 ∝m 4次元的 5次元的 BH最前線 柳 哲文 motivation So we focus on the a=0 case for simplisity. Then sigma can be calculated analytically. Focusing on the mass dependence of sigma, We can find it’s proportional to the mass in the mass Smaller than the size of extradimension. And proportional to the square of the mass in larger mass than square of The size of extra-dimension. This behavior can be easily understood by considering the dependence of Square of Schwarzschild radius in each dimension. In small m region the system is regarded as allmost 4-dimensional With KK reduction, then the cross-section is proportional to the mass squared. On the otherhand, in small m, it is fully 5-dimensional system and The cross section should be proportional to the mass itself. 5次元的 BH最前線 柳　哲文

σp Dtotal =10, その内二つ→10-2cm, 他の二つ→10-15cm, 残る二つ→10-17cm ６次元的 ８次元的 motivation σp Dtotal =10, その内二つ→10-2cm, 他の二つ→10-15cm, 残る二つ→10-17cm ６次元的 ８次元的 So we focus on the a=0 case for simplisity. Then sigma can be calculated analytically. Focusing on the mass dependence of sigma, We can find it’s proportional to the mass in the mass Smaller than the size of extradimension. And proportional to the square of the mass in larger mass than square of The size of extra-dimension. This behavior can be easily understood by considering the dependence of Square of Schwarzschild radius in each dimension. In small m region the system is regarded as allmost 4-dimensional With KK reduction, then the cross-section is proportional to the mass squared. On the otherhand, in small m, it is fully 5-dimensional system and The cross section should be proportional to the mass itself. １０次元的 BH最前線 柳　哲文

結論 • ハイパーフープ予想の拡張を行った • σp の質量依存性をモデルを指定せずに計算してみた． motivation 結論 • ハイパーフープ予想の拡張を行った “Black holes with horizons form when and only when a mass M gets compacted into a region whose n-dimensional circumferential surface volumes Vn (n=1…D-3) in every direction satisfy • σp の質量依存性をモデルを指定せずに計算してみた． Then these are conclusions • σp　の質量依存性が見れると楽しい． 　特に余剰次元サイズrhがコンパラなところが面白い（？） BH最前線 柳　哲文

motivation おわり BH最前線 柳　哲文

a = 0 case ∞ • The area of r = const. surface : A(r) motivation a = 0 case • The area of r = const. surface : A(r) • Horizon radius rh is given by • V2 can be calculated as In this case, r=constant surface become homogeneous surface and The area of r=constant surface is given by A(r). Then horizon radius rh can be given by this equation. Because the initial data is momentarily static. Then we can obtain the horizon radius analytically Then also V1 and V2 can be calculated as like this. These values become infinitely large in small l limit. ∞ m/l2 →∞ BH最前線 柳　哲文