Colliding Black Holes in AdS Hans Bantilan Queen Mary University of - PowerPoint PPT Presentation

Colliding Black Holes in AdS Hans Bantilan Queen Mary University of London July 1, 2016

Outline • Motivation • Setup • Simulations • Summary

Motivation Heavy ion collisions ◦ Why collisions? to probe the quark and gluon constituents of nuclei ◦ Why heavy ions? to get as many p + and n 0 as possible to hit each other The STAR, PHENIX experiments at RHIC, the ALICE, ATLAS, CMS experiments at LHC ◦ Strip gold ( 197 79 Au) or lead ( 208 82 Pb) nuclei of electrons ◦ Accelerate to speeds close to c ◦ Arrange for a collision ◦ Collision energies of 200[GeV] per nucleon at RHIC, 2.76[TeV] per nucleon at LHC

Motivation • A non-perturbative problem in QCD • Lattice QCD has no access to real-time dynamics • Experimental data are well described by relativistic viscous hydrodynamic simulations • But, several competing models for the pre-equilibrium stage that yield different initial energy density and flow velocity profiles for matching onto the hydrodynamic stage • Would be desirable to have a single model to describe both the pre-equilibrium stage and the hydrodynamic stage

Motivation Pre-equilibrium stage ◦ Duration: 0.2-0.4 fm/c ◦ A model: classical Yang-Mills dynamics of gluons ◦ Resulting energy density and flow velocity profiles are used to match onto a hydrodynamic form of the stress tensor in subsequent hydrodynamic stage Hydrodynamic stage ◦ Duration: 5-10 fm/c ( ↑ for higher collision energies) ◦ A model: relativistic viscous hydrodynamics ◦ Resulting hydrodynamic output is used to match onto particle distributions in subsequent hadronic stage Hadronic stage ◦ Duration: remaining evolution time ◦ A model: microscopic kinetic description

Motivation Figure: BH-BH collision in a Poincar´ e patch of AdS 5 , with black hole masses M 1 , M 2 , boosts γ 1 , γ 2 , and impact parameter b . Adapted from hep-th/0805.1551

Motivation AdS/CFT correspondence between an asymptotically AdS spacetime in d + 1 dimensions and a CFT in d dimensions Proposed use to find a gravity description of non-perturbative problems in QCD Major obstacle is the current lack of a gravity dual for QCD Possible approach: try to capture some features of QCD with a CFT toy model for which there is a known gravity dual N = 4 SYM 4 at strong coupling ← → AdS 5 classical gravity

Motivation AdS/CFT correspondence between an asymptotically AdS spacetime in d + 1 dimensions and a CFT in d dimensions Proposed use to find a gravity description of non-perturbative problems in QCD Major obstacle is the current lack of a gravity dual for QCD Possible approach: try to capture some features of QCD with a CFT toy model for which there is a known gravity dual N = 4 SYM 4 at strong coupling ← → AdS 5 classical gravity

Setup Classical gravity in d + 1 dimensions with cosmological constant Λ = d ( d − 1) / (2 L 2 ), coupled to real scalar field matter 1 : � 1 dx d +1 √− g � 16 π ( R − 2Λ) − 1 � 2 g αβ ∂ α ϕ∂ β ϕ − V ( ϕ ) S = The corresponding field equations take the local form 2 : dV ✷ ϕ = dϕ 2Λ � 1 � d − 1 T αα g µν R µν = d − 1 g µν + 8 π T µν − 1 We will use scalar field collapse as a convenient mechanism to form BHs ` 1 2 Real scalar field: T µν = ∂ µ ϕ∂ ν ϕ − g µν 2 g αβ ∂ α ϕ∂ β ϕ + V ( ϕ ) ´

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = R µν − d − 1 g µν − 8 π T µν −

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − R µν

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − − 1 2 g αβ g µν,αβ + g αβ g β ( µ,ν ) α + 1 2 g αβ,α ( g αβ,ν − g νµ,β + g βν,µ ) log √− g log √− g ,β Γ βµν − Γ ανβ Γ βαν � � � � − ,µν +

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − −∇ ( µ C ν ) − 1 2 g αβ g µν,αβ + g αβ g β ( µ,ν ) α + 1 2 g αβ,α ( g αβ,ν − g νµ,β + g βν,µ ) log √− g log √− g ,β Γ βµν − Γ ανβ Γ βαν � � � � − ,µν + C µ ≡ H µ − � x µ (physical solutions satisfy C µ = 0)

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − −∇ ( µ H ν ) + ∇ ( µ ✷ x ν ) − 1 2 g αβ g µν,αβ + g αβ g β ( µ,ν ) α + 1 2 g αβ,α ( g αβ,ν − g νµ,β + g βν,µ ) log √− g log √− g ,β Γ βµν − Γ ανβ Γ βαν � � � � − ,µν + C µ ≡ H µ − � x µ (physical solutions satisfy C µ = 0)

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − −∇ ( µ H ν ) + ✘✘✘✘ ∇ ( µ ✷ x ν ) ✘ − 1 ✭✭✭✭✭✭✭✭✭✭✭✭✭✭ 1 ✭ 2 g αβ g µν,αβ + ✘✘✘✘✘ g αβ g β ( µ,ν ) α + ✘ 2 g αβ,α ( g αβ,ν − g νµ,β + g βν,µ ) log √− g log √− g ✘ ,β Γ βµν − Γ ανβ Γ βαν − g αβ ✭ − ✘✘✘✘✘✘ � � ,µν + ✭✭✭✭✭✭✭✭ � � , ( µ g ν ) α,β C µ ≡ H µ − � x µ (physical solutions satisfy C µ = 0)

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − −∇ ( µ H ν ) + ✘✘✘✘ ∇ ( µ ✷ x ν ) ✘ − 1 ✭✭✭✭✭✭✭✭✭✭✭✭✭✭ 1 ✭ 2 g αβ g µν,αβ + ✘✘✘✘✘ g αβ g β ( µ,ν ) α + ✘ 2 g αβ,α ( g αβ,ν − g νµ,β + g βν,µ ) log √− g log √− g ✘ ,β Γ βµν − Γ ανβ Γ βαν − g αβ ✭ − ✘✘✘✘✘✘ � � ,µν + ✭✭✭✭✭✭✭✭ � � , ( µ g ν ) α,β C µ ≡ H µ − � x µ (physical solutions satisfy C µ = 0) choose some H µ = f µ ( g ) (this sets � x µ = f µ ( g ) as long as C µ = 0)

Setup µ, ν = 1 , ..., d + 1 2Λ � 1 � d − 1 T αα g µν 0 = − d − 1 g µν − 8 π T µν − 2 n ( µ C ν ) − (1 + κ 2 ) g µν n α C α � � −∇ ( µ H ν ) + ✘✘✘✘ ∇ ( µ ✷ x ν ) − κ 1 ✘ ✭ − 1 ✭✭✭✭✭✭✭✭✭✭✭✭✭✭ 1 2 g αβ g µν,αβ + ✘✘✘✘✘ g αβ g β ( µ,ν ) α + 2 g αβ,α ( g αβ,ν − g νµ,β + g βν,µ ) ✘ log √− g log √− g ✘ ,β Γ βµν − Γ ανβ Γ βαν − g αβ ✭ − ✘✘✘✘✘✘ � � ,µν + ✭✭✭✭✭✭✭✭ � � , ( µ g ν ) α,β C µ ≡ H µ − � x µ (physical solutions satisfy C µ = 0) choose some H µ = f µ ( g ) (this sets � x µ = f µ ( g ) as long as C µ = 0)

Setup Ingredients Evolution Equations Initial Data Boundary Conditions Gauge Choice

Setup Ingredients • Evolution Equations Initial Data Boundary Conditions Gauge Choice

Setup Evolution Equations − 1 2 g αβ g µν,αβ − g αβ 0 = , ( µ g ν ) α,β − H ( µ,ν ) + H α Γ αµν − Γ αβµ Γ βαν 2 n ( µ C ν ) − (1 + κ 2 ) g µν n α C α � � − κ 1 − 2Λ � 1 � d − 1 T αα g µν d − 1 g µν − 8 π T µν − ↓ 0 = E ( g µν ) ( d + 2)( d + 1) / 2 such equations, one for each g µν H µ = f µ ( g ) constraint damping terms ∼ κ 1 , designed to damp towards C µ = 0

Setup g tt dt 2 + 2 g tz dtdz + 2 g tx 1 dtdx 1 + 2 g tx 2 dtdx 2 + g µν dx µ dx ν = g zz dz 2 + 2 g zx 1 dzdx 1 + 2 g zx 2 dzdx 2 + g x 1 x 1 dx 2 1 + 2 g x 1 x 2 dx 1 dx 2 + g x 2 x 2 dx 2 2 g µν = g µν ( t, z, x 1 , x 2 )

Setup g tt dt 2 + 2 g tz dtdz + 2 g tx 1 dtdx 1 + 2 g tx 2 dtdx 2 + 2 g tx 3 dtdx 3 + g µν dx µ dx ν = g zz dz 2 + 2 g zx 1 dzdx 1 + 2 g zx 2 dzdx 2 + 2 g zx 3 dzdx 3 + g x 1 x 1 dx 2 1 + 2 g x 1 x 2 dx 1 dx 2 + 2 g x 1 x 3 dx 1 dx 3 + g x 2 x 2 dx 2 2 + g x 2 x 3 dx 2 dx 3 + g x 3 x 3 dx 2 3 + g µν = g µν ( t, z, x 1 , x 2 , x 3 )

Setup g tt dt 2 + 2 g tz dtdz + 2 g tx 1 dtdx 1 + 2 g tx 2 dtdx 2 + 2 g tx 3 dtdx 3 + g µν dx µ dx ν = g zz dz 2 + 2 g zx 1 dzdx 1 + 2 g zx 2 dzdx 2 + 2 g zx 3 dzdx 3 + g x 1 x 1 dx 2 1 + 2 g x 1 x 2 dx 1 dx 2 + 2 g x 1 x 3 dx 1 dx 3 + g x 2 x 2 dx 2 2 + g x 2 x 3 dx 2 dx 3 + g x 3 x 3 dx 2 3 + g µν = g µν ( t, z, x 1 = 0 , x 2 , x 3 ) with SO (2) in x 1 , x 2

Setup g tt dt 2 + 2 g tz dtdz + 2 g tx 1 dtdx 1 + 2 g tx 2 dtdx 2 + 2 g tx 3 dtdx 3 + g µν dx µ dx ν = g zz dz 2 + 2 g zx 1 dzdx 1 + 2 g zx 2 dzdx 2 + 2 g zx 3 dzdx 3 + g x 1 x 1 dx 2 1 + 2 g x 1 x 2 dx 1 dx 2 + 2 g x 1 x 3 dx 1 dx 3 + g x 2 x 2 dx 2 2 + g x 2 x 3 dx 2 dx 3 + g x 3 x 3 dx 2 3 + g µν = g µν ( t, z, x 1 = 0 , x 2 , x 3 ) with SO (2) in x 1 , x 2 L ξ g µν = 0 ∂ ∂ L ξ H µ = 0 ξ = x 2 − x 1 ∂x 1 ∂x 2 L ξ ϕ = 0

Setup Ingredients • Evolution Equations Initial Data Boundary Conditions Gauge Choice

Setup Ingredients Evolution Equations • Initial Data Boundary Conditions Gauge Choice

Setup Initial Data ( d ) R + K 2 − K ij K ij − 2Λ − 16 πρ 0 = D j K ji − D i K − 8 πj i 0 = ↓ 0 = E ( ζ µ ) ( d + 1) such equations, one for each ζ µ where 1 n µ = − α∂ µ t, ρ = n µ n ν T µν , j i = − g µi n ν T µν , K ij = − 1 2 L n g ij = − 1 2 α ( − ∂ t g ij + D i β j + D j β i ) 1 Here, α is the lapse function and β i is the shift vector