Black Hole Universe with Rotation 2016.01.02. Chan Park KAIST
Motivation
FLRW cosmology Highly symmetric assumption : spatial homogeneity and isotropy Metric ds 2 = dt 2 + a 2 t a t : scale factor Friedmann equation dr 2 a2 a 2 = 8π 3 ρ k a 2 k : conformal spatial curvature k = 1 : closed universe k = 0 : flat universe k = 1 : open universe 1 kr 2 + r2 dθ 2 + sin 2 θ dφ 2
The real universe Local inhomogeneity Clusters of galaxies with voids. Large scale homogeneity 100Mpc Local anisotropy Vorticity, shear and peculiar velocities. Large scale anisotropy CMB anisotropy SDSS / Wikipedia 9 years of WMAP data / Wikipedia
The lattice universe model Tiling 3-spaces with identical regular polyhedral cells. A black hole is located on the center of each cells. It mimic the galactic center. Elementary cell shape Number of cells at a lattice edge Background curvature Total cells in lattice Tetrahedron 3 + 5 Cube 3 + 8 Tetrahedron 4 + 16 Octahedron 3 + 24 Dodecahedron 3 + 120 Tetrahedron 5 + 600 Cube 4 0 Cube 5 - Dodecahedron 4 - Dodecahedron 5 - Icoshedron 3 - All possible lattices with constant curvature / T. Clifton and P. G. Ferreira (2009) Tiling 8 cubic cells / E. Bentivegna and M. Korzynski (2012)
The solutions of lattice universe model Approximate solutions Closed universe : R. W. Lindquist and J. A. Wheeler (1959) Flat/Open universe : T. Clifton and P. G. Ferreira (2011) Exact solutions Closed universe (contraction only) Initial data (analytically) 5 cells : J. A. Wheeler (1983) 8, 16, 24, 120 and 600 cells : T. Clifton et al. (2012) Evolution (numerically) 8 cells : E. Bentivegna and and M. Korzynski (2012) Flat universe Initial data (numerically) : C.-M. Yoo et al. (2012) Evolution (numerically) : C.-M. Yoo et al. (2013) Flat universe with cosmological constant Initial data / evolution (numerically) : C.-M. Yoo and H. Okawa (2014) Anisotropic flat universe Initial data (numerically) : C. Park and G. Kang (2015) R. W. Lindquist and J. A. Wheeler (1959)
Lattice universe with rotating black hole In order to give anisotropy globally, we consider the lattice universe model consist of identical rotating black holes with the flat tiling of cubic.
The Bianchi type universe For the 3-space that is homogeneous and anisotropic, we can apply Bianchi classification 9 classes of 3 dimensional real Lie algebra Classes that permit FRLW metric Flat : I, VII 0 Closed : IX Open : V, VII h As a counterpart of our model, we will consider Bianchi type I. (Kasner universe) ds 2 = dt 2 + t 2p 1dx 2 + t 2p 2dy 2 + t 2p 3dz 2 H 2 = 1 6 A2 + 8π 3 ρ
Initial data construction
Einstein constraint equations Initial data γ ij (induced metric) and K ij (extrinsic curvature) on initial hyper surface should satisfy the Einstein constraint equations in the 3+1 formalism. R + K 2 K ij K ij = 16πρ (Hamiltonian constraint) D j K j i D i K = 8πp i (Momentum constraints) Here, D : Levi-Civita connection associated with γ ij R : 3-dimensional Ricci scalar K : the trace of K ρ : the matter energy density p i : the matter momentum density
Example: FLRW metric R + K 2 K ij K ij = 16πρ (The Hamiltonian constraint) R = 6 k a 2 K ij = 1 γ 3 ijk K = 3H k a 2 + H2 = 8π 3 ρ (The Friedmann equation)
Conformal-transverse-traceless method Trace-traceless decomposition of K ij K ij = A ij + 1 3 γ ijk Conformal decomposition of γ ij and A ij γ ij = Ψ 4 γ ij A ij = Ψ 2 A ij Transverse-longitudinal decomposition of A ij TT A ij = LX ij + A ij LX ij = D i X j + D j X i 2 3 γ ij D k X k
Einstein constraint equations in CTT 1 RΨ D 8 i D i Ψ 1 8 2π EΨ 3 TT LX ij + A ij LX ij ij + A TT D j D j X i + 1 3 Di D j X j 2 3 Ψ6 D i K = 8π p i Here, R : 3-dimensional conformal Ricci scalar E = Ψ 8 E p i = Ψ 10 p i Ψ 7 + 1 12 K2 Ψ 5 =
Summary of CTT method Initial data Decomposition Physical meaning γ ij (6) γ ij (5) Ψ (1) X i (3) Coordinate freedom (3) Dynamical d.o.f (2) Given by constraint equations (4) K ij (6) K (1) Coordinate freedom (1) A TT ij (2) Dynamical d.o.f (2)
Assumption Vacuum spacetime E = 0 = p i Conformally flat Cartesian coordinate γ ij = δ ij R = 0 No transverse-traceless part of A ij A TT ij = 0
The vacuum Einstein constraints ΔΨ = 1 LX LX ij Ψ 7 + 1 8 ij 12 K2 Ψ 5 ΔX i = 1 3 i j X j + 2 3 Ψ6 i K Where LX ij = i X j + j X i 2 δ 3 ij k X k
The tiling of cube in flat 3-space Domain D = x, y, z L x, y, z L (0,0,0) Periodic boundary condition φ x + L, y, z = φ x, y, z φ x, y + L, z = φ(x, y, z) φ x, y, z + L = φ(x, y, z)
Choosing form of K K = 3H eff W r r = x 2 + y 2 + z 2 W r = 0 (r l)/σ 1 6 1 6 1 0 r l l < r < l + σ l + σ r l l + σ
Boundary conditions at the puncture Ψ = M + 1 + O r 2r X i = 1 r 2 εijk l j J k + O 1 Here, l j = x j r : longitudinal unit vector This is asymptotic black hole solutions for the vacuum Einstein constraints near the puncture where K = 0. (without peculiar motion of black hole) Parameters of the solution : L, H eff, M and J i
The volume integration on D Hamiltonian constraint : ΔΨ = 1 8 LX ij LX ij Ψ 7 + 1 12 K2 Ψ 5 8π M = 1 3 6 D LX LX ij Ψ 7 dx 3 + H 2 ij eff V Compare to Kasner universe : 8π 3 ρ = H2 1 6 A2 2 H eff where = 8π 3 M + M K V V = 8π 3 ρ BH + ρ K M K = 1 16π D LX ij LX ij Ψ 7 dx 3 V = D W 2 Ψ 5 dx 3 H eff and M can t be chosen independently.
The volume integration on D Momentum constraints : ΔX i = 1 3 i j X j + 2 3 Ψ6 i K 0 = D Ψ 6 i Wdx 3
Parity symmetries of Ψ and X i 0 = D Ψ 6 i Wdx 3 i W has the symmetry of odd parity i W x, y, z = i W(x, y, z) Ψ should have the symmetry of even parity Ψ x, y, z = Ψ(x, y, z) Momentum constraints : ΔX i = 1 3 i j X j 2H eff Ψ 6 i W For decomposition of X i = Xi i even + X odd ΔXi even j = 1 3 i j X even X even = constant for periodic boundary condition It set to be zero without changing any physical situation.
Singularity extraction Definition of ψ and Y i Ψ = M U + ψ 2r X i = X i J U + Y i Where U = 1 W X J i = 1 r 2 εijk l j J k Boundary condition for ψ and Y i as r 0 ψ 1 + O r Y i O 1
The final equation ψ = M 2r d 2 U U 7 1 LX ij LX du 2 8 ij ψ + M 2r + 3 4 H eff 2 W 2 ψ + M 5 2r U ΔY i = 1 3 i j Y j 1 r 2 εijk l j J k 2 r du + d2 U dr du 2 2H eff ψ + M 2r U 6 i W Where LX ij = LY ij + 1 r 2 du dr + 3U r l i ε jkl + l j ε ikl J k l l
Numerical construction
Numerical method 2 nd order finite difference method df = f i+1 f i 1 dx i 2h d 2 f dx 2 + O h 3 i = f i+1+f i 1 2f i h 2 + O h 3 Multigrid method Full weighting restriction Linear interpolation Gauss-Seidel relaxation Cubic interpolation for full multigrid method
Strategy Fix M and determine H eff by the effective Friedmann equation. Periodic boundary condition does not fix zero mode Y i : Don t care. Physical variables are depend only derivate of Y i ψ i : For each V cycle step fix zero mode to satisfy boundary condition near puncture (r 0). ψ = ψ ψ O + 1
Convergence Test Convergence factor α = e(n) e (n 1) e (n 1) e (n 2) = ψ(n) ψ (n 1) ψ (n 1) ψ (n 2) Where n : 2 n 3 is number of points on n the grid ψ (n) : numerical value of ψ using nth grid e (n) : discretization error using nth grid h (n) : grid spacing of nth grid p : the order of finite difference method h n p h n 1 p h n 1 p h n 2 p = 1 2 p
Ψ M/2 on z = 0
Ψ on z = L
Y i on z = 0
X i on z = L
Vector field of X i on z = 0 and z = L
Vector field of X i on x = L
Analysis
The parameter of size Effective number of black holes within cosmological horizon N BH = 4π 3 H eff 3 1 V
Convergence to Einstein de Sitter (EdS) universe 2 H eff = 8π 3 M V 1 + M K M 8π 3 M V
Effective scale factors 2 H eff = 8π 3 M a 3 EdS a P = 2LΨ 2 (L, L, L) L a L = L Ψ 2 L, L, z dz L a A = L Ψ 4 x, y, L dxdy
Anistopy of scale factor (x) L a L = L Ψ 2 x, L, z dx (y) L a L = L Ψ 2 L, y, L dy (z) L a L = L Ψ 2 L, L, z dz (x) L a A = L Ψ 4 L, y, z dydz (y) L a A = L Ψ 4 x, L, z dxdz (z) L a A = L Ψ 4 x, y, L dxdy
On the vertex of cube R ij = 2Ψ 1 i j Ψ + ΔΨδ ij 2 R = 6H eff + LX ij LX ij Ψ 12
Conclusion
Conclusion 1 We have numerically studied an initial data construction of the black hole universe model which consists of cubic lattice of identical rotating black holes. Adopting the periodic boundary conditions on each sides on a cubic domain and the conventional CTT (conformal transverse traceless) method, we obtained numerical solutions of two parameter families characterized by the ADM angular momentum of the black hole and the coordinate size of its cubic domain. Due to the boundary condition and the form of K imposed, the conformal factor ψ and the vector potential X i have symmetries of even and odd parities, respectively.
Conclusion 2 Our results are analyzed in terms of cosmological models. As the effective number of rotating black holes N BH (e.g., the separation of neighbouring black holes) increases, the relation between the Hubble expansion rate and the mass density of black holes becomes similar to the case of the Einstein de Sitter universe although the rotation slows down their agreement compared to the case of non-rotating black hole universe. The scale factors (i.e., a P, a L and a A ) reveal a sudden increase and decrease, quite differently from those in the non-rotating case, but show identical behaviors as N BH becomes large. The presence of rotation makes the geometry nearby cubic surfaces anisotropic. Also, the effective Firedmann equation for our model is well described by that of Kasner universe.
Future work
Future work Evolution and features of expansion. Geometrical properties near the black horizon. Relation to various rotating universe models.