Some nonstandard gravity setups in AdS/CFT Romuald A Janik Jagiellonian University Kraków M Heller, RJ, P Witaszczyk, 1103.3452, 1203.0755 RJ, J Jankowski, P Witkowski, work in progress / 24 Outline Introduction: Global AdS versus Poincare Patch Outer boundary conditions (in the bulk) – freezing the evolution Subtleties with ADM at the AdS boundary Dirac δ-like boundary conditions Conclusions / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 Global AdS versus Poincare Patch Global AdS5 ds = − cosh2 ρ dτ +dρ2 +sinh2 ρ dΩ23 Poincare patch ds = ηµν dx µ dx ν + dz z2 Poincare patch covers only a part of the global Anti-de-Sitter spacetime In AdS/CFT a crucial role is played by the boundary – in the global AdS case it is R × S – for the Poincare patch it is R1,3 This provides a quite different physical interpretation on the gauge theory side: – in the global AdS case we are dealing with N = SYM theory on R × S3 – for Poincare patch we are dealing with N = SYM theory on R1,3 / 24 III Dirac δ-like boundary conditions Numerically construct first an approximation of the Dirac-δ lattice at T = This can already be used for studying various physical questions Numerical problem — very large gradients!! Structure of the linearized scalar field for Dirac-δ lattice We use this information to modify the numerical grid We can get insight about the Dirac-δ solution 22 / 24 III Dirac δ-like boundary conditions Numerically construct first an approximation of the Dirac-δ lattice at T = This can already be used for studying various physical questions Numerical problem — very large gradients!! Structure of the linearized scalar field for Dirac-δ lattice We use this information to modify the numerical grid We can get insight about the Dirac-δ solution 22 / 24 III Dirac δ-like boundary conditions Numerically construct first an approximation of the Dirac-δ lattice at T = This can already be used for studying various physical questions Numerical problem — very large gradients!! Structure of the linearized scalar field for Dirac-δ lattice We use this information to modify the numerical grid We can get insight about the Dirac-δ solution 22 / 24 III Dirac δ-like boundary conditions Numerically construct first an approximation of the Dirac-δ lattice at T = This can already be used for studying various physical questions Numerical problem — very large gradients!! Structure of the linearized scalar field for Dirac-δ lattice We use this information to modify the numerical grid We can get insight about the Dirac-δ solution 22 / 24 III Dirac δ-like boundary conditions Numerically construct first an approximation of the Dirac-δ lattice at T = This can already be used for studying various physical questions Numerical problem — very large gradients!! Structure of the linearized scalar field for Dirac-δ lattice We use this information to modify the numerical grid We can get insight about the Dirac-δ solution 22 / 24 III Dirac δ-like boundary conditions We can observe evidence for logarithmic behaviour of the backreacted Dirac-δ lattice solution lim z z→0 ∂φ =0 ∂z work in progress 23 / 24 III Dirac δ-like boundary conditions We can observe evidence for logarithmic behaviour of the backreacted Dirac-δ lattice solution lim z z→0 ∂φ =0 ∂z work in progress 23 / 24 III Dirac δ-like boundary conditions We can observe evidence for logarithmic behaviour of the backreacted Dirac-δ lattice solution lim z z→0 ∂φ =0 ∂z work in progress 23 / 24 III Dirac δ-like boundary conditions We can observe evidence for logarithmic behaviour of the backreacted Dirac-δ lattice solution lim z z→0 ∂φ =0 ∂z work in progress 23 / 24 III Dirac δ-like boundary conditions We can observe evidence for logarithmic behaviour of the backreacted Dirac-δ lattice solution lim z z→0 ∂φ =0 ∂z work in progress 23 / 24 Conclusions The AdS/CFT correspondence provides a fascinating source of problems for numerical relativity These problems arise both in ‘global AdS’ and ‘Poincare patch’ contexts Sometimes one encounters the need for nonstandard techniques (like freezing of ADM evolution) and unexpected subtleties (non-Dirichlet boundary conditions at the AdS boundary) In other cases one encounters novel setups w.r.t conventional Numerical Relativity (e.g very localized Dirac-δ like sources/ boundary conditions) 24 / 24 Conclusions The AdS/CFT correspondence provides a fascinating source of problems for numerical relativity These problems arise both in ‘global AdS’ and ‘Poincare patch’ contexts Sometimes one encounters the need for nonstandard techniques (like freezing of ADM evolution) and unexpected subtleties (non-Dirichlet boundary conditions at the AdS boundary) In other cases one encounters novel setups w.r.t conventional Numerical Relativity (e.g very localized Dirac-δ like sources/ boundary conditions) 24 / 24 Conclusions The AdS/CFT correspondence provides a fascinating source of problems for numerical relativity These problems arise both in ‘global AdS’ and ‘Poincare patch’ contexts Sometimes one encounters the need for nonstandard techniques (like freezing of ADM evolution) and unexpected subtleties (non-Dirichlet boundary conditions at the AdS boundary) In other cases one encounters novel setups w.r.t conventional Numerical Relativity (e.g very localized Dirac-δ like sources/ boundary conditions) 24 / 24 Conclusions The AdS/CFT correspondence provides a fascinating source of problems for numerical relativity These problems arise both in ‘global AdS’ and ‘Poincare patch’ contexts Sometimes one encounters the need for nonstandard techniques (like freezing of ADM evolution) and unexpected subtleties (non-Dirichlet boundary conditions at the AdS boundary) In other cases one encounters novel setups w.r.t conventional Numerical Relativity (e.g very localized Dirac-δ like sources/ boundary conditions) 24 / 24 Conclusions The AdS/CFT correspondence provides a fascinating source of problems for numerical relativity These problems arise both in ‘global AdS’ and ‘Poincare patch’ contexts Sometimes one encounters the need for nonstandard techniques (like freezing of ADM evolution) and unexpected subtleties (non-Dirichlet boundary conditions at the AdS boundary) In other cases one encounters novel setups w.r.t conventional Numerical Relativity (e.g very localized Dirac-δ like sources/ boundary conditions) 24 / 24 ... complications/stumbling blocks in various setups within the Poincare patch context as encountered by an outsider in NR / 24 Global AdS versus Poincare Patch Sometimes one can interpret results from... complications/stumbling blocks in various setups within the Poincare patch context as encountered by an outsider in NR / 24 Global AdS versus Poincare Patch Sometimes one can interpret results from... complications/stumbling blocks in various setups within the Poincare patch context as encountered by an outsider in NR / 24 Global AdS versus Poincare Patch Sometimes one can interpret results from