Lorentzian Quantum Einstein Gravity
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1 Lorentzian Quantum Einstein Gravity Stefan Rechenberger Uni Mainz Phys.Rev.Lett. 106 (2011) with Elisa Manrique and Frank Saueressig Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
2 Outline 1 Motivation 2 CDT, Horava Gravity and Asymptotic Safety 3 Causal functional RG equation 4 Results 5 Conclusion Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
3 Motivation Classical GR reaches its limits close to space-time singularities Black Holes Big Bang Solution probably lies within a theory of Quantum Gravity Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
4 Motivation Classical GR reaches its limits close to space-time singularities Black Holes Big Bang Solution probably lies within a theory of Quantum Gravity different approaches to a theory of QG String Theory Loop Quantum Gravity Causal Dynamical Triangulations Horava Gravity Asymptotic Safety... Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
5 Motivation Classical GR reaches its limits close to space-time singularities Black Holes Big Bang Solution probably lies within a theory of Quantum Gravity different approaches to a theory of QG String Theory Loop Quantum Gravity Causal Dynamical Triangulations Horava Gravity Asymptotic Safety... which one is correct? Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
6 lack of experimental data nobody helps us to decide which is the best approach Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
7 lack of experimental data nobody helps us to decide which is the best approach Best thing to do: compare different approaches Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
8 Causal dynamical triangulations (arxiv: v1 [hep-th]) discretization of gravitational path integral Dg µν e isgrav summing over piecewise flat geometries modeling space-time geometries by gluing together simplices (higher dimensional generalizations of triangles) important: causal structure Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
9 Causal dynamical triangulations (arxiv: v1 [hep-th]) discretization of gravitational path integral Dg µν e isgrav summing over piecewise flat geometries modeling space-time geometries by gluing together simplices (higher dimensional generalizations of triangles) important: causal structure Horava Gravity ( arxiv: v2 [hep-th]) different scaling of space and time UV: Lorentz invariance is broken IR: Lorentz invariance reestablished maybe connection to CDT due to global time foliation ( arxiv: v2 [hep-th]) Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
10 Asymptotic Safety (Living Rev. Relativity 9, (2006)) non-trivial fixed point (for UV completion) finite dimensional critical surface (for predictability) Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
11 Asymptotic Safety (Living Rev. Relativity 9, (2006)) non-trivial fixed point (for UV completion) finite dimensional critical surface (for predictability) already found for pure Einstein Hilbert action f(r) gravity gravity coupled to a scalar field gravity with extra dimensions bimetric truncations... Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
12 Asymptotic Safety (Living Rev. Relativity 9, (2006)) non-trivial fixed point (for UV completion) finite dimensional critical surface (for predictability) already found for pure Einstein Hilbert action f(r) gravity gravity coupled to a scalar field gravity with extra dimensions bimetric truncations... strong evidence that nature might be asymptotically safe Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
13 Asymptotic Safety (Living Rev. Relativity 9, (2006)) non-trivial fixed point (for UV completion) finite dimensional critical surface (for predictability) already found for pure Einstein Hilbert action f(r) gravity gravity coupled to a scalar field gravity with extra dimensions bimetric truncations... so far only Euclidean space-time has been studied strong evidence that nature might be asymptotically safe Lorentzian space-times are necessary for comparison with CDT and HG Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
14 Causal functional RG equation Starting point: Einstein Hilbert action 1 S EH = d D x γ( R +2Λ) 16πG N G N... Newton constant D... space-time dimension (D = d +1) γ... metric R... curvature scalar of space-time Λ... cosmological constant Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
15 Causal functional RG equation Starting point: Einstein Hilbert action 1 S EH = d D x γ( R +2Λ) 16πG N G N... Newton constant D... space-time dimension (D = d +1) γ... metric R... curvature scalar of space-time Λ... cosmological constant g Λ Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
16 N a t2 split of space-time M D = M d S 1 Σ... spatial slices n a... vector orthonormal to Σ N... lapse function N a... shift vector N n a t1 Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
17 N n a N a t2 split of space-time M D = M d S 1 Σ... spatial slices n a... vector orthonormal to Σ N... lapse function N a... shift vector σ ij... spatial metric t1 ( ǫn γ µν = 2 +N i N i ) N j ds 2 = ǫn 2 dτ 2 +σ ij ( dx i +N i dτ )( dx j +N j dτ ) N i σ ij Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
18 geometric cutoff in time direction standard cutoff in spatial direction k k g k = β g (g,λ;m), k k λ k = β λ (g,λ;m) dim.less Newton constant: g dim.less cosmological constant: λ dim.less Kaluza-Klein mass: m = 2π Tk circumference of time circle: T Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
19 Results m = const.(e.g.2π) T 1 k flow eq. provide a fixed point in Euclidean and Lorentzian signature ǫ g λ g λ θ 1, ± 3.31i ± 3.10i Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
20 Results m = const.(e.g.2π) T 1 k flow eq. provide a fixed point in Euclidean and Lorentzian signature ǫ g λ g λ θ 1, ± 3.31i ± 3.10i g 0.35 g Λ Λ Euclidean Lorentzian Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
21 m dependence of the fixed point values Lorentzian (red) and Euclidean (blue) g Λ lim g = 0 m Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
22 dimensionless Newton constant in 3D: g 3 = g Tk Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
23 dimensionless Newton constant in 3D: g 3 = g Tk g Tk Λ Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
24 Conclusion FP for Euclidean and Lorentzian signature characteristics are similar also similar to covariant formulation time circle collapses toward UV signature does NOT matter in UV formulation prepares ground for comparison to other theories Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
25 Thank you for your attention! Questions? Stefan Rechenberger (Uni Mainz) Lorentzian Quantum Einstein Gravity / 14
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