Some simple exact solutions to d = 5 Einstein Gauss Bonnet Gravity

Transcription

1 Some simple exact solutions to d = 5 Einstein Gauss Bonnet Gravity Eduardo Rodríguez Departamento de Matemática y Física Aplicadas Universidad Católica de la Santísima Concepción Concepción, Chile CosmoConce, UBB, Concepción, Chile March 15 16, 2012

2 The Question we would like to address (Not answer, of course) Why Four?

3 The Question we would like to address To answer the question, we should begin by... We will never answer Why Four? if we assume from the outset that spacetime is four-dimensional Higher-dimensional theories (e.g. String, Supergravity) usually assume some kind of compactification Can we get an effectively four-dimensional spacetime emerging from a higher-dimensional theory?

4 Einstein Gauss Bonnet Gravity Usual Tensor Formulation d = 5 EGB Action in tensor notation S (5) EGB = d 5 x g [γ 0 + γ 1 R+ M +γ 2 ( R µν ρσr ρσ µν 4R µ νr ν µ + R 2)] d = 5 EGB Action explained The action includes three terms: a cosmological constant term the usual Einstein Hilbert term a curvature-squared Gauss Bonnet term

5 Einstein Gauss Bonnet Gravity Usual Tensor Formulation d = 5 EGB Action in tensor notation S (5) EGB = d 5 x g [γ 0 + γ 1 R+ M +γ 2 ( R µν ρσr ρσ µν 4R µ νr ν µ + R 2)] d = 5 EGB Action explained The action includes three terms: a cosmological constant term the usual Einstein Hilbert term a curvature-squared Gauss Bonnet term

6 Einstein Gauss Bonnet Gravity Usual Tensor Formulation d = 5 EGB Action in tensor notation S (5) EGB = d 5 x g [γ 0 + γ 1 R+ M +γ 2 ( R µν ρσr ρσ µν 4R µ νr ν µ + R 2)] d = 5 EGB Action explained The action includes three terms: a cosmological constant term the usual Einstein Hilbert term a curvature-squared Gauss Bonnet term

7 Einstein Gauss Bonnet Gravity Usual Tensor Formulation d = 5 EGB Action in tensor notation S (5) EGB = d 5 x g [γ 0 + γ 1 R+ M +γ 2 ( R µν ρσr ρσ µν 4R µ νr ν µ + R 2)] d = 5 EGB Action explained The action includes three terms: a cosmological constant term the usual Einstein Hilbert term a curvature-squared Gauss Bonnet term

8 Einstein Gauss Bonnet Gravity First-order Formulation d = 5 EGB Lagrangian in first-order formulation L (5) EGB = ε abcde Field Content e a = e a µdx µ : vielbein ( α 0 e a e b e c e d e e + α 1 R ab e c e d e e + α 2 R ab R cd e e). ω ab = ω ab µ dxµ : spin connection R ab = dω ab + ω a cω cb : Lorentz curvature T a = de a + ω a b eb : Torsion

9 An Open Problem in d = 5 EGB Gravity What is the vacuum of the theory? Field Equations for the EGB ε abcde (5α 0 e a e b e c e d + 3α 1 R ab e c e d + α 2 R ab R cd) = 0, ε abcde (3α 1 e c e d + 2α 2 R cd) T e = 0. Factorizing the Field Equations β 0 ε abcde (R ab β 1 e a e b) ( R cd β 2 e c e d) = 0. Relation between the α s and the β s 5α 0 + 3α 1 x + α 2 x 2 = β 0 (x β 1 ) (x β 2 )

10 Selecting the Coefficients The vacuum structure of the EGB theory depends strongly on the α s When β 1 and β 2 are real and distinct, there are two vacuum states with constant curvature. When β 1 = β 2 then there is a single vacuum state with constant curvature. This is a special case, because the action acquires a larger symmetry for this choice of coefficients and becomes the Chern Simons action for the (A)dS group. What happens when the β i are complex?

11 Selecting the Coefficients The vacuum structure of the EGB theory depends strongly on the α s When β 1 and β 2 are real and distinct, there are two vacuum states with constant curvature. When β 1 = β 2 then there is a single vacuum state with constant curvature. This is a special case, because the action acquires a larger symmetry for this choice of coefficients and becomes the Chern Simons action for the (A)dS group. What happens when the β i are complex?

12 Selecting the Coefficients The vacuum structure of the EGB theory depends strongly on the α s When β 1 and β 2 are real and distinct, there are two vacuum states with constant curvature. When β 1 = β 2 then there is a single vacuum state with constant curvature. This is a special case, because the action acquires a larger symmetry for this choice of coefficients and becomes the Chern Simons action for the (A)dS group. What happens when the β i are complex?

13 One theory, three different regimes Vacuum structure of the EGB theory parameterized by single constant χ It is convenient to parameterize the Lagrangian as L (5) EGB = κ ( l ε abcde R ab R cd 2χ 3l 2 Rab e c e d + 1 ) 5l 4 ea e b e c e d e e χ 2 > 1: two constant-curvature vacua χ 2 = 1: one constant-curvature vacuum χ 2 < 1: no constant-curvature vacua Why may be this last, pathological case be interesting?

14 Warped Product Warped product solutions as a means towards dynamical dimensional reduction General Warped Product ds 2 = f 2 (w) dt 2 + g 2 (w) dσ 2 + p 2 (t) q 2 (x, y, z) dw 2, where Σ is a constant-curvature 3-manifold: [ dσ 2 = 1 + K (x 2 + y 2 + z 2)] 2 ( dx 2 + dy 2 + dz 2). 4

15 Warped Product Plugging the ansatz in the field equations we find... The field equations imply the following: Simplified g (w) = 1, q (x, y, z) = 1, K = 1 χl 2. ds 2 = f 2 (w) dt 2 + dσ 2 + p 2 (t) dw 2.

16 Summary of Different solutions for the EGB theory with χ 2 < 1 Class f (w) p (t) Σ χ-range R, τ PH 1 hyp. K < 0 1 < χ < 0 l ξ FC circ. 1 K < 0 1 < χ < 0 l ξ PC+ 1 circ. K > 0 0 < χ < 1 l ξ FH+ hyp. 1 K > 0 0 < χ < 1 l ξ ξ = 1 2 ( χ 1 ) χ

17 Summary of Different solutions for the EGB theory with χ 2 > 1 Class f (w) p (t) Σ Range R, τ K PH+ 1 hyp. K > 0 EGB χ > 1 l 1 ξ GR Λ > 0 FC+ circ. 1 K > 0 EGB χ > 1 l 1 ξ GR Λ > 0 PC 1 circ. K < 0 EGB χ < 1 l 1 ξ GR Λ < 0 3 2Λ FH hyp. 1 K < 0 EGB χ < 1 l 1 ξ GR Λ < 0 3 2Λ 3 2Λ 3 2Λ χl 2 Λ 3 χl 2 Λ 3 χl 2 Λ 3 χl 2 Λ 3

18 Summary of A graphic summary of solutions and theories 2 1 PH /FC PH+/FC+ ξ 0 1 PC /FH PC+/FH χ

19 Circular Fifth Dimension Shrinking Fifth Dimension Circular Fifth Dimension An FC solution features a circular fifth dimension Line element for the FC ( spacetime ds 2 = cos 2 w ) dt 2 + dσ 2 + dw 2 R Features Compact (circular) fifth dimension of radius R Flow of time changes along the circle

20 Circular Fifth Dimension Shrinking Fifth Dimension Circular Fifth Dimension An PH solution features a shrinking fifth dimension Line element for the PH spacetime ds 2 = dt 2 + dσ 2 + e 2t/τ dw 2 Features Fifth dimension shrinks exponentially Effectively four-dimensional spacetime emerges dynamically after some time τ

21 Some Open Problems Or where this road might lead us next Circular Fifth Dimension Shrinking Fifth Dimension Are the solutions stable? Is this the vacuum for the EGB theory with χ 2 < 1? What happens in higher dimensions? Can we include matter and nontrivial torsion?