Bimetric Theory (The notion of spacetime in Bimetric Gravity)

Transcription

1 Bimetric Theory (The notion of spacetime in Bimetric Gravity) Fawad Hassan Stockholm University, Sweden 9th Aegean Summer School on Einstein s Theory of Gravity and its Modifications Sept 18-23, 2017, Sifnos

2 SFH & Mikica Kocic (arxiv: ) Also based on work with: Rachel. A. Rosen Angnis Schmidt-May Mikael von Strauss Anders Lundkvist Luis Apolo

3 Outline of the talk Motivation & Problems Gravity with an extra spin-2 field: Bimetric Theory Potential Issues Uniqueness and the local structure of spacetime Discussion

4 Bimetric gravity: Why study a theory of gravity with two metrics? (Gravitational metric coupled to extra spin-2 fields)

5 Outline of the talk Motivation & Problems Gravity with an extra spin-2 field: Bimetric Theory Potential Issues Uniqueness and the local structure of spacetime Discussion

6 Motivation GR + Λ + CDM : Very successful

7 Motivation GR + Λ + CDM : Very successful Observations and theory indicate new physics Dark matter, Dark energy (the cosmological constant problem), Origin of Inflation, The trans-planckian problem, Quantum gravity, Matter-Antimatter asymmetry

8 Motivation GR + Λ + CDM : Very successful Observations and theory indicate new physics Dark matter, Dark energy (the cosmological constant problem), Origin of Inflation, The trans-planckian problem, Quantum gravity, Matter-Antimatter asymmetry Theoretical (top down) approaches: GUTs, Susy, String Theory (with or without Multiverse), Quantum Gravity,

9 Motivation GR + Λ + CDM : Very successful Observations and theory indicate new physics Dark matter, Dark energy (the cosmological constant problem), Origin of Inflation, The trans-planckian problem, Quantum gravity, Matter-Antimatter asymmetry Theoretical (top down) approaches: GUTs, Susy, String Theory (with or without Multiverse), Quantum Gravity, But why gravity with extra spin-2 fields?

10 Recall: Spin (s) basic structure of field equations (Klein-Gordon, Dirac, Maxwell/Proca, Einstein)

11 Recall: Spin (s) basic structure of field equations (Klein-Gordon, Dirac, Maxwell/Proca, Einstein) Spin in known physics: Standard Model: multiplets of s = 0, 1 2, 1 fields Multiplet structure is crucial for the viability of SM

12 Recall: Spin (s) basic structure of field equations (Klein-Gordon, Dirac, Maxwell/Proca, Einstein) Spin in known physics: Standard Model: multiplets of s = 0, 1 2, 1 fields Multiplet structure is crucial for the viability of SM General relativity: single massless s = 2 field g µν Einstein-Hilbert for s = 2 Klein-Gordon for s = 0

13 Recall: Spin (s) basic structure of field equations (Klein-Gordon, Dirac, Maxwell/Proca, Einstein) Spin in known physics: Standard Model: multiplets of s = 0, 1 2, 1 fields Multiplet structure is crucial for the viability of SM General relativity: single massless s = 2 field g µν Einstein-Hilbert for s = 2 Klein-Gordon for s = 0 Low-energy String theory, etc: single massless spin-2 + plethora of spin 0, 1/2 and 1.

14 Recall: Spin (s) basic structure of field equations (Klein-Gordon, Dirac, Maxwell/Proca, Einstein) Spin in known physics: Standard Model: multiplets of s = 0, 1 2, 1 fields Multiplet structure is crucial for the viability of SM General relativity: single massless s = 2 field g µν Einstein-Hilbert for s = 2 Klein-Gordon for s = 0 Low-energy String theory, etc: single massless spin-2 + plethora of spin 0, 1/2 and 1. Unexplored corner: gravity with more spin-2 fields

15 Challenges in adding spin-2 fields to GR No multiple massless spin-2 fields Linear theory of massive spin-2 fields (5 helicities) [Fierz, Pauli (1939)]

16 Challenges in adding spin-2 fields to GR No multiple massless spin-2 fields Linear theory of massive spin-2 fields (5 helicities) [Fierz, Pauli (1939)] Ghosts in nonlinear theory (5 + 1 helicities) [Boulware, Deser (1972)] Ghosts in theories with multiple spin-2 fields

17 Challenges in adding spin-2 fields to GR No multiple massless spin-2 fields Linear theory of massive spin-2 fields (5 helicities) [Fierz, Pauli (1939)] Ghosts in nonlinear theory (5 + 1 helicities) [Boulware, Deser (1972)] Ghosts in theories with multiple spin-2 fields Can such theories exist or is GR unique? Consequences?

18 The Ghost Problem Ghost: A field with negative kinetic energy Example: L = T V = ( t φ) 2 (healthy) But L = T V = ( t φ) 2 (ghostly) Consequences:

19 The Ghost Problem Ghost: A field with negative kinetic energy Example: But Consequences: L = T V = ( t φ) 2 L = T V = ( t φ) 2 (healthy) (ghostly) Instability: unlimited energy transfer from ghost to other fields Negative probabilities, violation of unitarity in quantum theory

20 Outline of the talk Motivation & Problems Gravity with an extra spin-2 field: Bimetric Theory Potential Issues Uniqueness and the local structure of spacetime Discussion

21 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr

22 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr m 4 g V (g 1 f ) +

23 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr m 4 g V (g 1 f ) + No dynamics for f µν : Massive Gravity

24 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr m 4 g V (g 1 f ) + No dynamics for f µν : Massive Gravity Overcoming the ghost in massive gravity: [Creminelli, Nicolis, Papucci, Trincherini, (2005)] [de Rham, Gabadadze, Tolley (2010)] [SFH, Rosen (2011); SFH, Rosen, Schmidt-May (2011)]

25 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr m 4 g V (g 1 f ) + L(f, f )

26 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr m 4 g V (g 1 f ) + L(f, f ) what is V (g 1 f )? what is L(f, f )? proof of absence of the Boulware-Deser ghost

27 GR with a generic spin-2 field A dynamical theory for the metric g µν & spin-2 field f µν L = m 2 p gr m 4 g V (g 1 f ) + L(f, f ) what is V (g 1 f )? what is L(f, f )? proof of absence of the Boulware-Deser ghost V (g 1 f ) for a ghost-free theory: [SFH, Rosen (2011); de Rham, Gabadadze, Tolley (2010)]

28 Digression: elementary symmetric polynomials e n (S) Consider matrix S with eigenvalues λ 1,, λ 4. e 0 (S) = 1, e 1 (S) = λ 1 + λ 2 + λ 3 + λ 4, e 2 (S) = λ 1 λ 2 + λ 1 λ 3 + λ 1 λ 4 + λ 2 λ 3 + λ 2 λ 4 + λ 3 λ 4, e 3 (S) = λ 1 λ 2 λ 3 + λ 1 λ 2 λ 4 + λ 1 λ 3 λ 4 + λ 2 λ 3 λ 4, e 4 (S) = λ 1 λ 2 λ 3 λ 4.

29 Digression: elementary symmetric polynomials e n (S) Consider matrix S with eigenvalues λ 1,, λ 4. e 0 (S) = 1, e 1 (S) = λ 1 + λ 2 + λ 3 + λ 4, e 2 (S) = λ 1 λ 2 + λ 1 λ 3 + λ 1 λ 4 + λ 2 λ 3 + λ 2 λ 4 + λ 3 λ 4, e 3 (S) = λ 1 λ 2 λ 3 + λ 1 λ 2 λ 4 + λ 1 λ 3 λ 4 + λ 2 λ 3 λ 4, e 4 (S) = λ 1 λ 2 λ 3 λ 4. e 0 (S) = 1, e 1 (S) = Tr(S) [S], e 2 (S) = 1 2 ([S]2 [S 2 ]), e 3 (S) = 1 6 ([S]3 3[S][S 2 ] + 2[S 3 ]), e 4 (S) = det(s), e n (S) = 0 (for n > 4),

30 det(1 + S) = 4 n=0 e n(s)

31 det(1 + S) = 4 n=0 e n(s) Where: V (S) = 4 S = n=0 β n e n (S) g 1 f square root of the matrix g µλ f λν Real? Unique? How to make sense of it? (to be answered later)

32 Ghost-free bi-metric theory [SFH, Rosen ( , )] Ghost-free combination of kinetic and potential terms: L = m 2 g grg m 4 g 4 n=0 β n e n ( g 1 f ) + mf 2 f Rf bimetric nature forced by the absence of ghost

33 Ghost-free bi-metric theory [SFH, Rosen ( , )] Ghost-free combination of kinetic and potential terms: L = m 2 g grg m 4 g 4 n=0 β n e n ( g 1 f ) + mf 2 f Rf bimetric nature forced by the absence of ghost Hamiltonian analysis: 7 = nonlinear propagating modes, no BD ghost! C 1 = 0, C 2 = d dt C 1 = {H, C} = 0 Detailed analysis of constraints [SFH, Lundkvist (to appear)]

34 Mass spectrum & Limits Linear modes: f = c2ḡ, g µν = ḡ µν + δg µν, f µν = f µν + δf µν

35 Mass spectrum & Limits f = c2ḡ, g µν = ḡ µν + δg µν, f µν = f µν + δf µν Linear modes: ( ) Massless spin-2: δg µν = δg µν + m2 f δf mg 2 µν ) Massive spin-2 : δm µν = (δf µν c 2 δg µν (2) (5) g µν, f µν are mixtures of massless and massive modes

36 Mass spectrum & Limits f = c2ḡ, g µν = ḡ µν + δg µν, f µν = f µν + δf µν Linear modes: ( ) Massless spin-2: δg µν = δg µν + m2 f δf mg 2 µν ) Massive spin-2 : δm µν = (δf µν c 2 δg µν (2) (5) g µν, f µν are mixtures of massless and massive modes The General Relativity limit: m g = M P, m f /m g 0 (more later)

37 Mass spectrum & Limits f = c2ḡ, g µν = ḡ µν + δg µν, f µν = f µν + δf µν Linear modes: ( ) Massless spin-2: δg µν = δg µν + m2 f δf mg 2 µν ) Massive spin-2 : δm µν = (δf µν c 2 δg µν (2) (5) g µν, f µν are mixtures of massless and massive modes The General Relativity limit: m g = M P, m f /m g 0 (more later) Massive gravity limit: m g = M P, m f /m g

38 Can Bimetric be a fundamental theory? Similar to Proca theory in curved background, det g (Fµν F µν m 2 g µν A µ A ν + R g ) May need the equivalent of Higgs mechanism with the extra fields for better quantum or even classical behaviour

39 Some features 1) Ghost-free Matter couplings, same as in GR: L g (g, φ) + L f (f, φ)

40 Some features 1) Ghost-free Matter couplings, same as in GR: L g (g, φ) + L f (f, φ) 2) β 1, β 2, β 3 : effective cosmological constant at late times (protected by symmetry, like fermion masses) 3) Massive spin-2 particles: dark matter (stable enough)

41 Some features 1) Ghost-free Matter couplings, same as in GR: L g (g, φ) + L f (f, φ) 2) β 1, β 2, β 3 : effective cosmological constant at late times (protected by symmetry, like fermion masses) 3) Massive spin-2 particles: dark matter (stable enough) 4) Some blackhole and cosmological solutions explored 5) Gravitational waves? 6) Generalization to more than 2 fields

42 GR limit The General Relativity limit: m g = M P, α = m f /m g 0 Cosmological solutions in the GR limit: 3H 2 = ρ M 2 Pl 2 β1 2 m 2 α 2 β2 1 3 β 2 3β2 2 H 2 + O(α 4 ) The GR approximation breaks down at sufficiently strong fields

43 Outline of the talk Motivation & Problems Gravity with an extra spin-2 field: Bimetric Theory Potential Issues Uniqueness and the local structure of spacetime Discussion

44 Potential Issues (1) g, f : incompatible notions of space and time? (2) nonunique g 1 f : ambiguity in defining the action? Causality: local closed timelike curves (CTC s)? The initial value problem? Faster than light possible in the gravitational sector (good or bad?) Analogue of energy conditions and global bi hyperbolicity?

45 Problem of incompatible spacetimes Problem 1: gµν & fµν have Lorentzian signature (1, 3). But may not admit compatible 3+1 decompositions (a) (b) (c) Then, no consistent time evolution equations, no Hamiltonian formulation. (d)

46 Nonuniqueness, Reality and Covariance Problem 2: S = g 1 f is not unique, May not be real, covariant To properly define the theory (V (S)): (a) S µ ν needs to be specified uniquely, (b) Restrict g µν, f µν so that S is real, covariant What are the restrictions on g & f? Can they be imposed meaningfully & consistent with dynamics?

47 Outline of the talk Motivation & Problems Gravity with an extra spin-2 field: Bimetric Theory Potential Issues Uniqueness and the local structure of spacetime Discussion

48 Uniqueness and the local structure of spacetime Natural requirements: General Covariance: S µ ν must be a (1,1) tensor S must be real Implication : Problem 2 (reality, uniqueness) has a natural solution. This also solves Problem 1 (compatible spacetimes). [SFH, M. Kocic (arxiv: )]

49 Solution to the uniqueness problem of V (S) Matrix square roots: Primary roots: Max 16 distinct roots, generic Nonprimary roots: Infinitely many, non-generic (when eigenvalues in different Jordan blocks coincide)

50 Solution to the uniqueness problem of V (S) Matrix square roots: Primary roots: Max 16 distinct roots, generic Nonprimary roots: Infinitely many, non-generic (when eigenvalues in different Jordan blocks coincide) General Covariance: A µ ν = g µρ f ρν is a (1,1) tensor, x µ x µ A Q 1 AQ, for Q µ ν = x µ x ν

51 Uniqueness of S S µ ν = ( A) µ ν : Primary roots: Nonprimary roots: A Q 1 AQ = Q 1 A Q Q 1 AQ Q 1 A Q Step 1: General covariance only primary roots are allowed. A Consequence: Examples of backgrounds with local CTC s correspond to nonprimary roots and are excluded

52 Uniqueness of S S µ ν = ( A) µ ν : Primary roots: Nonprimary roots: A Q 1 AQ = Q 1 A Q Q 1 AQ Q 1 A Q Step 1: General covariance only primary roots are allowed. A Consequence: Examples of backgrounds with local CTC s correspond to nonprimary roots and are excluded Step 2: Only the principal root is always primary. Hence, S must be a principal root. (Nonprincipal roots degenerate to nonprimary roots when some eigenvalues coincide).

53 Reality of S Real S = g 1 f simple classification of allowed g, f configurations [SFH, M. Kocic (arxiv: )]

54 Reality of S Real S = g 1 f simple classification of allowed g, f configurations [SFH, M. Kocic (arxiv: )] Theorem: Real S = g 1 f exist if and only if the null cones of g and f (i) intersect in open sets, or, (ii) have no common space nor common time directions (Type IV) Type I Type IIa Type IIb Type III Type IV

55 Reality of S Real S = g 1 f simple classification of allowed g, f configurations [SFH, M. Kocic (arxiv: )] Theorem: Real S = g 1 f exist if and only if the null cones of g and f (i) intersect in open sets, or, (ii) have no common space nor common time directions (Type IV) Type I Type IIa Type IIb Type III Type IV *Types I-III: proper 3+1 decompositions, primary roots, allowed. *Type IV: only nonprimary real roots! (excluded).

56 Choice of the square root Reality + General Covariance * Real principal square root (unique), * Compatible 3+1 decomposition h µν = g µρ ( g 1 f ) ρ ν h null-cones for the principal root (except for the last one) Useful for choosing good coordinate systems, The specific, existing local CTC s in massive gravity rulled out.

57 Consistency with dynamics Type IV metrics arise as a limit of Type IIb metrics. But in the limit, S has a branch cut. Then, for the principal root, the variation δs is not defined at the cut Eqns. of motion not valid for Type IV. Hence cannot arise dynamically.

58 Consistency with dynamics Type IV metrics arise as a limit of Type IIb metrics. But in the limit, S has a branch cut. Then, for the principal root, the variation δs is not defined at the cut Eqns. of motion not valid for Type IV. Hence cannot arise dynamically. A simple mechanical example: A = dt (ẋ 2 /2 λ ) x 2, x 2 = x (1) ẍ = λ(x > 0), ẍ = λ(x < 0) (2) (no equation at x = 0) Implication for some acausality arguments (CTC s) in massive gravity

59 Outline of the talk Motivation & Problems Gravity with an extra spin-2 field: Bimetric Theory Potential Issues Uniqueness and the local structure of spacetime Discussion

60 Discussion The beginning of understanding theories of spin-2 fields beyond General Relativity.

61 Discussion The beginning of understanding theories of spin-2 fields beyond General Relativity. Superluminality? (yes, but not necessarily harmful, replacement for inflation?) Causality? (needs to be investigated further) Energy conditions, global bi-hyperbolicity A more fundamental formulation Application to cosmology, blackholes, GW, etc. Extra symmetries Modified kinetic terms? much less understood. [SFH, Apolo (2016)]

62 Thank you!

63 The Hojman-Kuchar-Teitelboim Metric General Relativity in 3+1 decomposition (g µν : γ ij, N, N i ): gr π ij t γ ij NR 0 N i R i Constraints: R 0 = 0, R i = 0. Algebra of General Coordinate Transformations (GCT): { R 0 (x), R 0 (y) } [ ] = R i (x) δ 3 (x y) R i (y) δ 3 (x y) x i y i { R 0 (x), R i (y) } = R 0 (y) δ 3 (x y) x i { Ri (x), R j (y) } [ ] = R j (x) δ 3 (x y) R x i i (y) δ 3 (x y) y j R i = γ ij R j, γ ij : metric of spatial 3-surfaces. Any generally covariant theory contains such an algebra. HKT: The tensor that lowers the index on R i is the physical metric of 3-surfaces.

64 The HKT metric in bimetric theory Consider g µν = (γ ij, N, N i ) and f µν = (φ ij, L, L i ), L g,f π ij γ ij + p ij φ ij M R 0 M i Ri On the surface of second class Constraints. GCT Algebra: { R0 (x), R 0 (y) } [ = Ri (x) δ 3 (x y) R ] i (y) δ 3 (x y) x i y i { R0 (x), R i (y) } = R 0 (y) δ 3 (x y) x i R i = φ ij Rj, φ ij : the 3-metric of f µν, or R i = γ ij Rj, γ ij : the 3-metric of g µν. The HKT metric of bimetric theory is g µν or f µν, consistent with ghost-free matter couplings [SFH, A. Lundkvist (to appear)]