A new look at Newton-Cartan gravity

Transcription

1 A new look at Newton-Cartan gravity Eric Bergshoeff Groningen University Memorial Meeting for Nobel Laureate Professor Abdus Salam s 90th Birthday NTU, Singapore, January

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3 Einstein (1905/1915) Élie Cartan (1923) Einstein achieved two things in 1915: He made gravity consistent with special relativity He used an arbitrary coordinate frame formulation

4 Geometry Riemann (1867) Einstein used Riemannian geometry General relativity Cartan used NC geometry NC gravity Newton-Cartan (NC) gravity is Newtonian gravity in arbitrary frame

5 why non-relativistic gravity?

6 gauge-gravity duality Motivation Liu, Schalm, Sun, Zaanen, Holographic Duality in Condensed Matter Physics (2015) Christensen, Hartong, Kiritsis Obers and Rollier ( ) condensed matter physics Son (2013), Can, Laskin, Wiegmann (2014), Gromov, Abanov (2015) Hořava-Lifshitz gravity, flat-space holography, etc. Hořava (2009); Hartong, Obers (2015); Duval, Gibbons, Horvathy, Zhang (2014) non-relativistic strings/branes Gomis, Ooguri (2000); Gomis, Kamimura, Townsend (2004)

7 How do we construct (Non-)relativistic Gravity? (1) gauging a (non-)relativistic algebra (2) taking a non-relativistic limit (3) using a nonrelativistic version of the conformal tensor calculus

8 Outline NC Gravity from gauging Bargmann

9 Outline NC Gravity from gauging Bargmann The Schrödinger Method

10 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion

11 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions

12 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions

13 Einstein Gravity In the relativistic case free-falling frames are connected by the Poincare symmetries: space-time translations: δx µ = ξ µ Lorentz transformations: δx µ = λ µ ν x ν In free-falling frames there is no gravitational force in arbitrary frames the gravitational force is described by an invertable Vierbein field e µ A (x) µ = 0,1,2,3; A=0,1,2,3

14 Non-relativistic Gravity In the non-relativistic case free-falling frames are connected by the Galilean symmetries: time translations: δt = ξ 0 space translations: δx i = ξ i i = 1,2,3 spatial rotations: Galilean boosts: δx i = λ i j x j δx i = λ i t In free-falling frames there is no gravitational force

15 Newtonian gravity versus Newton-Cartan gravity in frames with constant acceleration (δx i = 1 2 ai t 2 ) the gravitational force is described by the Newton potential Φ( x) Newtonian gravity in arbitrary frames the gravitational force is described by a temporal Vierbein τ µ (x), spatial Vierbein e µ a (x) plus a vector m µ (x) µ = 0,1,2,3; a=1,2,3 Newton-Cartan (NC) gravity

16 The Galilei Algebra versus the Bargmann algebra Einstein gravity follows from gauging the Poincare algebra The Galilei algebra is the contraction of the Poincare algebra does NC gravity follow from gauging the Galilei algebra? Can NC gravity be obtained by taking the non-relativistic limit of Einstein gravity? No! one needs Bargmann instead of Galilei and Poincare U(1)!

17 Gauging the Bargmann algebra cp. to Chamseddine and West (1977) [J ab,p c ] = 2δ c[a P b], [J ab,g c ] = 2δ c[a G b], [G a,h] = P a, [G a,p b ] = δ ab Z, a = 1,2,...,d symmetry generators gauge field parameters curvatures time translations H τ µ ζ(x ν ) R µν (H) space translations P a a e µ ζ a (x ν ) R a µν (P) Galilean boosts G a a ω µ λ a (x ν ) R a µν (G) spatial rotations J ab ab ω µ λ ab (x ν ) R ab µν (J) central charge transf. Z m µ σ(x ν ) R µν (Z)

18 Imposing Constraints R µν a (P) = 0, R µν (Z) = 0 : solve for spin-connection fields R µν (H) = [µ τ ν] = 0 τ µ = µ τ : foliation of Newtonian spacetime ( zero torsion ) R µν ab (J) 0 : restriction on-shell R 0(a,b) (G) 0 : Poisson equation on-shell

19 The Final Result The independent NC fields {τ µ,e µ a,m µ } transform as follows: δτ µ = 0, δe µ a = λ a be µ b +λ a τ µ, δm µ = µ σ +λ a e µ a The spin-connection fields ω µ ab and ω µ a are functions of e,τ and m There are two Galilean-invariant metrics: τ µν = τ µ τ ν, h µν = e µ ae ν bδ ab

20 The NC Equations of Motion Taking the non-relativistic limit of the Einstein equations Rosseel, Zojer + E.B. (2015) τ µ e ν ar µν a (G) = 0 e ν ar µν ab (J) = 0 after gauge-fixing and assuming flat space the first NC e.o.m. becomes Φ = 0 note: there is no action that gives rise to these equations of motion

21 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions

22 The Relativistic Conformal Method Conformal = Poincare + D (dilatations) + K µ (special conf. transf.) conformal gravity gauging of conformal algebra δb µ = Λ a K(x)e µ a, f µ a = f µ a (e,ω,b) Poincare invariant CFT of real scalar

23 An example P : e 1 L = 1 κ 2 R STEP 1 STEP 2 (e µ A ) P = κ 2 D 2 ϕ(eµ A ) C δϕ = Λ D ϕ, with δ(e µ A ) C = Λ D (e µ A ) C (e µ A ) C = δ µ A µ ξ ν +Λ νµ +Λ D δ µ ν = 0 make redefinition ϕ = φ 2 D 2, D > 2 CFT : L = 4 D 1 D 2 φ φ with δφ = ξµ µ φ 1 2 (D 2)Λ Dφ

24 from CFT back to P CFT : L φ φ δφ = ξ µ µ φ+wλ D φ STEP 1 replace derivatives by conformal-covariant derivatives e 1 L = 4 D 1 D 2 φ C φ STEP 2 gauge-fix dilatations by imposing φ = 1 κ P : e 1 L = 1 κ 2 R

25 Three Different Invariants 1. Kinetic terms Example: L φ φ e 1 L = R includes all CFT s with time derivatives 2. Potential terms Example: cosmological constant (κ = 1) e 1 L = Λ L = Λφ 2, w = D 2 3. Curvature terms Example: Weyl tensor squared e 1 L φ 2D 4 D 2 ( C µν AB ) 2 D 4

26 The Schrödinger Method The contraction of the conformal Algebra is the Galilean Conformal Algebra (GCA) which has no central extension! z = 2 Schrödinger = Bargmann + D (dilatations) + K (special conf.) [H,D] = zh, [P a,d] = P a z = 1: conformal algebra, z 2 : no special conf. transf.

27 Schrödinger Gravity Hartong, Rosseel + E.B. (2014) Gauging the z = 2 Schrödinger algebra we find that the independent gauge fields {τ µ,e µ a,m µ } transform as follows: δτ µ = 2Λ D τ µ, δe µ a = Λ a be µ b +Λ a τ µ +Λ D e µ a, δm µ = µ σ +Λ a e µ a The time projection τ µ b µ of b µ transforms under K as a a shift while the spatial projection b a e a µ b µ is dependent b a (e,τ) represents (twistless) torsion!

28 SFT s versus Galilean Invariants the Schrödinger action for a complex scalar Ψ with weights (w,m) SFT : S = dtd d xψ ( i 0 1 2M a a )Ψ is invariant under the rigid Schrödinger transformations δψ = ( b 2λ D t +λ K t 2) 0 Ψ+ ( b a λ ab x b λ a t λ D x a +λ K tx a) a Ψ +w ( λ D λ K t ) Ψ+iM ( σ λ a x a λ Kx 2) Ψ for w(ψ) = d/2 The corresponding Galilean invariant G has inconsistent E.O.M. s

29 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions

30 Case 1: zero torsion: b a = 0 Schrödinger method also works at level of E.O.M. s foliation constraint : µ (τ ν ) G ν (τ µ ) G = 0, Gal E.O.M. : (τ µ ) G (e ν a) G R µν a (G) = 0, (e ν a) G R µν ab (J) = 0. Schrödinger method leads to (Ψ = ϕe iχ ) SFT : 0 0 ϕ = 0 and a ϕ = 0 with w = 1

31 Case 2: twistless torsion: b a 0 foliation constraint is conformal invariant use the second compensating scalar χ to restore Schrödinger invariance: 0 0 ϕ 2 M ( 0 a ϕ) a χ+ 1 M 2( a b ϕ) a χ b χ = 0 Φ+ ˆτ µ µ K +K ab K ab 8Φb b 2ΦD b 6b a D a Φ = 0 plus e ν ar µν ab (J) = 0 Afshar, Mehra, Parekh, Rollier + E.B. (2015)

32 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions

33 New developments and Extensions relation to Hořava-Lifshitz gravity Hartong and Obers (2015) Afshar, Mehra, Parekh, Rollier + E.B. (2015) extension to z 2 and Galilean conformal symmetries matter-coupled NC gravity non-relativistic supergravity localization techniques Andringa, Rosseel, Sezgin + E.B. (2013) Knodel, Lisbao, Liu (2015)