Running at Non-relativistic Speed

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1 Running at Non-relativistic Speed Eric Bergshoeff Groningen University Symmetries in Particles and Strings A Conference to celebrate the 70th birthday of Quim Gomis Barcelona, September

2 Why always running fast?

3 why non-relativistic gravity? Newton-Cartan (NC) gravity is Newtonian gravity in arbitrary frame Cartan (1923)

4 non-relativistic strings/branes Motivation Gomis, Ooguri (2000); Gomis, Kamimura, Townsend (2004) condensed matter physics Son et al. ( ) gauge-gravity duality Christensen, Hartong, Kiritsis Obers and Rollier ( ) NC supergravity Pestun (2007); Festuccia, Seiberg (2011) Hořava-Lifshitz gravity Hořava (2009); Hartong, Obers (2015)

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

6 Outline NC Gravity from gauging Bargmann

7 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity

8 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity NC Gravity and a NR CFT

9 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity NC Gravity and a NR CFT Future Directions

10 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity NC Gravity and a NR CFT Future Directions

11 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

12 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

13 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

14 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)!

15 Gauging the Bargmann algebra [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)

16 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 (flat space) : optional R 0(a,b) (G) 0 : only non-zero components left

17 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

18 The NC Equation of Motion Case I: R ab µν (J) = 0 τ µ e ν ar a µν (G) = 0 invariant under local central charge transformations Case II: R µν ab (J) 0 τ µ( ) R a µa (G)+2M b a R µa b (J)+τ µ M b M c a R ba c (J) = 0 invariant under rigid central charge transformations after gauge-fixing NC e.o.m. becomes Φ = 0 Note: M a m a note: there is no action that gives rise to this equation of motion

19 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity NC Gravity and a NR CFT Future Directions

20 Can we obtain NC Gravity by taking a non-relativistic limit of Einstein gravity?

21 The Newtonian Limit R µν 1 2 g µνr = κt µν R µν = κ ( T µν 1 d 2 g µνt ) R 00 = κ d 3 d 2 T asumption: weak and slowly varying gravitational field Φ = κ d 3 d 2 ρ taking a non-relativistic limit and using an arbitrary frame are two different things!

22 Inönü Wigner Contraction [ PA,M BC ] = 2ηA[B P C], [ MAB,M CD ] = 4η[A[C M D]B] plus Z P 0 = 1 2ω H +ωz, Z = 1 2ω H ωz, P a = P a, M ab = J ab, M a0 = ωg a Taking the limit ω gives the Bargmann algebra: [ Pa,J bc ] = 2δa[b P c], [ Jab,J cd ] = 4δ[a[c J d]b], [ Ga,J bc ] = 2δa[b G c], [ H,Ga ] = Pa, [ Pa,G b ] = δab Z

23 Relativistic Gravity U(1) The independent fields {E µ A,M µ } of Poincare U(1) transform as follows: δe µ A = ξ ρ ρ E µ A +E ρ A µ ξ ρ +λ A BE µ B, δm µ = ξ ρ ρ M µ + µ ξ ρ M ρ + µ Λ, A = 0,1,2,...,d R µν A (E) = 2 [µ E ν] A 2Ω [µ A B E ν] B = 0 Ω µ AB = Ω µ AB (E) F µν (M) = 2 [µ M ν]

24 Taking the Limit Jensen, Karch (2014); Rosseel, Zojer + E.B. (2015) Inspired by the contraction of the Poincare U(1) to the Bargmann algebra we introduce a parameter ω and define E µ A = δ A 0 ( ωτµ + 1 2ω m ) µ +δ A a e a µ, M µ = ωτ µ 1 2ω m µ, where in the limit ω the fields {τ µ,e µ a,m µ } become the NC fields This indeed yields the transformation rules of NC gravity!

25 A subtlety one must take limit in δδφ and δφ Rosseel, Zojer + E.B. (2015) Some of the NC constraints follow directly by taking the non-relativistic limit of the Einstein constraints The remaining ones follow from the requirement that the limit of the transformation rules is well-defined orbit of constraints must be finite

26 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity NC Gravity and a NR CFT Future Directions

27 The Relativistic Conformal Tensor Calculus 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) rigid conformal symmetries follows from δe µ a = ξ λ (x) λ e µ a + µ ξ λ (x)e λ a +Λ ab (x)e µ b +Λ D (x)e µ a = 0 ξ µ (x) = a µ λ µν x ν λ D x µ +λ ν ( K x µ x ν 1 2 δµ ν x 2)

28 Poincare invariant CFT of real scalar Poincare gravity = conformal gravity + compensating scalar φ L = 1 2 φ µ µ φ, with δφ = wλ D φ L = 1 2 φ(da ) c (D a ) c φ φ=1,bµ=0 L = R L = R and (e µ a ) P = φ(e µ a ) C plus (e µ a ) C = δ µ a L = 1 2 φ µ µ φ

29 A Non-relativistic Conformal Tensor Calculus 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.

30 Schrödinger Gravity Hartong, Rosseel + E.B. (2014) Gauging the z = 2 Schrödinger algebra we find that the independent gauge fields {τ µ,e a µ,m µ } transform as follows: δτ µ = 2Λ D τ µ, δe a µ = Λ a be b µ +Λ a τ µ +Λ D e a µ, a δm µ = µ σ +Λ a e µ The connection fields ω µ ab,ω µ a and the K-gauge field f µ are dependent The time projection τ µ b µ of b µ transforms under K as a a shift while the spatial projection e µ a b µ is dependent: R 0a (H) = 0 is conventional constraint R µν (H) = [µ τ ν] b [µ τ ν] = 0 b a represents (twistless) torsion!

31 Rigid Schrödinger Transformations δτ µ = µ ξ 0 (x)+2λ D (x)δ µ0 = 0, δe µ a = µ ξ a (x)+λ ab (x)δ µ b +Λ a (x)δ µ0 +Λ D (x)δ µ a = 0 ξ 0 (x) = b 2λ D t +λ K t 2, ξ a (x) = b a λ ab x b λ a t λ D x a +λ K tx a

32 NC Gravity and a NR CFT Case I: R µν ab (J) = 0 and b a = 0 (zero torsion) τ µ e ν ar µν a (G) = 0 and [µ τ ν] = φ = 0 and a φ = 0 w = 1 Case II: R µν ab (J) 0 and b a = 0 (zero torsion) τ µ( R µa a (G) NC +2M b R µa a b (J) NC +τ µ M b M c R ba a c (J) NC) = φ = 0 and a φ = 0 second compensating scalar χ for central charge transfs. occurs via M µ m µ µ χ

33 NC Gravity with Torsion Case III: R µν ab (J) 0 and b a 0 (torsion) Use the second compensating scalar χ to restore the invariance under Galilean boosts: δ(λ a ) a χ Mλ a 0 0 ϕ 2 M ( 0 a ϕ) a χ+ 1 M 2( a b ϕ) a χ b χ = 0 τ µ f µ b a τ µ ω a µ 2M a τ µ D µ b a M a M b D a b b = 0

34 Outline NC Gravity from gauging Bargmann NC Gravity as a NR Limit of Einstein Gravity NC Gravity and a NR CFT Future Directions

35 Outlook 4D Newton-Cartan supergravity? supersymmetric Hořava-Lifshitz non-relativistic superspace techniques Applications!

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