Applied Newton-Cartan Geometry: A Pedagogical Review Eric Bergshoeff Groningen University 10th Nordic String Meeting Bremen, March 15 2016
Newtonian Gravity Free-falling frames: Galilean symmetries Earth-based frame: Newton potential no frame-independent formulation
Geometry Riemann (1867) for a frame-independent formulation you need geometry
Einstein 1915 Einstein achieved two things in 1915: He made gravity consistent with special relativity He used a frame-independent formulation
General Relativity Free-falling frames: Poincare symmetries arbitrary frames: metric tensor field
Newton-Cartan Gravity Elie Cartan 1923 Newton-Cartan (NC) gravity is Newtonian gravity in arbitrary frame NC gravity contains more gravitational fields than Newton potential Newtonian gravity is re-obtained by gauge-fixing
Newton-Cartan Geometry NC geometry is foliated with an absolute time
why non-relativistic gravity?
Gauge-Gravity Duality In special cases one finds NC gravity with twistless torsion Christensen, Hartong, Kiritsis, Obers and Rollier (2013-2015)
Condensed Matter Physics Effective Field Theory (EFT) coupled to NC gravity universal features Examples: Quantum Hall Effect, non-relativistic hydrodynamics compare to Coriolis force Son (2005, 2012), Can, Laskin, Wiegmann (2014), Jensen (2014), Gromov, Abanov (2015)
Hořava-Lifshitz gravity Other Motivations Hořava (2009); Hartong, Obers (2015) duality with Carroll symmetry Duval, Gibbons, Horvathy, Zhang (2014); Hartong (2015) flat-space holography Duval, Gibbons, Horvathy (2014) (non-relativistic) higher spins, entanglement entropy, etc. Leigh, Parrikar, Weiss (2014); Bagchi, Basu, Grumiller, Riegler (2015) non-relativistic strings/branes Gomis, Ooguri (2000); Gomis, Kamimura, Townsend (2004)
Outline Gauging Bargmann
Outline Gauging Bargmann NC Gravity with Torsion
Outline Gauging Bargmann NC Gravity with Torsion Future Directions
Outline Gauging Bargmann NC Gravity with Torsion Future Directions
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 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
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
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
Relativistic versus Non-relativistic Particle S relativistic = m dτ η µν ẋ µ ẋ ν µ = 0,1,2,3 Lagrangian is invariant under Poincare symmetries S non-relativistic = m 2 ẋiẋj δ ij dτ i = 1,2,3 ṫ Lagrangian is not invariant under Galilean boosts: δl non-relativistic = d dτ (mxi λ j δ ij ) modified Noether charge gives rise to central extension!
Gaugings, Contractions and Non-relativistic Limits Poincare U(1) gauging = General relativity U(1) contraction non-relativistic limit Bargmann gauging = Newton-Cartan gravity
Gauging the Bargmann algebra Andringa, Panda, de Roo + E.B. (2011); 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)
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 : absolute time ( zero torsion ) R µν ab (J) 0 : un-constrained off-shell R 0(a,b) (G) 0 : un-constrained off-shell
The Transformation Rules The independent NC fields {τ µ,e µ a,m µ } transform as follows: δτ µ = ξ λ λ τ µ + µ ξ λ τ λ, δe µ a = ξ λ λ e µ a + µ ξ λ e λ a +λ a be µ b +λ a τ µ, δm µ = ξ λ λ m µ + µ ξ λ m λ + 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
from Newton-Cartan back to Flat background flat background: τ µ = δ t µ, e t a = 0, e i a = δ a i, m µ = 0 Non-relativistic Killing equations µ ξ t = 0, t ξ i +λ i = 0, i ξ j +λ j i = 0, t σ = 0, i σ +λ i = 0 Galilean Symmetries ξ t (x µ ) = ζ, ξ i (x µ ) = ξ i λ i t λ i j x j, σ(x µ ) = σ λ i x i
The NC Equations of Motion The NC equations of motion are given by τ µ e ν ar µν a (G) = 0 1 e ν ar µν ab (J) = 0 a,(ab) 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
Outline Gauging Bargmann NC Gravity with Torsion Future Directions
Non-AdS Holography The simplest deviation from AdS is Lifshitz space-time: Christensen, Hartong, Kiritsis, Obers and Rollier (2013-2015) ds 2 = dt2 r 2z + 1 r 2 ( d 2 +d x 2) with z 1 characterizing the anisotropic scaling t λ z t and x λ x Towards the boundary (r 0) the lightcone flattens out (z > 1) Galilean lightcone at the boundary:
Torsion Torsionless [µ τ ν] = 0 τ µ = µ τ : absolute time Twistless Torsionτ [µ ν τ ρ] = 0 τ µ = ρ µ τ : foliated spacetime Christensen, Hartong, Kiritsis, Obers and Rollier (2013-2015) follows from non-relativistic scale invariance [µ τ ν] 2b [µ τ ν] = 0 with b a (e,τ) = e a µ τ ν [µ τ ν] Torsion τ µ is arbitrary unconstrained sources Gromov, Abanov (2014); Bradly, Read (2015)
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
From Poincare Invariant to CFT 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
from CFT back to Poincare Invariant 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 e 1 L = 1 κ 2 R
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.
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) = e a µ τ ν [µ τ ν] represents (twistless) torsion!
From SFT to Galilean Invariant 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 + 1 2 λ Kx 2) Ψ for w(ψ) = d/2 The corresponding Galilean invariant has inconsistent E.O.M. s
The Schrödinger Method can also be applied to the NC equations of motion that cannot be integrated to an action NC Gravity with Twistless Torsion
Case 1: zero torsion: b a = 0 foliation constraint : µ (τ ν ) G ν (τ µ ) G = 0, 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
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)
Outline Gauging Bargmann NC Gravity with Torsion Future Directions
Applications and Extensions stringy limits Hořava-Lifshitz gravity Hartong and Obers (2015) Afshar, Mehra, Parekh, Rollier + E.B. (2015) Galilean conformal symmetries Supersymmetry Pestun (2007), Festuccia, Seiberg (2011), Knodel, Lisbao, Liu (2015)
March 6-10, 2017: Save the Date! Simons Workshop on Applied Newton-Cartan Geometry organized by Gary Gibbons, Rob Leigh, Djordje Minic, Dam Thanh Son + E.B.