NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions 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 27 2016
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions ´ Einstein (1905/1915) Elie Cartan (1923) Einstein achieved two things in 1915: • He made gravity consistent with special relativity • He used an arbitrary coordinate frame formulation
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions 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
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions why non-relativistic gravity ?
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Motivation • gauge-gravity duality Liu, Schalm, Sun, Zaanen, Holographic Duality in Condensed Matter Physics (2015) Christensen, Hartong, Kiritsis Obers and Rollier (2013-2015) • condensed matter physics Son (2013), Can, Laskin, Wiegmann (2014), Gromov, Abanov (2015) • Hoˇ rava-Lifshitz gravity, flat-space holography, etc. Hoˇ rava (2009); Hartong, Obers (2015); Duval, Gibbons, Horvathy, Zhang (2014) • non-relativistic strings/branes Gomis, Ooguri (2000); Gomis, Kamimura, Townsend (2004)
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions 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
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Outline NC Gravity from gauging Bargmann
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Outline NC Gravity from gauging Bargmann The Schr¨ odinger Method
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Outline NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Outline NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Outline NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Einstein Gravity In the relativistic case free-falling frames are connected by the Poincare symmetries: δ x µ = ξ µ • space-time translations : δ x µ = λ µν x ν • Lorentz transformations : 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
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Non-relativistic Gravity In the non-relativistic case free-falling frames are connected by the Galilean symmetries: δ t = ξ 0 • time translations : δ x i = ξ i • space translations : i = 1 , 2 , 3 δ x i = λ i j x j • spatial rotations : δ x i = λ i t • Galilean boosts : In free-falling frames there is no gravitational force
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Newtonian gravity versus Newton-Cartan gravity • in frames with constant acceleration ( δ x i = 1 2 a i 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
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions 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) !
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Gauging the Bargmann algebra cp. to Chamseddine and West (1977) [ J ab , P c ] = − 2 δ c [ a P b ] , [ J ab , G c ] = − 2 δ 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 ) P a e µ a ζ a ( x ν ) R µν a ( P ) space translations G a ω µ a λ a ( x ν ) R µν a ( G ) Galilean boosts J ab ω µ ab λ ab ( x ν ) R µν ab ( J ) spatial rotations central charge transf. σ ( x ν ) R µν ( Z ) Z m µ
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions 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
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions The Final Result The independent NC fields { τ µ , e µ a , m µ } transform as follows: δτ µ = 0 , δ e µ a = λ ab e µ b + λ a τ µ , a δ m µ = ∂ µ σ + λ a e µ The spin-connection fields ω µ ab and ω µ a are functions of e , τ and m There are two Galilean-invariant metrics: h µν = e µ a e ν b δ ab τ µν = τ µ τ ν ,
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions The NC Equations of Motion Taking the non-relativistic limit of the Einstein equations ⇒ Rosseel, Zojer + E.B. (2015) τ µ e ν a R µν a ( G ) = 0 e ν a R 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
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Outline NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions The Relativistic Conformal Method Conformal = Poincare + D (dilatations) + K µ (special conf. transf.) conformal gravity ≡ gauging of conformal algebra a , a = f µ δ b µ = Λ a a ( e , ω, b ) K ( x ) e µ f µ Poincare invariant ⇔ CFT of real scalar
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions An example e − 1 L = 1 P : κ 2 R STEP 1 A ) P = κ 2 D − 2 ϕ ( e µ A ) C ( e µ with A ) C = Λ D ( e µ A ) C δϕ = − Λ D ϕ , δ ( e µ STEP 2 A ) C = δ µ ∂ µ ξ ν + Λ νµ + Λ D δ µ ν = 0 A ( e µ ⇒ 2 D − 2 , make redefinition ϕ = φ D > 2 ⇒ L = 4 D − 1 δφ = ξ µ ∂ µ φ − 1 CFT : D − 2 φ � φ with 2 ( D − 2)Λ D φ
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions from CFT back to P CFT : L ∼ φ � φ δφ = ξ µ ∂ µ φ + w Λ D φ STEP 1 replace derivatives by conformal-covariant derivatives ⇒ e − 1 L = 4 D − 1 D − 2 φ � C φ gauge-fix dilatations by imposing φ = 1 STEP 2 ⇒ κ e − 1 L = 1 P : κ 2 R
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions Three Different Invariants e − 1 L = R 1. Kinetic terms Example : L ∼ φ � φ ⇔ includes all CFT’s with time derivatives 2. Potential terms Example : cosmological constant ( κ = 1) L = Λ φ 2 , e − 1 L = Λ w = − D ⇔ 2 3. Curvature terms Example : Weyl tensor squared e − 1 L ∼ φ 2 D − 4 AB � 2 D − 2 � C µν D ≥ 4
NC Gravity from gauging Bargmann The Schr¨ odinger Method NC Gravity with Torsion Future Directions The Schr¨ odinger Method The contraction of the conformal Algebra is the Galilean Conformal Algebra (GCA) which has no central extension ! z = 2 Schr¨ odinger = Bargmann + D (dilatations) + K (special conf.) [ H , D ] = zH , [ P a , D ] = P a z = 1 : conformal algebra , z � = 2 : no special conf. transf.
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