Taking Limits of General Relativity Eric Bergshoeff Groningen - PowerPoint PPT Presentation

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Taking Limits of General Relativity Eric Bergshoeff Groningen University 11th Nordic String Theory Meeting 2017 Hannover, February 9 2017

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : why non-relativistic gravity ?

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : The Holographic Principle Gravity is not only used to describe the gravitational force!

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Effective Field Theory Examples: liquid helium, cold atomic gases and quantum Hall fluids Effective Field Theory (EFT) coupled to NC gravity ⇒ universal features compare to Coriolis force Greiter, Wilczek, Witten (1989), Son (2005, 2012), Can, Laskin, Wiegmann (2014), Jensen (2014), Gromov, Abanov (2015)

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Supersymmetry supersymmetry allows to apply powerful localization techniques to exactly calculate partition functions of (non-relativistic) supersymmetric field theories Pestun (2007); Festuccia, Seiberg (2011), This should also apply to the non-relativistic case !

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Non-relativistic Gravity • Free-falling frames: Galilean symmetries • Earth-based frame: Newtonian gravity/Newton potential Φ( x ) • no frame-independent formulation (needs geometry!)

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : General Frames • { τ µ , e µ a } a = 1 , 2 , 3; µ = 0 , 1 , 2 , 3 • { τ µ , e µ a } and m µ { τ µ , e µ a } • 3D: and m µ , s µ zero torsion : ∂ µ τ ν − ∂ ν τ µ = 0 → τ µ = ∂ µ τ τ ( x ) = t → τ µ = δ µ, 0

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Take Home Message Taking the non-relativistic limit is non-trivial and not unique !

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Outline { τ µ , e µ a } Galilei Gravity :

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Outline { τ µ , e µ a } Galilei Gravity : { τ µ , e µ a ; m µ } Newton-Cartan Gravity :

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Outline { τ µ , e µ a } Galilei Gravity : { τ µ , e µ a ; m µ } Newton-Cartan Gravity : { τ µ , e µ a ; m µ , s µ } 3D Extended Bargmann Gravity :

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Outline { τ µ , e µ a } Galilei Gravity : { τ µ , e µ a ; m µ } Newton-Cartan Gravity : { τ µ , e µ a ; m µ , s µ } 3D Extended Bargmann Gravity : Final Remarks

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Outline { τ µ , e µ a } Galilei Gravity : { τ µ , e µ a ; m µ } Newton-Cartan Gravity : { τ µ , e µ a ; m µ , s µ } 3D Extended Bargmann Gravity : Final Remarks

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Galilei 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 : [ 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 , [ J ab , J cd ] = δ c [ a J b ] d − δ a [ c J d ] b , a = 1 , 2 , 3

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : ‘Gaugings’, Contractions and Non-relativistic Limits ‘gauging’ Poincare = ⇒ General relativity ⇓ ⇓ contraction non-relativistic limit ‘gauging’ Galilei = ⇒ Galilei Gravity

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : In¨ on¨ u Wigner Contraction � � � � = 2 η A [ B P C ] , = 4 η [ A [ C M D ] B ] P A , M BC M AB , M CD 1 P 0 = 2 ω H , P a = P a , A = (0 , a ) M ab = J ab , M a 0 = ω G a Taking the limit ω → ∞ gives the Galilei algebra: � � P a , G b = 0

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : The Galilei Limit Our starting point is the Einstein-Hilbert action in first-order formalism: 1 � B R µν AB ( M ) S = − EE µ A E ν 16 π G N E 0 Ω 0 a = ω − 1 ω a µ = ωτ µ , µ , G N = ω G G ⇒ µ 1 � b R µν ab ( J ) S Gal = − e e µ a e ν 16 π G G accidental local scaling symmetry e µ a → λ ( x ) e µ a τ µ → λ ( x ) − ( D − 3) τ µ ,

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Constrained Geometry For D > 3 the e.o.m. for ω µ ab can be used to solve for ω µ ab ω µ ab = τ µ A ab + e µ c ω abc ( e , τ ) except for an antisymmetric tensor component A ab = − A ba of ω µ ab Furthermore, the e.o.m. lead to the following restriction on the geometry: τ ab ≡ e µ a e ν b ∂ [ µ τ ν ] = 0 : ( e µ a τ µ = 0) twistless torsion Using a second-order formalism the field A ab acts as a Lagrange multiplier enforcing the constraint τ ab = 0

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Carroll versus Galilei Gravity Gomis, Rollier, Rosseel, ter Veldhuis + E.B. (2017) Carroll gravity is the ultra-relativistic limit of Einstein gravity The Carroll algebra is similar to but not the same as the Galilei algebra • The Carroll action contains both a R µν ab ( J ) and a R µν a ( G ) term • Symmetric Lagrange multiplier S ( ab ) and constraint K ( ab ) = 0 • relation with strong coupling limit of Henneaux ? Henneaux (1979)

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Outline { τ µ , e µ a } Galilei Gravity : { τ µ , e µ a ; m µ } Newton-Cartan Gravity : { τ µ , e µ a ; m µ , s µ } 3D Extended Bargmann Gravity : Final Remarks

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : Bargmann Symmetries x i ˙ x j δ ij S non-relativistic ( massive ) = m � ˙ i = 1 , 2 , 3 d τ ˙ 2 t x i = λ i ˙ Lagrangian is not invariant under Galilean boosts δ ˙ t : δ L non-relativistic ( massive ) = d d τ ( mx i λ j δ ij ) ⇒ modified Noether charge gives rise to central extension: � � P a , G b = δ ab Z

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : ‘Gaugings’, Contractions and Non-relativistic Limits ‘gauging’ Poincare ⊗ U(1) = ⇒ GR plus ∂ µ M ν − ∂ ν M µ = 0 ⇓ ⇓ contraction non-relativistic limit ‘gauging’ Bargmann = ⇒ Newton-Cartan gravity

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : In¨ on¨ u Wigner Contraction � � � � = 2 η A [ B P C ] , = 4 η [ A [ C M D ] B ] Z P A , M BC M AB , M CD plus 1 Z = 1 P 0 = 2 ω H + ω Z , 2 ω H − ω Z , A = (0 , a ) P a = P a , M ab = J ab , M a 0 = ω G a Taking the limit ω → ∞ gives the Bargmann algebra including Z: � � P a , G b = δ ab Z

{ τ µ , e µ a } { τ µ , e µ a ; m µ } { τ µ , e µ a ; m µ , s µ Galilei Gravity : Newton-Cartan Gravity : 3D Extended Bargmann Gravity : The Newton-Cartan Limit I Dautcourt (1964) express relativistic fields { E µ A , M µ } in terms of non-relativistic STEP I: fields { τ µ , e µ a , m µ } E µ 0 = ω τ µ + 1 2 ω m µ , M µ = ω τ µ − 1 2 ω m µ , E µ a = e µ a ⇒ 1 � ω − 4 � E µ a = e µ a − 2 ω 2 τ µ e ρ a m ρ + O and similar for E µ 0