A Schrödinger approach to Newton-Cartan gravity Eric Bergshoeff Groningen University Miami 2015 A topical conference on elementary particles, astrophysics, and cosmology Fort Lauderdale, December 21 2015
why non-relativistic gravity? Newton-Cartan (NC) gravity is Newtonian gravity in arbitrary frame Cartan (1923)
Motivation condensed matter physics Son et al. (2011-2015) gauge-gravity duality Christensen, Hartong, Kiritsis Obers and Rollier (2013-2015) Hořava-Lifshitz gravity Hořava (2009); Hartong, Obers (2015) non-relativistic strings/branes Gomis, Ooguri (2000); Gomis, Kamimura, Townsend (2004)
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
Outline NC Gravity from gauging Bargmann
Outline NC Gravity from gauging Bargmann The Schrödinger Method
Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion
Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
Outline NC Gravity from gauging Bargmann The Schrödinger Method 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 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
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
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
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)!
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)
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
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
The NC Equations of Motion Taking the non-relativistic limit of the Einstein equations 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
Outline NC Gravity from gauging Bargmann The Schrödinger 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 δb µ = Λ a K(x)e µ a, f µ a = f µ a (e,ω,b) Poincare invariant CFT of real scalar
An example P 1 : 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 1 : L = 4 D 1 D 2 φ φ with δφ = ξµ µ φ 1 2 (D 2)Λ Dφ
from CFT 1 back to P 1 CFT 1 : L φ φ or L µ φ µ φ with δφ = ξ µ µ φ+wλ D φ STEP 1 require that δl = µ (ξ µ L) and replace derivatives by conformal-covariant derivatives LCFT 1 : e 1 L = 4 D 1 D 2 φ C φ STEP 2 gauge-fix dilatations by imposing φ = 1 κ P 1 : e 1 L = 1 κ 2 R
Three Different Invariants 1. Potential terms Example: cosmological constant (κ = 1) P 0 : e 1 L = Λ CFT 0 : L = Λφ 2, w = D 2 2. Kinetic terms Example: L φ φ e 1 L = R includes all CFT s with time derivatives 3. Curvature terms Example: Weyl tensor squared e 1 L φ 2D 4 D 2 ( C µν AB ) 2 D 4
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) represents (twistless) torsion!
Schrödinger Field Theories (SFT s) 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
Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
Case 1: zero torsion: b a = 0 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 E.O.M. : 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), to appear
Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
Outlook matter-coupled NC gravity extension to z 2 and Galilean conformal symmetries relation to Hořava-Lifshitz gravity Hartong and Obers (2015) Afshar, Mehra, Parekh, Rollier + E.B. (2015), to appear non-relativistic supergravity localization techniques Knodel, Lisbao, Liu (2015)