A new look at Newton-Cartan gravity Eric Bergshoeff Groningen - - PowerPoint PPT Presentation

NC Gravity from gauging Bargmann The Schr¨

• dinger 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¨

• dinger Method

NC Gravity with Torsion Future Directions

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

´ Elie Cartan (1923) Einstein (1905/1915)

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¨

• dinger 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¨

• dinger Method

NC Gravity with Torsion Future Directions

why non-relativistic gravity ?

NC Gravity from gauging Bargmann The Schr¨

• dinger 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¨

• dinger 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¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

NC Gravity from gauging Bargmann The Schr¨

• dinger 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 :

δxµ = ξµ

• Lorentz transformations :

δxµ = λµν xν In free-falling frames there is no gravitational force in arbitrary frames the gravitational force is described by an invertable Vierbein field eµA(x) µ = 0, 1, 2, 3; A=0,1,2,3

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Non-relativistic Gravity

In the non-relativistic case free-falling frames are connected by the Galilean symmetries:

• time translations :

δt = ξ0

• space translations :

δxi = ξi i = 1, 2, 3

• spatial rotations :

δxi = λi j xj

• Galilean boosts :

δxi = λit In free-falling frames there is no gravitational force

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Newtonian gravity versus Newton-Cartan gravity

• in frames with constant acceleration (δxi = 1

2ait2) 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 τµ(x), spatial Vierbein eµa(x) plus a vector mµ(x) → µ = 0, 1, 2, 3; a=1,2,3 Newton-Cartan (NC) gravity

NC Gravity from gauging Bargmann The Schr¨

• dinger 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!

• ne needs Bargmann instead of Galilei and Poincare ⊗ U(1) !

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Gauging the Bargmann algebra

• cp. to Chamseddine and West (1977)

[Jab, Pc] = −2δc[aPb] , [Jab, Gc] = −2δc[aGb] , [Ga, H] = −Pa , [Ga, Pb] = −δabZ , a = 1, 2, . . ., d symmetry generators gauge field parameters curvatures time translations H τµ ζ(xν) Rµν(H) space translations Pa eµa ζa(xν) Rµν a(P) Galilean boosts G a ωµa λa(xν) Rµν a(G) spatial rotations Jab ωµab λab(xν) Rµν ab(J) central charge transf. Z mµ σ(xν) Rµν(Z)

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Imposing Constraints

Rµνa(P) = 0 , Rµν(Z) = 0 : solve for spin-connection fields Rµν(H) = ∂[µτν] = 0 → τµ = ∂µ τ : foliation of Newtonian spacetime (‘zero torsion’) Rµνab(J) = 0 : restriction on-shell R0(a,b)(G) = 0 : Poisson equation on-shell

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

The Final Result

The independent NC fields {τµ, eµa, mµ} transform as follows: δτµ = 0 , δeµa = λab eµb + λaτµ , δmµ = ∂µσ + λa eµ

a

The spin-connection fields ωµab and ωµa are functions of e, τ and m There are two Galilean-invariant metrics: τµν = τµτν , hµν = eµaeνb δab

NC Gravity from gauging Bargmann The Schr¨

• dinger 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 µeνaRµν a(G) = eνaRµν ab(J) =

• 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¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

The Relativistic Conformal Method

Conformal = Poincare + D (dilatations) + Kµ (special conf. transf.) conformal gravity ≡ gauging of conformal algebra δbµ = Λa

K(x)eµ a ,

fµ

a = fµ a(e, ω, b)

Poincare invariant ⇔ CFT of real scalar

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

An example

P : e−1L = 1 κ2 R STEP 1 (eµ

A)P = κ

2 D−2 ϕ (eµ

A)C

with δϕ = −ΛDϕ , δ(eµ

A)C = ΛD(eµ A)C

STEP 2 (eµ

A)C = δµ A

⇒ ∂µξν + Λνµ + ΛDδµ

ν = 0

make redefinition ϕ = φ

2 D−2 ,

D > 2 ⇒ CFT : L = 4 D−1

D−2 φφ

with δφ = ξµ∂µ φ − 1

2(D − 2)ΛDφ

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

from CFT back to P

CFT : L ∼ φφ δφ = ξµ∂µ φ + wΛDφ STEP 1 replace derivatives by conformal-covariant derivatives ⇒ e−1L = 4 D−1

D−2 φCφ

STEP 2 gauge-fix dilatations by imposing φ = 1

κ

⇒ P : e−1L = 1 κ2 R

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Three Different Invariants

• 1. Kinetic terms

Example : L ∼ φφ ⇔ e−1L = R includes all CFT’s with time derivatives

• 2. Potential terms

Example : cosmological constant (κ = 1) e−1L = Λ ⇔ L = Λ φ2 , w = − D

2

• 3. Curvature terms

Example : Weyl tensor squared e−1L ∼ φ2 D−4

D−2

Cµν

AB2

D ≥ 4

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

The Schr¨

• dinger Method

The contraction of the conformal Algebra is the Galilean Conformal Algebra (GCA) which has no central extension ! z = 2 Schr¨

• dinger = Bargmann + D (dilatations) + K (special conf.)

[H, D] = zH , [Pa, D] = Pa z = 1 : conformal algebra , z = 2 : no special conf. transf.

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Schr¨

• dinger Gravity

Hartong, Rosseel + E.B. (2014)

Gauging the z = 2 Schr¨

• dinger algebra we find that the independent

gauge fields {τµ, eµa, mµ} transform as follows: δτµ = 2ΛDτµ , δeµa = Λab eµb + Λaτµ + ΛDeµa , δmµ = ∂µσ + Λa eµa The time projection τ µbµ of bµ transforms under K as a a shift while the spatial projection ba ≡ eaµbµ is dependent ⇒ ba (e, τ) represents (twistless) torsion!

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

SFT’s versus Galilean Invariants

the Schr¨

• dinger action for a complex scalar Ψ with weights (w, M)

SFT : S =

• dtddx Ψ⋆

i∂0 −

1 2M∂a∂a

• Ψ

is invariant under the rigid Schr¨

• dinger transformations

δΨ =

• b − 2λDt + λKt2

∂0Ψ +

• ba − λabxb − λat − λDxa + λKtxa

∂aΨ + w

• λD − λKt
• Ψ + iM
• σ − λaxa + 1

2λKx2 Ψ for w(Ψ) = −d/2 The corresponding Galilean invariant G has inconsistent E.O.M.’s

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Case 1: zero torsion: ba = 0

Schr¨

• dinger method also works at level of E.O.M.’s

foliation constraint : ∂µ(τν)G − ∂ν(τµ)G = 0 , Gal E.O.M. : (τ µ)G(eνa)GRµν a(G) = 0 , (eνa)GRµν ab(J) = 0 . Schr¨

• dinger method leads to (Ψ = ϕeiχ)

SFT : ∂0∂0ϕ = 0 and ∂aϕ = 0 with w = 1

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Case 2: twistless torsion: ba = 0

foliation constraint is conformal invariant → use the second compensating scalar χ to restore Schr¨

• dinger invariance:

∂0∂0ϕ − 2 M (∂0∂aϕ)∂aχ + 1 M2 (∂a∂bϕ)∂aχ∂bχ = 0 ⇓ − △Φ + ˆ τ µ∂µK + K abKab − 8 Φ b · b − 2 Φ D · b − 6 baDaΦ = 0 plus eνaRµνab(J) = 0

Afshar, Mehra, Parekh, Rollier + E.B. (2015)

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

Outline

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

NC Gravity from gauging Bargmann The Schr¨

• dinger Method

NC Gravity with Torsion Future Directions

New developments and Extensions

• relation to Hoˇ

rava-Lifshitz gravity

Hartong and Obers (2015) Afshar, Mehra, Parekh, Rollier + E.B. (2015)

• extension to z = 2 and Galilean conformal symmetries
• matter-coupled NC gravity
• non-relativistic supergravity → localization techniques

Andringa, Rosseel, Sezgin + E.B. (2013) Knodel, Lisbao, Liu (2015)