Newtonian Gravity and the Bargmann algebraSudhakar PandaHRI, Allahabad

W/w R. Andringa, E. Bergshoeff and M. de RooW/w R. Andringa, E. Bergshoeff and M. de RooClass. Quantum Grav. 28 (2011) 105011Class. Quantum Grav. 28 (2011) 105011

Overview

● NR-gravity: why bother?● Newtonian gravity● Einstein gravity from Poincaré● Newtonian gravity from Bargmann

Non-relativistic gravity: why bother?

● A lot of physics in our universe is NR● Cosmology● Holography and AdS/CFT!

Newtonian gravity

● Newton: space is flat, time is absolute

● Galilei group:

● EOM for particles:

● EOM for potential: , Poisson!

Einstein gravity from Poincaré

● Poincaré algebra generators: { P, J }

P: translations Gauge fields Curvatures

J: rotations/boosts

● R(P) = 0 → (1)

(2)

Einstein gravity from Poincaré II

● Remaining independent gaugefield: Vielbein● Symmetries: gct's and local Lorentz transfo's

● Vielbein postulate:

● Hilbert action:

● EOM, coupling to matter, ...

Newton-Cartan I

● Newton-Cartan: geometrized version of Newton (1923)

● Newtonian EOM → geodesic eqn

Newton-Cartan II

● Connection:

● Riemann tensor:

● Poisson eqn.: ,

● What about a metric?

Newton-Cartan III

● No 4-dimensional non-degenerate metric invariant under Galilei group!

● Lim c → oo gives two degenerate metrics!

Newton-Cartan IV

●

●

Newton-Cartan V

● Degenerate metric structure needed!

- spatial metric:

- temporal metric:

- ● Also and with

NOT uniquely defined; ambiguity in K!

Newton-Cartan VI

● So-called Trautman condition and Ehlers condition on Riemann tensor needed to constrain K and obtain Poisson

● -

-

● Newton-Cartan can be obtained as a GR-limit

Goal

● Reproduce Newton-Cartan by a gauging procedure, just like Poincaré gravity

● Naïve guess: Galilei algebra → won't work (spin connections can't be solved)

● Central extension needed → Bargmann algebra!

Newtonian gravity from Bargmann I

● Bargmann generators: {H,P,G,J,M}

Gauge fields Curvatures

Time transl.

Space transl.

Boosts

Space rot.

Central ext.

Newtonian gravity from Bargmann II

● R(M) = R(P) = 0: solve for

● R(H) = 0:

● Galilean Vielbein postulate:

● Ehlers and Trautman condition: R(J) = 0

Newtonian gravity from Bargmann III

● Only surviving component of Riemann tensor:

● Conclusion: Newton-Cartan can be obtained by gauging the Bargmann algebra

● Ehlers & Trautman condition: curvature constraints!

Future Work

● Action principle for Newton-Cartan ● Gauging Newton-Hooke algebras → holography!● Non-relativistic supergravity via

Super Bargmann algebra

QUESTIONS?

THANK YOU