Newtonian Gravity and the Bargmann algebra Sudhakar Panda...
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Newtonian Gravity and the Bargmann algebra Sudhakar Panda HRI, Allahabad W/w R. Andringa, E. Bergshoeff and M. de Roo W/w R. Andringa, E. Bergshoeff and M. de Roo Class. Quantum Grav. 28 (2011) 105011 Class. Quantum Grav. 28 (2011) 105011
Transcript of Newtonian Gravity and the Bargmann algebra Sudhakar Panda...
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
