Newtonian Gravity and the Bargmann algebra Sudhakar Panda...
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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
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Overview
● NR-gravity: why bother?● Newtonian gravity● Einstein gravity from Poincaré● Newtonian gravity from Bargmann
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Non-relativistic gravity: why bother?
● A lot of physics in our universe is NR● Cosmology● Holography and AdS/CFT!
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Newtonian gravity
● Newton: space is flat, time is absolute
● Galilei group:
● EOM for particles:
● EOM for potential: , Poisson!
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Einstein gravity from Poincaré
● Poincaré algebra generators: { P, J }
P: translations Gauge fields Curvatures
J: rotations/boosts
● R(P) = 0 → (1)
(2)
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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, ...
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Newton-Cartan I
● Newton-Cartan: geometrized version of Newton (1923)
● Newtonian EOM → geodesic eqn
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Newton-Cartan II
● Connection:
● Riemann tensor:
● Poisson eqn.: ,
● What about a metric?
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Newton-Cartan III
● No 4-dimensional non-degenerate metric invariant under Galilei group!
● Lim c → oo gives two degenerate metrics!
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Newton-Cartan IV
●
●
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Newton-Cartan V
● Degenerate metric structure needed!
- spatial metric:
- temporal metric:
- ● Also and with
NOT uniquely defined; ambiguity in K!
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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
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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!
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Newtonian gravity from Bargmann I
● Bargmann generators: {H,P,G,J,M}
Gauge fields Curvatures
Time transl.
Space transl.
Boosts
Space rot.
Central ext.
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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
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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!
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Future Work
● Action principle for Newton-Cartan ● Gauging Newton-Hooke algebras → holography!● Non-relativistic supergravity via
Super Bargmann algebra
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QUESTIONS?
THANK YOU