Well, perhaps one should consider reading The Hamiltonian formulation of General Relativity: myths and reality for further mathematical details. But I would like to remind to you with most constrained Hamiltonian systems, the Poisson bracket of the constraint generates gauge transformations.
For General Relativity, foliating spacetime M as R×Σ ends up producing diffeomorphism constraints Hi≈0 and a Hamiltonian constraint H≈0. Note I denote weak equalities as ≈.
This is first considered in Peter G. Bergmann and Arthur Komar's "The coordinate group symmetries of general relativity" Inter. J. The. Phys. 5 no 1 (1972) pp 15-28.
Since you asked, I'll give you a few exercises to consider!
Exercise 1: Lie Derivative of the Metric
The Lie derivative of the metric along a vector ξa is
Lξgab=gac∂bξc+gbc∂aξc+ξc∂cgab
Show that this may be rewritten as
Lξgab=∇aξb+∇bξa
where
∇ is the standard covariant derivative.
Exercise 2: Constraints generate diffeomorphisms
Recall that the Hamiltonian and momentum constraints are
H=16πG√q(πijπij−12π2)−√q16πG(3)R,Hi=−2Djπij
and
πij=116πG√q(Kij−qijK) with
Kij=12N(∂tqij−DiNj−DjNi). Let
H[ˆξ]=∫d3x[ˆξ⊥H+ˆξiHi]
Show that
H[ˆξ] generates (spacetime) diffeomorphisms of
qij, that is,
{H[ˆξ],qij}=(Lξg)ij
where
Lξ is the full spacetime Lie derivative and the spacetime vector field
ξμ is given by
ˆξ⊥=Nξ0,ˆξi=ξi+Niξ0
The parameters
{ˆξ⊥,ˆξi} are known as "surface deformation" parameters.
(Hint: use problem 1 and express the Lie derivative of the spacetime metric in terms of the ADM decomposition.)
Addendum: I'd like to give a few more references on the relation between the diffeomorphism group and the Bergmann-Komar group.
From the Hamiltonian formalism, there are a few references:
- C.J. Isham, K.V. Kuchar "Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories". Annals of Physics 164 2 (1985) pp 288–315
- C.J. Isham, K.V. Kuchar "Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics" Ann. Phys. 164 2 (1985) pp 316–333
The Lagrangian analysis of the symmetries are presented in:
- Josep M Pons, "Generally covariant theories: the Noether obstruction for realizing certain space-time diffeomorphisms in phase space." Classical and Quantum Gravity 20 (2003) 3279-3294; arXiv:gr-qc/0306035
- J.M. Pons, D.C. Salisbury, L.C. Shepley, "Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories". Phys. Rev. D 55 (1997) pp 658–668; arXiv:gr-qc/9612037
- J. Antonio García, J. M. Pons "Lagrangian Noether symmetries as canonical transformations." Int.J.Mod.Phys. A 16 (2001) pp. 3897-3914; arXiv:hep-th/0012094
For more on the hypersurface deformation algebra, it was first really investigated in Hojman, Kuchar, and Teitelboim's "Geometrodynamics Regained" (Annals of Physics 96 1 (1976) pp.88-135).
This post imported from StackExchange Physics at 2014-03-17 03:24 (UCT), posted by SE-user Alex Nelson