2001/05/31 by Josep M. Pons, J. M. Pons · 14 citations
Mathematics · Physics and Astronomy · #Action (physics) #Black Holes and Theoretical Physics #Boundary (topology) #Classical mechanics #Cosmology and Gravitation Theories #Differential geometry #Gauge symmetry #Gauge theory #General relativity #Inverse problem for Lagrangian mechanics #Lagrangian #Mathematical analysis #Mathematical physics #Mathematics #Noether's theorem #Noncommutative and Quantum Gravity Theories #Physics #Quantum mechanics #Symplectic geometry #TRACE (psycholinguistics) #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1023/a:1022329007805
published in General Relativity and Gravitation 35(2), 147-174 (Springer Science+Business Media) · 24 pages, some references added and a new paragraph in the conclusions
arxiv created 2002/03/07 · openalex publication_date 2003/02/01 · arxiv updated 2014/11/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We present the Lagrangian whose corresponding action is the trace K action for General Relativity. Although this Lagrangian is second order in the derivatives, it has no second order time derivatives and its behaviour at space infinity in the asymptotically flat case is identical to other alternative Lagrangians for General Relativity, like the gamma-gamma Lagrangian used by Einstein. We develop some elements of the variational principle for field theories with boundaries, and apply them to second order Lagrangians, where we stablish the conditions -- proposition 1 -- for the conservation of the Noether charges. From this general approach a pre-symplectic form is naturally obtained that features two terms, one from the bulk and another from the boundary. When applied to the trace K Lagrangian, we recover a pre-symplectic form first introduced using a different approach. We prove that all diffeomorphisms satisfying certain restrictions at the boundary -- that keep room for a realization of the Poincaré group -- will yield Noether conserved charges. In particular, the computation of the total energy gives, in the asymptotically flat case, the ADM result.