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Hamiltonian and Noether charges in first order gravity

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Abstract

We consider gravity in four dimensions in the vielbein formulation, where the fundamental variables are a tetrad \(e\) and a SO(3,1) connection \(\omega \). We start with the most general action principle compatible with diffeomorphism invariance which includes, besides the standard Palatini term, other terms that either do not change the equations of motion, or are topological in nature. For our analysis we employ the covariant Hamiltonian formalism where the phase space \(\Gamma \) is given by solutions to the equations of motion. We consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. For this extended action we study the effect of the topological terms on the Hamiltonian formulation. We prove two results. The first one is rather generic, applicable to any field theory with boundaries: The addition of topological terms (and any other boundary term) does not modify the symplectic structure of the theory. The second result pertains to the conserved Hamiltonian and Noether charges, whose properties we analyze in detail, including their relationship. While the Hamiltonian charges are unaffected by the addition of topological and boundary terms, we show in detail that the Noether charges do change. Thus, a non-trivial relation between these two sets of charges arises when the boundary and topological terms needed for a consistent formulation are included.

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Notes

  1. One should recall that the original Palatini action was written in terms of the metric \(g_{ab}\) and an affine connection \({\Gamma }_{bc}^a\) [3, 4]. The action we are considering here, in the so called “vielbein” formalism, was developed in [5–7] and in [8] in the canonical formulation.

  2. We should clarify our use of the name ‘topological term’. For us a term is topological if it can be written as a total derivative. This in turn implies that it does not contribute to the equations of motion. There are other possible terms that do not contribute to the equations of motion but that can not be written as a total derivative (such as the so called Holst term). For us, this term is not topological.

  3. See e.g. [14–16] and references therein for the asymptotically flat, isolated horizons and asymptotically AdS spacetimes respectively.

  4. Usually, a symplectic potential is defined as an integral of \(\theta \) over a spatial slice \(M\), see, for example, [18]. Here, we are extending this definition since, as we shall show, in order to construct a symplectic structure it is important to consider the integral over the whole boundary \(\partial \mathcal {M}\).

  5. Note that there have been some statements in the literature claiming that the topological terms do contribute to the symplectic structure when there are boundaries present [35–37].

  6. Nor to \(\delta a_\Delta \), for that matter.

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Acknowledgments

We would like to thank N. Bodendorfer, S. Deser, T. Jacobson and R. Olea for comments. We would also like to thank an anonymous referee for comments that helped to improve the manuscript. This work was in part supported by CONACyT 0177840, DGAPA-UNAM IN100212, and NSF PHY 1205388 Grants, the Eberly Research Funds of Penn State and by CIC, UMSNH.

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Authors and Affiliations

  1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, 58090, Morelia, Michoacán, Mexico

    Alejandro Corichi & Iraís Rubalcava-García

  2. Center for Fundamental Theory, Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, PA, 16802, USA

    Alejandro Corichi

  3. Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, Mexico

    Iraís Rubalcava-García

  4. Facultad de Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo, 58000, Morelia, Michoacán, Mexico

    Tatjana Vukašinac

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  1. Alejandro Corichi
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  2. Iraís Rubalcava-García
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Correspondence to Alejandro Corichi.

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Corichi, A., Rubalcava-García, I. & Vukašinac, T. Hamiltonian and Noether charges in first order gravity. Gen Relativ Gravit 46, 1813 (2014). https://doi.org/10.1007/s10714-014-1813-0

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