English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Using the Dirac constraint formalism, we examine the canonical structure ofthe Einstein-Hilbert action $S_d = \frac{1}{16\pi G} \int d^dx \sqrt{-g} R$,treating the metric $g_{\alpha\beta}$ and the symmetric affine connection$\Gamma_{\mu\nu}^\lambda$ as independent variables. For $d > 2$ tertiaryconstraints naturally arise; if these are all first class, there are $d(d-3)$independent variables in phase space, the same number that a symmetric tensorgauge field $\phi_{\mu\nu}$ possesses. If $d = 2$, the Hamiltonian becomes alinear combination of first class constraints obeying an SO(2,1) algebra. Theseconstraints ensure that there are no independent degrees of freedom. Thetransformation associated with the first class constraints is not adiffeomorphism when $d = 2$; it is characterized by a symmetric matrix$\xi_{\mu\nu}$. We also show that the canonical analysis is different if$h^{\alpha\beta} = \sqrt{-g} g^{\alpha\beta}$ is used in place of$g^{\alpha\beta}$ as a dynamical variable when $d = 2$, as in $d$ dimensions,$\det h^{\alpha\beta} = - (\sqrt{-g})^{d-2}$. A comparison with the formalismused in the ADM analysis of the Einstein-Hilbert action in first order form ismade by applying this approach in the two dimensional case with$h^{\alpha\beta}$ and $\Gamma_{\mu\nu}^\lambda$ taken to be independentvariables.Comment: 21 pages, published in Int. J. Mod. Phys. A, Vol. 21, 3401-3420  (2006

A CANONICAL ANALYSIS OF THE EINSTEIN–HILBERT IN FIRST ORDER FORM