[Paper Review] Conserved currents in the Cartan formulation of general relativity

Glenn Barnich, Pujian Mao+1 more|arXiv (Cornell University)|Nov 6, 2016

Black Holes and Theoretical Physics16 citations

TL;DR

This paper derives explicit expressions for local, on-shell closed co-dimension 2 forms—surface charges—in the Cartan formulation of general relativity using the variational bicomplex and BRST cohomology techniques. It proves their equivalence to the standard surface charge forms in the metric formulation, confirming consistency across first-order formulations and providing a covariant, field-redefinition-invariant framework for conserved currents in gravity.

ABSTRACT

We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Cartan formulation of general relativity and explicitly show their equivalence to those of the metric formulation.

Motivation & Objective

• To derive explicit expressions for local, on-shell closed co-dimension 2 forms (surface charges) in the Cartan formulation of general relativity.
• To establish their equivalence with the surface charge forms derived in the metric formulation of general relativity.
• To demonstrate that these conserved currents are invariant under field redefinitions and the elimination of generalized auxiliary fields, such as the Lorentz connection.
• To provide a covariant, geometric framework for surface charges using the variational bicomplex and reducibility parameters in ghost number -2 cohomology.
• To bridge first-order and metric formulations of general relativity by showing that conserved currents transform consistently across formulations.

Proposed method

• Uses the variational bicomplex formalism to classify local, on-shell closed co-dimension 2 forms via reducibility parameters in ghost number -2 BRST cohomology.
• Applies the contracting homotopy $ \rho_H $ to the Noether current form $ S_f $ to construct the conserved current $ k_f $, ensuring $ d_H k_f = 0 $ on-shell.
• Derives the explicit form of the current $ k^{ u au}_{ ilde{\xi},\omega} $ in terms of vielbein $ e_a^\mu $, Lorentz connection $ \Gamma^{ab}_\mu $, and gauge parameters $ \xi^\rho, \omega^{ab} $, using the Cartan structure equations.
• Performs a reduction to the metric formulation by eliminating the Lorentz connection as an auxiliary field and substituting $ \omega^{ab} \approx -e^{[a}_\rho \mathcal{L}_\xi e^{b]\rho} $, leading to the standard metric-formulation surface charge.
• Uses the identity $ \delta e_a^\mu e_a^\nu = \frac{1}{2} h_{\mu\nu} + \delta e_a^{[\mu} e_a^{\nu]} $ to express variations in terms of the metric perturbation $ h_{\mu\nu} $, enabling comparison with metric-based results.
• Verifies that the resulting current matches the known expression in the metric formulation, correcting a typo in a prior reference (equation (35) in [20]).

Experimental results

Research questions

• RQ1 How can conserved surface charges be systematically derived in the Cartan formulation of general relativity using first-order formalism?
• RQ2 Are the surface charge forms in the Cartan formulation equivalent to those in the standard metric formulation of general relativity?
• RQ3 How does the BRST cohomology in ghost number -2 classify local, on-shell closed co-dimension 2 forms in first-order gravity?
• RQ4 What is the role of auxiliary and generalized auxiliary fields—such as the Lorentz connection—in the construction and equivalence of conserved currents?
• RQ5 Can the surface charge current be expressed in a way that is manifestly invariant under field redefinitions and consistent across different formulations of general relativity?

Key findings

• The paper derives the explicit form of the conserved current $ k^{ u au}_{ ilde{\xi},\omega} $ in the Cartan formulation as a function of vielbein, Lorentz connection, and gauge parameters $ \xi^\rho, \omega^{ab} $, using the variational bicomplex and BRST cohomology.
• The derived current satisfies $ d_H k_{\tilde{\xi},\omega} = 0 $ on-shell, confirming it is a closed, conserved co-dimension 2 form when the vielbein and Lorentz connection satisfy the Einstein equations.
• Upon eliminating the Lorentz connection as an auxiliary field (torsionless case), the current reduces to the standard metric formulation surface charge, matching known results in the literature.
• The expression $ k^{ u\tau}_{\tilde{\xi},\omega} $ is shown to be equivalent to the metric-formulation current up to a trivial term (exterior derivative of an $ n-3 $-form), confirming consistency.
• The derivation corrects a typo in a prior reference: equation (35) in [20] should have $ \tilde{\xi}^\mu D_\sigma h^{\sigma\nu} $, not $ \tilde{\xi}^\mu D_\sigma h^{\sigma\mu} $, which is confirmed by the present analysis.
• The framework is invariant under field redefinitions and the elimination of generalized auxiliary fields, such as the Lorentz connection, ensuring robustness across formulations.