[Paper Review] New first order Lagrangian for General Relativity

Yannick Herfray, Kirill Krasnov|arXiv (Cornell University)|Mar 30, 2015

Noncommutative and Quantum Gravity Theories15 references3 citations

TL;DR

This paper introduces a new first-order Lagrangian for General Relativity in 4D spacetime, formulated using an SO(3,C)-connection and a Lie-algebra-valued two-form field, with no auxiliary Lagrange multiplier. The Lagrangian interpolates between topological gravity (α=0) and anti-self-dual gravity (λ=0), and for generic α,λ it describes full GR with a cosmological constant and a topological term, providing a unified first-order description that parallels Yang-Mills deformations.

ABSTRACT

We describe a new BF-type first-order in derivatives Lagrangian for General Relativity. The Lagrangian depends on a connection field as well as a Lie-algebra valued two-form field, with no other fields present. There are two free parameters, which translate into the cosmological constant and the coefficient in front of a topological term. When one of the parameters is set to zero, the theory becomes topological. When the other parameter is zero, the theory reduces to the (anti-) self-dual gravity. Thus, our new Lagrangian interpolates between the topological and anti-self-dual gravities. It also interprets GR as the (anti-) self-dual gravity with an extra quadratic in the auxiliary two-form field term added to the Lagrangian, precisely paralleling the situation in Yang-Mills theory.

Motivation & Objective

• To construct a first-order Lagrangian for General Relativity that avoids the need for a Lagrange multiplier field, unlike Plebanski's formulation.
• To unify the description of topological gravity and anti-self-dual gravity within a single Lagrangian framework.
• To provide a first-order formulation of GR that parallels the deformation structure seen in Yang-Mills theory.
• To enable a potential twistor formulation of full GR by building on the integrable structure of anti-self-dual gravity.
• To explore connections to spin foam quantization by linking the topological and anti-self-dual sectors to known quantum gravity models.

Proposed method

• The Lagrangian is constructed as $\mathcal{L}_{\text{GR}}[B,A] = B^i \wedge F^i + \frac{\alpha}{2}\left(\text{Tr}(\sqrt{B^i \wedge B^j})\right)^2 - \frac{\lambda}{2}B^i \wedge B^i$, with parameters α and λ.
• The first term is the standard BF term; the second introduces a non-polynomial, square-root structure in the two-form field; the third term is a cosmological term.
• The theory reduces to topological gravity when α=0 and to anti-self-dual gravity when λ=0, with the full GR limit for generic α,λ ≠ 0.
• The two-form field is integrated out to derive a pure connection formulation, yielding $\mathcal{L}_{\text{GR}}[A] = \frac{\alpha}{2\lambda(\lambda - 3\alpha)}\left(\text{Tr}(\sqrt{F^i \wedge F^j})\right)^2 + \frac{1}{2\lambda}F^i \wedge F^i$, confirming the GR dynamics.
• The parameter λ controls the cosmological constant, while α controls the topological term, with the Newton constant set to unity.
• The limit λ→0 is shown to enforce the anti-self-dual condition on the curvature, confirming the ASD gravity limit.

Experimental results

Research questions

• RQ1 Can a first-order Lagrangian for General Relativity be formulated without a Lagrange multiplier field, while still capturing the full dynamics of GR with a cosmological constant?
• RQ2 How does the inclusion of a non-polynomial, square-root term in the two-form field relate to the structure of GR and its deformations?
• RQ3 Can the anti-self-dual gravity sector of GR be naturally embedded within a larger Lagrangian framework that also includes the full GR and topological gravity?
• RQ4 Is there a consistent way to unify the topological and anti-self-dual sectors of gravity in a single first-order action with a single set of fields?
• RQ5 Can this new Lagrangian framework support a twistor formulation of full GR, analogous to the known twistor actions for self-dual Yang-Mills theory?

Key findings

• The proposed Lagrangian $\mathcal{L}_{\text{GR}}[B,A]$ describes General Relativity with a non-zero cosmological constant and a topological term for generic values of the parameters α and λ.
• When α=0, the theory reduces to a known topological field theory, and when λ=0, it reduces to anti-self-dual gravity, confirming the interpolation between these limits.
• The pure connection formulation derived by integrating out the two-form field matches the known pure connection action for GR with a cosmological constant, confirming consistency.
• The λ→0 limit of the pure connection Lagrangian diverges unless the eigenvalues of the curvature matrix become degenerate, which corresponds to the anti-self-dual condition.
• The structure of the Lagrangian parallels that of Yang-Mills theory, where self-dual YM is deformed by a quadratic term in the self-dual two-form field.
• The new formulation opens a path to a twistor description of full GR by starting from the integrable anti-self-dual sector and adding a deformation term, analogous to the MHV rules in Yang-Mills.