[Paper Review] (2+1)-dimensional Chern-Simons bi-gravity with AdS Lie bialgebra as an interacting theory of two massless spin-2 fields

Siamak Hoseinzadeh, A. Rezaei-Aghdam|arXiv (Cornell University)|Jun 7, 2017

Advanced Differential Geometry Research3 citations

TL;DR

This paper proposes a novel (2+1)-dimensional Chern-Simons bi-gravity theory based on an AdS Lie bialgebra, formulating two interacting massless spin-2 fields via two dreibein fields rather than metrics. The model is ghost-free, has no local degrees of freedom, and yields a new black hole solution with two horizons and two curvature singularities, distinct from previous solutions, along with a homogeneous, isotropic cosmological solution with decelerating expansion.

ABSTRACT

We introduce a new Lie bialgebra structure for the anti de Sitter (AdS) Lie algebra in (2+1)-dimensional spacetime. By gauging the resulting extit{AdS Lie bialgebra}, we write a Chern-Simons gauge theory of bi-gravity involving two dreibeins rather than two metrics, which describes two interacting massless spin-2 fields. Our ghost-free bi-gravity model which has no any local degrees of freedom, has also a suitable free field limit. By solving its equations of motion, we obtain a extit{new black hole} solution which has two curvature singularities and two horizons. We also study cosmological implications of this massless bi-gravity model.

Motivation & Objective

• To construct a new Lie bialgebra structure for the (2+1)-dimensional anti-de Sitter (AdS) Lie algebra so(2,2).
• To formulate a Chern-Simons gauge theory based on this bialgebra that describes two interacting massless spin-2 fields via two dreibein fields.
• To develop a ghost-free bi-gravity model without local degrees of freedom, with a consistent free field limit.
• To derive new exact solutions, including a black hole with two horizons and two curvature singularities, distinct from known BTZ-type solutions.
• To explore cosmological implications by finding homogeneous and isotropic Friedmann-Robertson-Walker solutions in both metrics.

Proposed method

• Construct a new Lie bialgebra from the AdS algebra so(2,2) in (2+1) dimensions using a specific classical r-matrix.
• Utilize the resulting Manin triple structure to define a Chern-Simons gauge theory with two independent dreibein fields.
• Formulate the action as a sum of two Chern-Simons terms with gauge algebra so(2,2)⊕so(2,2), ensuring gauge invariance and topological character.
• Solve the equations of motion by assuming a BTZ black hole ansatz for one metric, leading to a new black hole solution in the second metric.
• Derive cosmological solutions by assuming spatially homogeneous and isotropic FRW metrics in both gμν and fμν, solving for scale factors A(t) and ā(t′).
• Perform coordinate transformations to express the FRW solutions in standard form, analyzing Hubble and deceleration parameters to characterize expansion dynamics.

Experimental results

Research questions

• RQ1 Can a new Lie bialgebra structure be constructed from the AdS algebra so(2,2) in (2+1) dimensions to describe interacting massless spin-2 fields?
• RQ2 Does the resulting Chern-Simons bi-gravity model remain ghost-free and free of local degrees of freedom, even in the interacting regime?
• RQ3 What are the exact black hole solutions in this model, and how do they differ from known solutions like the BTZ black hole or previous bi-gravity solutions?
• RQ4 Does the model admit cosmological solutions with homogeneous and isotropic spatial sections, and what is the nature of the expansion (accelerating or decelerating)?
• RQ5 How do the two metrics in the model behave asymptotically, and what is the physical interpretation of their distinct asymptotic structures?

Key findings

• The model yields a new black hole solution with two horizons and two curvature singularities, differing from the BTZ black hole and previous bi-gravity solutions that had only coordinate singularities.
• The black hole solution is not asymptotically AdS, indicating a distinct asymptotic structure compared to standard three-dimensional gravity models.
• The cosmological solution exhibits two distinct scale factors: one oscillating (in the gμν metric) and one growing but decelerating (in the fμν metric), both with positive deceleration parameters.
• The Hubble parameter for the oscillating solution is H(ût) = (1/ℓ) cot(ût/ℓ), indicating periodic expansion and contraction.
• The Hubble parameter for the second solution is H(t′) = b² / [t′(b² + ℓ⁻²(ab−1)²t′²)], showing a decelerating expansion that slows over time.
• The model is exactly soluble, ghost-free, and has no local degrees of freedom in both the free and interacting limits, confirming its consistency as a topological field theory.