[Paper Review] Conserved Charges of Minimal Massive Gravity Coupled to Scalar Field

M. R. Setare, H. Adami|arXiv (Cornell University)|Apr 25, 2016

Black Holes and Theoretical Physics3 references4 citations

TL;DR

This paper extends minimal massive gravity to include non-minimal coupling to a scalar field with non-zero torsion, deriving a generalized off-shell ADT formalism for quasi-local conserved charges. It computes energy, angular momentum, and entropy of the BTZ black hole via both direct integration and Virasoro algebra methods, finding exact agreement between results, confirming consistency with the first law and Cardy formula.

ABSTRACT

Recently, the theory of Topologically massive gravity non-minimally coupled to a scalar field has been proposed which comes from Lorentz-Chern-Simons theory \cite{1}. That theory is a torsion free one. We extend that theory by adding an extra term which makes torsion to be non-zero. The extended theory can be regarded as an extension of Minimal massive gravity such that it is non-minimally coupled to a scalar field. We obtain equations of motion of extended theory such that they are expressed in terms of usual torsion free spin-connection. We show that BTZ spacetime is a solution of this theory when scalar field is constant. We define quasi-local conserved charge by the concept of generalized off-shell ADT current which both are conserved for any asymptotically Killing vector field as well as a Killing vector field which is admitted by spacetime everywhere. Also we find general formula for entropy of stationary black hole solution in the context of considered theory. We apply the obtained formulas on BTZ black hole solution, to calculate energy, angular momentum and entropy of this soultion. We obtain the central extension term, the central charges and the eigenvalues of the Virasoro algebra generators for the BTZ black hole solution. We calculate energy and angular momentum of BTZ black hole using the eigenvalues of the Virasoro algebra generators. Also we calculate the entropy of BTZ black hole by using the Cardy formula. We find that obtained results using two different ways are exactly matched as we expected.

Motivation & Objective

• To generalize topologically massive gravity with a scalar field to include non-zero torsion by adding a new term to the Lagrangian.
• To derive a generalized off-shell ADT current and charge formalism for conserved quantities in first-order formalism.
• To compute energy, angular momentum, and entropy of the BTZ black hole solution in the extended theory.
• To verify consistency between results from the ADT formalism and the Virasoro algebra approach using the Cardy formula.
• To confirm that the derived quantities satisfy the first law of black hole mechanics.

Proposed method

• Introduce a new term to the torsion-free Lagrangian of [27] to allow non-zero torsion, resulting in a non-minimally coupled minimal massive gravity with scalar field.
• Derive equations of motion in terms of a torsion-free spin connection, showing BTZ spacetime with constant scalar field satisfies them under specific parameter conditions.
• Define a generalized off-shell ADT current and use Poincaré lemma to construct a quasi-local conserved charge for asymptotically Killing vector fields.
• Establish a general formula for black hole entropy in stationary solutions via Noether charge methods, extending Tachikawa's approach to non-covariant theories.
• Apply the formalism to the BTZ black hole, compute conserved charges, central charges, and Virasoro generator eigenvalues.
• Compare results from ADT charge computation with those from the Virasoro algebra and Cardy formula, confirming exact agreement.

Experimental results

Research questions

• RQ1 How does the inclusion of a non-minimal coupling to a scalar field with non-zero torsion modify the structure of minimal massive gravity?
• RQ2 Can a generalized off-shell ADT formalism be constructed to compute conserved charges in first-order gravity with Chern-Simons terms and scalar fields?
• RQ3 Do the energy, angular momentum, and entropy of the BTZ black hole computed via the ADT method match those derived from the Virasoro algebra and Cardy formula?
• RQ4 What are the central charges and eigenvalues of Virasoro generators for the BTZ black hole in this extended theory?
• RQ5 Is the first law of black hole thermodynamics satisfied by the derived conserved quantities in this model?

Key findings

• The BTZ spacetime with constant scalar field is a solution of the extended theory when parameters satisfy conditions (34)–(36).
• The generalized off-shell ADT current is conserved for both asymptotically Killing and globally Killing vector fields, enabling consistent charge definition.
• The entropy of a stationary black hole is given by a general formula (75), derived via Noether charge method in non-covariant gravity.
• Energy and angular momentum of the BTZ black hole are computed as E = (1/8G)[(φ₀ - αβ/μ)(r₊² + r₋²)/l² - 2r₊r₋/(μl³)] and j = (1/8G)[(φ₀ - αβ/μ)(2r₊r₋)/l - (r₊² + r₋²)/(μl²)], matching the first law.
• The central charges are c± = (3l/2G)(φ₀ ± 1/(μl) - αβ/μ), and the eigenvalues of Virasoro generators are l±₀ = (l/16G)(φ₀ ± 1/(μl) - αβ/μ)(r₊ ∓ r₋)²/l².
• The entropy computed via the Cardy formula, S = (π/2G)[(φ₀ - αβ/μ)r₊ - r₋/μ], exactly matches the result from the ADT method and the general entropy formula.