[Paper Review] Holographic Aspects of a Higher Curvature Massive Gravity

Shahrokh Parvizi, Mehdi Sadeghi|arXiv (Cornell University)|Apr 3, 2017

Black Holes and Theoretical Physics47 references4 citations

TL;DR

This paper studies the holographic dual of a higher curvature massive gravity theory including Gauss-Bonnet and cubic quasi-topological terms. It derives the boundary energy-momentum two-point function, establishes an a-theorem under the null energy condition, and computes the shear viscosity to entropy ratio as a rate of entropy production due to strain, showing that graviton mass reduces viscosity. The results are consistent with causality and unitarity constraints on mass parameters.

ABSTRACT

We study the holographic dual of a massive gravity with Gauss-Bonnet and cubic quasi-topological higher curvature terms. Firstly, we find the energy-momentum two-point function of the 4-dimensional boundary theory where the massive term breaks the conformal symmetry as expected. An $a$-theorem is introduced based on the null energy condition. Then we focus on a black brane solution in this background and derive the ratio of shear viscosity to entropy density for the dual theory. It is worth mentioning that the concept of viscosity as a transport coefficient is obscure in a nontranslational invariant theory as in our case. So although we use the Green-Kubo's formula to derive it, we rather call it the rate of entropy production per the Planckian time due to a strain. Results smoothly cover the massless limit.

Motivation & Objective

• To investigate the holographic dual of a higher curvature massive gravity theory including Gauss-Bonnet and cubic quasi-topological terms.
• To analyze the boundary field theory's conformal structure, particularly the energy-momentum two-point function, and identify the presence of a massive operator.
• To formulate an a-theorem based on the null energy condition in the presence of mass deformation.
• To derive the shear viscosity to entropy density ratio in a black brane background, interpreting it as entropy production rate under strain.
• To impose physical constraints such as unitarity, absence of ghosts, and causality on the mass parameters.

Proposed method

• Constructing a higher curvature massive gravity action with Gauss-Bonnet and cubic quasi-topological terms in AdS spacetime.
• Computing the boundary energy-momentum two-point function using holographic methods, revealing a massive operator with Yukawa-like behavior.
• Defining an a-function based on the null energy condition to probe renormalization group flow and establish an a-theorem.
• Finding an exact black brane solution in the bulk theory and deriving its temperature and entropy via standard holographic techniques.
• Applying the Kubo formula to compute the shear viscosity to entropy ratio, reinterpreted as the rate of entropy production per Planckian time due to strain.
• Analyzing causality, unitarity, and temperature positivity to constrain the mass parameters $ m_1, m_2, m_3 $, finding $ m_i ightarrow 0 $ or negative for stability.

Experimental results

Research questions

• RQ1 How does the inclusion of massive gravity with higher curvature terms affect the conformal structure of the dual boundary CFT?
• RQ2 Can an a-theorem be established in this massive higher curvature gravity framework using the null energy condition?
• RQ3 What is the shear viscosity to entropy density ratio in the dual field theory, and how does it differ from the massless case?
• RQ4 How do physical constraints like causality, unitarity, and non-negative temperature constrain the mass parameters of the theory?
• RQ5 What is the role of the graviton mass in modifying transport properties such as viscosity in a non-translational-invariant theory?

Key findings

• The boundary energy-momentum two-point function exhibits a massive operator with a Yukawa-like fall-off, indicating spontaneous breaking of conformal symmetry.
• An a-function is constructed that monotonically decreases under the null energy condition, supporting an a-theorem in the massive gravity setup.
• The shear viscosity to entropy density ratio is derived as $ \eta/s = \text{const} \times \phi_0(r_0)^2 $, with $ \phi_0(r_0)^2 < 1 $, indicating a reduction in viscosity due to the graviton mass.
• Causality constraints require $ m_1 \leq 0 $, and if $ m_1 = 0 $, then $ m_2 \leq 0 $, and so on, with $ m_3 \leq 0 $ in the $ m_1 = m_2 = 0 $ limit.
• Temperature positivity imposes a lower bound on the mass parameters: $ m_1 + 2m_2 + 2m_3 \geq -4 $, ensuring physical black brane solutions.
• The results smoothly reduce to the massless limit, confirming consistency with known results in Einstein and Gauss-Bonnet gravity.