[Paper Review] Solutions in IR modified Horava-Lifshitz Gravity

Tae-Kyung Kim, Chong Oh Lee|arXiv (Cornell University)|Feb 3, 2010

Black Holes and Theoretical Physics61 references3 citations

TL;DR

This paper investigates infrared (IR)-modified Hořava-Lifshitz gravity with a soft violation of detailed balance, showing that coupling to a global monopole yields a spherically symmetric solution with a deficit solid angle. The solution reproduces a key feature of general relativity—deficit angle due to a scalar field source—confirming consistency with GR in the low-energy limit.

ABSTRACT

In order to allow the asymptotically flat, we consider Hořava-Lifshitz gravity theory with a soft violation of the detailed balance condition and obtain various solutions. In particular, we find that such theory coupled to a global monopole leads to a solution representing a space with deficit solid angle, which is well matched with genuine feature of GR.

Motivation & Objective

• To explore whether IR-modified Hořava-Lifshitz gravity with soft detailed balance violation can reproduce known general relativity results.
• To investigate the role of matter fields, particularly global monopoles, in generating spacetime geometry in this modified gravity framework.
• To determine whether the resulting spacetime exhibits a deficit solid angle, a hallmark feature of GR solutions with point-like sources.
• To verify consistency with GR by analyzing the low-energy limit and comparing with known solutions involving global monopoles.
• To assess the viability of this gravity model for cosmological applications by examining its physical consistency.

Proposed method

• Adopting a (3+1)-dimensional ADM decomposition with static, spherically symmetric metric ansatz to simplify the field equations.
• Deriving the IR-modified Hořava-Lifshitz action with soft violation of detailed balance, including UV and IR terms with parameters κ, λ, μ, ν, Λ, ω.
• Applying a hedgehog ansatz for the global monopole scalar field ψa = r̂aψ(r), reducing the system to a radial effective Lagrangian.
• Solving the equations of motion for the metric function F(r) and the radial profile ψ(r), imposing boundary conditions ψ(0) = 0 and ψ(∞) = v.
• Using the resulting metric to compute the solid angle deficit and identifying conditions under which a black hole horizon forms.
• Rescaling coordinates to simplify the final metric form and extract physical observables like the deficit angle and horizon radius.

Experimental results

Research questions

• RQ1 Does IR-modified Hořava-Lifshitz gravity with soft detailed balance violation reproduce the deficit solid angle feature of general relativity?
• RQ2 What is the nature of the spacetime geometry generated by a global monopole in this modified gravity framework?
• RQ3 Can the source of the deficit angle be identified as a scalar field rather than an electric field, as in standard GR?
• RQ4 Under what conditions does a black hole horizon form in this solution, and how does it depend on model parameters?
• RQ5 Does the low-energy limit of this theory yield results consistent with known GR solutions, particularly for global monopoles?

Key findings

• The solution exhibits a deficit solid angle of 4πΔ = 8πv² / (κ²μ²√[ω(ω−2Λ)]) for 0 < 2v²/(κ²μ²√[ω(ω−2Λ)]) < 1, matching GR's prediction for global monopoles.
• The source of the deficit angle is a scalar field (global monopole), not an electric field, confirming a key GR feature.
• The metric function F(r) is found to be F(r) = 1 + [(ω−Λ)±√(ω(ω−2Λ))]r² − 2v²/(κ²μ²√[ω(ω−2Λ)]), describing a spacetime with curvature due to the monopole.
• A black hole horizon forms at rH = [2v²/(κ²μ²√[ω(ω−2Λ)]) − 1] / √[(ω−Λ)±√(ω(ω−2Λ))], provided the deficit parameter exceeds unity.
• The radial profile ψ(r) is zero inside the monopole core (r ≤ 1/(v√λₘ)) and approaches v at infinity, consistent with a global monopole configuration.
• The solution confirms that soft violation of detailed balance is necessary to achieve consistency with GR, particularly in cosmological contexts.