[Paper Review] Static Charged Black Hole Solutions in Horava-Lifshitz Gravity

Jin-Zhang Tang|arXiv (Cornell University)|Nov 19, 2009

Black Holes and Theoretical Physics23 references3 citations

TL;DR

This paper derives static charged black hole solutions in Ho’rava-Lifshitz gravity with and without the projectability condition, using a generalized action coupling electromagnetism to the gravitational sector. It identifies (A)dS-Reissner-Nordström solutions in Painlevé-Gullstrand coordinates in the IR region and de-Sitter solutions in the UV region, with new charged black hole solutions emerging when the projectability condition is relaxed, consistent with prior results in the literature under specific parameter limits.

ABSTRACT

In the present work, we search static charged black hole solutions to Hořava-Lifshitz gravity with or without projectability condition. We consider the most general form of action which electromagnetic field couples with Hořava-Lifshitz gravity. With the projectability condition, we find dS-Reissner-Nordstrom black hole solution in Painlevé-Gullstrand type coordinates in the IR region and a de-Sitter space-time solution in the UV region. Without the projectability condition, in the IR region, we find an especial static charged black hole solution.

Motivation & Objective

• To derive static charged black hole solutions in Hoýrava-Lifshitz gravity coupled to electromagnetism, extending previous work on uncharged solutions.
• To investigate the role of the projectability condition in determining black hole solution structure in the IR and UV regimes.
• To explore how the anisotropic scaling and modified gravity dynamics affect the existence and form of charged black hole solutions.
• To compare solutions under different assumptions (projectable vs. non-projectable lapse function) and identify consistency with known results in the literature.
• To examine the impact of electromagnetic coupling on black hole geometry, particularly in the presence of cosmological-like terms and curvature corrections.

Proposed method

• Uses the ADM formalism with a metric ansatz in Painlevé-Gullstrand-type coordinates: $ ds^2 = -dt^2 + (dr + N^r dt)^2 + \frac{1}{f(r)}(dr + N^r dt)^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2) $, where $ N $ is either a function of $ t $ (projectable) or $ r $ (non-projectable).
• Derives field equations from a generalized Hoýrava-Lifshitz action including kinetic, potential, and electromagnetic coupling terms, with $ \lambda $ as the dynamical critical exponent.
• Applies the projectability condition ($ N = N(t) $) to derive solutions in the IR ($ \lambda = 1 $) and UV ($ \lambda \neq 1 $) regimes, solving the resulting differential equations for $ f(r) $, $ N^r(r) $, and $ N(t) $.
• For the non-projectable case ($ N = N(r) $), solves the field equations under $ \lambda = 1 $ (IR) and $ \lambda \neq 1 $ (UV), yielding new solutions involving integration constants $ \beta, c $ and parameters $ \Omega, \Lambda_W $.
• Imposes Gauss's law for electromagnetism: $ \int_S \tilde{g}_{em} \frac{Q_e}{r^2} d\vec{\sigma} = \pm Q_e $, with $ \tilde{g}_{em} = 1 $ for physical charged black holes.
• Performs coordinate transformations to compare solutions with known forms (e.g., Reissner-Nordström) and validates consistency with prior works (e.g., RongCai-2009, Lu2009).

Experimental results

Research questions

• RQ1 What are the static charged black hole solutions in Hoýrava-Lifshitz gravity when the projectability condition is imposed?
• RQ2 How do the solutions differ in the IR and UV regimes under the projectability condition?
• RQ3 What new charged black hole solutions emerge when the projectability condition is relaxed (i.e., $ N = N(r) $)?
• RQ4 How do the electromagnetic coupling and curvature terms affect the structure of the black hole solutions in both IR and UV limits?
• RQ5 Are the derived solutions consistent with known solutions in general relativity and prior Hoýrava-Lifshitz studies (e.g., RongCai-2009, Lu2009) under specific parameter limits?

Key findings

• With the projectability condition, the paper finds an (A)dS-Reissner-Nordström black hole solution in Painlevé-Gullstrand coordinates in the IR region ($ \lambda = 1 $), with $ f(r) = 1 - \Lambda_W r^2 - \sqrt{c r + 2\tilde{g}_{em} \Lambda_W (Q_e^2 - Q_m^2)} $ and $ N^r = 0 $.
• In the UV region ($ \lambda \neq 1 $), a de-Sitter space-time solution is found when $ f = 1 $ and $ Q_e^2 = Q_m^2 $, with $ N^r = \pm \sqrt{\frac{\Lambda_W}{3\lambda - 1}} r $, consistent with earlier work.
• Without the projectability condition, in the IR ($ \lambda = 1 $), a new static charged black hole solution is derived with $ f(r) = 1 - \Lambda_W r^2 - \sqrt{c r + 2\tilde{g}_{em} \Lambda_W (Q_e^2 - Q_m^2)} $ and $ N_r^2 = \frac{\beta}{r f} $, matching RongCai-2009 for $ Q_m = 0 $.
• In the UV region ($ \lambda \neq 1 $), when $ N_r = 0 $, the paper obtains a solution with $ f = 1 - \Lambda_W r^2 - \alpha r^{\frac{2\lambda \pm \sqrt{6\lambda - 2}}{\lambda - 1}} $ and $ N = a r^{-\frac{1 + 3\lambda \pm 2\sqrt{6\lambda - 2}}{\lambda - 1}} \sqrt{f} $, consistent with Lu2009.
• When $ Q_e^2 = Q_m^2 $, the UV solution reduces to a de-Sitter geometry, confirming the consistency of the model under electromagnetic neutrality.
• The solutions are validated through coordinate transformations and comparison with known Reissner-Nordström and Schwarzschild-de-Sitter forms, confirming physical consistency.