[Paper Review] On black hole solutions in model with anisotropic fluid

H. Dehnen, В. Д. Иващук+1 more|ArXiv.org|Nov 13, 2002

Black Holes and Theoretical Physics9 references3 citations

TL;DR

This paper presents a new family of exact spherically symmetric black hole solutions in a model with a one-component anisotropic fluid, characterized by a parameter $ q > 0 $ relating radial pressure to energy density. Using a Toda-like system reduction and Lagrangian formalism, the authors derive solutions that generalize Reissner-Nordström and M2/M5 brane metrics for integer $ q $, and compute post-Newtonian parameters for the 4D section, showing consistency with solar system tests for $ q=1 $. The key contribution is a unified framework linking anisotropic fluid models to $ p $-brane solutions via specific equations of state.

ABSTRACT

A family of spherically symmetric solutions in the model with 1-component anisotropic fluid is considered. The metric of the solution depends on a parameter q > 0 relating radial pressure and the density and contains n -1 parameters corresponding to Ricci-flat ``internal space'' metrics. For q = 1 and certain equations of state the metric coincides with the metric of black brane solutions in the model with antisymmetric form. A family of black hole solutions corresponding to natural numbers q = 1,2, ... is singled out. Certain examples of solutions (e.g. containing for q =1 Reissner-Nordström, M2 and M5 black brane metrics) are considered. The post-Newtonian parameters beta and gamma corresponding to the 4-dimensional section of the metric are calculated.

Motivation & Objective

• To construct exact spherically symmetric solutions in a model with one-component anisotropic fluid, extending known black hole metrics.
• To identify conditions under which these solutions describe black holes, particularly for integer values of $ q $.
• To establish a correspondence between the anisotropic fluid model and $ p $-brane solutions in higher-dimensional gravity with antisymmetric forms.
• To compute post-Newtonian parameters $ \beta $ and $ \gamma $ for the 4-dimensional spatial section of the metric to test compatibility with solar system observations.

Proposed method

• Formulate the Einstein equations with anisotropic fluid energy-momentum tensor on a product manifold $ \mathbb{R} \times S^{d_0} \times \mathbb{R} \times \prod_{i=2}^n M_i $, where $ M_i $ are Ricci-flat manifolds.
• Introduce equations of state relating radial pressure $ p_r $, tangential pressure $ p_0 $, and other pressures $ p_i $ to energy density $ \rho $, parameterized by $ q > 0 $, $ q \neq 1/2 $, and arbitrary $ U_i $ for $ i > 1 $.
• Apply a minisuperspace reduction to the field equations, leading to a Lagrangian system with metric components $ X^i(u) $, where $ u $ is a radial variable, and derive the corresponding Euler-Lagrange equations.
• Use the harmonic time gauge and impose zero-energy constraint to solve the system, obtaining solutions in terms of hyperbolic, trigonometric, or logarithmic functions depending on constants $ C_\alpha $.
• Introduce a radial coordinate transformation $ r(u) $ via $ \exp(-2\bar{\mu}u) = 1 - 2\mu / r^d $, mapping the solution to a standard black hole-like form.
• Derive the metric components $ J_i = H^{-2U^i/(U,U)} $, with $ H $ defined via $ f_1(u - u_1) $, and express energy density $ \rho $ in terms of $ H $, $ r $, and constants.

Experimental results

Research questions

• RQ1 For which values of the parameter $ q $ do the anisotropic fluid solutions describe black holes?
• RQ2 How do the derived solutions relate to known $ p $-brane solutions such as Reissner-Nordström, M2, and M5 branes?
• RQ3 What are the post-Newtonian parameters $ \beta $ and $ \gamma $ for the 4-dimensional spatial section of the metric, and do they satisfy solar system constraints?
• RQ4 Can the anisotropic fluid model reproduce $ p $-brane solutions via a specific equation of state?
• RQ5 What is the role of the vector $ U_i $ and the inner product $ (U,U) $ in determining the structure of the solutions?

Key findings

• A new family of exact spherically symmetric solutions is derived for a one-component anisotropic fluid with equations of state parameterized by $ q > 0 $, $ q \neq 1/2 $.
• For integer $ q = 1,2,\ldots $, the solutions describe black holes, with the horizon condition enforced via specific integration constants $ \bar{c}^i = 0 $, $ c^i $ proportional to $ U^{(\alpha)i} $.
• When $ q = 1 $, the solution reduces to the Reissner-Nordström metric for $ d_0 = 3 $, and to M2 and M5 brane metrics in higher dimensions, confirming the correspondence with $ p $-brane solutions.
• The post-Newtonian parameters for the 4D section are computed as $ \beta = 1 $, $ \gamma = 1 $, consistent with general relativity and solar system observations.
• The energy density is given by $ \rho = \frac{(2q-1)(dq)^2 P(P+2\mu)(1 - 2\mu r^{-d})^{q-1}}{2(U,U)H^2 J_0 r^{2d_0}} $, showing explicit dependence on $ q $, $ \mu $, and $ P $.
• The solution structure depends crucially on the inner product $ (U,U) > 0 $, which is proven to be positive under the condition $ U_0 = 0 $, ensuring the spacelike nature of the vector $ U $.