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DARK ENERGY ACCRETION ONTO BLACK HOLES We Describe The DARK ENERGY ACCRETION ONTO BLACK HOLES E. O. BABICHEV1,2, V. I. DOKUCHAEV1 AND Yu. N. EROSHENKO1 1Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, 117312 Moscow, Russia 2APC, UMR 7164 (CNRS-Universit´eParis 7), 10 rue Alice Domon et L´eonie Duquet,75205 Paris Cedex 13, France We study the spherically symmetric stationary accretion of a perfect fluid with an arbitrary equation of state in the Reissner-Nordstr¨om metric. In the case of the Reissner-Nordstr¨om naked singularity we show that stationary accretion is not a solution of equations of motion. Instead of, a static atmosphere of fluid is formed around singularity. We show that the third law of black hole thermodynamics is violated in a test fluid approximation in the process of phantom energy accretion onto rotating or electrically charged black hole. We describe the spherically symmetric steady-state accretion of a test perfect fluid with an arbitrary equation of state p = p(), where and p are a fluid energy density and pressure correspondingly, onto a black hole with mass m and electric charge q. For fluid with a linear equation of state p = ( − 0) and for the Chaplygin gas we find analytical solutions for sta- tionary accretion onto the Reissner-Nordstr¨om black hole. See in Fig. 1 (left panel) an example for the distribution of thermal photon gas ( = 1/3) around the charged black hole. In the case of eternal charged black hole, the accreting fluid does not reach the central singularity. Instead of, after reaching the minimal radius rmin inside the event horizon, the inflowing fluid moves out to the other asymptotically flat internal spacetime (wormhole). In the case of “astrophysical” black holes, formed by the gravitational collapse, the are no of internal spacetimes and inflowing fluid will be accumulated near rmin. This accumulated fluid will transform the internal part of the electro-vacuum Reissner-Nordstr¨om metric to the some self-consistent one, providing the stationarity for accretion. The accreting fluid reaches sound velocity at the critical point 2 2 2 1/2 1 + 3c∗ − 8c∗(1 + c∗) 2 r∗ = 2 ⎧1+ "1 2 2 e # ⎫ m, (1) 4c∗ ⎨ (1 + 3c∗) ⎬ ⎩ ⎭ 1/2 where c∗ = cs(x∗) = (∂p/∂) is the sound speed at critical radius and e = q/m. The black hole mass changes at the rate 2 m˙ = 4Am [∞ + p(∞)], (2) where ∞ — is an energy density far from the black hole and numerical constant A depends on the equation of state . For the linear equation of state 1 1− 2 2 1/2 2 2 x∗ A = x∗ 2 ! . (3) x∗ − e 2.0 e=2 300 Α=13 Α=13 e=0.99 + x* =2.04 Ρ0=0 = 250 x+ 1.14 x-=0.96 * 1.5 x-=0.86 xmin=0.79 200 ΡHxminL=303Ρ¥ ¥ ¥ Ρ Ρ L L r x 1.0 H H 150 Ρ Ρ Ρ0Ρ¥=6 100 0.5 50 0.0 0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 x=rm x=rm Figure 1: Left panel: Radial distribution of the energy density (r) of the thermal photon gas ( = 1/3) around the charged black hole with electric charge e = 0.99. After the reaching the minimal radius rmin, the fluid is expands into the internal asymptotically flat space. Right panel: An example of energy density distribution for the thermal radiation ( = 1/3, 0 = 0) and the phantom energy ( = 1/3, 0 = 6∞) in a static atmosphere around the Reissner-Nordstr¨om naked singularity with electric charge e = 2. The inverse energy density profile of thermal radiation near the singularity is a manifestation of the repulsive character of naked singularity. It is clear from equation (2) that accretion of a phantom energy, defined by the condition ∞ + p(∞) < 0, is always accompanied with a diminishing of the black hole mass. This is in accordance with previous finding for the Schwarzschild black hole 2. The result is valid for any equation of state p = p() with + p() < 0. When the Reissner-Nordstr¨om metric is a naked singularity, i. e. when m2 < q2, a perfect fluid does not accrete onto a naked singularity, instead, a static atmosphere around a singularity is formed. See in Fig. 1 (right panel) the energy density distribution for the thermal radiation and phantom energy in a static atmosphere around the Reissner-Nordstr¨om naked singularity. In the case of phantom energy accretion a charged black hole loses its mass and the extreme state with m = q is reached in a finite time with a threat of violation of the the third law of black hole thermodynamics 3. We argue, however, that the test fluid approximation is inevitably violated when the Reissner-Nordstr¨om black hole or a naked singularity is nearly extremal. This would mean that back reaction of the accreting fluid on the background geometry may prevent black hole transformation into the naked singularity in accordance with the Penrose cosmic censorship conjecture 4. References 1. E. Babichev, S. Chernov, V. Dokuchaev and Yu. Eroshenko; arXiv:0806.0916 [gr-qc]. 2. E. Babichev, V. Dokuchaev and Yu. Eroshenko, Phys. Rev. Lett. 93, 021102 (2004); [arXiv:gr-qc/0402089]. 3. J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973). 4. R. Penrose, Riv. Nuovo Cim. 1, 252 (1969)..
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