Black rings in large D membrane paradigm at the first order

Abstract

Black rings are the black objects found in D spacetime dimensional gravity when D ≥ 5. These have event horizon topology SD−3× S1. In this work the solutions of the large D membrane paradigm dual to stationary black rings in Einstein-Maxwell theory with or without cosmological constant are studied. It is shown that the first order membrane equations can only admit static asymptotically flat black rings, and the equilibrium angular velocity for the asymptotically AdS black rings at large D was obtained. The thermodynamic and dynamic stability of the asymptotically flat black ring solutions is studied. The apparent shortcomings of some of these results are argued to be curable within the large D membrane paradigm framework.

De Sitter static black ring in large D membrane paradigm at the second order

“…This class of solutions was discovered in [15,16] for D = 5 and then for any D > 4 [17] within 'thin ring' approximation. Later in [1] The asymptotically AdS black rings were found from the first order stationary membrane paradigm, and also the equilibrium rotational speed of the asymptotically flat black ring was determined from the second order equation, which matched with that reported in [2].…”

“…However, [2] also has an asymptotically de Sitter black ring solution which is static at an equilibrium 'ring radius'. This solution was absent from the set of solutions of the stationary membrane paradigm of [1] because the parameters for the reported solution lied outside the domain of validity of [14]. The author indicated that consideration of the second order membrane paradigm should recover this solution, and this note does exactly that.…”

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It was shown in [1] that the effective stationary membrane equations from the large D membrane paradigm at the first order admit black ring solutions in flat and AdS cases, but the de Sitter solution obtained in [2] lies outside the domain of their applicability. In this short note the static de Sitter black ring is obtained from the second order membrane paradigm, and it satisfies the equilibrium condition for the thin ring solution of [2].

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de Sitter Static Black Ring in Large $D$ Membrane Paradigm at the Second Order

“…To recap [1], for stationary axisymmetric configurations the (t, r, θ, s, {χ a }) coordinate system, fondly known as r − s coordinate system, is constructed by splitting the background…”

“…n s s needs to be calculated up to O (ǫ1 )n r = G rr n r + G rs n s = −N g ′ + O ǫ 1 = r − b β + O ǫ 1 , (B.11) n s = G rs n r + G ss n s = N s − ǫ N s L2 (2g − rg ′ ) + O ǫ 2 , (g − (r − b)g ′ ) r n r + ∂ s n s = 2 β + O ǫ 1 , (B.13) Thus from (B.10), (B.12), (B.11) and (B.13), becomes using (B.9) and (B.14), (r − b)g′ − g = β β 2 )(r − b) 2 2 L2 . (B.15)This is a first order ODE which can be easily solved to getg(r) =2β 2 − b) ln(r − b) + C(r − b), (B.16)Where C is the integration constant.…”