Comparison between fluid-gravity and membrane-gravity dualities for Einstein-Maxwell system

Abstract

Derivative expansion and large-D expansion are two perturbation techniques, which are used to generate dynamical black-brane solutions to Einstein's equations in presence of negative cosmological constant. In this note we have compared these two techniques and established the equivalence of the gravity solutions generated by these two different techniques in appropriate regime of parameter space up to first non-trivial order in both the perturbation parameters for Einstein-Maxwell systems, generalizing the earlier works of [1,2] for non-charged systems. An one-toone map between dynamical black-brane geometry and AdS space, which also exists at finite number of dimensions, has also been established. Contents 1 Introduction : 1.1 Strategy 2 Review of Hydrodynamics from charged black-branes in arbitrary dimensions : 2.1 Scalars at first order 2.2 Vectors at first order 2.3 Tensors at first order 2.4 The global metric and gauge field at first order 2.5 The boundary stress tensor and the charge current 2.6 Hydrodynamic metric and gauge field up to first order in derivative 3 The large D metric, gauge field and membrane equations: 3.1 The dual system 4 Comparing fluid-gravity and membrane-gravity dualities : 4.1 The split of the hydrodynamic metric 4.2 Membrane data in terms of fluid data 4.2.1 Determining ψ 4.2.2 Determining U A 4.2.3 Determining Q 4.2.4 Relevant derivatives of the basic data 4.3 Comparing the metrics and gauge fields 4.3.1 Comparing the gauge fields 4.3.2 Comparing the metric 4.4 Comparing the evolution of two sets of data 5 Conclusons and future directions : A The Large-D limit of the integrations appearing in hydrodynamic metric A.1 Analysis of the integral in the function F 1 A.2 Analysis of the integral in the function F 2 B The inverse of the background metric and christoffel symbols w.r.t background metric C Notation 34

The Large D Limit of Einstein's Equations

“…For this same reason, it also applies to black holes of finite extent, including the Schwarzschild-Tangherlini solution, and not only to black branes or ultraspinning black holes. In general, the expansions in 1/D and in gradients (powers of k) have overlapping but non-coincident ranges of applicability; see (Bhattacharyya et al, 2019a,b;Patra, 2019) for a detailed comparison.…”