ner_tags: O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Model O O O O O O O B-Model I-Model O O O O O O O O O O O O O B-Model I-Model O B-Model I-Model O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-Citation I-Citation O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O
tokens: of the total stress-energy tensor , namely which yields Here, γ αβ = g αβ + U α U β is the projection tensor, and α is the lapse function. Latin indices take values 1, 2, 3 and Greek ones 0, 1, 2, 3. Vectors and tensors with tilde are purely spatial. For an ideal fluid with density ρ, 3-velocity , and pressure p we have We will also assume an equation of state p = p (ρ) from which we deduce the ‘speed of sound’, Finally, we will also assume ideal MHD conditions, namely In order to investigate the Rayleigh–Taylor instability in an astrophysical context, we will now consider the special case of a Kerr space–time. 2.1 Kerr space–time In Boyer–Lindquist (BL) coordinates the Kerr metric reads where M is the mass of the black hole, a is the angular momentum per unit mass (0 ≤ a ≤ M ), and (Cowling 1941 ). Notice that we work in geometrical units in which c = G = 1. For our further study we need the components of the 4-velocity of fiducial observers, now identified as zero angular momentum observers (ZAMOs), namely where In the Kerr space–time ( 11 ) with 4-velocity U μ given by equation ( 13 ), the expansion θ vanishes, the shear has two non-zero components e.g. the σ 13 and σ 23 but σ αβ γ αβ = 0 (see Thorne Macdonald 1982 , paper I, equation 2.5). The acceleration a μ is given by γ ij is the spatial metric on the space-like hypersurface x 0 ≡ t = const., with normal vector n α : where β i = γ ij g 0 j . 2.2 The perturbed MHD equations We consider only small perturbations of physical quantities as and keep only linear terms of the perturbations. In this case and In the Cowling approximation of a fixed Kerr space–time: The first-order perturbed MHD equations now become 3 The static equatorial disc We will now restrict our analysis to the investigation of a static equatorial distribution of matter of thickness h ≲ r . By ‘static’ we mean that the disc fluid is initially at rest with respect to ZAMOs, i.e. that . Our disc configuration only vaguely mimics astrophysical accretion discs. We acknowledge that neglecting the Keplerian disc rotation is an important simplification that we apply only to make some progress with the complex general relativistic formalism. However, our results may be physically relevant in MADs in which rotation plays a secondary role (see Discussion section). The problem we have in mind is a distribution of matter consisting of two regions inside and outside some radius r 0 . We will thus only consider perturbations in the immediate neighbourhood of r 0 of the form in the equatorial plane θ = π/2, where m takes integer values 1, 2, 3...For simplicity, we will henceforth ignore the index ‘0’ from the zeroth-order terms. In this case, We will also assume dipolar symmetry in the magnetic field, namely B r (π − θ) = − B r (θ), B θ (π − θ) = B θ (θ), and B φ (π − θ) = − B φ (θ). On the equatorial plane in particular, and B r , θ ∼ h 1 B θ / r , B φ, θ ∼ h 2 B θ / r , with h 1 , h 2 ≈ const. In what follows, we will set for