Special relativistic spherically symmetric Lagrangean hydrodynamics from General Relativity

We re-derive hydrodynamical equations in General Relativity (GR) in the comoving reference frame for spherical symmetry and obtain from them the well-known but not explicitly derived Lagrangean equations in Special Relativity (SR), that is, for the case of weak gravitational fields. Explicit formulae are presented which relate General Relativistic independent variables, Lagrangean mass coordinate and variables in the lab-frame. Conversion of one set of variables into another requires knowing the solution to the set of equations. These formulae allow one to translate the solution to the exact set in General Relativity into the form in which Special Relativistic solutions are usually obtained. This is applicable for comparison of SR-numerical simulations of collapses, GRBs and Supernovae explosions with more precise GR-simulations. e-mail: [email protected] 1 General relativistic hydrodynamics in the comoving reference frame A detailed derivation of the hydrodynamical equations for the spherically symmetric motion in 3+1 dimensions can be found in the paper [5] (Liebendörfer, Mezzacappa & Thielemann), along with relativistic Boltzmann equation and its forms. However, the relation of General Relativistic hydrodynamical equations with Special Relativistic ones, such as used without derivation by Daigne & Mochkovitch [2], has remained unclear. We fill this blank with a comprehensive derivation of the basic equations in [2] starting from the very Einstein equations. We will work in the reference frame moving with (baryonic) matter and suppose that the metric tensor has been diagonalised, adopting the following (already spherically symmetric) form of the interval: ds = edt − edR − rdΩ, (1) where dΩ ≡ dθ + sin θdφ is a solid angle differential. The quantities φ, λ, r are the functions of independent variables R, t, which are correspondingly the arbitrarily chosen radial and universal time coordinates. We have already set c to be 1. Jacobian of these 4-coordinates is √−g = eφ+λr2| sin θ|, and jacobian of spatial coordinates equals √ γ = eλr2| sin θ|, hence, a 3-dimensional infinitesimal volume element integrated over φ, θ equals dV = e4πrdR, (2) and r(R, t) has the geometrical sense that the length of circle of radius r is 2πr. Now we have to put some physical sense to the coordinate R in order to fix its definition (one can apply a conversion R → R̃ staying in the comoving frame and only changing λ → λ̃, so R is not fixed). We relate this coordinate with the baryon density, so that dR/G ≡ dmb is the baryon mass inside the layer dR: ρbdV = ρbe 4πrdR = dmb = 1 G dR. (3) In the following we put G = 1 (so that mass and length are of the same dimension) and denote by prime the ∂/∂R derivative. In our hands are the baryon number conservation law (ρbu );μ = 0 and the Einstein equations Rμν − 1 2 Rgμν = 8πTμν . In our comoving reference frame the 4-velocity equals u = (e−φ,~0), and using the formula for Γνμ = ∂ln √−g ∂xν we get (ρbu );μ = (ρbu ),μ + Γ μ νμρbu ν = e−φ(ρ̇+ ρλ̇ + 2ρ ṙ r ) = 0, that is, ρ̇ ρ = −(λ̇ + 2 ṙ r ). (4) (We omit the index b of ρb and mb). The Ricci tensor for the diagonal metrics gik = eie δik, ei = (1,−1,−1,−1), can be easily calculated using formulae Rii = ∑ l 6=i [ Fi,iFl,i − F 2 l,i − Fl,i,i + eieleil(Fl,lFi,l − F 2 i,l − Fi,l,l − Fi,l ∑