General Relativity

Governing equations

Let g_μν be the components of a stationary metric with time coordinate x⁰, so ∂₀g_μν = 0. We can define the lapse α = (-g⁰⁰)^-1/2. Then the unit future-pointing timelike normal to surfaces of constant x⁰ has components n_μ = -αδ_0μ. Projection onto the spatial part of the tangent space can be accomplished with the components j_μν = g_μν + n_μn_ν.

When working with general relativity, the primitive variables in Athena++ are taken to be

• comoving rest mass density ρ,
• gas pressure p_gas, and
• spatial components of the projected 4-velocity ũⁱ = jⁱ_μu^μ.

In MHD there will also be the magnetic field components Bⁱ = (★F)ⁱ⁰, where F is the electromagnetic field tensor. Note that these are not the projected variables ℬⁱ = -jⁱ_μn_ν(★F)^μν = αBⁱ.

The following quantities are useful to define:

• projected field components b^μ = u_ν(★F)^νμ, in particular
  • b⁰ = g_iμBⁱu^μ,
  • bⁱ = (Bⁱ + b⁰uⁱ)/u⁰;
• magnetic pressure p_mag = b_μb^μ/2;
• total pressure p_tot = p_gas + p_mag;
• adiabatic index Γ;
• gas enthalpy w_gas = ρ + Γ/(Γ−1) × p_gas;
• magnetic enthalpy w_mag = 2p_mag;
• total enthalpy w_tot = w_gas + w_mag;
• stress-energy components T^μν = w_totu^μu^ν + p_totg^μν − b^μb^ν;
• metric determinant g;
• connection coefficients Γ^σ_μν = 1/2 × g^σλ(∂_μg_νλ + ∂_νg_μλ − ∂_λg_μν).

The evolution equations are

• ∂₀ ((-g)^1/2ρu⁰) + ∂_j ((-g)^1/2ρu^j) = 0,
• ∂₀ ((-g)^1/2T⁰_μ) + ∂_j((-g)^1/2T^j_μ) = (-g)^1/2T^ν_σΓ^σ_μν,
• ∂₀ ((-g)^1/2(★F)ⁱ⁰) + ∂_j ((-g)^1/2(★F)^ij) = 0.

Athena++ considers the conserved variables to be

• conserved density D = ρu⁰,
• energy density E = T⁰₀, and
• momentum density M_i = T⁰_i,

as well as the magnetic fields Bⁱ. In particular, the factors of (-g)^1/2 are not included in the output data, nor should they be included when setting conserved variables in problem generators.