Relativistic Hydrodynamics 2025, 12 - Einstein-Euler Equations

For computable EEQs, an approach is to decompose spacetime as in the 3+1-decomposition of spacetime. Given a constant-time hypersurface Σ(t) of the foliation Σ, introduce a timelike 4-vector n normal to the hypersurface at each event with dual 1-forms Ω parallel to the gradient of the coordinate. The specification of the resulting normal vector n defines the metric associated to each hypersurface. n_μ = (-α, 0, 0, 0) n^&mu = 1/α (1, -β) The line element in 3+1 decomposition is ds² = -(α²β_iβⁱ)dt² + 2β_idxⁱdt + γ_ijdxⁱdx^j ↠ v = (γ⋅u)/(-n⋅u)

The choice of coordinates will narrow down the particulars of any formulation, and can be almost directly inserted into the definition of the line element. A formulation that has the EEQs in terms of purely spatial tensors, integrable forward after applications of constraints is the ADM formulation. It projects the standard covariant derivative onto the space orthogonal to the normal vector. Following the standard Riemann-tensor shenanigans of general relativity, then ∂_tγ_ij = -2αK_ij + L_βγ_ij = -2αK_ij + D_iβ_j + D_jβ_i D_j(K^ij - γ^ijK) = 8πSⁱ

By starting from the weakly hyperbolic formulation of the Maxwell equations and replacing the time evolution of the electric field component, a formulation for the EEQs emerges that is conformal and traceless. For this, a conformal transformation of the 3-metric is imposed, before obtaining the spatial volume element. A conformal factor for the theory then factors into the field definitions. There are several differently problematic formulations of this type, usually chosen depending on which errors one expects during computation.

3+1 formulations have two additional DOF to be specified, before they're closed. These are the slicing condition and spatial shift condition. The choice of these gauge conditions is arbitrary, but should avoid singularities, counteract coordinate distortions and be as computationally inexpensive as possible. The most common slicing condition to approximate this is ∂_tα - β^k∂_kα = -f(α)α²(K - K₀) and the most common shift condition is ∂_tβⁱ - β^j∂_jβⁱ = 3Bⁱ/4 ∂_tBⁱ - B^j∂_jBⁱ = ∂_tΓⁱ - β^j∂_jΓⁱ - ηBⁱ These choices are the "generalized harmonic formulation".