### 6.1 Riemann problems in locally-Minkowskian coordinates

In [320], a procedure to integrate the general-relativistic hydrodynamics equations (as formulated in Section 2.1.3), taking advantage of the multitude of Riemann solvers developed in special relativity, was presented. The approach relies on a local change of coordinates in terms of which the spacetime metric is locally Minkowskian. This procedure allows, for 1D problems, the use of the exact solution of the special relativistic Riemann problem [239].

Such a coordinate transformation to locally Minkowskian coordinates at each numerical interface assumes that the solution of the Riemann problem is the one in special relativity and planar symmetry. This last assumption is equivalent to the approach followed in classical fluid dynamics, when using the solution of Riemann problems in slab symmetry for problems in cylindrical or spherical coordinates, as the solution breaks down near the singular points (e.g., the polar axis in cylindrical coordinates). In analogy to classical fluid dynamics, the numerical error depends on the magnitude of the Christoffel symbols, which might be large whenever huge gradients or large temporal variations of the gravitational field are present. Finer grids and improved time-advancing methods will be required in those circumstances.

Following [320], we illustrate the procedure for computing the second flux integral in Equation (76), which we call . We begin by expressing the integral on a basis with and forming an orthonormal basis in the plane orthogonal to with normal to the surface and and tangent to that surface. The vectors of this basis verify with the Minkowski metric (in the following, caret superscripts will refer to vector components in this basis).

Denoting by the coordinates at the center of the interface at time , we introduce the following locally-Minkowskian coordinate system , where the matrix is given by , calculated at . In this system of coordinates the equations of general-relativistic hydrodynamics transform into the equations of special relativistic hydrodynamics, in Cartesian coordinates, but with nonzero sources, and the flux integral reads

(the caret symbol representing the numerical flux in Equation (76) is now removed to avoid confusion) with , where we have taken into account that, in the coordinates , is described by the equation (with ), where the metric elements and are calculated at . Therefore, this surface is not at rest but moves with speed .

At this point, all the theoretical work developed in recent years on special relativistic Riemann solvers can be exploited. The quantity in parentheses in Equation (103) represents the numerical flux across , which can now be calculated by solving the special relativistic Riemann problem defined with the values at the two sides of of two independent thermodynamic variables (namely, the rest mass density and the specific internal energy ) and the components of the velocity in the orthonormal spatial basis ().

Once the Riemann problem has been solved, we can take advantage of the self-similar character of the solution of the Riemann problem, which makes it constant on the surface , simplifying the calculation of the above integral enormously:

where the superscript (*) stands for the value on obtained from the solution of the Riemann problem. Notice that the numerical fluxes correspond to the vector fields , , , , and linearized Riemann solvers provide the numerical fluxes as defined in Equation (103). Thus, the additional relation has to be used for the momentum equations. The integral in the right-hand side of Equation (104) is the area of the surface and can be expressed in terms of the original coordinates as
which can be evaluated for a given metric. The interested reader is directed to [320] for details on the testing and calibration of this procedure.