### 6.2 Characteristic fields in the 3+1 conservative Eulerian formulation of Section 2.1.3

This section collects all information concerning the characteristic structure of the general-relativistic
hydrodynamics equations in the Eulerian formulation of Section 2.1.3. As explained in Section 3.1.2, this
information is necessary in order to implement approximate Riemann solvers in HRSC finite-difference
schemes.
We present only the characteristic speeds and fields corresponding to the -direction. Equivalent
expressions for the other two directions can be obtained easily from symmetry considerations. The
characteristic speeds (eigenvalues) of the system are given by:

where denotes the local sound speed, which can be obtained from , with and
. We note that the Minkowskian limit of these expressions is recovered properly (see [101]) as well
as the Newtonian one (, ).
A complete set of right eigenvectors is given by (superscript denotes transpose):

where the following auxiliary quantities are used:
Finally, a complete set of left eigenvectors is given by:
where the following relations and auxiliary quantities have been used:
These two sets of eigenfields reduce to the corresponding ones in the Minkowskian limit [101].