The solution to this problem is self-similar, because it only depends on the two constant states defining the discontinuity and , where , and on the ratio , where and are the initial location of the discontinuity and the time of breakup, respectively. Both in relativistic and classical hydrodynamics the discontinuity decays into two elementary nonlinear waves (shocks or rarefactions) which move in opposite directions towards the initial left and right states. Between these waves two new constant states and (note that and in Figure 1) appear, which are separated from each other by a contact discontinuity moving with the fluid. Accordingly, the time evolution of a Riemann problem can be represented as
As in the Newtonian case, the compressive character of shock waves (density and pressure rise across the shock) allows us to discriminate between shocks () and rarefaction waves ():p is the pressure, and subscripts a and b denote quantities ahead and behind the wave. For the Riemann problem and for and , respectively. Thus, the possible types of decay of an initial discontinuity can be reduced to
Across the contact discontinuity the density exhibits a jump, whereas pressure and normal velocity are continuous (see Figure 1). As in the classical case, the self-similar character of the flow through rarefaction waves and the Rankine–Hugoniot conditions across shocks provide the relations to link the intermediate states () with the corresponding initial states . They also allow one to express the normal fluid flow velocity in the intermediate states ( for the case of an initial discontinuity normal to the axis) as a function of the pressure in these states.
The solution of the Riemann problem consists in finding the intermediate states and , as well as the positions of the waves separating the four states (which only depend on , , , and ). The functions and allow one to determine the functions and , respectively. The pressure and the flow velocity in the intermediate states are then given by the condition
In the case of relativistic hydrodynamics, the major difference to classical hydrodynamics stems from the role of tangential velocities. While in the classical case the decay of the initial discontinuity does not depend on the tangential velocity (which is constant across shock waves and rarefactions), in relativistic calculations the components of the flow velocity are coupled by the presence of the Lorentz factor in the equations. In addition, the specific enthalpy also couples with the tangential velocities, which becomes important in the thermodynamically ultrarelativistic regime.
The functions are defined by
The fact that one Riemann invariant is constant across any rarefaction wave provides the relation needed to derive the function . In differential form, the function reads
Considering that in a Riemann problem the state ahead of the rarefaction wave is known, the integration of Equation (19) allows one to connect the states ahead () and behind the rarefaction wave. Moreover, using Equation (21), the EOS, and the following relation obtained from the constraint , that holds across the rarefaction wave,p. Let us point out that the integration of Equation (19) is along an adiabat of the EOS.
In the limit of zero tangential velocities, , the function g does not contribute. In this limit and in the case of an ideal gas EOS one has. The equation can be then integrated to give 
The family of all states , which can be connected through a shock with a given state ahead of the wave, is determined by the shock jump conditions. One obtainsp. In the case of ideal gas EOS with constant adiabatic index, the post-shock density can be easily eliminated, and the post-shock enthalpy is the (unique) positive root of the quadratic equation 
Finally, the tangential velocities in the post-shock states can be obtained from 
Figure 2 shows the solution of a particular mildly relativistic Riemann problem for different values of the tangential velocity. The crossing point of any two lines in the upper panel gives the pressure and the normal velocity in the intermediate states. The range of possible solutions in the ()-plane is marked by the shaded region. While the pressure in the intermediate state can take any value between and , the normal flow velocity can be arbitrarily close to zero in the case of an extremely relativistic tangential flow. The values of the tangential velocity in the states and are obtained from the value of the corresponding functions at in the lower panel of Figure 2. The influence of initial left and right tangential velocities on the solution of a Riemann problem is enhanced in highly relativistic problems. We have computed the solution of one such problem (see Section 6.2.2 below, Problem 2) for different combinations of and . The initial data are = 103, = 1, = 0; = 10–2, = 1, = 0, and the 9 possible combinations of = 0, 0.9, 0.99. The results are given in Figure 3 and Table 1, and a complete discussion can be found in .
|0.00||0.00||9.16 × 10–2||1.04 × 10+1||1.86 × 10+1||0.960||0.987||–0.816||+0.668|
|0.00||0.90||1.51 × 10 –1||1.46 × 10+1||4.28 × 10+1||0.913||0.973||–0.816||+0.379|
|0.00||0.99||2.89 × 10 –1||4.36 × 10+1||1.27 × 10+2||0.767||0.927||–0.816||–0.132|
|0.90||0.00||5.83 × 10–3||3.44 × 10+0||1.89 × 10–1||0.328||0.452||–0.525||+0.308|
|0.90||0.90||1.49 × 10–2||4.46 × 10+0||9.04 × 10–1||0.319||0.445||–0.525||+0.282|
|0.90||0.99||5.72 × 10–2||7.83 × 10+0||8.48 × 10+0||0.292||0.484||–0.525||+0.197|
|0.99||0.00||1.99 × 10–3||1.91 × 10+0||3.16 × 10–2||0.099||0.208||–0.196||+0.096|
|0.99||0.90||3.80 × 10–3||2.90 × 10+0||9.27 × 10–2||0.098||0.153||–0.196||+0.094|
|0.99||0.99||1.29 × 10–2||4.29 × 10+0||7.06 × 10–1||0.095||0.140||–0.196||+0.085|
Finally, let us note that the procedure to obtain the pressure in the intermediate states is valid for general EOS. Once has been obtained, the remaining state quantities and the complete Riemann solution,RIEMANN (Section 9.4.1) and RIEMANN-VT (Section 9.4.2), which allow one to compute the exact solution of an arbitrary special relativistic Riemann problem for an ideal gas EOS with constant adiabatic index, both with zero and non-zero tangential speeds using the algorithm discussed above.
Solving a Riemann problem involves the solution of an algebraic equation for the pressure (Equation (17)). Moreover, the functional form of this equation depends on the wave pattern under consideration (see expressions (16). In a recent paper , Rezzolla and Zanotti have presented a procedure, suitable for implementation into an exact Riemann solver in one dimension, which removes the ambiguity arising from the wave pattern. That method exploits the fact that the expression for the relative velocity between the two initial states is a (monotonic) function of the unknown pressure, , which determines the wave pattern. Hence, comparing the value of the (special relativistic) relative velocity between the initial left and right states with the values of the limiting relative velocities for the occurrence of the wave patterns (16), one can determine a priori which of the three wave patterns will actually result (see Figure 4). In  the authors extend the above procedure to multi-dimensional flows.
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