## 4. Testing approximative equations
As mentioned above, the pseudo-Newtonian potential has been used
frequently in various situations in which general relativistic effects
on the motion of material (either test particles or fluids) are
essential but exact calculations are too difficult. It is however
impossible to estimate, It is quite straightforward to check whether the pseudo-Newtonian potential is suitable for treating the motion of test particles and fluids when one deals with a spherically symmetric system. What appears more difficult is to develop an analogous pseudo-Kerr theory, describing the relativistic effects near a rotating compact object. Now we will propose a systematic approach to estimate the quality of a particular model by computing a sufficiently large number of trajectories with different initial conditions and comparing results with exact calculations of geodesic motion. The test we suggest determines the rate of divergence of trajectories (as given by approximative versus exact equations). We propose that this type of check should be carried out for each particular set of approximative equations before they are applied to astrophysical situations. (Until now, only rather restricted tests on a relatively small number of trajectories have been applied. As a consequence, one cannot be sure about predictions based on approximative equations of such models because there is little or no control of their overall precision.) For illustration, we then submit the approximative equations of Sect. 3 to our test. ## 4.1. Criterion of accuracy of approximative solutionsOur criterion is motivated by the definition of the Lyapunov characteristic coefficients which have been introduced in order to characterize chaotic systems (see, e.g., Chapt. 5 of Lichtenberg & Lieberman (1983)). The most important features of the Lyapunov coefficients relevant for the present work are summarized in the Appendix. We will now introduce a parameter which is analogous to the maximum
Lyapunov characteristic exponent .
One should however realize the basic difference between the standard
formulation of the problem of chaotic system and our present
situation. While characterizes the
rate of divergence of two neighbouring (initially close to each other)
trajectories, in the present work one always starts with In analogy with standard definition (A.2), we introduce a critical parameter with Now: -
(i) two particles start with identical initial conditions; the first one evolves according to geodesic equations (1)-(3), while the second one according to approximative equations (Eqs. (20)-(22), for example); -
(ii) is determined as a time interval during which the two trajectories remain close to each other and the initial separation is an arbitrary small number (typically, is of the order of the Keplerian orbital period at the initial radius); -
(iii) Eq. (35) is evaluated and it is determined, numerically, whether convergence has been reached for large *n*with a pre-determined accuracy. The final value, , characterizes the rate of divergence of the two trajectories with the same initial conditions: corresponds to separation increasing exponentially while corresponds to a polynomial (i.e. much slower) increase of the separation.
In order to estimate the accuracy of the approximative equations,
one needs to follow a large number of trajectories with randomly
chosen initial conditions. By averaging over
corresponding to different initial
conditions, one obtains a value which shows whether the approximative
equations describe, ## 4.2. An example: test of the pseudo-Kerr equationsWe submitted our approximative equations to the above-described test on . Relative position of the two test particles is determined by difference equations: with The suffix "Kerr" indicates
that the trajectory is a geodesic in the Kerr spacetime while the
suffix "approx" corresponds to pseudo-Newtonian equations. We should
stress in this place that comparing processes in different spacetimes
We have integrated difference Eqs. (36) using the Bulirsch-Stoer scheme (Press et al. 1994). Initial conditions cover the parameter space of Typically, we set . As expected, most of the particles with escape quickly to large distances where both the exact and approximative equations give identical results. It is thus relevant to investigate trajectories with lower energies which often make a number of revolutions around the black hole. Some of these trajectories tend to diverge in a chaotic-type manner, as we will see below. Each run, characterized by the value of , resulted in about values of . For each set of initial conditions (37) within the run, the separation of the two corresponding trajectories was periodically rescaled down to a pre-determined value after a fixed interval . We have typically chosen which is comparable to the Keplerian orbital period near the horizon. The integration was terminated when one of the following conditions had been satisfied: (i) converged to a finite positive value (numerically, we checked that the relative change of during the last 30 rescalings did not exceed 0.5 %); (ii) the upper limit of on the number of rescalings was exceeded, while was oscillating or decreasing monotonically to negative values; (iii) the trajectory was captured by a black hole, The case (iii) was excluded from further consideration. Fig. 1 illustrates the evolution of for two sets of initial conditions which both fall under item (i) above. Apparently, both cases shown there converge to positive , although the numbers of rescalings were different. In this respect, it is interesting to note that the separation of nearby geodesics in the Kerr spacetime never increases exponentially. This fact is a consequence of the existence of the fourth (Carter's) constant of motion, as discussed in Karas & Vokrouhlický (1992).
Next we illustrate the mean value
as a function of the constants of motion,
© European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |