Numerical Experiments

The first example we would like to show is that of the Precesion of an orbit, which corresponds to the weak field limit. In figure (1.5) we show a geodesic with a $L^2=100$, $r= 1000$, $\dot{r}=0.0$, $d\tau= 100$ and $\tau_f= 1000000$. Most experimental tests of general relativity involve the motion of test particles in the solar system, and hence geodesics of the Schwarzschild metric. Einstein suggested three tests: the deflection of light, the precession of perihelia, and gravitational redshift. The deflection of light is observable in the weak-field limit, andtherefore is not really a good test of the exact form of the Schwarzschild geometry. Observations of this deflection have been performed during eclipses of the Sun, with results which agree with the GR prediction (although it is s not an especially clean experiment).

The precession of perihelia reflects the fact that noncircular orbits are not closed ellipses; to a good approximation they are ellipses which precess, describing a flower pattern.

In the very weak fiel limit (Newtonian physics) we expect a closed elliptical orbit. In figure (3) we show a precessing elliptical geodesic, which corresponds to a strong field limit, and the one in the right is a newtonian orbit.

In figure (1.5) we have two unstable orbits. In the first one the particl falls into the event horizon; in the second one the particle breaks away after completing a few orbits around the compact object in the center. The outcome depends on the small radial velocity we are applying; the one in the right is $0.001$ and the one in the left, $2.5x10^{-6}$, so the first one is larger and that is why the particle goes away.

Next, in figure (1.5) we have another unstable ``runaway'' orbit, and to the right we have another precessing orbit. These precession is appreciated as we approach the strong field regime, and it is one of the main differences with respect to Keplerian classical orbits.

In figure (6) we managed to construct a unstable circular orbit. to the right we appreciate a particle falling into the eventhorizon, where the integrator stops due to the dingularity of the equations in that point.

Plots in (8) show an interesting orbit where the kinetic energy of the particle is not enoguh to escape, and maybe it will follow a precession patter in the future after this ``swing by''. precession pattern; anyway numerical errors will prevent us to study the orbit in the long term. Next to this geodesic we show the only future for a particle inside the horizon: falling into the singularity, trapped forever.

Figure 2: Precession of an orbit in the weak field regime.

Figure 3: A geodesic in the strong field regime (left)and another in the weak limit(right): in the short term is a closed elliptical orbit, but as we can see it is really a precessing geodesic. The particle's mass is $m=0.5$ in Geometric units

Figure 4: Two unstable orbits. In the first one the particl falls into the event horizon; in the second one the particle breaks away after completing a few orbits around the compact object.

Figure 5: The precession of an orbit is evident in the strong field regime (right), and it is one of the main differences with respect to Keplerian classical orbits.

Figure 6: These circular orbit shown to the left is was carefully chossen, and it survives due to these numerical fine tunning.

Figure 7: Strong field regime examples of orbits.

Figure 8: .More examples of strong orbits.

Figure 9: Plot of the radius vs proper time of the particle. Inspired by spacetime diagrams we were looking for some patters, but they do not seem very interesting.