Summary

1. Summary N-body problem Globular Clusters Jackiw-Teitelboim Theory Poincare plots Chaotic Observables Symbolic Dynamics Some quick math Different Orbits Conclusions

2. N-body problem Method of describing N-body gravitational interactions Only N=2 is known in closed form (Newtonian) N>2 can only be approximated numerically In general relativity N=2 is still not known in closed form Applications of this problem are quite necessary for cosmic study.

3. Globular clusters One mentioned application of N-body problem Newtonian system One defines a globular cluster as gravitationally bound concentrations of approximately 1E4 – 1E6 stars within a volume of 10-100 light years radius

4. Relativistic 1D Self Gravitation (ROGS) This paper tackles ROGS' in 1+1 spacetime This models 3+1 spacetime using R=T theory That is it includes dilaton theory This theory is consistent with nonrelativistic theory Also reduces to Jackiw-Teitelboim Theory

5. Jackiw-Teitelboim Theory 2D action for gravity coupled to matter couples a dilatonic scalar field to the curvature

6. Poincare Section A surface of section as a way of presenting a trajectory of n-dimensional phase space in an (n-1)-dimensional space. One selects a phase element to be constant and plotting the values of the other elements each time the selected element has the desired value, an intersection surface is obtained.

7. Chaotic Observables Winding Number – a method of tracking a trajectory around phase space If the winding number is not rational we have a chaotic orbit

8. Winding Number “R” • The Winding Number is the average rotation angle per drive cycle. • The black line in the above picture displays a winding number of 2/5, since it is rational the trajectory is periodic • The winding number is defined as the asymptotic limit over the entire trajectory

9. Chaotic Observables Lyapunov Exponent The Lyapunov exponent (or index) measures the rate of divergence between a trajectory with 2 different initial conditions

10. Lyapunov Exponent • The Lyapunov index measures the rate of divergence between a trajectory with 2 different initial conditions l > 0 Divergent l = 0 Unchanging l < 0 Convergent

11. Logistic Equation and Maps L

12. Symbolic Dynamics A novel method of attempting to find periodic orbits One partitions the return map or poincare section and labels it appropriately Then one observes the location of the points during a cycle or orbit If the orbit is periodic or quasiperiodic one will receive a perfectly periodic set of symbols describing the trajectory

13. Symbolic Dynamics The partitioning of the return map A resulting trajectory in symbol space LRLRRRRLR…

14. Symbolic Dynamics a = 3.9 xo = 0.30001 a = 3.9 xo = 0.29999

15. Attractors Chaotic systems are said to have space filling trajectories These trajectories always fall on what are known as chaotic attractors It is a slice through one of these attractors which comprises the Poincare section

16. Periodic, Quasi-Periodic, Chaotic Periodic orbits -- exactly repeat their trajectories with no deviations Quasi-Periodic orbits – exhibit small to large deviations from a perfectly periodic trajectory however when looking at their symbolic dynamics they do exhibit periodic behaviour Chaotic orbits do not ever repeat themselves, they may come very close to repeating

17. Bifurcation Diagrams A simple test for chaos to exist occurs in bifurcation diagrams In regions where one finds single trajectories no chaos is expected

18. 3-body ROGS with L No known nonrelativistic analogue L-- induces expansion or contraction of spacetime competing with gravitational self interaction Large and positive L overcomes gravity but ?loses causality?

19. EoM We start with the well known action:

20. EoM That leaves us with the following equations of motion: And the stress energy for the point masses:

21. Some change of variables Using the ADM formalism, and canonical variables the action may be re-written as: This leads to a longer set of first order field equations Then finally reducing the problem further we get a nice simple action with 2 constraint equations

22. Conjugate Momenta With the Hamiltonian in the action we can calculate the conjugate momenta for the system: p_i = diff(L,x_i) Rearranging the canonical variables and corresponding conjugate momenta we have a system with sixfold symmetry (find this symmetry) Since Z is arbitrary (chooses a plane) and p_Z =0 in the center of inertia frame, we are left with a 4D phase space

23. Potential Well The relativistic potential well is defined as the difference between the Hamiltonian and the relativistic kinetic energy For low momenta the potential wall becomes that of the non-relativistic system

24. Annulus Orbits Particles Never cross same bisector twice in succession (re-word) Their claim is periodic orbits are difficult to find Insert figure 4 and description on page 19

25. Pretzel Orbits Particles oscillate around a bisector corresponding to a stable or quasistable bound subsystem of 2 particles (classical analogue) Found Characteristsic are similar at different energies Change of cosmo const also changes these orbits as they did in the annulus case

26. Chaotic Orbits Particles wander between A and B motion in an irregular fashion (direct quote) Poincare section shows dark regions Chaos exhibits space filling. Claims to occure in transition regions between annuli and pretzel orbits These depend strongly on cosmo const These orbits are hard to find due to sensitivity to IC's (no kidding) Increase/Decrease cosmo const expand/shrink phase space stretch Most significant change occurs when cosmo const goes negative

27. Conclusions Not much was really concluded, general relationships between the chaos exhibited and the cosmological constant were drawn, but nothing quanitative.

28. Comments Looking for periodic orbit theory one can “easily”determine full chaotic constants for the system.