COMPUTATIONS IN GENERAL RELATIVITY SEMINAR AND DISCUSSION 9:00 – 11:00 a.m. Thursdays

1. COMPUTATIONS IN GENERAL RELATIVITY • SEMINAR AND DISCUSSION • 9:00 – 11:00 a.m. • Thursdays • Spring Semester 2002 • Technology Hall – N276 • Observation Data [Bridle, 1994] , 1-17 • Relativity [Ohanian and Ruffini, 1994] , 18-60 • Plasma Physics [Stix, 1992]; Black Holes[Thorn et al,1986] , 61-89 • Computational Issues [Chung, 2002] , 90-92 • Neutron Star Magnetospheres [Michel, 1991] , 93-106 • Physics of Black Hole Gravitohydromagnetics [Punsly, 2001] , 107-186 a

2. COMPUTATIONS IN GENERAL RELATIVITY • No. Date _____________________ Subject*_______________________________________ • 1 24-Jan Introduction • 2 31-Jan Relativistic plasma physics, discussion, and numerical simulation • 3 7-Feb Particle trajectories in the ergosphere, discussion, and numerical simulation • 4 14-Feb Vacuum electrodynamics, discussion, and numerical simulation • 21-Feb The horizon electromagnetic boundary condition, discussion, and numerical simulation • 28-Feb Magnetically dominated time-stationary MHD jets, discussion, and numerical simulation • 7-Mar Winds and waves in ergosphere, discussion, and numerical simulation • 8 14-Mar Ergosphere driven winds, discussion, and numerical simulation • 9 21-Mar Ergosphere disk dynamics, discussion, and numerical simulation • 10 4-Apr Winds from event horizon magnetospheres, discussion, and numerical simulation • 11 11-Apr Winds from event horizon magnetospheres, discussion, and numerical simulation • 12 18-Apr Extragalactic radio sources, discussion, and numerical simulation • 13 25-Apr Non-pulsed black holes, discussion, and numerical simulation • 14 2-May Non-pulsed black holes, discussion, and numerical simulation • *Contributed seminar topics are not necessarily related to the main subjects. Please provide hard copy information of your talk. We need at least one volunteer at each session. Contributed Seminar (15-40 min.)

3. Trapped In a Black Hole Speaking of black holes, here is my story: Sometime ago, I was attracted to a black hole by supergravity and superstrings, and sucked hopelessly deep into a singularity. I was desperate, trying to get out. I had heard that black holes occasionally leak out some radiation, contrary to the story that what goes in never gets out, not even light. To get out, you must have the right frequency and right wave length. Fortunately I had a pocket size supercomputer and found the right wave length. That’s how I was able to escape from the deadly hell and here I am back on earth, waking up from the bad dream. So I made up my mind to go back and investigate the monster. The rumor is that black holes are hanging around gobbling up everything they can get their hands on. According to some reliable sources, however, that’s not always so. There are some friendly black holes. They conceal the secrets of the Universe; but most importantly, they contain the most precious treasure called “antimatter,” which disappeared after the Big Bang, but is probably hidden in the fifth dimension. This is the gold mine of the black hole. Whoever gets this antimatter is to strike rich overnight, by selling it to NASA for next generation rocket propulsion. Did you know there are six more dimensions hidden somewhere, making a total of eleven dimensions? No telling how many more treasures are hidden behind them. So it is worth studying the black hole. This is a treasure hunting--not on the bottom of the ocean--but inside the black hole. Unfortunately, you have to do so much more work, so many extra studies. You need to have a working knowledge in plasma physics, magnetohydrodynamics, quantum mechanics, quantum field theory, particle physics, string theory, shock waves, turbulence, radiation and, most important, general relativity. This is because all of these physics reside in the black hole. The Large Hadron Accelerator being built in Geneva, Switzerland, to be completed in 2005, can hardly duplicate the mighty black hole.  

4. You need to have a tremendously powerful telescope gazing right down to the black hole, billions of light years away, to find out what’s going on in there. If you don’t have such a telescope, then what other choices do you have? Numerical simulations in the comfort of your room. But, numerical simulations of the universe are not a trivial task. Observations and measurements in astrophysics have been in progress for the last six centuries, but numerical simulations began only thirty years ago. I attended the first UK Computational Astrophysics Conference at University of Leicester, several months ago. Many of the attendees were postdocs and graduate students. One of them sent me some movies simulating the dark matter halo and a neutron star merging into a black hole. These were his Ph. D. dissertation. ---movies here---- (1) Numerical simulation of the formation and evolution of galaxies dominated by cold dark matter, using an N-body simulation of 30 million particles for SPH with inviscid gas physics. A zoom into the center of very high resolution dark matter halo is shown, which is rotated while the zoom takes place. (2) This is a neutron star merging into a black hole, forming the accretion disk. No viscous effects are taken into account. Simulations are based on the 3+1 formulation with general relativity. No turbulence, MHD, or radiation is included in this analysis. No jet formations were observed due to crude approximations made in this simulation. Curvature distortions due to angular momentum were not taken into account. Earlier this month I attended the 199th AAS meeting in Washington DC. I saw many people making presentations of the results of their observations and measurements. Side by side they showed some pretty pictures of someone else’s numerical simulations, but they seemed to have no idea how many simplifications had to be made for the numerical computations to work. What if both measurements and computations are wrong? This is why we are here to study what we can do to improve our skills to explore the universe, specifically to study physics, mathematics, and numerical simulations for the next fourteen weeks. I hope we can stick around together to the end, exchanging ideas, sharing knowledge, searching for truth. So stay tuned.

7. The hot, turbulent plasma that is distributed in a region over 700,000 light years in extent in the FRII radio galaxy, 3C 353, is a signature of the violent world of black hole gravitohydromagnetics. The VLA image is provided courtesy of Alan Bridle.

8. This image is an overlay of the radio (red) and optical (blue) emission that shows jets of magnetized relativistic plasma emerging from the center of the elliptical galaxy in the radio source 3C 31. This picture was generously provided by Alan Bridle.

9. Fig. 1.1. An HST image of the central disk in the elliptical galaxy NGC 4261. Note the bright central feature, possible accretion disk radiation from the region of the active nucleus shining through the dusty gaseous disk. The disk is approximately 250 pc across and a gas kinematical estimate of the central black hole mass is 4.5 X 108MΘ. Photograph provided courtesy of Laura Ferrarase.

10. Fig. 1.2. Inserts of the large scale FR I radio structure of NGC 4261 and the small parsec scale VLBA jet that appears to emanate from the bright spot in the center of the disk (which is featured more prominently in Fig. 1.1). Photograph provided courtesy of Laura Ferrarese.

11. Fig. 1.3. The central disk of the elliptical galaxy NGC 7052 is revealed in this HST image. The disk is 1000 pc in diameter and the orbital kinematics imply a central black hole mass of 3x108MΘ. Notice the bright central region that shines through the disk as in NGC 4261. This is a weak radio source and the VLA jet is misaligned with the symmetry axis of the disk, as in NGC 4261. The photograph is provided courtesy of Roeland van der Marel.

12. Fig. 1.4. The optical jet is emanating from the center of the inner disk in this deep HST image of M87. The disk is 20 pc across and the orbital motion indicates a central black hole mass of 3x109MΘ. The photograph is provided courtesy of Holland Ford.

13. Fig. 1.5. A 5 GHz deep VLA image of a prototypical FR I radio source 3C 296. The jets are very bright compared to the diffuse lob emission. Image provided courtesy of Alan Bridle.

14. Fig. 1.6. A deep VLA image of Cygnus A at 5 GHz. The lobes are separated by 180 kpc (HO = 55km/sec/Mpc,qO = 0). Notice the strong hot spots at the end of each lobe where most of the luminosity resides. A highly collimated low surface brightness jet extends into the eastern lobe from a faint radio core. There are suggestions of a counter jet in the image. The counter jet is more pronounced in Fig. 1.10. The VLA image was provided courtesy of Rick Perley.

15. Fig. 1.7. A deep 5 GHz VLA image of the radio loud quasar 3C 175. Notice the morphological similarity to Cygnus A. The jet is more pronounced relative to the lobe emission than Cygnus A, and there is no hint of a counter jet. This is anecdotal evidence for mildly relativistic flows in kiloparsec scale jets. Image provided courtesy of Alan Bridle.

16. Fig. 1.8. This deep 5 GHz VLA image of the FR II radio galaxy 3C 219 shows a strong jet and a knot in a counter jet. It is overlaid on the diffuse (blue) optical image of the host elliptical galaxy. Image provided courtesy of Alan Bridle.

17. Fig. 1.10. The jet in Cygnus A is mapped from scales on the order of 50 kpc down to less than a light year in this series of inserts. The VLBI maps indicate that the central engine is less than a light year in diameter. The images are from Krichbaum et al. 1998.

18. Computations in General Relativity Our Immediate goal is to determine what causes jets to emerge from compact objects. To do this, we must model the black hole physics. This research will require: (1) Combine the quantum gravity field theory into general relativity, (2) Solve the resulting governing equations numerically on the computer. But the quantum field theory is incomplete, some difficult problems unresolved. Numerical solutions may help in revising existing theories and redeveloping more perfect theories by examining the numerical results. Eventually, we may be able to help extrapolate back to the origin of the Universe and help predict the fate of the Universe into a distant future.

19. Past Achievements and Future Goals • Curved spacetime geometries have been computed numerically using standard CFD schemes. • Cosmological singularities have been numerically modeled using the Mixmaster or Gowdy Cosmology Model. • Quantum gravity equations have been numerically solved. • The above three problems are involved in the black hole physics that may hold the secrets of the universe. • Can we numerically simulate any and all of the black hole physics ? This is what we wish to explore. • (6) Can we someday model the entire universe with all the physics taken into account? This is what we wish to explore.

20. Computations in General Relativity for Black Holes with Singularities • Particle Physics (Nachtmann, 1990), • Quantum Mechanics (Hecht, 2000), • Quantum Field Theory (Kaku, 1993), • Plasma Physics and MHD (in Punsly, 2001), String Theory, Vol 1&2 (Polchenski, 1998) • Celestial Mechanics (Taft, 1985) • Gravitation and Spacetime (Ohanian & Ruffini, 1994) • Physics of Black Holes (Novikov & Frolov, 1989) • Cosmology and Particle Astrophysics (Bergström & Goobar, 1999) • Turbulence (Lesieur, 1997) • Shock Waves and Radiation (in Miharas & Miharas, 1984) • Flat and Curved Spacetimes (Ellis and Williams, 1988) • Relativity and Scientific Computing (Hehl, Puntigam & Ruder, 1996) • Computational Fluid Dynamics (Chung, 2002)

21. Why do we need the theory of relativity in Particle Physics? Consider a typical reaction: P+P—>P+P+Π++Π- • The kinetic energy of the original protons is converted into rest mass of new particles • The greater the energy of the protons, the larger the number and mass of the particles that can be produced • Collisions between fast, relativistic particles are important in this process

22. Why do we need the theory of relativity in quantum gravity, supersymmetry, supergravity and superstrings? • Equivalence Principle: The laws of physics in a gravitational field are identical to those in a local accelerating frame --Einstein • Construct a theory that is invariant under general coordinate transformations, that is, a theory in which one can choose coordinates such that the gravitational field vanishes locally • Construct tensors of arbitrary rank or indices and their covariant derivatives

23. ROLE PLAYED BY QUANTUM EFFECTS IN BLACK HOLE PHYSICS: • Quantum effects significant for black holes of mass smaller than solar mass with a spacetime singularity • Virtual particles are constantly created, interact with one another, and are annihilated in a vacuum • In an external field, some virtual particles may acquire sufficient energy for becoming real • The result is the effect of quantum creation of particles from vacuum by an external field

24. Special Relativity in Flat Space ds2=gαβdxαdxβ α,β=(0,1,2,3) gαβ=metric tensor (Minkowski tensor) Spacelike metric ds2 = -(cdt)2 +dx2 + dy2 + dz2 gαβ Timelike metric ds2 = (cdt)2 – dx2 – dy2 – dz2 gαβ

32. GENERAL RELATIVITY IN CURVED SPACETIME

33. = Curvilinear coordinates gα = Covariant tangent vectors (undeformed) gα = Contravariant tangent vectors (undeformed) Initial curvature Deformed curvature due to angular momentum and torsional deformations Gα= Covariant tangent vectors (deformed) Gα= Contravariant tangent vectors (deformed) xα = Reference Cartesian coordinates gα . gβ = δβ α

34. Covariant metric tensor (undeformed) Covariant metric tensor (deformed) Contravariant metric tensor (undeformed) Contravariant metric tensor (deformed) Squared line segments (undeformed ) Squared line segments (deformed)

35. General Relativity Riemann curvature tensor in general relativity Ricci tensor Einstein equation Torsionally strained line segments gα;βα = Curvature of tangent vectors (vector curvature) = scalar curvature

36. THE SCHWARZSCHILD SOLUTION • This is applicable to the gravitational collapse of any • nonrotating, electrically neutral star. • For rotating black holes in a magnetic field we require • Kerr-Neumann spacetime, Boyer-Lindquist coordinates • (Reissner-Nordström Solution).

37. THE KERR BLACK HOLE SOLUTION ۰ Boyer-Lindquist Coordinates ۰ The available extractible energy is the reducible energy (Mred)max = 0.29M

38. BLACK HOLES AND GRAVITATIONAL COLLAPSE • Large relativistic effects are found in the gravitational field in the neighborhood of an extremely compact mass, . • For , the gravitational fields are so strong that nothing can escape from this grip.

40. Null Surface t/rs Worldlines by integrating r/rs r<rs Spacelike r>rx Timelike Fig. 8.3 The forward light cones near and inside a black hole. As r -> , the light cone assumes its usual shape and direction, that is, dr/dt = +/- 1. The curve AB, BC is the worldline of an ingoing light signal.

42. Timelike Spacelike Interior region Exterior region (No singularity at r = rs) Fig. 8.5 The maximal Schwarzschild spacetime in Kruskal coordinates.

44. HORIZONS AND SINGULARITIES IN THE ROTATING BLACK HOLES

47. THE MAXIMAL KERR GEOMETRY The worldline begins outside of the black hole, crosses the horizons r = r+ and r = r- , and approaches near r = 0, moving in the outward direction and approaching the surface r = r- from the inside. The surface r = r- belongs to a white hole rather than a black hole. It is one way out rather than one way in. This implies an infinite sequence of universes. Fig 8.14 A possible worldline for a particle moving along the axis of a black hole.