Rational numbers and black hole formation paradox Igor V. Volovich

1. Rational numbers and black hole formation paradox Igor V. Volovich Steklov Mathematical Institute, Moscow International Workshop "p -Adic Methods for Modelling of Complex Systems“ ZiF, Bielefeld University, Bielefeld, Germany April 15–19, 2013

2. PLAN Rational, real and p-adic numbers in natural sciences Non-Newtonian Classical Mechanics Functional (random) quantum theory Functional Probabilistic General Relativity and Black Holes

3. I. V. , “Randomness in classical mechanics and quantum mechanics”,  Found. Phys., 41:3 (2011), 516. • M. Ohya, I. Volovich, “Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems”, Springer, 2011. I.V. “Bogoliubov equations and functional mechanics”, TMF (2012) 35, 17.

4. Newton Equation Phase space (q,p), Hamilton dynamical flow

5. Newton`s approach: • Empty space • (mathematical, real numbers) and • point-like (mathematical) particles. • Reductionism to mechanics: • In physics, biology, economy, politics (freedom, liberty, market agents,…)

6. Why Newton`s mechanics can not be true? • Newton`s equations of motions use real numbers while one can observe only rationals. • Classical uncertainty relations • Time irreversibility problem • Singularities in general relativity

7. Real Numbers • A real number is an infinite series, which is unphysical:

8. Classical Uncertainty Relations

9. Time Irreversibility Problem The time irreversibility problem is the problem of how to explain the irreversible behaviour of macroscopic systems from the time-symmetric microscopiclaws: Newton, Schrodinger Eqs –- reversible Navier-Stokes, Boltzmann, diffusion, Entropy increasing --- irreversible

10. Time Irreversibility Problem Boltzmann, Maxwell, Poincar´e, Bogolyubov, Kolmogorov, von Neumann, Landau, Prigogine, Feynman, Kozlov,… Poincare, Landau, Prigogine, Ginzburg, Feynman: Problem is open. We will never solve it (Poincare) Quantum measurement? (Landau) Black hole information loss (Hawking). Lebowitz, Goldstein, Bricmont: Problem was solved by Boltzmann I.Volovich

11. Boltzmann`s answers to: Loschmidt: statistical viewpoint Poincare—Zermelo: extremely long Poincare recurrence time Coarse graining • Not convincing, since the time reversable symmetry is unbocken.

12. Ergodic Theory • Boltzmann, Poincare, Hopf, Kolmogorov, Anosov, Arnold, Sinai, Kozlov,…: • Ergodicity, mixing,… for deterministic mechanical and dynamical systems • However, the time symmetry is unbrocken.

13. Proposal of how to solve the time irreversibility problem • Probabilistic formulation of equations of mechanics • Violation of time symmetry at microlevel

14. Singularities in general relativity • Penrose – Hawking theorems • Black holes. • Black hole information loss (Hawking). • Black hole formation paradox. • Cosmology. Big Bang.

15. Try to solve these problems by developing a new, non-Newtonian mechanics. • This approach was motivated by p-adic mathematical physics

16. Functional formulation of non-Newtonian classical mechanics • Here the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space. The fundamental equation of the microscopic dynamics in the proposed "functional" approach is not the Newton equation but the Liouville or Fokker-Planck-Kolmogorov (Langevin, Smoluchowski) equation for the distribution function of the single particle. • .

17. States and Observables inFunctional Classical Mechanics

18. States and Observables inFunctional Classical Mechanics Not a generalized function. Mean values are rational numbers

19. Fundamental Equation inNon-Newtonian Classical Mechanics Looks like the Liouville equation which is used in statistical physics todescribe a gas of particles. But here we use it to describe a single particle.(moon,…). Or Fokker-Planck eq. Or… Instead of Newton equation. No trajectories!

20. Non-Newtonian dynamical systems

21. Free Motion

22. Cauchy Problem for Free Particle

23. Average Value and Dispersion

24. Newton`s Equation for Average

25. Delocalization

26. Comparison with Quantum Mechanics Wigner

27. Liouville and Newton. Characteristics

28. Corrections to Newton`s EquationsNon-Newtonian Mechanics

29. Corrections to Newton`s Equations

30. Corrections to Newton`s Equations

31. Corrections E. Piskovsky, A. Mikhailov, O.Groshev, Problem: h is fixed: Inoue, Ohya, I.V. Log dependence on time

33. Stability of the Solar System? • Kepler, Newton, Laplace, Poincare,…,Kolmogorov, Arnold,…---stability? • Solar System is unstable, chaotic (J.Laskar) • an error as small as 15 metres in measuring the position of the Earth today would make it impossible to predict where the Earth would be in its orbit in just over 100 million years' time. • Asteroids. Chelyabinsk. Apophis, 2029, 2036.

34. Fokker-Planck-Kolmogorov versus Newton

35. Einstein equations

36. BLACK HOLES in GR. Schwarzschild solution Asymptoticaly flat Birkhoff's theorem:Schwarzschild solution is the unique spherically symmetric vacuum solution Singularity

38. Gravitational collapse Oppenheimer – Snyder solution Uniform perfect sphere of fluid of zero pressure (dust) The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius (infinite time is required to each event horizon).

39. Applications of this non-Newtonian functional mechanics to the black hole formation paradox : It is believed that observations indicate that many galaxies, including the Milky Way, contain supermassive black holes at their centers. However there is a problem that for the formation of a black hole an infinite time is required as can be seen by an external observer. And that is in contradiction with the finite time of existing of the Universe.

40. Fixed classical spacetime? • A fixed classical background spacetime does not exists (Kaluza—Klein, Strings, Branes). There is a set of classical universes and a probability distribution which satisfies the Liouville equation (not Wheeler—De Witt). Stochastic inflation?

41. Functional General Relativity • Fixed background . Geodesics in functional mechanics Probability distributions of spacetimes • No fixed classical background spacetime. • No Penrose—Hawking singularity theorems • Stochastic geometry?

42. Geodesics in Functional Mechanics

43. Example

44. BH Complementarity or Firewalls? `t Hooft, Preskill, Susskind: `BH complementarity. AdS/CFTholography. Postulate: The process of formation and Hawking evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. Almheiri, Marolf, Polchinski, and Sully: once a black hole has radiated more than half its initial entropy (the Page time), the horizon is re- placed by a “firewall" at which infalling observers burn up. Non-deterministic functional general relativity. Probabilistic scenario.

45. Newton`s approach: Empty space (vacuum) and point particles. • Reductionism: For physics, biology economy, politics (freedom, liberty,…) • This non-Newtonian approach: • There is no empty space. • No classical determinism. • Probability distribution. Collective phenomena.Subjective.

46. Single particle (moon,…)

47. Conclusions Arbitrary real numbers are unobservable. One can observe only rational numbers. Non-Newtonian classical mechanics: distribution function instead of individual trajectories. Fundamental equation: Liouville or FPK for a single particle. Irreversibility. Newton equation—approximate for average values. Corrections to Newton`s law. Stochastic geometry, general relativity,…

49. Arbitrary real numbers are unobservable. Therefore the widely used modeling of physical phenomena by using differential equations, which was introduced by Newton, does not have an immediate physical meaning. What to do? One suggests that the physical meaning should be attributed not to an individual trajectory in the phase space but only to the probability distribution function. This approach was motivated by p-adic mathematical physics and the time irreversibility problem.

50. Even for the single particle the fundamental dynamical equation in the proposed "functional" approach is not the Newton equation. In functional mechanics the basic equation is the Liouville equation or the Fokker - Planck - Kolmogorov equation. Langevin eq. for a single particle. Random parameters. The Newton equation in functional mechanics appears as an approximate equation for the expected values of the position and momenta. Corrections.