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
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
• 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 biosystems”, Springer, 2011.
I.V. “Bogoliubov equations and
Newton Equation
2
d
m 2 x  F ( x),
dt
x  x(t ),
Real numbers
( M  R , t )
2n
Phase space (q,p), Hamilton dynamical flow
•
•
•
•
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,…)
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
Real Numbers
• A real number is an infinite series,
which is unphysical:
1
t   an n , an  0,1,...,9.
10
n
2
d
m 2 x(t )  F
dt
Classical Uncertainty Relations
q  0, p  0
t  0
Time Irreversibility Problem
The time irreversibility problem is the problem
of how to explain the irreversible behaviour
of macroscopic systems from
the time-symmetric microscopic laws: t  t
Newton, Schrodinger Eqs –- reversible
Navier-Stokes, Boltzmann, diffusion,
Entropy increasing --- irreversible
u
 u.
t
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
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.
Ergodic Theory
• Boltzmann, Poincare, Hopf,
Kolmogorov, Anosov, Arnold,
Sinai, Kozlov,…:
• Ergodicity, mixing,… for
deterministic mechanical and
dynamical systems
• However, the time symmetry is
unbrocken.
Proposal of how to solve the time
irreversibility problem
• Probabilistic formulation of equations
of mechanics
• Violation of time symmetry at
microlevel
Singularities in general relativity
• Penrose – Hawking theorems
• Black holes.
• Black hole information loss (Hawking).
• Black hole formation paradox.
• Cosmology. Big Bang.
• Try to solve these
problems by developing a
new, non-Newtonian
mechanics.
• This approach was motivated
by p-adic mathematical physics
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.
States and Observables in
Functional Classical Mechanics
States and Observables in
Functional Classical Mechanics
Not a generalized function.
Mean values are rational numbers
Fundamental Equation in
Non-Newtonian Classical Mechanics
Looks like the Liouville equation which is used
in statistical physics to describe 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!
Non-Newtonian dynamical systems
( M ,  t ,  , A)
M - space (rational, real, p - adic),
 t - automorphi sm,
 - very special measure,
A - very special observable s.
Study special mean values.
Free Motion
Cauchy Problem for Free Particle
Average Value and Dispersion
Newton`s Equation for Average
Delocalization
Comparison with Quantum
Mechanics
Wigner
Liouville and Newton.
Characteristics
Corrections to Newton`s Equations
Non-Newtonian Mechanics
Corrections to Newton`s Equations
Corrections to Newton`s Equations
Corrections
E. Piskovsky, A. Mikhailov, O.Groshev,
Problem: h is fixed:
Inoue, Ohya, I.V. Log dependence on time
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.
Fokker-Planck-Kolmogorov
versus Newton
f
 f
( pf )
 {H , f }   2  
t
p
p
2
Einstein equations
(M , g)
1
R  g  R  T
2
BLACK HOLES in GR. Schwarzschild solution
 rS
ds    1 
r

2
1
 2  rS 
2
2
2
dt

1

dr

r
d

2



r


• Asymptoticaly flat
• Birkhoff's theorem: Schwarzschild solution
is the unique spherically symmetric vacuum solution
• Singularity
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).
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.
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  ( M , g  )
which satisfies the Liouville equation
(not Wheeler—De Witt).
Stochastic inflation?
Functional General Relativity
• Fixed background ( M , g ).
Geodesics in functional mechanics
 ( x, p )
Probability distributions of spacetimes
(M , g)
• No fixed classical background
spacetime.
• No Penrose—Hawking singularity
theorems
• Stochastic geometry?
Geodesics in Functional
Mechanics
 


( M , g ), g  u u  m ,    ( x , u ),
2
  
  
u    u u  0,

x
u
    1

Example

x0
x  x , x(t ) 
singular
1  x0t
2
x
2
2
 ( x, t )  C exp{ (
 q0 ) /  },
1  xt
nonsingula r
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.
• 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.
Single particle (moon,…)
   ( q, p , t )

p  V 


t
m q q p
1
( q  q0 ) 2 ( p  p0 ) 2
 |t 0 
exp{ 

}
2
2
ab
a
b
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,…
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.
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.
In the functional approach to
general relaivity one deals with
stochastic geometry of spacetime
manifolds.
It is different from quantum gravity or
string theory.
Probability of formaion of a
black hole for the external observer in
finite time during collapse is estimated.
Newton`s Classical
Mechanics
Motion of a point body is described by the
trajectory in the phase space.
Solutions of the equations of Newton or
Hamilton.
Idealization: Arbitrary real
numbers—non observable.
• Newton`s mechanics deals with
non-observable (non-physical)
quantities.
Rational numbers. P-Adic numbers
Vladimirov, I.V., Zelenov,
Dragovich, Khrennikov, Aref`eva,
Kozyrev,,…Witten, Freund, Frampton,…
Vladimirov, Volovich, Zelenov “p-Adic
analysis and mathematical physics”
Springer, 1994.
Journal: “p-Adic Numbers,…”(Springer)
• No classical determinism
• Classical randomness
• World is probabilistic
(classical and quantum)
Compare: Bohr, Heisenberg,
von Neumann, Einstein,…