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

PLANRational, 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 bio-systems”, Springer, 2011.

I.V. “Bogoliubov equations and

functional mechanics”, TMF (2012) 35, 17.

Newton Equation

),(

numbers Real

),(

),(

2

2

2

tnRM

txx

xFxdt

dm

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:

.9,...,1,0,10

1 n

nnn aat

Ftxdt

dm )(

2

2

Classical Uncertainty Relations

0,0 pq

0t

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:

Newton, Schrodinger Eqs –- reversible

Navier-Stokes, Boltzmann, diffusion, Entropy increasing --- irreversible

tt

.ut

u

Time Irreversibility ProblemBoltzmann, 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 inFunctional Classical Mechanics

States and Observables inFunctional Classical Mechanics

Not a generalized function.

Mean values are rational numbers

Fundamental Equation inNon-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

values.mean specialStudy

s.observable special very - A

measure, special very -

sm,automorphi -

adic),-p real, (rational, space -

),,,(

t

t

M

AM

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 EquationsNon-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

p

pf

p

ffH

t

f

)(

},{ 2

2

Einstein equations

TRgR 2

1

),( gM

BLACK HOLES in GR. Schwarzschild solution

• Asymptoticaly flat• Birkhoff's theorem: Schwarzschild solution

is the unique spherically symmetric vacuum solution

• Singularity

12 2 2 2 2

21 1S Sr rds dt dr r d

r r

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 thatmany 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 withthe 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

which satisfies the Liouville equation

(not Wheeler—De Witt).

Stochastic inflation?

),( gM

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?

),( gM

),( gM

),( px

Geodesics in Functional Mechanics

1

,0

),,(,),,( 2

uuu

ux

uxmuuggM

Example

singular1

)(,0

02

tx

xtxxx

rnonsingula

},/)1

(exp{),( 220 q

xt

xCtx

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,…)

})()(

exp{1

|

),,(

2

20

2

20

0 b

pp

a

qq

ab

pq

V

qm

p

t

tpq

t

ConclusionsArbitrary 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 physicalphenomena 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 andthe 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 ablack 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,…