A new approach to the problem of black hole and cosmological

singularities

Igor V. Volovich

Steklov Mathematical Institute, Moscow

QUARKS-2010 16th International Seminar on High Energy Physics Kolomna, Russia, 6-12 June, 2010.

PLAN• Non-Newtonian Classical

Mechanics

• Functional Probabilistic General Relativity

• Singularities

Cosmological Singularity Problem

0,)(

))(( 22222

ttta

dzdydxtadtds

Friedmann cosmology

Density of matter and curvature tensor go to infinity as 0tWhether anything existed before If not, then where did the universe come from?

?0t

Black Hole Singularity

222122 )21()21( drdrrMdt

rMds

Singularity 0r

Geodesics. Compare: Newton`s equations

General relativity is based on Newton`s mechanics and special relativity.

Newton`s mechanics can not be true(not because of relativistic or quantum

corrections).

Try to change Newton`s approach to mechanics and therefore Einstein`s approach to special and general relativity.

Why Newton`s mechanics can not be true?

• Newton`s equations of motions use real numbers.

• Classical uncertainty relations

• Time irreversibility problem

• Singularities

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

.9,...,1,0,101 n

nnn aat

Ftxdtdm )(2

2

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.

Classical Uncertainty Relations

0,0 pq

• We can observe only rational numbers, fractions,

,NMх

(M, N – integers)

With some error .0x

Rational numbers. p-adic numbers

Vladimirov, Zelenov, Khrennikov,Kozyrev, Dragovich,…

Witten, Freund, Frampton, Parisi,…

• Journal: “p-Adic Numbers, Ultrametric Analysis and Applications” (Springer)

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, Bolzmann, diffusion, Entropy increasing --- irreversible

Expansion of Universe after Big Bang (?)

Time Irreversibility Problem

Boltzmann, Maxwell, Poincar´e, Bogolyubov, Kolmogorov, von Neumann, Landau, Prigogine,Feynman,…

Poincar´e, Landau, Prigogine, Ginzburg,Feynman: Problem is open.

We will never solve it (Poincare)

Quantum measurement? (Landau)

Lebowitz, Goldstein, Bricmont: Problem was solved by Boltzmann

Loschmidt paradox

•  From the symmetry of the Newton equations upon the reverse of time it follows that to every motion of the system on the trajectory towards the equilibrium state one can put into correspondence the motion out of the equilibrium state if we reverse the velocities at some time moment.

• Such a motion is in contradiction with the tendency of the system to go to the equilibrium state and with the law of increasing of entropy.

Poincare – Zermelo paradox

• Poincar´e recurrence theorem: a trajectory of a bounded isolated mechanical system will be

many times come to a very small neighborhood of an initial point.

• Contradiction with the motion to the equilibrium state.

Boltzmann`s answers to:

• Loschmidt: statistical viewpoint

• Poincare — Zermelo: extremely long Poincare recurrence time

• Not convincing…

Boltzmann Great Fluctuation ConjectureTo explain entropy increasing

• Compare: Friedmann gravitational picture of the Big Bang

• Hawking Black Hole Information Paradox

• Compare: Black Body.

Our low-entropy world is a fluctuationin a higher-entropy universe

Functional Formulation of Classical Mechanics

Usual approaches to the irreversibility problem (Bogolyubov):

Start from Newton Eq. Gas of particlesDerive Boltzmann Eq.

This talk: Irreversibility for one particleModification of the Newton approach to Classical

mechanics: Functional formulation

Functional formulation of 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 in "functional" approach is not the Newton equation but the Liouville equation for the distribution function of the single particle.

States and Observables inFunctional Probabilistic

Mechanics

States and Observables inFunctional Classical Mechanics

Fundamental Equation inFunctional 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.

Instead of Newton equation. No trajectories!

• Solutions of the Liouville equation have the property of delocalization which corresponds to irreversibility.

• The Newton equation in this approach appears as an approximate equation describing the dynamics of the expected value of the position and momenta for not too large time intervals.

• Corrections to the Newton equation are computed

Single particle (moon,…)

})()(exp{1|

),,(

2

20

2

20

0 bpp

aqq

ab

pqV

qmp

t

tpq

t

• No classical determinism• Classical randomness

• World is probabilistic (classical and quantum)

Compare: Bohr, Heisenberg,von Neumann, Einstein,…

Average Value and Dispersion

Free Motion

DelocalizationIrreversibility

Comparison with Quantum Mechanics

Liouville and Newton. Characteristics

• arXiv: 0907.2445• Foundations of Physics (2010) …..

• Bogolyubov, Krylov (1934), Koopman, Born, Blokhintsev, Prigogine

Corrections to Newton`s Equations

Corrections to Newton`s Equations

Corrections to Newton`s Equations

Corrections

,Ht

Irreversibility in Functional mechanics

Particle in Box: Not Maxwell

Strategies• Fixed background . Geodesics in functional mechanics

Probability distributions of spacetimes

• No fixed classical background spacetime.

• No Penrose—Hawking singularity theorems

),( gM

),( gM

),( ux

Geodesics in Functional Mechanics

0

),,,(

uuu

ux

ux

Example

singular1

)(,0

02

txxtxxx

rnonsingula

},/)1(exp{),( 22

0 qxtxCtx

Fixed classical spacetime?• A fixed classical background spacetime does not exist (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

ConclusionsFunctional probabilistic formulation of classical mechanics: distribution function instead ofindividual trajectories.

Fundamental equation: Liouville even for a single particle. Irreducible classical randomness.

Newton equation—approximate for average values.Corrections to Newton`s trajectories.

Attempts to extend the functional approach to general relativity

Information Loss in Black Holes

• Hawking paradox.

• Particular case of the Irreversibility problem.

• Bogolyubov method of derivation of kinetic equations -- to quantum gravity.

• Th.M. Nieuwenhuizen, I.V. (2005)