Здесь могла бы быть ваша реклама . Maxim Eingorn & Alexander Zhuk Living In A World Without Multidimensional Kaluza-Klein Models

Song: The Rasmus “Living In A World Without You” Theodor Franz Eduard Kaluza

(November 9, 1885 – January 19, 1954)

“In April, 1919 Kaluza noticed that when he solved Albert Einstein’s equations for general relativity using five dimensions, then James Clark Maxwell’s equations for electromagnetism emerged spontaneously.”

(http://en.wikipedia.org/wiki/Kaluza)

[1] Kaluza Th. Zum Unitätsproblem in der Physik. // Sitzungsber. Preuss. Akad. Wiss. – 1921. – P. 966-972.

Song: The Rasmus & Anette Olzon “October And April” Oskar Benjamin Klein

(September 15, 1894 – February 5, 1977)

“In 1926 Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of very small radius.”

(http://en.wikipedia.org/wiki/Kaluza-Klein_theory)

[2] Klein O. Quantentheorie und fünfdimensionale Relativitäts- theorie. // Zeitschrift für Physik. – 1926. – Vol. 37. – P. 895–906.

An alternative “Oscar” Multidimensional Kaluza-Klein Models

“Kaluza-Klein theory (KK theory ) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism.”

“The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set and the phenomenon of having a space-time with compact dimensions is referred to as compactification .” “Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the experimental physics and astrophysics communities. A variety of predictions, with real experimental consequences, can be made (in the case of large extra dimensions/warped models). For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension(s). If a spatial extra dimension is of radius R, the invariant

mass of such standing waves would be Mn=nh/(Rc) with n an integer... This set of possible mass values is A table lamp “The Eiffel Tower” often called the Kaluza-Klein tower .” 55 cm, 269 € Nima Arkani-Hamed

(April 5, 1972)

The paradigm (the ADD-model) of “large extra dimensions” (with G.Dvali and S.Dimopoulos).

[3] Arkani-Hamed N., Dimopoulos S. and Dvali G. The hierarchy problem and new dimensions at a millimeter. // Phys. Lett. B. – 1998. – Vol. 436. – P. 263–272; arXiv:hep-ph/9803315.

“A leading Canadian-American theoretical physicist with interests in high-energy physics, string theory and cosmology.”

(http://en.wikipedia.org/wiki/Nima_Arkani-Hamed) Savas Dimopoulos

(1952)

“A Greek particle physicist well-known for his work on constructing theories beyond the Standard Model, which are currently being searched for and tested at particle colliders and in other experiments.”

(http://en.wikipedia.org/wiki/Savas_Dimopoulos) MSSM = Minimal Supersymmetric Standard Model

[4] Arkani-Hamed N., Dimopoulos S. and Dvali G. Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity. // Phys. Rev. D. – 1999. – Vol. 59, 086004; arXiv:hep-ph/9807344. Georgi (Gia) Dvali

(1964)

“Best known for the ADD-model, which he proposed together with N.Arkani-Hamed and S.Dimopoulos in 1998. It is a scenario inspired by string theory to explain the relative weakness of gravity to other forces, in which the Standard Model fields are confined to a (3+1)-dimensional membrane but gravity can also propagate in additional transverse spatial dimensions that are compact but may be as large as one-tenth of a millimeter. In this framework quantum gravity, string theory and black holes may be experimentally investigated at the Large Hadron Collider.”

(http://en.wikipedia.org/wiki/Georgi_Dvali)

[5] Antoniadis I., Arkani-Hamed N., Dimopoulos S. and Dvali G. New dimensions at a millimeter to a fermi and superstrings at a TeV. // Phys. Lett. B. – 1998. – Vol. 429. – P. 257–263; arXiv:hep-ph/9804398. http://arxiv.org

Arkani-Hamed N. Dimopoulos S. Dvali G.

[3] The hierarchy problem and new dimensions at a millimeter

Abstract

Eric Theodore Cartman from “South Park” [4] Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity

Abstract

[5] New dimensions at a millimeter to a fermi and superstrings at a TeV

Abstract

Kenny McCormick from “South Park” Song: Слот “Они убили Кенни ” [6] Arkani-Hamed N., Dimopoulos S., Kaloper N. and Sundrum R. A small Cosmological Constant from a Large Extra Dimension. // Phys. Lett. B. – 2000. – Vol. 480. – P. 193-199; arXiv:hep-th/0001197.

Abstract

[7] Arkani-Hamed N., Dimopoulos S., Kaloper N. and March-Russell J. Early inflation and cosmology in theories with sub-millimeter dimensions. // arXiv:hep-ph/9903239.

Abstract

Mr. Mackey from “South Park” [8] Arkani-Hamed N., Dimopoulos S., Dvali G. and Kaloper N. Infinitely large new dimensions. // Phys. Rev. Lett. – 2000. – Vol. 84. – P. 586-589; arXiv:hep-th/9907209.

Abstract

[9] Arkani-Hamed N., Dimopoulos S., Kaloper N. and March-Russell J. Rapid asymmetric inflation and early cosmology in theories with sub-millimeter dimensions. // Nucl. Phys. B. – 2000. – Vol. 567. – P. 189-228; arXiv:hep-ph/9903224.

Abstract Plotting paper String Theory

“String theory is a developing theory in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a candidate for the theory of everything (TOE), a manner of describing the known fundamental forces and matter in a mathematically complete system. The theory has yet to make testable experimental predictions, which a theory must do in order to be considered a part of science.”

“String theory mainly posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines (“strings”). The earliest string model, the bosonic string, incorporated only bosons, although this view developed to the superstring theory , which posits that a connection (a “supersymmetry”) exists between bosons and fermions. String theories also require the existence of several extra, unobservable, dimensions to the universe, in addition to the usual four spacetime dimensions.”

Jim Parsons/Sheldon Cooper (http://en.wikipedia.org/wiki/String_theory) in “The Big Bang Theory”. “Five major string theories were formulated. The main difference between each of them was the number of dimensions. Then a unification of all previous superstring theories, called M-theory, was proposed, which asserted that strings are really 1-dimensional slices of a 2-dimensional membrane vibrating in 11-dimensional space.

As a result of the many properties and principles shared by these approaches, their mutual logical consistency and the fact that some easily include the standard model of particle physics, some mathematical physicists believe that string theory is a step towards the correct fundamental description of nature. Nevertheless, other prominent physicists have criticized string theory for not providing any quantitative experimental predictions.”

“String theory can be formulated in terms of an action principle which describes how strings move through space and time. In the absence of external interactions, string dynamics are governed by tension and kinetic energy, which combine to produce oscillations. The quantum mechanics of strings implies these oscillations take on discrete vibrational modes, the spectrum of the theory.”

theory of everything (TOE) vs. king of nothing (KON), an adaption of the age old saying “Jack of all trades, master of none” “Since the string theory is widely believed to be a consistent theory of quantum gravity, many hope that it correctly describes our universe, making it a theory of everything.”

“On distance scales larger than the string radius, each oscillation mode behaves as a different species of particle, with its mass, spin and charge determined by the string’s dynamics. Splitting and recombination of strings correspond to particle emission and absorption, giving rise to the interactions between particles.” “An analogy for strings’ modes of vibration is a guitar string’s production of multiple but distinct musical notes. In the analogy, different notes correspond to different particles. The only difference is the guitar is only 2-dimensional; you can strum it up, and down. In actuality the guitar strings would be every dimension, and the strings could vibrate in any direction, meaning that Marco Leonardi, José Antonio Domínguez Banderas and Enrique the particles could move through not Miguel Iglesias Preysler in “Once Upon a Time in Mexico” only our dimension, but other Songs: Enrique Iglesias “Not In Love” and dimensions as well.” Antonio Banderas “Cancion Del Mariachi” String Theory and extra dimensions

“Two different ways have been proposed. The first is to compactify extra dimensions. 6 or 7 extra dimensions are so small as to be undetectable by present day experiments.”

“Another possibility is that we are “stuck” in a (1+3)-dimensional (three spatial dimensions plus the time dimension) subspace of the full universe.”

“Gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza’s early work demonstrated that general relativity in five dimensions actually predicts the existence of electromagnetism.”

Superstring Theory

“Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.”

(http://en.wikipedia.org/wiki/Superstring_theory) “Our physical space is observed to have only three large dimensions... However, nothing prevents a theory from including more than 4 dimensions. In the case of string theory consistency requires spacetime to have 10 (1+3+6) dimensions. The conflict between observation and theory is resolved by making the unobserved dimensions compactified. Our minds have difficulty visualizing higher dimensions because we can only move in 3 spatial dimensions. One way of dealing with this limitation is not to try to visualize higher dimensions at all, but just to think of them as extra numbers in the equations that describe the way the world works. This opens the question of whether these “extra numbers” can be investigated directly in any experiment.”

“Superstring theory is not the first theory to propose extra spatial dimensions; the Kaluza-Klein theory had done so previously.”

(http://en.wikipedia.org/wiki/Superstring_theory)

Christopher D’Olier Reeve (September 25, 1952 – October 10, 2004) in “Superman” Classical tests of multidimensional gravity: negative result [10] Eingorn M. and Zhuk A. Classical tests of multidimensional gravity: negative result. // Class. Quant. Grav. – 2010. – Vol. 27, 205014; arXiv:gr-qc/1003.5690.

Classical gravitational tests: General Relativity is in good agreement with experiments. What about Kaluza-Klein models? 1. the frequency shift; 2. the perihelion shift; =1 × = 1 × × D−3 3. the deflection of light. Topology: M R MD R M 3 T

the external (or “our”) 3-dimensional space the internal (D-3)-dimensional compact space

2=i k = 0 2 + Metrics: ds gdxdxik g00 ( dx ) +0 β α α + 2 αβg0α dxdx g dxdx

Alexander Nestor Haddaway Song: “What about me?” Three assumptions:

1. If there is no matter source, then the spacetime is the Minkowski spacetime : δ η η ==η == ==− g00 001, αβ αβ g α 0α 0 0, g

2. The weak field limit : a) the gravitational field is weak; b) velocities of test bodies are small.

3. The energy-momentum tensor of N moving point masses:

N − 1 i k ik D 2 dx dx cdt Tm=() −1 g  δ () rr − p ∑ p   p=1 dt dt ds r= (, xx1 2 ,… , x D ) The Einstein equation: 2S G ɶ 1  R=D D  T − gT  ikc4  ik D −1 ik  the gravitational constant in the multidimensional spacetime square of the (D-1)-dimensional sphere of a unit radius Metric coefficients: 222 N D 1 N g ϕ≈++1 ϕ ϕ ϕ ϕ () r2 () r +′ ′() rr −+ vrr2 ′ () − 00 2 4 4 ∑pp− 4 ∑ pp cc cp=1 Dc2 p = 1

ϕ ′ The nonrelativistic gravitational p is the potential in the point r p produced potential φ satisfies the D- by all particles, except of the p-th. dimensional Poisson equation: ∆ ρϕ = ϕ′(r− r ) is the potential in the point r SD G p D produced by the p-th particle: The rest mass N density: δρ ≡m() r − r δ∆ϕ ′ =( − ) ∑ p p SGmDD p r r p p=1 N ϕϕ ()=′ − r∑ () r r p ∆f = ϕ ( r ) p=1 For D=3: (106.13) and (106.14) in [11] Ландау Л.Д., Лифшиц 2(D − 1 ) 1N 1 ∂2 f g≈− vrrϕ′() −− Е.М. Теоретическая α 0α 3∑ p p 3 α физика : Учебное пособие . Dc−2 p=1 ctx ∂ ∂ В 10 т. Т. II. Теория поля . – М.: Наука , 1988. – 512 с. Metric coefficients in the case of one particle at rest:   δ ϕ ≈+22 ϕ ϕ +2 ≈ ≈−− 12 g1() rrgg() ,0, αβ α  1 () r  αβ 00 cc2 4 0  Dc− 2 2  ∂2ϕ δ ∆ϕ δ =αβ = SGm() r ∂x βα ∂ x D D

[12] Eingorn M. and Zhuk A. Nonrelativistic limit of multidimensional gravity: exact solutions and applications. // Class. Quant. Grav. – 2010. – Vol. 27, 055002; arXiv:gr-qc/0910.3507.

2 1/2  G m+∞ +∞ d  k   π ξ ϕ ξ (,,,)r …=−N ⋯ exp2 −i  r  × 3 1 d ∑ ∑ ∑    3  rk=−∞ k =−∞ i = 1  a  3 1 d i   2GN m 2 r = 2π k  2π k  G m c r g 2 ×cos ξ 1 ξ ⋯ cos d ≈−=− N g c 1  d  a1   ad  rr32 3 = r3| r 3 | ≫ … r3 aa 1, 2 , , a d S G in the Solar System π = D D 4 GN d ∏ a periods of tori i=1 i 21 Metrics in isotropic 3-D spherical coordinates:

r r 2  – the asymptotic form ds2≈−+1 g g  c 2 dt 2 − of metrics for the delta- 2  r32 r 3  shaped matter source. 1 r  −+1g ()dr22222 ++ r θ ψ dθ r sin d 2 − −  3 3 3 D2 r 3  Three classical 1 rg  42 5 2D 2 −+1  (()()()dx + dx ++… dx ) gravitational D− 2 r 3  tests

The frequency shift 1 2

ω ϕ ωω ()()1/2 =  1/2  ω ϕ 1, 1 1g 00 2 g 00 2, 2  1   2 ϕ ϕ−  ω ω≈1 + 1 2  and there is no deviation from General Relativity. 2 1 c2  The perihelion shift The deflection of light

The observed value, 43.11± 0.21 The observed value, arcsec: 1.75 arcsec per century:

2 2 2 − Dπ m′ cr D 1 rg δψ = g δψ = 2()D− 2 M 2 D − 2 ρ

General General Relativity Relativity

D = 3⇒ 42.94 ′′ D = 3⇒ 1.75 ′′

D = 4⇒ 28.63 ′′ D = 4⇒ 1.31 ′′ D = 9⇒ 18.40 ′′ D = 9⇒ 1.00 ′′

Suddenly: multidimensional Kaluza-Klein models contradict experimental data. Conclusions

1. Metric coefficients in the weak field limit were obtained for Kaluza-Klein multidimensional models in the case of the delta-shaped matter source.

2. Formulas for the perihelion shift and the deflection of light were found.

3. These formulas demonstrate good agreement with experimental data only in the case of ordinary 3-dimensional space.

4. This result does not depend on the size of extra dimensions.

Alternative models:

1. brane worlds; 2. black strings.

[13] Eingorn M. and Zhuk A. Kaluza-Klein models: can we construct a viable example? // arXiv:gr-qc/1010.5740 Thank you for attention!

Pallas’s cat (Otocolobus manul or Felis manul) , also known as the Manul, a wild cat of Central Asia