Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "REPULSIVE GRAVITATIONAL INTERACTION...

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REPULSIVE GRAVITATIONAL INTERACTION BETWEENMATTER AND ANTIMATTER

Pier Francesco Roggero, Michele Nardelli1,2

, Francesco Di Noto

1 Dipartimento di Scienze della Terra

Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10

80138 Napoli, Italy

2 Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”

Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie

Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract:

In this paper we examine in detail the hypothesis that matter and

antimatter instead of attract each other have a gravitational repulsion.

Furthermore, we have also described some new possible mathematical

connections with some sectors of Number Theory and String Theory



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Index:

1. ANTIMATTER.................................................................................................................................... 3 2. UNIVERSAL GRAVITATIONAL CONSTANT G........................................................................... 4 3. UNIVERSAL GRAVITATIONAL FORCE ....................................................................................... 5 4. CONSEQUENCES ON CURRENT THEORIES ............................................................................... 6 5. MATHEMATICS OBSERVATIONS...............................................................................................10 5.1 FINE STRUCTURE CONSTANT ..................................................................................................12 6. REFERENCES ..................................................................................................................................18



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1. ANTIMATTER

Antimatter is the set of particles, corresponding to the mass of the particles that make up

ordinary matter, but with opposite charge. For example, an atom of "antihydrogen" is

composed of a negatively charged antiproton, around which orbit a positron

(antielectron) positively charged.

If the particle and antiparticle are in contact they are transformed into high-energyphotons (gamma rays) or other particle-antiparticle pairs, such that the sum of the total

energy, preceding and following the impact, remains constant.

The antimatter has short life and cannot be stored, as it annihilates the first contact with

the matter. We can only generate very small quantities in the physics laboratory forparticles.



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2. UNIVERSAL GRAVITATIONAL CONSTANT G

The universal gravitational constant is the constant of proportionality that appears in

the law of universal gravitation, formulated in the late seventeenth century by Newton,but also in the general theory of relativity of Einstein.

The universal gravitational constant is a nature constant, independent of any physicalproperty of the interacting bodies.

Its value experimentally determined in the international system turns out to be equal to:

( )2

3111000067,067428,6

sKg

mG

⋅⋅±=

−

This constant is often approximated to:

2

3111067,6

sKg

mG

⋅⋅=

−

This constant has a positive value when applied to systems where they are only

attractive forces between matter and matter and antimatter-antimatter.

In case, however, we consider the interactions between matter-antimatter the value is

negative.

This assumption is considered to be a postulate, such as the maximum speed given by

light.



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3. UNIVERSAL GRAVITATIONAL FORCE

The gravitational interaction (gravity in common parlance) is one of the four

fundamental interactions.

In classical physics is interpreted as a force of attraction between bodies conservative

agent, whose most obvious manifestation in everyday experience is the weight force.

In general relativity the gravitational interaction is seen as a consequence of thecurvature of space-time created by the presence of bodies with mass or energy.

The law of universal gravitation states that two point masses attract each other with a

force of intensity directly proportional to the product of the masses of individual bodiesand inversely proportional to the square of their distance.

2r

GMmF =

where G is the universal gravitational constant, which is worth about 6,67 × 10-11

Nm²/kg2, M and m are the masses of the two bodies, r the distance between the two

bodies.

This law has the value G positive if we want the attraction between two bodies of

matter-matter or if we are the attraction between two bodies of antimatter-antimatter.

If, however, we mean the interaction between matter and antimatter, G takes a negative

value and the law becomes:

2r

GMmF −=

So we have a repulsive force between matter M and antimatter m.



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4. CONSEQUENCES ON CURRENT THEORIES

1) Violation of CP symmetry would make no sense

In the theory of the Big Bang, matter and antimatter in the universe initial had to

be present in equal proportions. The process of annihilation NOT occurredbecause the matter and antimatter repel each other with a repulsive gravitational

force that prevents their mutual destruction.

Currently it is believed that there has been a slight imbalance in favor of matter(known as CP violation) so that it would not be completely annihilated by

antimatter, making possible the formation of the universe in which we live

through the process of baryogenesis.

But this theory is not sufficient to explain the real observed asymmetry today.

2) Lack of local antimatter

Antimatter and matter repel each other gravitationally, forming separate galaxies

of matter and antimatter, which for more, they tend to repel each other, thus

avoiding possible collisions and annihilation.

It’s for this reason that we cannot have, eg. antimatter in our galaxy.

3) Expanding universe and its acceleration without introducing the theory ofdark energy

This same galactic repulsion is valid for the observation of an acceleratinguniverse. If gravity is always attractive, one might expect that the expansion ofthe universe slows and eventually contracts in a big crunch. Using the

observations of the red shift, astronomers and physicists estimate that, instead, the

size of the universe is expanding and the rate of expansion is accelerating at a rate

roughly constant.

The galaxies consist of matter and those consist of antimatter moved away from

them and avoid clash and destroy each other.



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Several theories have been proposed to explain this observation in the context of

a gravity always attractive.The most accredited is relative to dark energy, as plausible explanation would

make no more sense to exist. The acceleration, according to this theory implies

that the universe could be made up to a 75% dark energy which has never been

observed directly and that would also be evenly distributed in the universe.

On the other hand, if the antigravity between matter and antimatter is mutually

repulsive, equal amounts of matter and antimatter precisely compensate any

attraction. The galaxies moving away from us have a redshift and are simply made up ofantimatter. This could even imply that there is no expansion and acceleration

even with it’s necessary a more accurate and timely in-depth review of the theory

of the Big Bang.

4) It is not necessary to have a cosmological constant Λ

From Einstein's relativistic cosmological model characterized by a volume space

finite but unbounded, such a finite universe is destined to collapse in on itself

because of its own gravity. Einstein, believing that the universe should be

globally static, needed a repulsive force to prevent their collapse.

He realized that in order to express the equations of relativity in their most

general form, it was to introduce an additive constant which, if greater than zero,

it acts in terms of a Newtonian repulsive force and that, assuming a critical value,compensates implosion of the universe. It was for this reason that Λ was

introduced.

With the repulsive force between matter and antimatter the cosmological constant

Λ is no longer needed in the theory of general relativity.

5) The electromagnetic radiation passes through both matter and antimatter

Since the photon is its own antiparticle explains why electromagnetic radiation

can pass through both galaxies made up of matter and galaxies made up of

antimatter without being absorbed.



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The antiphoton would be absorbed (or destroyed and not having mass it cannot berejected by the gravitational force) when passing through a galaxy made up of

matter and also the photon would be absorbed when passing through a galaxymade up of antimatter.

Since there is an experimental evidence, the photon is also its own antiparticle

and it can cross both galaxies of matter and galaxies of antimatter.

In this way only if there are galaxies consist of matter the antiphoton would be

absorbed and the photon would pass through. But then one wonders why exists

also the antiphoton? It is a "indirect" test that must exist also antimatter galaxies!

Besides the electromagnetic radiations as well as also the light are deflected fromtheir paths from the masses consist of matter or antimatter in the same way.

6) The gravitational interaction is attractive and repulsive as the other threefundamental forces

There are four fundamental forces or interactions, which are the basis of energy

exchanges between the particles, which are responsible for the structure of the

universe: the gravitational interaction, the electromagnetic interaction, the strongnuclear force and the weak nuclear force.

If matter and antimatter repel would explain that the gravitational interaction is

repulsive and like the other 3 forces can be both attractive to repulsive.

Therefore we have a SYMMETRY also in the formulas describing the four

fundamental forces, which in the current theory does not exist where the

gravitational force is always attractive.

7) The Higgs boson is also its own antiparticle, and also gives the mass to theantimatter

The Higgs boson is the bearer of strength of the Higgs field , which according to

current theory permeates the universe giving mass to elementary particles.

Being also its own antiparticle also gives rise to the mass of the antiparticle and

then to antimatter.



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8) The gravitational field equation of general relativity for the interactionbetween matter and antimatter is negative and therefore repulsive

The form of the field equation is:

µν µν µν µν

π T

c

Gg Rg R

4

8

2

1=Λ+−

But with negative G, we have:

Rµυ

2

1− gµυR µν

π T c

G4

8−=

It must therefore be considered in the solutions of the field equation even thosewith negative G and which give rise to a repulsive gravitation.

The cosmological constant Λ

has been suppressed as described in point 4 .



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5. MATHEMATICS OBSERVATIONS

From the mathematical point of view, we can observe that the value of G = 6.67428:

( )2

3111000067,067428,6

sKg

mG

⋅⋅±=

−

corresponds to about 1,6180334 = 6,854085…, with very small difference

6,85408 – 6,67428 = 0.1797 ≈ (√1,618033 + √√1,618033)/2 -1

=(1,27 + 1,12)/2 -1 = 0,19 ≈ 0,1797 real difference

So a possible connection with the golden ratio seems possible in the formula of

gravitational constant.

The universal constants must often rely on mathematical ones, and the latter often also

prime numbers, with some connection to the number e, of Euler, 2,728…, π =3,14, but

also by Φ =1,618…An example we find in the formula the strength of the Casimir effect:

The Casimir force per unit area ( ) AF c / , in the ideal case of perfectly conductive metal

plates between which it was created the vacuum, is calculated as:

where we have the symbol π.

So it might also be possible that Φ =1,618 is connected to the universal

gravitational constant, as indicated above, that is,



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1,6180334 = 6,854085…, with very small difference 6,85408 – 6,67428 = 0.1797 ≈ 0,18≈ (√1,618033 + √√1,618033)/2 -1

=(1,27 + 1,12)/2 -1 = 0,19 ≈ 0.1797 ≈ 0,18 real difference, to the nearest 1 cent = 0,19-

0,18 = 0.01 between the two values.

Also in the formula of the fine structure constant will find π as we describe in the

following Section.



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5.1 FINE STRUCTURE CONSTANT

The fine structure constant, denoted by the Greek letter α, is a parameter that correlates

the main physical constants of electromagnetism. It expresses the coupling constant thatcharacterizes the intensity of the electromagnetic.

The fine structure constant was introduced by Arnold Sommerfeld in 1916 as a measure

of the deviation in the relativistic spectral lines than the Bohr model. For this reason it isalso called Sommerfeld constant.

The fine structure constant in MKS, is defined as:

c

ke

hc

e

c

e

hh

2

0

2

0

2

24===

ε πε α

where:

• e is the electric charge of the electron = -1,6 × 10-19

[C]

• 0ε is the electric permittivity of vacuum = 8,852 × 10-12

[C]2[m]

-2[N]

-1

• h is Planck's constant = 6,626075 × 10-34

[J][s]

• π 2

h=h

• c is the speed of light in vacuum = 299 792 458 [m][s]-1

• k is Coulomb's constant [N] [m]

2 [C]

-2.



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The formula and its value are:

( )( )44035999074,137

110242973525698,7

4

3

0

2

=×== −

πε α

c

e

h

Shortly after the value calculations were completed, it was discovered a small error in

the data, but also it was decided to publish this result in 2008 waiting for more precise

measurements expected for the beginning of 2011.

As the value 7.2973, we note that it is very near to the square of the Euler number:

e = 2,7182818284… We have that: e2 = 7,3890560989…, with

DIFFERENCE BETWEEN THE TWO VALUES EQUAL TO 0.09170 ≈ 0.1PRACTICALLY A TENTH OF UNIT

and with ratio 7,3890/7,2973 = 1,01256 ≈ 1,015150 ≈ 32 618,1

1.015746 ≈ 64 718,2 .

The fine structure constant could therefore approach to

α = e2

/ 64 718,2 = 7,3890560989 / 1, 01574606300 = 7,2745111874,

with difference 0,022 = two cents of the actual value 7,2973 …, or more precisely 22

thousandths.

Now, we note that the number 8, and thence the numbers 2864 = and 8232

2×= , are

connected with the “modes” that correspond to the physical vibrations of a

superstring by the following Ramanujan function:



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( )

++

+

⋅

=

−

∞−

∫

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wt itwe

dxe x

txw

anti

w

wt

w x

φ

π

π

π

π

.

This shows how the physical constants depend on those mathematics as π , Φ and e,

and these in turn by the primes, as we will see later with a work in progress on thenumerical series and mathematical constants, which then appear in the physical: as we

have seen in the formula of the Casimir force and the constant fine structure appears π ,and in the number 6,85352 could get in

Φ=1,618, being very near to its fourth power, i.e. 6,85352

With ratio 6,85352 / 6,67428 = 1,026856 ≈ 1,03053 = 16 618,1 …

then the number 6.67428 could approximate to

1,6184 / 16 618,1 … = 6,85352… / 1,03053…= 6,65222 ≈ 6,67428,

with difference 6,67428 -6,65222 = 0,022, the same 22 thousandths

encountered for the APPROXIMATION OF CONSTANT FINE STRUCTURE IS

THIS SOMETHING PURELY RANDOM?

We note that also here, 16 = 8 * 2, thence we have the mathematical connection with the

number 8, i.e. with the “modes” that correspond to the physical vibrations of a

superstring by the following Ramanujan function:



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( )

++

+

⋅

=

−

∞−

∫

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wt itwe

dxe x

txw

anti

w

wt

w x

φ

π

π

π

π

.

A CONNECTION BETWEEN OUR MOST NOTES MATHEMATICS CONSTANT

IS 'THE FOLLOWING:

Our possible connection between all three constants e, Ф e π :

φ

e ≈ 64π (1)

In fact,φ

e =

618033,1

718,2 = 1,0189665

64π = 1,0180472

A more precise formula is:

φ

e ≈ 64π /

2

20481024 π π +

which gives as a result 1,0181126, nearest to 1,0180472, with a difference of0,0000656, that is, of only 656 ten millionths.



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Also here, the values 1024 = 83 * 2 and 2048 = 83 * 22 are connected with the number 8i.e. with the “modes” that correspond to the physical vibrations of a superstring by

the following Ramanujan function:

( )

++

+

⋅

=

−

∞−

∫

42710

421110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wt itwe

dxe x

txw

anti

w

wt

w x

φ

π

π

π

π

.

ANOTHER CONNECTION, ALREADY KNOWN, BUT ONLY BETWEEN e and π,

is the following:

e is connected to π by Euler's formula

eix

= cos (x) + i sen(x) (2)

which becomes

eiπ

+ 1 = 0 (3)

in the particular case x = π (Euler's identity).

These mathematical constants are in some way related to the numbers also according tothe first harmonic series, (as we will see in the next work) then appear in some physical

constants, such as the formula for the fine structure constant and many others, and this

is how the prime numbers eventually adjust some physical laws of nature, gravitation

included, see Einstein's equation:

Let's review the formulas of field equation:



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The field equation is:

µν µν µν µν

π T

c

Gg Rg R

4

8

2

1=Λ+−

But with negative G we have:

Rµυ

2

1− gµυR µν

π T c

G4

8−=

Note that π is present here as well as its multiples (in the latter case as 8π) equivalent to

powers of 2 in these formulas: 2 π, 4 π, 8 π, in the various formulas in this paper.

Why not 3 π, 5 π, or 6 π?

It’s a case, or will there be a specific physical / mathematical reason that prefer the

power of 2, i.e. 2, 4 and 8? More research are necessary.

Besides the fine structure constant is increasingly important in string theories, and so

consequently also in the future TOE, or Theory of Everything.



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6. REFERENCES

1)”Wormhole”Teoria delle stringhe

Ing. Pier Francesco Roggero, Dott. Michele Nardelli, P.A. Francesco Di Noto

2) Note in Italian L'Ams scova positroni nei raggi cosmici."Potrebbero essere tracce di materia oscura"

Sul sito

http://www.repubblica.it/scienze/2013/04/03/news/caccia_antimateria-55861299/ L'Ams scova positroni nei raggi cosmici. "Potrebbero essere tracce di materia

oscura"

L'annuncio al Cern: dal 2011 lo spettrometro posizionato sulla ISS ha catturatoquasi 400 mila di queste "strane particelle" con una precisione senza precedentinella tempesta che spazza l'universo. "Ora serve capire qual è la loro origine" di

ELENA DUSI



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Come un retino, Ams va a caccia di strane particelle nello spazio. Simile a unsatellite, ma fissato a un braccio della Stazione Spaziale Internazionale, oggil'occhio della fisica a 400 chilometri di altezza chiamato "Alpha Magnetic

Spectrometer" ha messo in mostra il suo primo raccolto di "farfalle". Da quando isuoi rivelatori sono entrati in funzione - nel maggio 2011 - Ams ha catturato400mila positroni: elettroni con carica positiva che fanno parte di quel regnodell'antimateria che sempre nell'universo è indizio di fenomeni interessanti e

misteriosi.

I positroni di Ams non fanno eccezione. Raccolti con una precisione senza

precedenti in mezzo alla tempesta di raggi cosmici che spazza costantementel'universo, sono stati contati uno a uno e selezionati a seconda della loro energia. Iprimi risultati (la raccolta dei dati dovrebbe durare ancora un decennio) sono statiannunciati oggi al Cern, alla Nasa e pubblicati su Physical Review Letters. Leparticelle di antimateria sono piovute negli strumenti di Ams da ogni direzione. Unaspetto interessante è che la quantità di positroni aumenta all'aumentare della loro

energia (almeno fino a un certo livello, poi il loro numero cessa di crescere). "Ma lanostra curiosità ora è capire l'origine di queste particelle" spiega RobertoBattiston, il fisico dell'Istituto Nazionale di Fisica Nucleare e dell'università diTrento che di Ams è numero due. "Può darsi che provengano da un tipo di stellachiamato pulsar all'interno della nostra galassia. Ma potrebbero anche essere unatraccia di quel fenomeno ancora sconosciuto che è la materia oscura".

Di materia oscura è composto oltre il 20% dell'universo. Ma da quali particelle siacomposta resta un mistero totale. "Ipotizziamo che siano piuttosto pesanti -spiega Battiston - che non interagiscono con la luce. Ogni tanto, al centro della

galassia, dove la materia è più densa, potrebbero scontrarsi e annichilirsi,liberando energia e positroni". I frammenti di antimateria raccolti da Ams - èuna delle ipotesi - nascono dalla fusione della materia oscura. "Come la cenere

può darci informazioni sul legno da cui proviene, così questi positroni sono forsemessaggeri di notizie importanti sulla materia oscura che li ha prodotti" spiega ilfisico italiano.Esperimenti sulla materia oscura sono in corso anche nei laboratori del GranSasso e nello spazio con i due satelliti Pamela e Fermi. Ams è superiore perdimensioni e precisione "in un campo - aggiunge Battiston - in cui il dettaglio



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delle misure è tutto, perché i fenomeni che analizziamo sono debolissimi".

Costato 2 miliardi di dollari, pesante 7 tonnellate e realizzato con il contributoitaliano dell'Istituto Nazionale di Fisica Nucleare e dell'Agenzia Spaziale Italiana,Ams è il più importante esperimento a bordo della Stazione SpazialeInternazionale (spesso accusata di non aver mantenuto le sue promesse in fatto dicontributi alla scienza). Alla raccolta e all'analisi dei suoi dati lavorano 600 fisici eingegneri di 16 paesi.

Il "retino" spaziale è considerato uno strumento complementare a Lhc. Il LargeHadron Collider, l'acceleratore di particelle del Cern di Ginevra, fa scontrareprotoni ad altissima energia per studiare i frammenti delle collisioni. Ams nellospazio svolge un ruolo simile, sfruttando le energie ancora più estreme raggiuntedai raggi cosmici al di fuori dell'atmosfera terrestre. Per costruirlo ci sono voluti18 anni pieni di difficoltà. Il progetto, guidato dal premio Nobel per la fisica

Samuel Ting, è stato dato più volte per spacciato e solo il colpo di scena di un volodello Shuttle programmato dalla Nasa solo per lui (l'ultima missionedell'Endeavour prima del pensionamento) gli ha permesso due anni fa di attivare il

suo potente magnete: l'occhio di Ams capace di scrutare nella parte oscuradell'universo.

"I dati di AMS - osserva Fernando Ferroni, presidente Istituto Nazionale di FisicaNucleare (INFN) - sono una interessantissima conferma, con precisione maggiore,dei dati rilevati dagli esperimenti spaziali Pamela e Fermi in questi ultimi anni.Una conferma che non risolve certo il rebus dell'antimateria in eccesso. Ma cheindica come i dati dei due esperimenti (anche questi a fortissima presenza italiana)avessero visto giusto nel rivelare questa anomalia. Che sia o meno materia oscuranon può che dirlo un ulteriore sforzo per produrre nuovi dati e analizzarli. Unlavoro che, come giustamente sottolinea il team AMS, richiede ancora del tempo edella prudenza".

(03 aprile 2013) “



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3) Note in Italian

Cern, una nuova anomalia nell’universo

Lunedì, 29 aprile 2013 - 08:22:00

Un gruppo di scienziati - tra cui alcuni italiani - impegnati in sofisticate esperienze conil potente anello LHC – large hadron collider, la stessa macchina del Cern di Ginevra

con la quale è stato scoperto, sempre da un’équipe coordinata da ricercatori italiani il

bosone di Higgs nel luglio scorso-, ha osservato un fenomeno che potrebbe portare achiarire la tanto discussa asimmetria tra materia e antimateria. l’Universo,

immediatamente dopo il Big Bang, doveva essere costituito da eguali quantità di

materia e antimateria.

Come mai invece l’universo che conosciamo è fatto di materia, dove è andata afinire l’antimateria? Il fatto che successivamente l’antimateria sia completamente

scomparsa è uno degli enigmi più affascinanti della scienza. Ma oggi,proprio grazie amacchine come LHC, siamo in grado di fabbricare artificialmente pezzetti di

antimateria come positroni, -antiparticella dell’elettrone-, e antiprotoni, -antiparticella

del protone-, che hanno la stessa massa delle loro gemelle, ma carica elettrica opposta.(

Tra l’altro è proprio grazie all’antimateria che siamo oggi in grado di curare alcune

forme tumorali con la Pet, letteralmente Positron emission tomography) Così, grazie

alle collisioni che si realizzano all’interno di un acceleratore di particelle, qual è LHC,

oggi i fisici sono in grado di produrre quotidianamente migliaia di miliardi di particelledi antimateria.

E possono studiare con precisione il loro comportamento, confrontandolo conquello delle gemelle di materia. Se particelle e antiparticelle avessero proprietà

esattamente speculari non riusciremmo a giustificare la scomparsa dell’antimateria

dall’Universo. Viceversa negli esperimenti si osservano piccole asimmetrie di

comportamento. L’asimmetria rilevata nell’esperimento del Cern è abbastanza

significativa :circa il 30 per cento degli eventi registrati viola la simmetria. “ L’entità di

questa asimmetria di comportamento è molto grande – spiega Vincenzo Vagnoni

dell’INFN di Bologna e uno dei firmatari dell’articolo pubblicato su Physics Review



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Letter - e la misura effettuata apre un nuovo settore d’indagine che potrà portare unamigliore comprensione dei fenomeni alla base della violazione della simmetria e

spiegare il mistero della scomparsa di antimateria dal nostro Universo”.

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