Black Holes, Neutrinos And Gravitational Proprieties Of Antimatter (www.oloscience.com)

Black holes, neutrinos and gravitational proprieties of antimatter Dragan Slavkov Hajdukovic CERN, CH-1211 Geneva 23 and Cetinje, Montenegro [email protected] Abstract The gravitational proprieties of antimatter are still a secret of nature. One outstanding possibility is that there is gravitational repulsion between matter and antimatter (in short we call it antigravity). We argue that in the case of antigravity, the collapse of a black hole doesn’t end with singularity and that deep inside the horizon, the gravitational field may be sufficiently strong to create (from the vacuum) neutrino-antineutrino pairs of all flavours. The created antineutrinos (neutrinos) should be violently ejected outside the horizon of a black hole composed from matter (antimatter). Our calculations suggest that both, the supermassive black hole in the centre of our Galaxy (Southern Sky) and in the centre of the Andromeda Galaxy (Northern Sky) may produce a flux of antineutrinos measurable with the new generation of the neutrino telescopes; like the IceCube Neutrino Detector under construction at the South Pole, and the future one cubic kilometre telescope in the Mediterranean Sea. A by the way result of our consideration, is a conjecture allowing determination of the absolute neutrino masses. In addition, we predict that in the case of microscopic black holes which may be eventually produced at the Large Hadron Collider at CERN, their thermal (Hawking’s) radiation should be dominated by a non-thermal radiation caused by antigravity.

1. Introduction A huge majority of physicists thinks that particles and antiparticles (for instance protons and antiprotons) have the same acceleration in the gravitational field of the Earth. It may be correct, but at the present stage of our knowledge, it is just a conviction, not an established experimental fact. Simply, as to this point of time, i.e. end of 2007, there is no any experimental evidence concerning gravitational proprieties of antimatter. The conviction that gravitational proprieties of matter and antimatter are the same has the ground in the Weak Equivalence Principle (WEP); the oldest and the most trusted principle of contemporary physics. Its roots go back to the time of Galileo and his discovery of “universality of the free fall” on the Earth. A deeper understanding was achieved by Newton who explained “universality of the free fall” as a consequence of the equivalence of the inertial mass m I and the gravitational mass mG . Unlike many other principles, which lose importance with time, the WEP has even increased its importance and is presently the cornerstone of Einstein’s General Relativity and of modern Cosmology. It is amusing that physicists use the word “weak” for a principle having such a long and successful life and such great names associated with it (Galileo, Newton, and Einstein). It should rather be called “the principle of giants”. Speculations concerning possible violation of the WEP may be divided into two groups. The first group consists of a number of different theoretical scenarios for minimal violation of the WEP. The universality of gravitational attraction is not being questioned (so, there is no room for antigravity), but in some cases gravitational and 1

inertial mass may slightly differ (See [1] and References therein). In the present paper we are exclusively interested in the second group of speculations, predicting gravitational repulsion between matter and antimatter, i.e. antigravity as the most dramatic violation of the WEP. There are three new lines of argument in favour of antigravity: The intriguing possibility that CP violation observed in the neutral Kaon decay may be explained by the hypothesis of a repulsive effect of the earth’s gravitational field on antiparticles [2]. The Isodual theory of antimatter [3].The growing evidence (since 1998) that the expansion of the universe is accelerating [4] rather than decelerating, with antigravity as a possible explanation of this accelerated expansion of the universe [5]. Just to add a little bit of humour to the subject, I will point out one more argument. Looking at all the natural beauties and wonders on our planet and in the Universe, we must conclude that at the time of Creation, God was a child – because only a child can have such an imagination. And, if at that time God was a child, antigravity must exist! There are two complementary approaches to search signatures of antigravity. The first approach is laboratory experiments strictly devoted to the study of gravitational proprieties of antimatter, as for instance a proposed test with antiprotons [6] and future spectroscopy of antihydrogen [7] that may be mastered in ten-twenty years from now. Such experiments are presently possible only at CERN, but CERN is completely absorbed by the development of the colossal LHC project, presumably the most sophisticated scientific project in human history. So, experiments devoted to test existence of antigravity (or only small violations of WEP) are hardly compatible with the current priorities of CERN. Conditions for such experiments will essentially improve after construction of FAIR (Facility for Antiproton and Ion Research) at Darmstadt, Germany [8]. The second approach is to assume existence of antigravity and to predict some effects that may be detected by astronomical observations or at LHC. The advantage of this approach is to look for signatures of antigravity in already existing experiments, without need to design new ones. In Section 2 we suggest that under assumption of antigravity, a gravitational field deep inside the Schwarzschild radius, may produce particle-antiparticle pairs in the same way as in Quantum Electrodynamics a classical external electric field creates electron-positron pairs from the (Dirac) vacuum [9]. If for instance, the black hole is made from ordinary matter, created particles must stay confined inside the horizon of the black hole, while antiparticles (because of gravitational repulsion) are violently ejected outside the horizon. In the particular cases of the supermassive black hole in the centre of our galaxy (Southern Sky) and in the centre of Andromeda galaxy (Northern Sky), we estimate that the flux of eventually ejected antineutrinos is sufficiently high to be detected with the new generation of the neutrino telescopes; like the IceCube Neutrino Detector under construction at the South Pole, and the future one cubic kilometre telescope in the Mediterranean Sea.. In Section 3 we turn towards mini black holes (MBH) predicted by theories with large or warped extra dimensions (for a topical Review, see [10] and references therein). CERN is fully prepared not only to detect eventual creation of MBH at LHC, but also to study their decay. We argue that instead of expected decay through thermal Hawking’s radiation, MBH may decay through a dominant non-thermal radiation caused by antigravity. Appendix A contains a conjecture allowing determination of absolute neutrino masses, while Appendix B drafts a proposal for testing existence of

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antigravity with antiprotons, as a complementary research to observation of (supermassive and microscopic) black holes. Thus, if there is antigravity, the first signatures of it may be seen at the new generation of the neutrino telescopes and at LHC.

2. Brief summary of Black Holes and Hawking radiation It is appropriate to start with a very short (and superficial) overview of black holes and Hawking radiation. There is convincing evidence that there are two types of black holes in the Universe: stellar and supermassive black holes. Stellar black holes are black holes of stellar masses which are the result of the gravitational collapse after the end of the process of nuclear fusion in a sufficiently massive star. A necessary condition for a star to become a black hole is that at the very end of its evolution, it has a mass greater than about three actual solar masses ( 6 × 10 30 kg ). Roughly speaking the heaviest stellar black holes are never more than 100 times more massive than the lightest ones. The supermassive black holes exist in centres of galaxies and may have masses as large as 1010 solar masses. Let’s consider a black hole with mass M. For further presentation we will need expressions for the Schwarzschild radius RS, and the gravitational acceleration a S , at the surface of the Schwarzschild sphere: RS =

2GM c2

, aS =

GM R S2

≡

c2 2 RS

(1)

The famous result obtained by Hawking is that a black hole radiates as a black body with temperature TH, and surface area A, given respectively by: TH =

hc 3 , A = 4πR S2 8πGMk

(2)

where k is the Boltzmann constant, G the gravitational constant, h reduced Planck constant and c speed of light. Consequently, the total energy radiated from a black hole in unite time is determined by the Stefan-Boltzmann law, i.e. equals to σ TH4 A , where σ = π 2 k 4 60h 3 c 2 is the Stefan-Boltzmann constant. Hence, the rate of mass loss of a black hole can be estimated as: 4πTH4 R S2 1 hc 4 1 dM ≈ = (3) 15360π G 2 M 2 dt H c2 By using this estimate, one can conclude that the lifetime of a black hole with respect to the process of thermal radiation is: t H ≈ 5120π

G2 hc

4

M3

(4)

Let’s note that in “deriving” Equation (3), for simplicity we have omitted a multiplicative factor depending on the number of states and species of particles that are radiated. As the mass of the black hole decreases in the process of thermal radiation, its temperature as well as he number of species of particles that can be emitted grow. In fact, for real black holes, the temperature (2) is negligibly small. In particular, for a black hole having the same mass as the Sun, TH is only about 6 × 10 −7 K , while for the black hole in the centre of our galaxy TH ≈ 2 × 10 −3 K . It is obvious that at such low temperatures only massless particles can be emitted. For

3

instance, thermal emission of electrons and positrons is possible only when M < 1014 kg (what is 16 order of magnitude smaller than the mass of the Sun). Black holes of smaller mass can emit heavier “elementary” particles as well. The key point is that, roughly speaking, a particle can be emitted only if its reduced Compton wavelength is greater than the Schwarzschild radius of the black hole. Hence, particles and antiparticles of mass m (neutrinos, electrons and so on) can be emitted only if the mass M of the black hole is less than a critical mass M Cm : M Cm =

M P2 2m

(5)

where MP is the Planck mass: MP =

hc G

(6)

While the existence of the stellar and supermassive black holes is considered as an established fact, the other possible types of black holes, like primordial and microscopic black holes are still a theoretical speculation. But, if they exist they must be subject to Hawking radiation.

3. Astronomical black holes and antigravity 3.1 Rudimentary theoretical consideration Let’s start this section with an illuminating example coming from Quantum Electrodynamics: creation of electron-positron pairs from the (Dirac) vacuum by an external (classical i.e. unquantized), constant and homogenous electrical field E. In this particular case of the uniform electric field, the particle creation rate per unite volume and time is known [9] exactly: 2 dN e + e − 4 c ⎛ E ⎞ ∞ 1 ⎛ nπ E cr ⎞ ⎟⎟ ∑ 2 exp⎜ − (7) = 2 4 ⎜⎜ ⎟ dtdV π D e ⎝ E cr ⎠ n =1 n ⎝ 2 E ⎠ where 2me2 c 3 h (8) De = and E cr = me c eh are respectively reduced Compton wavelength of the electron, and critical electric field. In fact we have slightly transformed the original result [9] to the form given by Equation (1) which is more appropriate for our further discussion. It is evident that particle creation rate is significant only for an electric field greater than the critical value Ecr. The above phenomenon is due to both, the complex structure of the physical vacuum in QED and the existence of an external field. In the (Dirac) vacuum of QED, short-living “virtual” electron-positron pairs are continuously created and annihilated again by quantum fluctuations. A “virtual” pair can be converted into real electronpositron pair only in the presence of a strong external field, which can spatially separate electrons and positrons, by pushing them in opposite directions, as it does an electric field E. Thus, “virtual” pairs are spatially separated and converted into real pairs by the expenditure of the external field energy. For this to become possible, the potential energy has to vary by an amount eEΔl > 2mc 2 in the range of about one Compton wavelength Δl = h mc , which leads to the conclusion that the pair creation occurs only in a very strong external field E, greater than the critical value Ecr in Equation (8). By the way, let’s note that when electric field E is less than the critical 4

value Ecr, instead of pair creation, there is the well known phenomenon of vacuum polarization, i.e. the vacuum in which “virtual” electron-positron pairs are present, behaves up to some extent, as a usual polarisable medium. Now, let’s assume that there is gravitational repulsion between matter and antimatter (in short antigravity), while usual gravitational attraction stays valid for both, matter-matter and antimatter-antimatter interactions. It is evident, that in the case of antigravity, a uniform gravitational field, just as a uniform electric field tends to separate “virtual” electrons and positrons, pushing them in opposite directions, which is a necessary condition for pair creation by an external field. But while an electric field can separate only charged particles, gravitation as a universal interaction may create particle-antiparticle pairs of both charged and neutral particles, like for instance ν 1ν 1 ,ν 2ν 2 ,ν 3ν 3 , e − e + ,..., π 0π 0 ,..., pp ... pairs. Here ν 1 , ν 2 , ν 3 denotes known types of neutrinos, from the lightest ν 1 to the heaviestν 3 . In the case of a uniform gravitational field, characterized with acceleration a, Equations (7) and (8), trivially transform to

dN m 4 c ⎛ a ⎞ ⎟ = 2 4 ⎜⎜ dtdV π D m ⎝ a cr ⎟⎠

2

∞

1

∑n n =1

2

⎛ nπ a cr ⎞ exp⎜ − ⎟ ⎝ 2 a ⎠

(9)

where Dm =

h 2mc 3 2c 2 and a cr = = mc h Dm

(10)

are respectively, reduced Compton wavelength of particle with mass m, and critical acceleration. Let’s stress two important consequences of Eq. (9). First, the particleantiparticle creation rate is significant only for an acceleration a greater than the critical acceleration acr. Second, in the case a > a cr (and we are interested only in this case) the infinite sum in Eq. (9) has numerical value not too much different from 1. So, a simple, but good approximation is given by: 2

dN m 4 c ⎛ a ⎞ ⎟ = 2 4 ⎜⎜ (11) dtdV π D m ⎝ acr ⎟⎠ It is obvious that if a > a cr , in all points of a volume V, than putting a=acr in Eq. (11) gives a crude lower bound (lb) for the particle rate creation per unite time: 4 c ⎛ dN m ⎞ (12) ⎜ ⎟ = 2 4 V ⎝ dt ⎠ lb π D m In order to get further peace of information, let’s compare accelerations in Eq. (1) and (10). It reveals that critical acceleration is much larger than acceleration at the Schwarzschild sphere. So, the conclusion is that creation of particle-antiparticle pairs by gravitational field is eventually possible only deep inside the horizon of black holes. An immediate consequence is that if (for instance) a black hole is made from ordinary matter, produced particles must stay confined inside the Schwarzschild sphere, while antiparticles should be violently ejected because of gravitational repulsion. In general, a black hole made of matter ejects antiparticles and just opposite to it, a black hole made of antimatter ejects particles. Now, the question arises, how deep inside the horizon the creation of pairs becomes possible. As a very simplified model, let's assume spherical symmetry of the black hole and consider it as a miniscule ball with radius RH (we will come back to this question later). In addition let's define a critical radius RCm, defined as the

5

distance at which gravitational acceleration has the critical value acr. Combining Eq. (10) with the Newton's law of gravitation leads to 2 = RCm

D m RS M ≡ L2P 4 2m

(13)

where RS is the Schwarzschild radius and L P = hG c 3 is the Planck length. The critical radius RCm, defines a critical sphere SCm, which divides the vacuum (surrounding the black hole) in two regions. The first region is a sphere shell with the inner radius RH and the outer radius RCm, i.e. the volume enclosed by the „surface” of the black hole and the critical sphere SCm. This region should be a „factory” for creation of particle-antiparticle pairs with mass m. In the second region, the space outside the critical sphere SCm, there is no significant pair creation, but during their short lifetime, „virtual“ pairs of mass m, behave as „gravitational dipoles“ in the gravitational field of the black hole, which must result in vacuum polarization, in analogy with corresponding phenomenon in QED. In the present paper we are interested only in the creation of particle-antiparticle pairs by black hole, i.e. in the region inside the critical sphere SCm. It is obvious that for every kind of particle, corresponds a critical sphere SCm, defined by the corresponding critical radius RCm. So, there is a series of decreasing critical radiuses: RCν1 , RCν 2 , RCν 3 , RCe , RCμ , RCπ 0 ,..., RCp ,... (14) corresponding respectively to known types of neutrinos, electron (e) muon ( μ ), π 0 meson … proton (p), and so on. In order to get some quantitative estimate of the number of created particles, let us adapt Eq. (11). This equation addresses the idealized model with uniform acceleration, while in reality particle-antiparticle pairs are created by the gravitational field of a black hole, which varies in magnitude and direction. However, if the linear size of a small volume V is much smaller than its distance R from the centre of the black hole, acceleration inside such a volume is fairly uniform with magnitude a = GM R 2 . Having this in mind and using Eq. (13), Eq. (11) transforms to: 2

2

4

dN m 4 c ⎛R ⎞ c ⎛ RS ⎞ 1 c ⎛ Mm ⎞ 1 ⎜ ⎟⎟ 4 = 2 ⎜⎜ 2 ⎟⎟ 4 = 2 4 ⎜ Cm ⎟ = (15) 2 ⎜ dtdV π D m ⎝ R ⎠ π ⎝ MP ⎠ R 4π ⎝ D m ⎠ R Or, after integration over the volume of the sphere shell with inner radius RH and outer radius RCm 2

2

dN m c ⎛ RS ⎞ RCm − RH 4c ⎛ Mm ⎞ RCm − RH ⎟ (16) = ⎜⎜ = ⎜⎜ 2 ⎟⎟ dt π ⎝ D m ⎟⎠ RH RCm π ⎝ M P ⎠ RH RCm In the above approximate formulae, the radius RH of the black hole is not known. General Relativity teaches us that if the mass of the collapsing body is larger than a critical value (estimated to be a few times the mass of the Sun), the collapse can’t be stopped. Having reached the Schwarzschild radius the body will continue to collapse, with all of its particles arriving at the centre within a finite time. So, if General Relativity is right, RH must be zero; collapse ends with singularity. However, if there is antigravity, a radically different picture of the collapse inside the horizon may be expected. During the collapse, the surface of the contracting body passes through a succession of critical surfaces defined by their critical radiuses (14). The production of ν 1ν 1 pairs starts when the first critical surface ( RCν1 ) is

6

reached, i.e. when RH becomes smaller than RCν1 . When the second critical surface is reached (i.e. RH becomes smaller than RCν 2 , it is time for the beginning of the creation of ν 2ν 2 pairs and (in principle) so on following the series (14). In general, as a consequence of decrease of RH, creation of more massive particle-antiparticle pairs becomes possible, resulting in faster “evaporation” of the black hole. A trivial numerical study, based on the use of lower bound (12), reveals that for astronomical black holes, if RH is smaller than the critical radius of the neutron (RCn), the rate of the mass production per unite time is close to the mass of the black hole. Thus, black holes as long-living objects can’t exist for RH
7

In addition to the number of neutrino-antineutrino pears produced, it is important to have an estimate for the energy of the antineutrinos ejected from the horizon of a black hole made by matter. As pairs are created in the vicinity of RH=RCe, as an estimate of energy, it is possible to use the following lower bound: Eν =

GMmν = RCe

RS

mν c 2

(19)

λe Let's end this section with the remark that creation of neutrino-antineutrino pairs of a certain type should stop when the Schwarzschild radius becomes smaller than the Compton wavelength of the corresponding neutrinos. It is a consequence of the fact, known from Quantum Electrodynamics [9], that a sufficiently strong external field is a necessary but not the sufficient condition for creation of particle-antiparticle pairs from the vacuum. In addition to its strength, the external field must extend to a sufficiently large space volume. Roughly speaking, the cube of the Compton wavelength is the minimal needed volume, and it is obvious that this additional condition can’t be satisfied when RS is smaller than the Compton wavelength. In other words, neutrino-antineutrino pairs with mass m, can be produced only by a black hole which has a mass M greater than the critical value defined by Equation (5). It is amusing that the same critical mass (5) is upper bound in the case of thermal radiation and lower bound in the case of “antigravitational” radiation.

3.2 Numerical results The formulas (18) and (19) allow estimating the energy and the number of antineutrinos eventually ejected from the horizon of a black hole during a certain period of time. If such a phenomenon exists in nature, it may be eventually revealed by observing supermassive black holes. The best evidence for the presence of supermassive black holes at the centres of galaxies (see [11] for a short review) comes from the observations of the Galactic Centre of the Milky Way. Let’s point out that the black hole in the centre of our galaxy is the best situated for observation of eventual antineutrino emission caused by antigravity. Firstly, it is the nearest supermassive black hole. Secondly, the centre of our galaxy does not belong to the family of active galactic nuclei, so that neutrinos produced by other mechanisms are reduced to a minimum. The second best choice is the supermassive black hole in the centre of the Andromeda (M31) Galaxy. It is in fact, the next nearest supermassive black hole. Contrary to the Milky Way, Andromeda has an active galactic nucleus. The best estimates [11] of the mass, the Schwarzschild radius and the distance for these supermassive black holes are given in the following table: Table 1: Numerical data for black holes in the centre of Milky Way and Andromeda Milky Way Andromeda 6 Mass M = 3.67 ×10 M Sun = M = 1.4 ×10 8 M Sun = = 7.3 ×10 36 kg

The Schwarzschild radius

distance

= 2.8 × 10 38 kg

RS = 1.1×1010 m

RS = 4.2 ×1011 m

d = 8kpc = 2.4 ×10 20 m

d = 760kpc = 2.3 ×10 22 m

In order to calculate creation rates for all known types (ν e ,ν μ , ν τ ) of neutrinos, we need to know their absolute masses, but, at the present stage of

8

knowledge, we are uncertain about the masses of neutrinos. We have chosen to use the value mν = 0.0727eV / c 2 for the mass of the heaviest neutrino. This value is the result of our calculations in Appendix A, and lies in the expected interval [12] between 0.04eV/c2 and 0.2eV/c2. Using Equations (18) and (19), together with the value mν 3 = 0.0727eV / c 2 , leads to the energies and numbers of ejected neutrinos presented in Table 2. Table 2: The energy and the number of the heaviest neutrinos ejected by the corresponding supermassive black hole Milky Way Andromeda 40 Ejected per second 5.3 × 10 1.3 × 10 43 Ejected per year 1.7 × 10 48 4 × 10 50 Energy (GeV) 12.2 75 The above numbers show that creation of neutrino-antineutrino pairs is quite significant. In fact the number of produced neutrinos by a supermassive black hole in the Galactic Centre should be much bigger than the number of neutrinos emitted by the Sun. As we lie at a distance of d = 8kpc = 2.4 × 10 20 m from the centre of our galaxy and the surface of the corresponding sphere is: 2 41 2 35 2 S d = 4πd = 7.2 × 10 m = 7.2 × 10 km , a detector the size of one kilometre cube, should be “visited” by about 2.3 × 1012 antineutrinos per year, coming from the centre of the Milky Way. For the Andromeda Galaxy S d = 4πd 2 = 6.6 × 10 39 km 2 and only 6 × 1010 antineutrons may “visit” the IceCube during a year. This smaller number (compared to the number of neutrinos from the centre of our galaxy) could be compensated by a bigger cross-section and presumably detected, but an unwanted complication is that Andromeda Galaxy has an active galactic nucleus. These numbers of “visitors” should be sufficient to detect between a few tens and a few hundreds antineutrinos per year.

3.3 Concerning the lifetime of black holes Let’s end this section with a simple estimation of the lifetime of black holes. In fact, from Equation (18), it is easy to obtain: 2 RS 0 − Rs 3π ⎛ M P ⎞ D ν ⎜⎜ ⎟⎟ De (20) t= 8 ⎝ mν ⎠ c RS 0 RS where mν is the mass of the most massive neutrino, while RS0 and RS are respectively the Schwarzschild radius at time t=0 and after some time t. In the derivation of (20), in addition to Equation (18), we have used the following obvious results: c2 dM = dRS , dM ν = mν dNν , dM = − dM ν (21) 2G The first one is the proportionality between the Schwarzschild radius and the mass of a black hole, the second one is the proportionality between the mass and the number of ejected antineutrinos, while the third one states that the decrease of the

9

mass of the black hole has the same absolute value as the increase of the mass of ejected antineutrinos. For instance formula (20) predicts that, after about 1036 seconds from now, as result of emission of antineutrinos, the mass of the black hole in the centre of the Milky Way will become as small as the actual mass of the Sun. However if instead of antineutrinos, positrons are emitted, the same change of mass should happen in about 1015 seconds, less than a billion years, which is shorter than the lifetime of a star and unrealistically short for a black hole. If we naively apply relation (20) to photons and antiphotons having the same wavelength as electrons, the same unrealistically short lifetime follows. It suggests that even in the case of existence of antigravity, the photon is its own antiparticle. So, it seems that photons must be attracted by both matter and antimatter. A possible alternative is that a large portion of created antiphotons is somehow absorbed during their travel inside the horizon of a black hole, which consequently allows longer lifetimes for black holes. The formula (21) suggests that the lifetime of stellar and supermassive black holes is nearly independent of their initial mass. In fact, in the limit RS 0 >> RS , the ratio of lifetimes of different black holes tends to 1, independently of their initial masses. In particular, all black holes are reduced to the Schwarzschild radius equal to the reduced Compton wavelength of the heaviest neutrino, D ν = h mν c = 2.7 × , i.e. to the critical mass (2) in the time 3π tA = 8

⎛ MP ⎜⎜ ⎝ mν

2

⎞ h ⎟⎟ ≈ 1.1 × 10 41 sec 2 c m m ⎠ ν e

(22)

Let’s note that the corresponding critical mass (2) equals to 1.8 × 10 21 kg .

3.4 Comparison between Hawking and “antigravitational” radiation It is important to compare the mass loss of a black hole caused by Hawking (thermal) radiation with the loss caused by the gravitational repulsion between matter and antimatter. We will show that, for all masses, the “antigravitational” radiation is stronger than the Hawking radiation. As noticed before, the Schwarzschild radius can be considered, as the lower bound for wavelengths of thermal radiation and the upper bound for wavelengths of “antigravitational” radiation. Hence, putting D m = RS and V = 4πRS3 / 3 in Equation (12) leads to the following lower bound for mass loss caused by antigravity (Alb): dM dt

= Alb

4 hc 4 1 hc 4 1 0 . 424 ≈ 3π G 2 M 2 G2 M 2

(23)

Comparison between results (3) and (23) shows that dM dt Alb > 2 × 10 4 dM dt H . In fact, as (23) is only a lower bound, the mass lost caused by antigravity should be much bigger. In the case of supermassive and stellar black holes, instead of a lower limit (19) we may use a much more accurate result (18) in order to get: dM dt

=

4 2

c2 me h

⎛ mν ⎜⎜ ⎝MP

π Now, the quotient of the values (24) and (3) is dM / dt dM / dt

A

A H

≈ 8.7 × 10

4

⎛m me ⎜⎜ ν ⎝MP

10

3

⎞ ⎟⎟ M 3 / 2 ⎠

(34)

3

⎞ M 7/2 ⎟⎟ 4 ⎠ MP

(25)

It is evident that this ratio grows with mass, but already for a mass as small as a solar mass, the numerical value is 9.4 × 10 38 , 34 orders of magnitude larger than it can be concluded from the lower bound estimation! However the lower bound estimation is important in the case of small masses when formulae (8) and consequently (24) can’t be used. Let’s note that the fourth-root of the quotient (25) shows how many times the temperature of a black hole must be greater than its Hawking temperature, in order to have a thermal radiation as strong as the “antigravitational” radiation. In the above numerical example, when the mass equals a solar mass, the fourth-root of the ratio is 5.5 × 10 9 . So, instead of a Hawking temperature of TH ≈ 6 × 10 −7 oK , the black hole should have 3300 o K in order to produce the same mass loss as antigravity.

3.5 A new interpretation of the Planck length The Hawking temperature (2), and consequently the spectrum of radiation, depend only on the mass M of the black hole and are independent of the mass distribution inside the Schwarzschild sphere. Whatever is the final result of the collapse, a singularity or a “ball” with a finite radius RH < RS . , the thermal radiation is the same. However, and it is in complete contrast with the thermal radiation, the “antigravitational” radiation depends on both M and RH. As the radius RH of a black hole decreases, the more massive particle-antiparticle pairs can be created; consequently more massive (anti)particles are emitted and hence the lifetime of the black hole decreases. From the purely mathematical point of view, the maximal possible mass of a particle-antiparticle pair is equal to the mass of the black hole, what, at least mathematically, corresponds to a division of the black hole in a pair of black holes; one composed from matter and the other from antimatter. The question arises: what is the critical radius, for which, the gravitational field is sufficiently strong to split a black hole into a pair of black holes. The answer comes from Equation (13). This ultimate critical radius is equal to the Planck length. Hence, if the collapse of a black hole results in a value RH smaller than the Planck length, even a supermassive black hole will decay immediately. So, RH=0 (singularity) is excluded as a possibility. Of course a black hole can be a long living object only if RH is many orders of magnitude larger than the Planck length; presumably equal to RCe, as argued in the section 3.1.

4. Microscopic black holes and antigravity Recently developed theories with large or warped extra dimensions suggest that it would be possible to produce microscopic black holes (MBH) in the Large Hadrons Collider (LHC), at the European Centre for Nuclear Research, CERN (for a topical review see [10] and references therein). So, detection of the first MBH, created by human activity, may become reality already next year (i.e. 2008). Unambiguous detection of eventually formed MBH would be in its own a major scientific achievement. Of course, the goal is more ambitious; not only to produce and detect MBH, but also to study their proprieties. The cornerstone of the planned studies is the Hawking radiation, i.e. fast decay of a MBH by emitting elementary particles with a black body energy spectrum. For instance, Hawking radiation is expected to be a sensitive probe of the dimensionality of extra space. To be more specific, the characteristic Hawking temperature (in natural system of units, ћ=c=kB=1) is given [10] by:

11

1

TH

⎛ ⎞ d +1 ⎜ ⎟ M d +2 ⎟ d +1 = M D ⎜⎜ D ⎟ M ⎛ d + 3⎞ 4 π ⎜⎜ BH 8Γ⎜ ⎟ ⎟⎟ ⎝ 2 ⎠⎠ ⎝

(26)

where, Г is the gamma function, d is the number of extra dimensions, MBH stays for the mass of the black hole and MD (expected to be about 1TeV) is Planck mass in (4+d) dimensional space-time. The mass, MBH, of the black hole can be reconstructed from the total energy of decay products, while Hawking temperature may be determined from the energy spectrum of emitted particles. Now, after taking a logarithm of both sides of Equation (26), the dimensionality of extra space, d, may be determined from the slope of a straight-line fit to the log10(TH/1TeV) versus log10 (MBH/1TeV) data. It is evident that in the above reasoning the Hawking radiation is taken for granted and that the possible existence of other mechanisms of decay of a black hole is neglected. The Hawking process was studied by many authors and (using different methods and approaches) they have confirmed the same original theoretical result [13] that black holes decay by black body radiation. So, with such a high level of consensus, the plans to use this phenomenon in the future study of MBH seem quite reasonable. However, in spite of it, we believe it is both worthy and necessary to think under which circumstances, the use of this method may become more difficult or even inconclusive. Just as an illuminating example (independent of our assumption of antigravity) let’s point out the case of a charged black hole. Firstly, thermal radiation of a charged black hole depends on both its mass M and its electric charge Q (see [14] and references therein). Consequently, a simple formula (17) is not more valid, what makes the use of the above method more difficult (but not impossible). Secondly, in addition to the radiation of thermal nature, a charged black hole emits particles through a non-thermal mechanism as well [14]. In fact, particle production by charged (Reissner-Nordstrem) black holes was predicted simultaneously with, or even somewhat earlier [14] than, the famous Hawkins’s thermal radiation. This nonthermal radiation may be easily understood in the framework of Quantum Electrodynamics, where, as well known [9], in a sufficiently strong electric field (in this case the electric field of a charged black hole), the (Dirac) vacuum becomes unstable and decays leading to a spontaneous production of electron-positron pairs. What we can learn from this example is that decay of a black hole may be caused by both, thermal and non-thermal radiation and that, at least in principle, there are circumstances when thermal radiation is dominated by a non-thermal one. The most striking is the example of an extremal black hole (satisfying condition Q 2 = 4πε 0 GM 2 ). Such a black hole has a zero Hawking temperature (and accordingly gives no thermal radiation) but it still radiates [14] through the mechanism of creation of particle-antiparticle pairs. So, the extremal black hole is an example of purely non-thermal radiation and, in such a case, the proposed method based on the assumption of domination of thermal radiation, simply can’t work. It may be eventually argued that in the physical world such black holes presumably don’t exist, or if they exist, they are a rarity and a large fraction of MBH that can be created in CERN will decay following Hawkins’s law. However, if there is antigravity, non-thermal radiation (caused by pair creation) is inevitable.

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The important difference between an astronomical black hole and a mini black hole is that an astronomical black hole should emit only neutrinos, while a mini black hole may decay through the emission of much heavier particles. It is a consequence of the fact that in the moment of creation, a mini black hole has a very small Schwarzschild radius RS (10-19m or less) and even a smaller RH. As we have seen in the previous section, the collapse of an astronomical black hole is prevented at RH=RCe which is, for instance, in the case of the black hole in the centre of our galaxy, more than 1018 times bigger than RH of a mini black hole. Without recourse to any quantitative estimation, it is immediately clear that non-thermal radiation caused by antigravity must dominate Hawking's radiation. In fact, the pair production of the most massive particle-antiparticle pairs is happening deep inside the horizon (i.e. inside the spherical shell determined with radiuses RH and RCm) while Hawking radiation comes from the close vicinity of the Schwarzschild radius RS which is significantly larger than both RH and RCm. So, non-thermal radiation corresponds to shorter wavelengths (i.e. higher frequencies, energies and masses). Shortly, Hawking radiation should be dominated. Thus, if the decay of (eventually produced) mini black holes is dominated by non-thermal radiation, it should be considered as a signature of antigravity.

5. Comments Just to be clear, in the present paper we do not advocate the existence of antigravity (but of course we will be very pleased if it exists). It is wrong to say that antigravity exists, but it is also wrong to say that antigravity doesn’t exist. Simply, in the absence of the experimental evidence, we do not know what the gravitational proprieties of antimatter are. To prove or disprove the existence of antigravity is a very important task that deserves the full attention of the scientific community. Best would be to design special experiments devoted to it [6,7], but as the present situation is not favourable to it, we have proposed to look for the eventual signature of antigravity in two extreme cases: supermassive and microscopic black holes. Our semi-classical approach is highly simplified, but even so, it presents the main ideas, demonstrates how new and rich physics may result from the existence of antigravity and predicts some consequences that may be testable with new generation Neutrino Telescopes and at the LHC at CERN. We hope our rudimentary approach will provoke further interest and deeper studies. The eventual discovery of antigravity would be one of the biggest scientific surprises and achievements with fundamental implications in physics and a deep impact on the human mind as whole. However, unambiguous evidence that the predicted effects of gravitational repulsion between matter and antimatter do not exist, would also be a significant result (as a hint that there is no antigravity). Whatever the answer is, it is important to know it. This paper is dedicated to my father Slavko, and my children Ivan and AnjaMilica

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Appendix A: Concerning the rest mass of neutrinos The recent investigations of neutrinos from the sun and of neutrinos created in the atmosphere by cosmic rays, have given strong evidence for neutrino oscillations (i.e. phenomenon when neutrinos change from one flavour to another). Neutrino flavour change implies that neutrinos have masses. To determine these masses remains one of the most challenging tasks of contemporary physics, bearing fundamental implications to particle physics, astrophysics and cosmology. In the present paper we have calculated the rest mass of neutrinos by using the existing experimental data and a new, theoretical assumption. Let’s consider the case when there are three neutrino flavour eigenstates (ν e , ν μ , ν τ ) and three neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) with corresponding

masses (m1 , m 2 , m3 ) . In such a case neutrino spectrum contains two mass eigenstates 2 separated by the splitting ΔmSun = m22 − m12 (needed to explain the solar data) and a third eigenstate separated from the first two by the larger splitting 2 Δmatm ≈ m32 − m22 ≈ m32 − m12 (needed to explain the atmospheric data).

From recent experiments, we know with a reasonable accuracy, only mass squared differences Δmik2 = mk2 − mi2 and sometimes only their absolute value. As there are only two independent squared mass differences [12], we only have two equations available: 2 (A1) Δm Sun = Δm122 = m22 − m12 2 2 Δmatm ≈ Δm23 = m32 − m22

(A2)

These two equations are obviously not sufficient to determine three masses m1 , m2 , m3 . A third relation between masses is needed. The most recent proposal [15] is to use, as a third equation, geometric mean neutrino mass relation m2 = m1 m3 for normal spectrum and m1 = m2 m3 for the inverted spectrum. This proposal is inspired by the geometric mean mass relation used previously for quarks. A weak point of the proposal lies in the well known fact [12] that leptonic mixing (characterized with large mixing angles) is very different from its quark counterpart, where all the mixing angles are small. Our second objection is that geometric mean mass relation has a form which is quite different from Equations (A1) and (A2) while it is desirable to have a third equation with similar form. As we will show, such equation is a consequence of the results of section 2.1. In order to be definite, let’s consider a black hole composed from matter which emits antineutrinos. Let the total number of emitted antineutrinos to be N = N 1 + N 2 + N 3 where N 1 < N 2 < N 3 , correspond respectively to the masses m1 < m2 < m3 of the normal neutrino mass spectrum. It is obvious that whatever the numbers N 1 , N 2 , N 3 are, there is always a positive real number a ( 0 < a < 2 ) so that a relation between the numbers of produced neutrinos can be written as N 1 + N 2 = aN 3 (A3) However, as follows from Equation (13), the number of created and emitted neutrinos during a certain period of time t, is proportional to its squared mass: Nν Rs 2 2 c3 (A4) = Rs mν ≡ Amν2 2 t π h De Thus Equations (A3) and (A4) lead to: 14

m12 + m22 = am32 (A5) Equation (A5) may be used as a third relation, in the case of the normal spectrum, while in the case of the inverted mass spectrum ( m3 < m1 < m2 ) we have: (A6) m12 + m32 = bm22 ; 0 < b < 2 If we consider fractions w1 = N 1 / N , w2 = N 2 / N , w3 = N 3 / N as classical probabilities, with additional simplification N 1 = N 2 , relation (12) trivially leads to: 1 a 1 (A7) w1 = w2 = , w3 = 2 a +1 a +1 Thus, “probabilities” w1 , w2 , w3 describe the initial beam emitted by a black hole. It is evident that creation of neutrino-antineutrino pairs from vacuum is not “democratic”; the production of the heaviest pairs is favoured. The ratio w3 / w2 = 2 / a shows that production is more democratic for the larger values of a. However, compared with the other sources of neutrinos, a black hole (according to section 2.) is both a universal and quite “democratic” source. For instance, a star, like our Sun, creates and emits only electron neutrinos, while a black hole creates neutrino-antineutrino pairs of all flavours. In fact Equation (A3) can be written only in the case of a universal source. Because of neutrino oscillations, the initial beam will change on its way to the detector. The expected (anti)neutrino flavour ratios at Earth may be estimated by using, for instance, “tri-bimaximal” model [16]. An ideal experiment that measures N (anti)neutrino events will correctly partition these events into N 1 , N 2 , N 3 flavour bins. At least in principle it allows experimental determination of a. Now, let’s calculate masses m1 , m2 , m3 as the functions of the parameter a. From Equations (A1), (A2) and (A5) follow: a a −1 2 m12 = Δm23 + Δm122 (A8) 2−a 2−a a 1 2 (A9) m22 = Δm23 + Δm122 2−a 2−a 2 1 2 (A10) m32 = Δm23 + Δm122 2−a 2−a with the allowed values of a satisfying condition 2 Δm12 2 2 Δm12 + Δm 23

≤a<2

(A11)

Relation (A11) is a simple consequence of the obvious conditions m12 ≥ 0 ; m32 > m 22 . 2 Because Δm32 ≅ 30Δm122 , the lower bound for a is about 1/30.It is trivial to deduce that masses increase with a. The most elegant solution (but we do not know if nature has the same aesthetic taste as we have) is a = 1 . In this particular case 2 2 2 2 2 m12 = Δm 23 , m 22 = Δm 23 + Δm12 , m32 = 2Δm 23 + Δm12

(A12)

The use of the numerical data [12, 17] coming from the experiments: 2

⎛ eV ⎞ ⎛ eV ⎞ 2 2 2 Δm12 = ΔmΘ2 ≈ 8 × 10 −5 ⎜ 2 ⎟ ; Δm 23 ≈ Δm atm ≈ 2.6 × 10 −3 ⎜ 2 ⎟ ⎝c ⎠ ⎝c ⎠

leads to the following values

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2

(A13)

m1 = 0.05099

eV c

2

; m 2 = 0.05176

eV c

2

; m3 = 0.07266

eV c2

(A14)

If we trust (model depending) constraints coming from cosmology consideration, the sum of masses m1 + m2 + m3 , must be [1] less than 0.2 − 0.4eV / c 2 or even [17] less than 0.17eV / c 2 . Thus, in the general solution given by (A8), (A9) and (A10), a can be only slightly larger than 1. For completeness of the presentation, let’s note that in the case of inverted mass spectrum m3 < m1 < m2 , the basic equations are: 2 Δm122 = m22 − m12 , Δm32 = m22 − m32 , m12 + m32 = bm22

(A15)

Appendix B: Testing existence of antigravity with antiprotons B.1 The basic idea for experimental testing of antigravity In order to grasp the idea, let us start from the equation of motion of a charge q in an electromagnetic field r r r d r p = qE + qv × B (B1) dt r The solution of this equation for a particle with non-relativistic velocity v , in r r crossed, constant and uniform electric ( E ) and magnetic field ( B ) may be found in any good book of electrodynamics (see for example Ref. [18]). The crucial point for r further analysis is that in the direction perpendicular to the common plane of fields E r and B , a particle moves with a velocity which is a periodic function of time with an average value, drift velocity r r r E×B v drift = (B2) B2 r r In the particular case E =0, v drift = 0 and the orbit of the particle is a helix, r with its axis parallel to B (in fact the well known cyclotron motion). r Let us consider the case E =0, but instead there is a constant and uniform r r gravitational field. That instead of qE there is a term m I a in the equation of the r motion, a being the gravitational acceleration. So, gravitation is equivalent to the r r existence of an electric field E = m I a / q in Eq. (B1). The corresponding expression for drift velocity caused by gravitation is r r mI a × B r (B4) v drift = q B2 In principle, Eq. (B3) can serve as basis for determination of the gravitational acceleration of antiprotons. Let us note that in the case of the inhomogeneous magnetic field, there is r magnetic gradient force F = grad ( μB ) , where μ (the magnetic moment associated with a cyclotron orbit) is a constant of the motion [19]. So, the corresponding drift velocity is r r μ gradB × B v drift = (B5) q B2

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So, in the general case, there are drift velocities (B2), (B3) and (B4) caused respectively by an electric field, a gravitational field and a magnetic field gradient. Of course, from the point of view of the gravitational experiment, non-gravitational drifts (B2) and (B4) are just unwanted complication. The ideal experimental configuration is a horizontal, sufficiently long, metallic, r cylindrical, drift tube, with constant and uniform magnetic field B (the only nongravitational field!) collinear with the axis of the cylinder. However, drift velocity (B3) is extremely small (in a field B=1T it is about 0.1 micrometer/sec, i.e. ten thousand seconds are needed for drift of one millimeter!). Consequently, a sufficiently long drift tube (a few tens of kilometers!) can’t be realized in practice. So, instead of a very long tube, it is necessary to confine the movement of the particle in a short cylindrical tube. But this confining of the particles demands additional nongravitational fields (electric field or inhomogeneous magnetic field). Additional fields therefore assure, crucial, axial confinement of the particles, but there is high price to pay for it: Instead of a purely gravitational drift, there is a superposition of gravitational drift (B3) and non-gravitational drifts (B2) and/or (B4), caused by the confining fields. In short, the key non-trivial idea is to study gravitational properties of antiprotons, measuring effects of the gravitational drift on particles confined in a trap (or a quasi-trap). This idea was firstly suggested seventeen years ago [20], but at that time (1989) antigravity was not considered a credible idea [21] and the interest for such an experiment was very low. Different choices of confining electromagnetic fields lead to different realizations of the same basic idea. The open question is: which is the most appropriate choice of the confining fields for the purpose of the gravitational experiment, i.e. how to reduce to a minimum, the inevitable perturbations of the tiny effect of the gravitational drift? One promising choice is to use confining fields with cylindrical symmetry, so that symmetry is violated only by gravitation. In general, if ρ , ϕ , z are cylindrical coordinates, cylindrical symmetry means, that potential V and the intensity of do not depend on ϕ . So, V = V ( ρ , z ) and B = B( ρ , z ) . magnetic field B r Consequently, electric field ( E ) and the magnetic field gradient ( gradB ) have only axial and radial components. Thus, in absence of gravity, non-gravitational drift velocities (B2) and (B4) are always tangential to a circle perpendicular to the geometrical axis of the cylinder and having centre on that axis. The result is a circular precession (known as magnetron motion) around the symmetry axis. Effects of gravity may be literally switched on and off, putting the drift tube respectively in horizontal and vertical position. When gravity is “switched on” (horizontal drift tube), magnetron motion is slightly perturbed. In fact, the resultant r r force F (the vector sum of the gravitational force m I a , and radial non-gravitational r force Fρ ) is not radial for a circle with centre on the geometrical axis, but for a circle

with the centre shifted above the geometrical axis in the case of gravity, and below the axis in the case of antigravity. The new centre is the point at which the resultant force r F vanishes. The corresponding distance Δ between axis and the new centre is determined by Δ mI a = (B5) ρ Fρ

17

So, direction of the shift Δ contains information if there is antigravity or not. Magnetron motion (modified by gravity or antigravity) starts with injection of the particle. Figure 1 provides a schematic view of the modified magnetron motion of a particle injected along the axis of the cylinder. The black and red circles correspond respectively to gravity and antigravity. Full lines with an arrow represent corresponding position after some time (in fact a few hours). So, if after some time the particle is detected below the horizontal plane passing through the axis, it proves that there is antigravity. If there is no antigravity, but only violation of the WEP, both circles will have the centre above the axis but with different radii. Fig. 1: Schematic view of the magnetron motion modified by gravity or antigravity. For a particle injected along the axis of the cylinder, black and red circles correspond respectively to gravity and antigravity. For simplicity only the cross section of the cylinder is shown, because axial motion is not essential for the understanding of the principle: if after some time a particle is detected below the horizontal plane passing through the axis, there is evidence of antigravity. A simple violation of the WEP corresponds to two circles with centre above the axis but with different radii.

B.2 Standard electromagnetic traps for charged and neutral particles Electromagnetic traps for charged and neutral particles, based on the use of electric or magnetic multipole fields, have a few decades of very successful applications in experimental physic [22]. It would be ideal if one of these routinely used traps can serve as realisation of our basic idea. The assessment of existing traps shows that only two of them, the Penning trap and the magnetic bottle, may be appropriate. In a cylindrical Penning trap confinement is achieved with a constant and homogenous magnetic field collinear to the axis of the cylinder, and a quadrupole electrostatic field giving a parabolic potential near the centre of the trap. In a magnetic bottle only an inhomogeneous magnetic field is used. Our basic idea was studied in detail for both a Penning trap [23] and a magnetic bottle [24]. The comparison between these analyses favours the magnetic bottle. In fact the gravitational effect is completely negligible at typical experimental conditions for Penning traps. The “Gravitational Penning trap” must be much larger and the applied voltage six order lower than in a standard trap. In a magnetic bottle, gravitational and magnetic gradient forces are comparable at typical experimental conditions and consequently a standard trap may be used However, as we will argue in the following section, there are configurations (of course quite different than a Penning trap), when confinement is achieved with an electrostatic field, that may have advantages compared with a magnetic bottle(see also Ref. [6]).

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B.3 Minimal confining configuration As it may be concluded by the general formula (B5), and in particular from the study of the Penning trap, the quotient m I a Fρ mustn’t be too small. In other words the radial electrical force must be very small (close to the value of the gravitational force: 1.6 × 10 −26 N ). However, electric multipole fields in traditional traps (as Penning trap) are generated by shaped electrodes extended in space, demanding in principle a distribution of charges that can be considered as continuous. So, the inherent limitation of these configurations is that the number of charged particles producing a confining field must be big, causing trouble with an electric field too strong compared with gravitation. A single way to escape from this limitation is to attack the problem from the opposite side, when the confining field is not generated by electrodes, but by a low number of charged particles. Just one particle may produce a confining electric field, but in order to preserve cylindrical symmetry, a minimum of two particles must be used. So, as we will demonstrate, the “minimal confining configuration”, consists of two charged particles, producing an electric field for the confinement of a third particle (an antiproton in our case). Our basic apparatus is a metallic, cylindrical, maximum two meters long drift r tube, with constant and uniform magnetic field B collinear with the axis of the r cylinder. The field B assures both radial confinement of particle and existence of gravitational drift described by Equation (B3). It is necessary to add an electric field r E in order to assure axial confinement (i.e. oscillations of particle between ends of the cylinder). In order to grasp the idea let’s imagine that two antiprotons are somehow “fixed” on the axis of the cylinder at the distances z = L and z = − L . It is evident, that the electric field produced by these two antiprotons, together with magnetic r field B , can assure confinement of a third antiproton. One antiproton may be “fixed” on the axis (i.e. forced to oscillate with small amplitude) by using an appropriate circular charged ring with uniform linear charge density. This ring must be hidden behind a metal wall, in such a way that its field is shielded from the area where the antiproton used for the gravitational measurement, is moving. So, the antiproton testing antigravity, moves only in the electric field of two “fixed” antiprotons. Fig. 2 shows a possible experimental realization. Figure 2: Schematic view of the experimental configuration

Antiproton for testing antigravity (●) moves in a uniform and homogenous magnetic field collinear with the axis of the metallic cylinder, and within the electric field of two antiprotons (■), that are “fixed” on the axis by using the corresponding circular charged ring (■) with uniform linear charge density. Rings are hidden behind metal walls, so that their electric field is shielded from the central part of the cylinder where the antiproton moves.

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The cylinder is divided into three parts. The central part is partially separated from the other two by metal walls, having in the centre a circular hole. Both non-central parts of the cylinder have a charged ring for “fixing” one antiproton on the axis near the entrance of the central part. The circular metal wall, separating central part of the cylinder, serves to hide the ring. Without detailed studies, including Monte Carlo simulations, it is impossible to say if this is a feasible configuration or not. For more details see Ref. [6]. REFERENCES [1] M.M. Nieto and T. Goldman, Physics Reports 205, 221 (1991) [2] G. Chardin, Nuclear Physics A 558, 477c (1993) G. Chardin, Hyperfine Interactions 109, 83 (1997) D.V. Ahluwalia, ArXiv: hep-ph/9901274, 1999 - arxiv.org [3] R.M. Santilli, International Journal Modern physics A 14, 2205 (1999) R.M. Santilli, Foundations of Physics 33, 1373 (2003) R.M Santilli http://www.i-b-r.org/docs/Iso-Geno-Hyper-paper.pdf (2002) [4] A.G. Reiss et al., The Astrophysical Journal 560, 49 (2001) S. Perlmutter, Int. J. Modern Physics A 15S1, 715 (2000) [5] Guang-Jiong Ni, ArXiv: physics/0308038, 2003 – arxiv.org J.M Ripalda, ArXiv: gr-gc/9906012, 1999 – arxiv.org [6] D.S. Hajdukovic, ArXiv: gr-gc/0602041, 2006 – arxiv.org [7] M. Amoretti et al. , ArXiv: hep-ex/0503034, 2005-arxiv.org M. H. Holzscheiter et al. , Hyperfine Interactions, 109, 1 (1997) T.J Philips, Hyperfine Interactions 109, 357 (1997) [8] FAIR (Facility for Antiproton and Ion Research) http://www.gsi.de/fair/index_e.html [9] J.S. Schwinger, Phys. Rev. 82 (1951) 664-679 W. Greiner, B. Müller, J. Rafaelski, Quantum Electrodynamics of Strong Fields (1985),Springer-Verlag, Berlin V.P.Frolov and I.D.Novikov, Black Hole Physics, (1998), Kluwer, Dordrecht [10] Greg Landsberg, J. Phys. G: Nucl. Part. Phys. 32 (2006) R337-R365 [11] R. Bender and R.P. Saglia, C.R. Physique 8, 16 (2007) [12] Particle Data Group, J. Phys. G 33 (2006), 156 [13] S.V. Hawking, Commun. Math. Phys. 43 (1975), 199 [14] I. B. Khriplovich, Physics of Atomic Nuclei, 65 (2002) 1259 [15] Xiao-Gang He and A. Zee, http://arxiv.org/abs/hep-ph/0702133v1 [16] Thomas J. Weiler, Journal of Physics: Conference Series 60 (2007) 26-33 [17] G.Fogli at al, Phys.Rev. D 75 (2007) 053001 [18] L.D Landau and E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1975) [19] J.D Jackson, Classical Electrodynymics (Wiley, New-York, 1962) [20] D. Hajdukovic, Physics Letters B 226, 352 (1989) [21] T.E.O. Ericson and A.Richter, Europhysics Letters 11, 295(1990) [22] Wolfgang Paul, Reviews of Modern Physics 62, 531 (1990) [23] V. Logomarsino et al. , Physical Review A 50, 977(1994) [24] F. M. Huber et al. , Classical and Quantum Gravity 18 2457(2001)

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