NUCLEAR PHYSICS A

G. Chardin

DAPNIA/SPP, Centre d’lltudes de !&clay, F-91191 Gif/Yvette Cedex, France

Several arguments favour gravitation as the ‘7%4aster Arrow af Time”. Providing an explanation in terms of gravitation for the CP (and T) violation observed in the neutral kaon system, which provides the only example of a microscopic “arrow of time”, is therefore extremely tempting. We show qualitatively that ~~antigravity’~ would pravide just the amount of anomalous regeneration that we call CP violation and that a consequence of antigravity would be that any massive body would evaporate with the timescale of the Hawking radiation of black holes, which is linked to the generalized Second Law of Thermodynamics. Tests of this idea are proposed.

Perhaps one of the most fascinating and unresolved questions in Physics is the origin of time-as~metry. The various arrows of time which can be observed or conjectured in Nature -the decay of the V-meson, quantum-mechanical observations, general entropy increase, retardation of radiation, psychological time, cosmological expansion and black holes versus white holes puzzle--may appear quite disparate or of a too different nature to be unified in a common context. The reluctance of most physicist to deal with this complex problem can be synthesized in the comment by von Neumann that when you want to end a discussion you just have to speak about entropy.. .

But of the handful of physicists who have tried to gather in a unified context the various arrows of time 13-51, it is a remarkable fact that ail of them estimate that gravitation must be considered as the Master Arrow of Time. For examples Davies [3] notes: “We have reached a remarkable canclusion. The origin of all thermodynamic irreversibility in the real universe depends ultimately on gravitation. Any gravitating universe that can exist and contains more than one type of interact~g material must be asy~et~c in time, both globally in its motion, and locally in its thermodyn~mics.~~ Most high-energy physicists are unaware of the developments of the last twenty years about time-asymmetry: gravitational entropy @I, Hawking radiation of biack holes 6% time-asymmetric character of the singularities in general relativity I41 and quantization of time I.54 It seems therefore opportune to reconsider in the light of these developments the

478c G. Chardin I CP violation and antigravity

possibility that the only evidence for a microscopic time-asymmetry -the so- called CP violation in the neutral kaon system-could be related to the other “arrows of time” through gravitation.

It is rather ironical that, on the contrary, the very existence of the neutral kaon system has been used until now as a means to constrain the difference of gravitational interaction between matter and antimatter [8,9]. The small amount of CP violation in the neutral kaon system is usually considered as evidence for the violation of the discrete T symmetry, as opposed to the violation of the continuous time symmetry observed in all macroscopic phenomenon: under the demoralizing influence of Kobayashi and Maskawa, the “explanation” of CP violation in the standard model has been essentially reduced to the counting of free parameters in a 3x3 unitary matrix, and most authors have considered the neutral kaon system as irrelevant to the question of time asymmetry, with a few notable exceptions.

In the following, we will review the link between gravitation and time- asymmetry, and explore the phenomenological consequences of antigravity where, typically, a positron would “fall up” and an e+e- pair would levitate (Morrison’s antigravity [lo]). Although largely self-consistent, this paper can also be understood as complementary to [ill.

2. GRAVITATION AND TIME-ASYMMETRY

What are the arguments for insisting that Gravitation is at the origin of the Master Arrow of Time ? The first argument comes from a general feature of the evolution of massive bodies which dissipate their energy while collapsing; when quantum effects are taken into account, it can be seen that this dissipation will only end when the massive body will have completely evaporated into photons. The example is well-known of the satellite which increases its speed when being dragged by the atmosphere -a consequence of the surprising “minus” sign in the expression of the virial theorem-; this results in the contraction of stars and the release of dissipated gravitational energy into photons. Chandrasekhar 1121 was the first to demonstrate that sufficiently massive bodies will contract without limit, resulting in “black holes”. It may seem at first sight that we have found just the opposite of our mechanism of complete dissipation: the classical picture of a black hole provides us with an object where no dissipation is possible since even light is trapped in it. But when quantum effects on curved spacetime are taken into account, even black holes must evaporate by thermal emission -the so- called Hawking radiation [7].

The fate of very massive structures may seem to be of little relevance to the future of most celestial bodies, which are of a smaller mass than the Chandrasekhar limit, but two mechanisms make this fate almost unavoidable in an open or critical universe. Under the cooling by the expansion, huge black holes will be created from the coalescence of celestial bodies, which will then evaporate due to Hawking’s radiation. Even if this coalescence did not occur, quantum mechanics would allow the tunneling, albeit incredibly slow, of any massive body into a black hole and its subsequent evaporation into thermal,

G. Chardin I CP violation and antigravity 479c

structureless radiation. This relation between gravitation and time-asymmetry is exemplified by the generalized Second Law of Thermodynamics formulated by Bekenstein [13].

A second argument is provided by the work of Penrose [141 and Hawking 1151: if we assume the validity of general relativity, the existence of a past singularity in our universe is unavoidable, whereas the character of future singularities is far less obvious. Penrose has conjectured through his ‘Weyl curvature hypothesis” [4] that any past singularity -the Big Bang- is associated with a low gravitational entropy which Penrose associates with a vanishing Weyl curvature, whereas future singularities like black holes would be associated with a huge Weyl curvature and entropy.

A further comment concerns the CPT theorem: most theorists would not pay any attention to your pet theory if it violated CPT symmetry. But as should be well known, the possibility of demonstrating the CPT theorem depends on the topology of spacetime: the unavoidable existence of past singularities (assuming the validity of general relativity) makes it very doubtful that the CPT theorem can be demonstrated without modification for gravitation. In other words, boundary conditions cannot be eliminated. The problems raised by a local quantum expression of gravitation can be illustrated by the following remark: the dimensionless quantities entering the quantization of electroweak and strong interactions are local quantities -the coupling constants; as is well known, no such local expression has been found in the case of gravitation (the theory is non renormalizable). Instead, the only dimensionless quantity is here a non-local or global quantity : entropy or information (the presence of the Boltzmann constant in the usual expression of entropy is irrelevant; its physical content is expressed in bits, and can be measured (at least in principle) through the reversible coupling of a black hole with a gas of photons contained in a reflecting box [16]). In a fundamental sense, the quantization of gravity may then have been expressed twenty years ago through the Bekenstein entropy relation : S = M* relating entropy and mass of a black hole.

Finally, Bekenstein 1171 has also demonstrated that the very notion of baryon number loses its significance in general relativity. This is due to the loss of information associated with the interior of a black hole. As the internal structure of any object which has fallen inside a black hole cannot influence any more, except for a very short time, the outside region of the black hole, an outside observer may then choose to dump in a black hole a given mass of matter and antimatter in any proportion: the physics outside the black hole will be exactly the same whatever that proportion. And since, from the Hawking radiation process, the black hole will subsequently evaporate in photons (except maybe for the very last stage of its life) which are their own antiparticle, the baryon number of the Universe can be changed at will by an observer having access to a black hole. The absence of definition of the baryon number in gravitation is a further indication that gravitation may be responsible for the predominance of matter over antimatter in the Universe, also related to the problem of CP violation.

480~ G. Chardin / CP violation and antigravity

b)

Figure 1.a) The conventional representation of the Schwarzschild solution of a spherically symmetric matter configuration is that of an asymptotically flat spacetime gradually deformed as the mass of the central object is increased. b) The construction of the complete Schwarzschild solution shows that a better representation is that of two asympotically flat spacetimes connected by a wormhole.

G. Chardin I CP violation and antigravity 481c

3. WHITE HOLES AND REPULSIVE GRAVITY IN GENERAL RELATIVITY

1s it possible to find in general relativity some indications justifying the notion of antigravity for antiparticles? First of all, in general relativity, gravitation can be expressed as a manifestation of the curvature of spacetime: the trajectories of free particles are then defined as geodesics on this curved spacetime. In these conditions, the Equivalence Principle appears of an unavoidable consequence of the theory: for a given curved spacetime, the geodesics are unique and there is no mention in the theory about particles or antiparticles.

In reality, the situation is a bit more complex than this: firstly, all particles do not follow the same geodesics, if they have internal degrees of freedom; particles with non zero spin do not follow the same geodesics than particles with zero spin, although this can also be incorporated in the geometrical description of general relativity. A second and more fundamental caveat concerns the fact that the Schwarzschild geometry is more complicated than it seems at first glance.

For a massive body like the Sun, the geometry of such a body with (almost) zero charge and angular momentum is usually represented as a minor deformation (of the order of 10-S) of a flat spacetime. Starting from a flat spacetime, it then seems natural to represent such a geometry as in Figure la where the curvature effects have been greatly exaggerated. The spacetime curvature is gradually increased when we get closer to the star and redecreases gradually until getting to zero at the centre of the star.

When the mass of the star is gradually increased, the curvature effects are larger and larger until the star can no longer sustain the gravitational attraction. Starting from this representation, we are then led to the description of a black hole where the curvature well at the centre is infinitely deep and surrounding the singularity, hidden at the centre from outside observers. The usual represen- tation, which is found even today in most textbooks on general relativity, encompasses two regions: the black hole interior, where gravity is so strong that it is not possible, even for a photon, to resist the attraction towards the central singularity, and the black hole exterior, where it is possible to escape to asymptotically flat regions of spacetime. In fact the construction of maximal and complete solutions in general relativity has led to realize that this representation contains only half the Schwarzschild geometry. A better representation is schematically depicted in Figure lb. Instead of an asymptotically flat spacetime with an infinitely deep hole at the centre, we have now two asymptotically flat universes connected by a kind of “wormhole”. In fact, the communication between the two outside regions is not possible although outside observers can send scouts on a journey of no return who will meet in the inside region of the black hole: the wormhole is pinched before the traveller can reach the other exterior.

The topology of the geometry can be represented more clearly on a compact diagram, named after Roger Penrose (Figure 2a): in addition of the two exterior spacetimes, there are also two interior regions; the first one is the black hole region described above and in which it is not possible to escape the attraction of

482~ G. Chardin I CP violation and antigravity

our universe

AOOthW universe

An antigravity universe

Another universe

n antigravity universe

Another universe

Space a) b)

Figure 2.a) A Penrose diagram of the Schwarzschild geometry exhibits the symmetry under time reversal of the complete geometry: in addition to the black hole interior from which particles cannot escape being crushed by the singularity, there also exists a “white hole” interior where particles cannot escape being ejected from the singularity. b) a Penrose diagram of the complete Kerr solution of a rotating black hole includes an infinite number of alternated attractive and repulsive gravitation regions. These regions are connected by interior regions where a traveler may avoid falling on the singularity. Disregarding the stability of the solution, communication between the various exterior regions is now possible. Note that attractive and repulsive gravity are present on the same complete solution.

G. Chardin I CP violation and antigravity 483~

the singularity. The second is, rather improperly, called a zulzite hole interior, a region where it is impossible not to be ejected from the singularity!

Communication between the two asymptotically flat regions is impossible as the wormhole is pinched before an adventurous traveller may manage to reach the other side [18]. Retrospectively, it is easy to understand the reason for the second half of the geometry: since general relativity respects time-symmetry in its solutions, once we have obtained the black hole part of the solution, there must exist somewhere the time reversed of this solution, which is the white hole solution. The surprising aspect of the full geometry is that both regions, black hole and white hole, are present on the same complete geometry. Have we found the antigravity solution that we are looking for ? A priori not since it is easy to demonstrate that the black hole and the white hole parts of the solution are isometric, which means in particular that gravity is attractive in both regions.

In order to get repulsive gravity in general relativity, it is necessary to study the Kerr geometry of a rotating black hole (in fact, we need only give our black hole a very small amount of spin in order to study the features of the Kerr geometry). The geometry of a rotating black hole, represented on the Penrose diagram on Figure 2b, is much more complicated than the Schwarzschild geometry. It can be seen that the Kerr geometry is constituted of an infinite number of alternated attractive and repulsive gravitation regions. It should be stressed that repulsive gravity is really present in this solution and is not just a slingshot effect due the spin of the black hole; in particular, an observer in an antigravity universe would measure a negative mass for the central object. This geometry has been studied in detail by Carter [19,20] who showed the disturbing features related to the causality violations present in this solution.

This rather lengthy (and yet cursory) description of the Schwarzschild and Kerr geometry has been made to exhibit the curious unexplained features present in the solutions of general relativity and their relation to time reversal symmetry. Obviously, it is here that we may find a classical solution of antigravity where one sheet of the complete solution would be assigned to matter and another sheet to antimatter. As we shall see in the following, the different metric on each of the two sheets would almost necessarily generate an evaporation of the massive structure not unlike Hawking’s radiation of black holes.

4. REVISITING THE GOOD ARGUMENT

In the following, we will be defending the provocative idea that Cl’ violation as observed in the neutral kaon system can be entirely due to a violation of the Equivalence Principle. As a starting point, let us use Morrison’s antigravity [lo], a gross violation of the equivalence principle where antimatter is assumed to “fall up”, the total force on a static e+e- pair, e.g., being zero. Note that such a definition for antigravity cannot be extended to ultrarelativistic particles : a photon is its own antiparticle (if we disregard its polarisation) and is known [21,22] to follow the equivalence principle to a high precision.

It is “well known” that antigravity violates the sacrosanct CPT symmetry, contradicts the results of the Eotvos-Dicke-Panov experiments, excludes the

484~ G. Chardin ! CP violation and antigravity

existence of the long-lived component in the neutral kaon system in the presence of the Earth gravitational field, violates energy conservation and implies vacuum instability. Except for the last one, these arguments have been reviewed critically by Nieto and Goldman [231 and, although we disagree with the solutions proposed by these authors, we refer the reader to their recent review for a critical discussion of these impossibility arguments (see also [ll]). Here, we will only insist on the Good argument and on the vacuum instability induced by antigravity.

In 1961, three years before the first experimental observation of CP violation, Good [81 observed that antigravity would impose that the KL, a linear combination of K” and F, would regenerate a KS component. Good estimated the phase shift which would develop between the K” and p components from the energy difference due to the gravitational potential @G; he supposed that the phase factor between the two components would oscillate as

exp(2i ~K#G t fh) .

(In 1964, when Cl’ violation in the neutral kaon system was discovered, Bell and Perring 1241, together with Bernstein, Cabibbo and Lee 1251 have also used a vector interaction and a potential to attempt recovering the Cl? symmetry; the energy dependence predicted by a vector coupling has since been excluded by experiments at various energies).

Since the potential energy of a kaon in the Earth gravitational field is co.4 eV, which is 105 larger than the energy splitting between the KS and KL eigenstates, Good concluded that antigravity was excluded. But, as Good himself had noticed, there is no obvious reason why one should use the Earth potential; why not use instead the Sun, or the Galactic potential which would give even more stringent limits on the difference of acceleration between matter and antimatter ?

In fact, gauge theories teach us that potential themselves are not observable, but only potential differences. Let us then try to restate the Good argument independently of absolute potentials. A kaon has a rest energy of -500 MeV and is constituted from a quark s (or S) of mass ~200 MeV, and of a much lighter quark Z (or d) of mass a few MeV. In these conditions, the quark s is semirelativistic and the quark Z ultrarelativistic; this system is strongly bound and its size must then be = tr/mK c = 0.5 fm. Reasoning for a matter of simplicity in terms of neutral kaons, suppose there is a mechanism which would separate the K” from the TT” by more than this size of 0.5 fm during the mixing time AT introduced by weak interactions, which continuously transform s t) S (and d ++ A).

As can be seen from Figure 3, the mixing time is AZ = x WAmc2) = 6 TV. Under this hypothesis of a fast separation between the K” and the p components, the kaon would be reprojected either on a K“ or a F state so that we no longer observe the KL as a propagation eigenstate. If the relative displacement between the K” and the F is of the order or smaller than the kaon size, the phase shift between the two components will be approximately proportional to the displacement.

G. Chardin I CP violation and antigravity 485c

Assuming antigravity, let us estimate the time needed for the s quark to separate from the S to a distance that would induce a regeneration of the KS component such as observed in CP violation.

0.2

0

-0.2

-0.4

-0.6

-0.8

_ q+_ = (2.30 !z O.lO(stat.)

f O.O4(sys.))l o-3

_ p+_ = 44.9 * 2.5(stat.) f 0.5(sys.)

_ X’/ndf= 0.75

I Dashed curve is fit assuming no background

I I I I I I I I I I I , I I I I I I L

0 2 4 6 8 10 12 14 16 ie Neutral Koon Eigen Lifetime (Units of 7%)

Figure 3. The CP-LEAR experiment has used tagged K” and F to determine the CP-violating n+_ parameter in the charged two-pion decay. These kaons, originating from the annihilation at rest of a proton and an antiproton, provide a nearly isotropic kaon beam in the momentum range 11400, 7001 MeV/c. In addition, the figure clearly shows the oscillation time (~%/Amc2) induced by weak interactions between the sd and the sd states (reproduced from [26]).

486~ G. Chardin / CP violation and antigravity

This time is given by the equation

8t2 = & X (k&m size) = & X fi/r?ZK C.

where E is the CP violating parameter. This gives numerically

t = 3 lo-lo s = 1.7%/Amc2

(1)

In other words, fhe time needed for antigravity to generate the amount of regeneration observed in CP violation is just about equal to the mixing time imposed by weak interactions. Expressed independently of absolute potentials, the Good argument, far from excluding the possibility of antigravity, provides an indication that CP violation may be explained by some kind of antigravity ! From equation (l), we are then led to the following approximate expression for the E parameter

E = Q(l)%WKg / Am2c3 = O(1) x 0.88 10-S (2)

where O(1) is a constant of order unity and which is the only dimensionless quantity linear in the kaon mass which can be built from the above quantities. This expression has a simple physical interpretation : it means that the energy to lift a kaon by AZ =Ii/Amc is of the order of E Amc 2. It is difficult to believe that Good himself would not have proposed the previous expression for E, had CP violation been observed at the time when he devised his argument. For a different expression of the argument leading to equation (21, see also [271. Fischbach [28] had also noted that the quantityhmKg /Am2c3 is of the order of the experimental value of e.

5. VACUUM INSTABILITY AND BLACK HOLE RADIATION

A classical argument against Morrison’s antigravity, devised by Morrison himself [lo], is that energy conservation appears to be violated under such a process. But if, as we suggest, the evidence for CP violation in the neutral kaon system is linked, through antigravity, to a failure of time invariance under continuous time translations, instead of the discrete symmetry usually considered in this context, we may expect some difficulty in building a conserved quantity such as energy: from Noether’s theorem, energy as a conserved quantity can only be defined whenever invariance under time translations is respected. Quite generally, any process defining an “arrow of time” will, by definition, forbid the construction of energy as a conserved quantity. As we shall see in the following, entropy will instead become the relevant quantity.

Let us illustrate the preceding remarks about energy and entropy by the following gedanken experiment: we imagine the following process to extract energy from the “vacuum” in the vicinity of a massive body. Let us get a source of electrons and positrons at the surface of the Earth. If one had antigravity, the total force on an e+e- pair at rest would be zero, by definition; it would therefore be possible to carry adiabatically an e+e- pair at some arbitrary height AZ above the

G. Chardin / CP violation and antigravity 48lC

initial level at no cost. Letting the e+e- pair annihilate at the higher level into two photons, we would propagate downwards the two photons to the lower level and there recreate the e+e- pair. But we know from the experiments of Pound and Rebka [21] and Vessot and Levine [22] that photons respect the equivalence principle to a high precision (=lOA), which implies that each of the two photons has gained an energy ~&AZ, where me is the mass of an electron, and g is the gravitational acceleration for matter at the surface of the Earth (all energies are measured by local observers at rest with respect to the Earth). Using these photons, it is then possible to recreate the e+e- pair with a little bonus of energy equal to 2megA.z.

It then seems that we have obtained “something for nothing” and that we have realized the construction of a perpetual engine. Considering the tremendous difficulties to manipulate antimatter in a matter environment, we should not expect to solve the energy crisis using such a process. More fundamentally, as we shall see in the following, there is some irreducible cost to manipulate matter and antimatter on curved spacetime.

But even if there are no “real” particles present, virtual creation-annihilation processes occur continuously in the vacuum, which must then become unstable in the presence of a gravitational field. Let us then determine the characteristics of such an evaporation of the massive structure responsible for the gravitational field. From Heisenberg inequalities, a virtual creation-annihilation process involving a particle of mass m will probe spacetime over a length scale

AZ -$/me

where m is the mass of the propagated particle. Assuming antigravity, the typical energy of a photon which will be produced by the gravitational field is then:

AE = mgAz =ftgg/c

From the exponential suppression which results from the propagator, it is reasonable to assume that the vacuum will radiate with a thermal spectrum and a temperature of the order of the typical energy that we have just found:

The careful reader may have noticed that this is just the expression of the Hawking radiation of a black hole, expressed in terms of its surface gravity (up to a factor of order unity). Let us recall the main relations for this radiation. For a (non rotating and uncharged) black hole of mass M, the internal energy is U = Mc2, and its entropy S = 47tM2 in units of Planck mass. From this, we can derive the expression for the temperature:

and find that the evaporation time is :

tevap = M3

488~ G. Chardin I CP violation and antigravity

again in units of Planck mass. For a solar mass black hole, this time is = 1066 years, much larger than the age of the Universe and 33 orders of magnitude beyond the best limits obtained by the extremely sensitive nucleon decay experiments.. .

How would these expressions be modified in the case of thermal radiation induced by antigravity ? It should be noticed that the wavelength of the evaporation radiation is always larger than the distance to the centre of the massive body. This means that the emission will be suppressed compared to the emission by a blackbody. Typically, the region which will contribute the most to the emission due to the larger values of g extends over a distance R where R is the radius of the massive body. It is a standard result that the emission of radiation of wavevector k through an aperture of radius a (k >> a> is reduced by a factor (ak)4 compared to the geometrical section xa2 [29]. This means that, compared to the radiation power emitted by a black hole of the same mass M, a massive body of roughly uniform density and extension R will radiate at a rate:

where PBH is the power radiated by a black hole of mass M and rs is its Schwarzschild radius (the first term between brackets comes from the increase in surface of radiation, the second from the smaller temperature at equal mass in Stefan’s law, and the third from the quenching of the emission of radiation discussed above). For the Sun, the timescale of this evaporation process is =I095 years, and for the Earth =10*07 ans, obviously not a strong experimental constraint.

The main result of our discussion is that the vacuum insfability induced by antigravity then appears very similar to fhe very effect which introduces the second law and time-asymmetry in the realm of general relativity. If black-hole evaporation is acceptable, why should we reject a priori a similar effect induced by antigravity ? Expressed in yet another way, antigravity is the tidal force which must be exerted on the vacuum in order that any massive body, and not just black holes, radiate with a characteristic temperature of order N/c.

6. MAXWELL’S DEMON, ANTIGRAVITY AND THE SECOND LAW

In the previous paragraph, we have shown qualitatively that antigravity would imply the (usually very slow) evaporation of any massive structure. We have introduced antigravity because it may provide an explanation of CP violation in the neutral kaon system and because the evaporation of a massive body is typically of the same order as the Hawking radiation of black holes which leads to the generalized Second Law of Thermodynamics. Of course, we should make sure that it is not possible to use the violation of the Equivalence Principle that we have introduced in order to violate the Second Law. It seems at first sight that by carefully following the trajectories of the annihilation photons of the positronium, the previous gedanken experiment can be reversed and the process used to gradually remove photons initially present in a box. If it were possible to

G. Chardin I CP violation and antigravity 489c

hide such photons at no cost, the Second Law of Thermodynamics would be violated.

The fundamental point of the following discussion is that when the information cost of following the annihilation photons on curved spacetime is taken into account, it appears that the process cannot be used to violate the Second Law.

Let us then introduce a Maxwell demon trying to violate the Second Law using antigravity. He (or she) intends to hide photons in the vacuum at a minimal cost. Our demon must realize his experiment in a reflecting box in order not to loose the annihilations photons. The demon will then try to remove photons from the enclosure where he is operating in the most economical way : if he measures too often the position and momentum of the annihilation photons during his attempt to rebuild the e+ e- pair, the information cost of such a measurement will more than compensate the disappearance of a third spectator photon. He will then use a passive focusing system such as an ellipsoid and set the source of positronium at the lower focus and attempt to recreate the e+ e- pair at the upper focus of the ellipsoid. The demon must then face a first constraint. If he tries to make his task easier by using a small enclosure where the photons would be better confined and easier to track, Heisenberg inequalities impose that the energy of the fundamental state be inversely proportional to the size of the enclosure. Our demon may not deal with photons of arbitrarily low energy in a box of given dimensions. On the other hand, we have seen that the energy of the photons which can be created or hidden using the difference of “gravitational charge” between matter and antimatter is proportional to the difference of height between the emission and reception of the photons. If our demon is carrying electron and positron over a height AZ, the net gain or loss of energy is :

AE=m AgAz

where Ag is the difference of acceleration between electron and positron (i.e. 2g in the case of “pure antigravity” which we have used as an example). Such a photon has a wavelength :

which must be smaller than the size of the box. This implies a constraint on the size of the box, noted Az~+i~ :

where rs is the Schwarzschild radius and rC the Compton wavelength of the electron.

Once the constraint on the size of the ellipsoid is met, it seems that the demon can perform his work completely passively and at no cost. But a second constraint comes from the fact that in order to recreate an e+e- pair from two annihilation photons, the demon must aim very precisely: typically, the volume of interaction during the pair creation process is (arc)3 and the collision must be almost exactly head on if the reaction is just above threshold. On the other hand, if the demon

49oc G. Chardin I CP violation and antigravity

is trying to make disappear a photon of precisely the energy m Ag Az, this photon is spread over the whole volume of the box, which is of order AzHds3.

The demon must then follow the annihilation photons over a very long distance in order to have a reasonable chance to achieve the three photon collision which will result in the disappearance of such a low energy photon. If there is no irreducible information cost in the process, so much the better for the demon who has just to wait until the reaction finally occurs.

However, on a surface with negative curvature, the information cost is not zero and the demon must pay a price in order to follow the annihilation photons

Figure 4. A Maxwell demon trying to violate the Second Law of Thermodynamics by using the violation of the Equivalence Principle in order to hide photons in the “vacuum” must pay an irreducible information cost due to the divergence of photon trajectories induced by the curvature of spacetime. This information cost is always larger than the expected gain of his manipulation.

with the constant precision needed to recreate the e+e- pair. The rate of divergence of a beam of photons initially parallel determines the information cost that the demon must pay (in general relativity, the phase space measure of a beam of photons remains constant in the vacuum, and when there is divergence in one direction, it means that there must be convergence in another; however, information cost goes only one way : information is not gained when there is convergence, only lost when there is divergence (Figure 4), another illustration of Murphy’s Law).

The characteristic distance for the divergence of the photons is obtained from the positive components of the Riemann tensor, homogeneous to the inverse of

G. Chardin I CP violation and antigravity 491c

the square of a curvature radius. Denoting this characteristic distance ~~~~~ in reference to the Lyapounov exponent [30], we have :

AzLyap = ff$ -I’* ( ) Finally, if we review the constraints that our demon must fulfil in order to

have a reasonable chance to suppress a photon in the fundamental state of the box, he will have to pay a price, expressed in bits :

k Heis & Heis) 1 _ r 5’2 , 1 - - -&ar~~ rcrs3”a3

The first term is the distance between two photon reflections, the second term is the number of collisions needed to absorb with a reasonable chance a photon spread over the whole box, and the third term is the length for which the demon must pay a price of one bit.

f 7 f \

matter - curvature I I (negative) - dynamical entropy curvature production

\ J L. /

matter f--) physical entropy production

Figure 5. Einstein’s fundamental equation of general relativity is equating the curvature tensor to the stress-energy tensor. Kolmogorov showed the relation between negative curvature and the entropy which must be paid to keep track of the state of a system with a constant precision. It is then natural to conjecture that matter is in turn associated with a production of entropy which defines time-asymmetry.

Since obviously the demon must make his experiment at a distance of the centre of attraction greater than the Schwarzschild radius (it would be otherwise impossible to achieve a stationary experiment), and greater than the Compton wavelength of an electron (it would be otherwise impossible to speak of individual electrons and positrons), it appears that the information needed for the demon to hide a photon is always greater than the information hidden in the vacuum. The demon is therefore unable to operate efficiently enough to be able to violate the Second Law of Thermodynamics.

492~ G. Chardin I CP violation and anrigravity

Clearly, we are not proposing here a complete demonstration but mainly a dimensional analysis of the information cost that the Maxwell’s demon would have to fight in his endeavour to violate the Second Law. Our argument is still unsatisfying: in particular, the notion of quantum chaos itself is only very partially understood, and our argument is semiclassical. But despite its semiclassical character, it can be considered as a strong indication that it is not possible to achieve such a violation using antigravity. Figure 5 summarizes the links between matter, curvature and entropy production.

The argument suggests, in particular : - that the Second Law can be preserved when the dynamical information cost

is taken into account - and that the fundamental quantity in the problem is entropy or

information, a notion first stressed by Bekenstein [6,13]. This leads us to the expected variation for the E parameter for a system where all the mass would be carried by the quarks. In our analysis, the E parameter is proportional to the rate of separation of a quark and an antiquark, which is in turn related in our entropy balance to the characteristic curvature radius given by the square root of the positive components of the Riemann tensor. In a suitable basis, the components of this tensor are equal to f 1 or 2 GM/c*+ and this gives the order of magnitude of the effect that we might expect in a non relativistic system.

7. EXPERIMENTAL TESTS

In contradiction to a widely held belief, there is no experimental constraint on the gravitational mass of antiparticles, except for the ultrarelativistic limit of the photon, which is its own antiparticle [21,22] and, within some hypothesis, of the neutrino [31]. The PS-200 experiment [32], at the Low-Energy Antiproton Ring (LEAR) at CERN, is currently the only such experiment in the stage of installation that proposes to measure the difference between the gravitational masses of the proton and the antiproton. This experiment intends to reach a statistical precision of 1% on Am$mp. For a discussion about the experimental problems raised by the measurement of the gravitational mass of charged antiparticles;see [33].

The next generation of experiments will require using atomic systems and, in the first place, atomic antihydrogen. A consensus seems to emerge among the communities of atomic and particle physicists that the goal of making ultracold atomic antihydrogen could be achieved at LEAR before the end of the decade. As atomic interferometers have recently been used as gravitometers and may soon reach a precision Ag/g of lo-10 or even better [34];this program offers the perspective to measure the gravitational mass of antihydrogen at this level (see also [35]).

From our reanalysis of the Good argument and the derivation of the E parameter in terms of antigravity, it appears that the neutral kaon system should also be used as a test of the Equivalence Principle. In particular, it is interesting to note that all the precision measurements of the E parameter have involved ultrarelativistic horizontal kaon beams, i.e. always in the same direction of

G. Chardin I CP violation and antigravity 493c

spacetime relative to the Earth, with one notable exception. The only experiment for which the energy difference associated with the change in altitude of a kaon during the mixing time of weak interactions is comparable to E Am ~2, i.e. the energy difference between the K” and the T(o needed to generate the amount of anomalous regeneration that we call CP violation, has led to a determination of the n+- parameter equal to (2.09 f 0.02) 10-3, in disagreement by 9 standard deviations from the world average [36]. In this last experiment, the neutral beam used made an angle of 6.25 10-S rad with the horizontal, at a typical energy of 70 GeV. It seems therefore crucial to test whether this non standard result is just due to a coincidence or whether the E parameter really depends on the direction in spacetime of the neutral kaon beam relative to the Earth. In this respect, the PS- 195 experiment [37], also at LEAR, should be able to determine with a precision better than 1% the E parameter for an isotropic kaon “beam” and using kaons in the momentum range [400, 7001 MeV/c, generated from the annihilation at rest of a proton with an antiproton. An experiment to measure E with an inclined kaon beam at Brookhaven is also being considered [38].

The variation of g at the surface of the Earth (excluding of course the centrifugal component...) could also be used as a test if E is measured at the 10-S level or better since the maximum variation Ag/g (between the pole and the equator) is only 5 10-S.

Since, for the kaon system, antigravity is equivalent to a regeneration effect, the E’ parameter must be zero. Any confirmed non zero measurement of the E’ parameter would then exclude the antigravity hypothesis. Finally, we can also use the m/Am2 scaling of our approximate expression for the E parameter to predict that the E parameter in the B system must be of the order of 10-5-10-6, which is hopelessly small to be measured as a non zero value in a foreseeable future. Since these two predictions differ from the Standard Model predictions, they provide two further tests of our hypothesis.

8. CONCLUSIONS

In this paper, we have tried to demonstrate that the arguments against antigravity should be reconsidered. We have shown that the Good argument on the neutral kaon system, when reexpressed in the most natural way, does not exclude antigravity but, on the contrary, that antigravity would provide just the amount of anomalous regeneration that is observed in the neutral kaon system and that we call Cl’ violation.

Perhaps the most disturbing aspect of antigravity is the thermal evaporation of any massive structure that it implies. We have stressed, however, that the timescale for this process is much larger than the age of the universe and that an extremely similar thermal evaporation occurs in the Hawking radiation of black holes. As in all situations related to time-asymmetry, we have shown that the key quantity is entropy or information and we have sketched a demonstration of the impossibility to violate the Second Law using antigravity.

Looking for a relativistic classical expression of antigravity, we have considered the features of the Schwarzschild and of the Kerr geometries; these

494c G. Chardin I CP violation and antigravity

geometries incorporate regions of spacetime where repulsive gravity and time- reversed solutions of black hole may provide some indications on the way to get the relativistic expression of the E parameter in the kaon system assuming CP violation is really a manifestation of antigravity.

Experimentally, three sets of experiments are available to test the idea of antigravity. The direct tests on charged antiprotons, with the possibility to calibrate the experiment using H- ions, and in the next few years, the interferometric gravitometer using antihydrogen, provide the two most direct measurements of the Equivalence Principle for antimatter. As we have shown, another interferometric experiment can be provided by the neutral kaon system, and the C&LEAR experiment will provide the first precision measurement of the E parameter using an isotropic kaon beam (in the momentum range [400, 7001 MeV/c), testing its possible variation. With the measurement of the CP-violating parameter E using inclined beams, these experiments are essential tests of general relativity.

9. ACKNOWLEDGEMENTS

It is a pleasure to thank the organizers of this conference for their invitation, and especially Carlo Guaraldo for having set up a very pleasant and fruitful conference. Part of this work has been done with J.M. Rax.

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