Lucie Ducloué
email: lucie.ducloue@manchester.ac.uk

Olivier Pitois

Laurent Tocquer

Julie Goyon

Guillaume Ovarlez

Yielding and flow of foamed metakaolin pastes

Keywords: metakaolin, bubbles, rheology, yield stress, viscosity

Metakaolin is a broadly used industrial raw material, with applications in the production of ceramics and geopolymers, and the partial replacement of Portland cement. The early stages of the manufacturing of some of these materials require the preparation and processing of a foamed metakaolinbased slurry. In this study, we propose to investigate the rheology of a foamed metakaolin-based fresh paste by performing well-controlled experiments. We work with a non-reactive metakaolin paste containing surfactant, in which we disperse bubbles of known radius at a chosen volume fraction. We perform rheometry measurements to characterize the minimum stress required for the foamed materials to flow (yield stress), and the dissipation occurring during flow. We show that the yield stress of the foamed samples is equal to the one of the metakaolin paste, and that dissipation during flow increases quadratically with the bubble volume fraction. Comparison with yielding and flow of model foamed yield stress fluids allows us to understand these results in terms of coupling between the bubbles' surface tension and the metakaolin paste's rheology.

Introduction

Metakaolin is a dehydroxylated clay (Bich et al., 2009) with a high pozzolanic activity, making it widely used as a raw industrial material. The alkali activation of alumina-and silica-rich metakaolin leads to the formation of a three-dimensional aluminosilicate network. The resulting product, called a geopolymer [START_REF] Davidovits | Geopolymers[END_REF] is a strong and durable cementitious material which hardens at moderate temperature, thus being relatively low energy consuming. For that reason, geopolymers are suggested as a sustainable alternative to other materials that require processing at very high temperature, among which some ceramic parts [START_REF] Davidovits | Geopolymers[END_REF] and Portland cement, which is the most common binder for mortar and concrete. Metakaolin has also been successfully used as a partial replacement for Portland cement in regular construction materials, with benefits including an increase in the early age strength, enhanced durability and a better resistance to chemical attacks compared to ordinary Portland cement-based materials [START_REF] Sabir | Metakaolin and calcined clays as pozzolans for concrete: a review[END_REF].

In the context of a general industrial effort to develop building materials with lower environmental impact, foamed geopolymer concrete combines the advantages of a low energy consuming geopolymer matrix with the benefits of adding gas bubbles which will make the resulting concrete lightweight and thermally insulating [START_REF] Ramamurthy | A classification of studies on properties of foam concrete[END_REF][START_REF] Zhang | Geopolymer foam concrete: an emerging material for sustainable construction[END_REF]. There are two main routes to produce foamed geopolymer concrete: gas bubbles can either be generated by an in-situ chemical reaction [START_REF] Prud'homme | In situ inorganic foams prepared from various clays at low temperature[END_REF][START_REF] Masi | A comparison between different foaming methods for the synthesis of light weight geopolymers[END_REF] or can be added mechanically, either through strong agitation that entraps air in the paste [START_REF] Cilla | Geopolymer foams by gelcasting[END_REF] or by mixing with a foam concentrate [START_REF] Zhang | Geopolymer foam concrete: an emerging material for sustainable construction[END_REF]. For high temperature applications, this foamed geopolymer can then be sintered, leading to the formation of a foamed ceramic (Bell and Kriven, 2009) which can be used to filter hot gases or to make thermal insulating tiles for extreme conditions [START_REF] Montanaro | Ceramic foams by powder processing[END_REF].

Whether producing a foamed geopolymer or a foamed ceramic, the size, volume fraction and distribution of the gas bubbles in the final foamed clay material are crucial to the application the material is designed for [START_REF] Glad | Highly porous geopolymers through templating and surface interactions[END_REF]. Therefore, the incorporation and the preservation of the gas bubbles in the mineral paste when elaborating the material is essential. From a rheological point of view, the paste is a complex fluid, which only flows if the stress applied to it is larger than a critical value: the yield stress of the paste. Above that yield stress, the paste flows with a viscosity which depends on the flow rate applied. The interplay of the complex rheology of the mineral paste with the bubbles will determine the stability of the bubbles entrapped in the paste, but also the rheology of the foamed material. As the development of materials such as geopolymer foamed concrete is currently in the relatively early stages of technological maturity, understanding this behavior is necessary to elaborate optimized processes and materials. Those stability and rheological issues should be tackled as a complement to the study of the chemical reactions and functional properties.

In this paper, we propose to shed light on the flow properties of just mixed foamed clay pastes by investigating the rheological behavior of a metakaolin paste containing air bubbles. We work with a non-reactive material containing surfactant to stabilize the mechanically added air bubbles, and focus on how the behavior of the paste is modified by the addition of bubbles of a known radius at a given volume fraction. This question is for instance of interest to design efficient processes to cast or pump a geopolymer foamed concrete. The first section of this paper presents in more detail our experimental system and the rheometry procedure used to characterize it. The following section presents the results we have obtained for two rheological properties relevant for bubble stability and material workability: the minimal stress required to induce flow and the dissipation during flow of the foamed metakaolin pastes. In the third and last section, those results are discussed in the light of previous experiments done on foamed samples of simple yield stress fluids.

Experimental procedure

We prepare model samples of foamed metakaolin paste by dispersing bubbles of well defined radius at a chosen volume fraction in a concentrated metakaolin paste. To achieve such a level of control on our systems, we prepare a very concentrated metakaolin paste containing no bubbles, which we then mix with a separately produced monodisperse aqueous foam. This procedure, which is presented below after the details of the materials used, allows us to control the bubble size and the gas volume fraction of the foamed metakaolin samples throughout the process of mixing and characterization. The volume fraction of entrapped air, φ, is varied in the range 0-50%, which is the void volume fraction commonly found in the production of foamed geopolymer concrete by mechanical agitation or mixing with a foam [START_REF] Zhang | Geopolymer foam concrete: an emerging material for sustainable construction[END_REF]. Sample preparation and characterization is done at 25 • C.

Metakaolin paste

The metakaolin paste which will be mixed with the foam ("stock paste") is a dispersion of a fine metakaolin (Argical M1200-S, AGS minéraux -Imerys group) powder at 46.4wt% in an aqueous solution containing 10wt% of a nonionic surfactant (Tween 20 R , from Fisher Chemicals) in deionized water. The presence of this surfactant at the same concentration in the stock paste and in the aqueous foam ensures that the bubbles' surface will not be deprived of surfactants once the foam is mixed with the paste, keeping the bubbles stable to coalescence in the resulting foamed sample. A micrograph of a dilute suspension of the metakaolin powder in water in shown in figure 1. The grains are lamellar, and of a very small size: the median spherical equivalent diameter is 1.3 µm (data from AGS). Because of this micrometric size, there is no sedimentation nor creaming of the metakaolin suspensions in water at the concentrations we are using for the pastes. In the foamed samples, the micrometric size of the metakaolin particles will be much smaller than both the bubble radius and the typical distance between neighboring bubbles in the paste, meaning that the metakaolin paste can be described as a continuous medium embedding the bubbles.

Monodisperse foam

A monodisperse foam is produced separately from the paste. We use the same aqueous solution (10wt% of Tween 20 R in deionized water) as the one used to prepare the stock metakaolin paste, so that the mixing of the stock paste and the foam is easy. We have used two different techniques to produce monodisperse foams with two different bubble radii. A foam containing bubbles of radius R b = (100±20)µm has been obtained by blowing nitrogen at a few milliliters per minute through a porous glass frit. A second series of foamed suspensions containing smaller bubbles, of radius R b = (35± 10)µm has been prepared thanks to a millifluidic device (made of a T junction followed by a porous medium). In all foams, a very small amount of a gas Figure 2: Schematic of the timed experimental protocol (bubbles not to scale). The metakaolin powder is first dispersed in water under strong agitation, then the rotational speed is decreased and the surfactant is slowly added to the mixture. The resulting paste ("stock paste") is mixed with a separately produced monodispersed foam to get the foamed samples, which are then characterized by rheometry.

with very low solubility in water, perfluorohexane, is added to the nitrogen to reduce coarsening [START_REF] Gandolfo | Interbubble gas diffusion and the stability of foams[END_REF].

Preparation of the foamed samples

The protocol we use to prepare the samples and then measure their flow properties has been designed to ensure that sample preparation and characterization is reproducible. The metakaolin paste surrounding the bubbles is non-reactive, but it slowly evolves with time (like many other clays, it is thixotropic [START_REF] Ryan | Properties of ceramic raw materials[END_REF]: although no chemical bonds are created nor destroyed, the physical structure of the paste evolves slowly at rest, changing its rheological properties in response). For that reason, the preparation and measurement sequence follows a precisely timed procedure, which is detailed below and summarized schematically in figure 2.

To prepare the stock paste in a reproducible way and without entrapping unwanted air bubbles, we first disperse the powder by adding it little by little into deionized water under strong agitation. The vigorous stirring of the metakaolin suspension in water minimizes the time evolution of the resulting metakaolin paste [START_REF] Coussot | Mudflow Rheology and Dynamics[END_REF]. At the end of the 9 minutes necessary for the addition of metakaolin in water, we obtain a smooth and bubble-free homogeneous metakaolin paste. The speed of agitation is then decreased, and the surfactant is gradually added in the paste via a syringe. This addition of surfactant in the paste makes it prone to air entrapment, which is greatly limited by the use of a low rotation speed: measurement of the density of the prepared stock paste shows that it contains less than 1% of air. At the end of the 3 minutes necessary to add and blend the surfactant into the paste, the paste is mixed with the separately produced foam. This mixing brings in more aqueous solution, which will lower the final solid content in the paste of the foamed samples. The mass of added foam is always 6% of the mass of prepared stock paste, so that all the foamed metakaolin samples have the same solid content (44wt%) in the suspending paste surrounding the bubbles ("suspending paste"). The gas volume fraction in the foamed samples is varied by changing the wetness of the aqueous foam prior to mixing. The incorporation of the bubbles in the paste is done by blending for 2 minutes with a spoon, to ensure that stirring occurs both in the vertical and the azimuthal directions. After mixing, the foamed metakaolin sample is poured in a tared rheometry cup of known volume (61.6 mL). Weighting of the full cup allows for the determination of the density of the samples, and hence the volume fraction of entrapped air. The cup is then set in the rheometer and 4 minutes after pouring the sample in the cup, the mechanical measurements start.

Rheometry procedure

The characterization of the flow properties of the samples is done by using a commercial controlled-stress rheometer (Bohlin C-VOR 200), which works at either set torque or set rotational velocity (thanks to a torque feedback loop). The rheometer is fitted with a six-bladed vane tool (radius R i = 12.5 mm) in a cup (radius R o = 18 mm) geometry. The gap R o -R i is larger than several bubble diameters, which ensures that the bulk properties of the foamed samples are measured. Fine sand paper is glued inside the cup to avoid slippage of the metakaolin paste at the cup wall [START_REF] Coussot | Rheometry of pastes, suspensions, and granular materials: applications in industry and environment[END_REF]. Because of the wide gap of the geometry we are using, the shear stress varies along the radius r in the gap. We compute the shear stress and shear rate at r = R i from the torque and rotational velocity assuming that the rotating vane tool is equivalent to a cylinder of same radius R i . We subtract the contribution of the end effects at the bottom of the virtual cylinder defined by the vane, and take into account the localization of shear when computing the flow curve [START_REF] Coussot | Rheometry of pastes, suspensions, and granular materials: applications in industry and environment[END_REF]. Throughout the sequence, the free surface of the sample is visible, allowing us to check that the bubbles do not coalesce nor cream during the measurement.

After the cup is set in the rheometer, the vane tool is slowly inserted, so that the last significant flow undergone by the sample before the start of the rheometry procedure is the one generated when pouring it into the cup. Because of the random nature of this flow, the sample is assumed to be homogeneous and isotropic prior to the start of the measurement sequence. The critical stress to make the samples flow after a 4 min rest (static yield stress τ ys ) is first measured by initiating flow from rest: a small and constant shear rate (0.01 s -1 ) is applied to the samples. This small velocity ensures that the contribution of viscous effects to the measured stress is negligible. The stress-strain curve obtained during this measurement on the suspending paste is shown in figure 3. For very small deformation, the stress first increases very abruptly with the deformation, which shows that the critical deformation to exit the linear elastic regime of the metakaolin paste is very small. As the deformation increases, the stress rapidly reaches a peak value τ ys , at which point the material starts to flow. As flow goes on, the stress relaxes to a plateau value. The large stress overshoot at the start of flow is typical of the flow start-up of thixotropic materials, in which a microstructure has built up during the resting time. The slow decrease in stress once flow is established corresponds to the destructuration of the material under flow, which lowers its apparent viscosity [START_REF] Coussot | Rheometry of pastes, suspensions, and granular materials: applications in industry and environment[END_REF]Barnes, 1997;[START_REF] Ovarlez | Influence of shear stress applied during flow stoppage and rest period on the mechanical properties of thixotropic suspensions[END_REF].

The constant shear rate is applied for 60 s, followed by a rest (zero stress) of 10 s, after which the flow curve of the samples is measured by imposing a shear stress ramp, first increasing from 0 Pa to a maximum stress of the order of twice the static yield stress, and then decreasing back to 0 Pa. For this measurement, it is necessary to work at imposed stress because the metakaolin paste starts flowing very abruptly at yielding, and the feedback loop of the rheometer is then unable to impose the chosen shear rate ramp. The flow curve of the suspending paste during the decreasing shear stress ramp is shown in figure 4 (a). At the beginning of the ramp, the stress is high and the paste flows. When stress is decreased, the paste stops flowing at a finite value of applied stress: this value defines the dynamic yield stress τ yd of the paste. For our metakaolin paste, the dynamic yield stress is much lower than the static yield stress. This can be understood by the fact that it is easier to keep the destructured material flowing than break the structure to initiate flow from rest (Barnes, 1997). Above the dynamic yield stress, after a smooth transition at low shear rates, the stress seems to increase linearly with the shear rate. At high shear rates, however, the behavior of the paste deviates from the linear regime and a faster increase in the stress seems to indicate that the metakaolin paste is shear thickening. To better evidence this phenomenon, we fit the whole flow curve to a polynomial function and compute its derivative dτ /d γ, which is the apparent plastic viscosity of the material. The evolution of this apparent plastic viscosity with the shear rate is plotted in figure 4 (b). A plateau in the apparent viscosity at intermediate shear rates allows us to define the region for which the behavior of the paste is linear. This plateau is followed by a sharp increase, clarifying that the paste is apparently shear thickening. This apparent shear-thickening is likely due to inertial effects: because the paste's viscosity is rather low, its flow in a Couette-like geometry is prone to develop instabilities. In a Couette cell, the first instability to appear in the flow of low viscosity Newtonian fluids is the Taylor-Couette instability, characterized by the dimensionless Taylor number

T a = Ω 2 a 3 R m ν 2 (1)
where Ω is the rotation rate of the inner cylinder, a is the gap between the two cylinders, R m is the mean radius of the two cylinders and ν is the kinematic viscosity of the fluid. T a is only rigorously defined for a Newtonian fluid flowing in a Couette cell, in which case the instability develops above a critical Taylor number of 1712 [START_REF] Guyon | Physical hydrodynamics[END_REF]. By using this critical Taylor number to get an order of magnitude of the importance of inertial effects in our Bingham fluids flowing in a vane geometry, we can compute a critical shear rate at which the instability is expected. This value is around 40 s -1 for the metakaolin paste. We do not expect the agreement to be quantitative, but the order of magnitude for this critical shear rate is compatible with the onset of shear thickening observed in the paste. This confirms our hypothesis that this apparent shear-thickening, also observed in the foamed samples, is of inertial origin and we will thus not consider it when determining the flow behavior of our samples. We fit the linear regime of the flow curve with a Bingham law

τ ( γ) = τ c + k 1 γ (2)
from which we extract the plastic viscosity k 1 describing the dissipation in the samples during flow. This law for the suspending paste is plotted with τ ys (0)

τ yd (0) k 1 σ R b 2σ/R b (Pa) (Pa) (Pas) (mN/m) (µm)
(Pa) series 1 25 ± 2 10.2 ± 0.5 0.082 ± 0.002 35 ± 1 35 ± 10 2000 ± 500 series 2 25 ± 2 10.2 ± 0.5 0.082 ± 0.002 35 ± 1 100 ± 20 700 ± 100

Table 1: Properties of the suspending metakaolin paste and the bubbles added into it. σ is the surface tension of the aqueous phase. a dashed line in figure 4 (a). Because the flow curves of the paste and the foamed samples deviate from the linear behavior at low shear rates, we measure the dynamic yield stress τ yd directly from the experimental curve, and do not use the value τ c provided by the linear fit at higher shear rates.

Table 1 summarizes the relevant physical quantities to quantify the rheological behavior of the metakaolin suspending paste in our samples, and the bubbles added into it.

Experimental results

Critical stress to induce flow

The suspending metakaolin paste is, as we have shown along the rheometry procedure, a yield stress fluid: a minimum stress τ ys is required to make it flow. We first focus on flow start-up of the foamed samples. The stress-strain curves obtained by imposing a small and constant velocity to the samples after a rest of 4 minutes are presented for the two series of foamed samples studied in figure 5. We observe that the foamed samples exhibit the same behavior as the suspending metakaolin paste, with a sharp increase in stress at small deformation, followed by a stress overshoot and a subsequent relaxation. The peak value of stress is the same for all the samples within experimental reproducibility, and is equal to the one of the suspending paste. The introduction of bubbles does not seem to affect the static yield stress of the samples. The dynamic yield stress of the two series of samples, obtained from the flow curves, is plotted on the right axis of each subfigure in figure 5. It is much lower than the static yield stress of the samples, and comparable to τ yd of the suspending paste, with a scatter but no trend in its dependence on the gas volume fraction φ. The dynamic yield stress, too, seems to be insensitive to the presence of the bubbles within the range of experimental parameters investigated here. To better explicit the dependence of the yield stress, static τ ys (φ) and dynamic τ yd (φ), on φ, we now analyze the data by plotting the static yield stress (respectively the dynamic yield stress) of the samples, scaled by τ ys (0) (resp. τ yd (0)), as a function of φ. The result of this operation is shown for the two series of foamed samples by the square symbols (resp. the triangular symbols) in figure 6.

As expected from the experimental curves obtained during the measurements, both the dimensionless static yield stress and the dimensionless dynamic yield stress are approximately constant, equal to 1. In the range of bubble sizes and void volume fraction studied, the static and dynamic yield stress of the foamed samples are imposed by the suspending metakaolin paste. ) as a function of the gas volume fraction, for the two series of suspensions studied. The full line is a micromechanical estimate for the dimensionless static yield stress of suspensions of monodisperse bubbles in model yield stress fluids [START_REF] Ducloué | Rheological behaviour of suspensions of bubbles in yield stress fluids[END_REF].

0 1 0 2 0 3 0 4 0 5 0 0 5 1 0 1 5 2 0 ( a ) ( a ) τ ( P a ) d γ/ d t ( s -1 ) 1 3 % 1 7 % 2 4 % 2 8 % 3 9 % 4 0 % 4 2 % 4 6 % 0 % 1 0 2 0 3 0 4 0 5 0 0 5 1 0 1 5 2 0 ( b ) τ ( P a ) d γ/ d t ( s -1 ) 0 % 1 2 % 3 0 % 3 3 % 4 3 %

Dissipation during flow

For stresses higher than the yield stress, the suspending metakaolin paste flows, with a relationship between the shear rate and shear stress which is non-linear and has been described in section 2.4. The flow curve obtained during the decreasing ramp of shear stress on the suspending metakaolin paste and the foamed samples is shown for the two series of suspensions in figure 7. For both series of suspensions, the general shape of the curve is not modified by the presence of the bubbles: there is a finite dynamic yield stress, followed by a linear regime at higher shear rates which can be fitted to a Bingham model. We observe that the plastic viscosity k 1 increases with the gas volume fraction.

The presence of the air bubbles, although of little or no effect on the dynamic yield stress, clearly increases the plastic viscosity of the foamed samples during flow. To quantify this increase in the flow dissipation, we now plot in figure 8 the plastic viscosity k 1 (φ) of the foamed samples scaled by the one of the suspending metakaolin paste k 1 (0). As expected from the flow curves of the samples, the dimensionless plastic viscosity strongly increases with φ: 40% of bubbles in the paste is enough to triple the plastic viscosity.

Discussion

Yield stress

The somewhat surprising result that the introduction of 50% of bubbles in the paste does not change its yield stress can be understood in the light of work done on model foamed materials. The behavior of suspensions of bubbles in model yield stress fluids has been experimentally studied, in the same range of void volume fraction as here, by using suspensions of bubbles in concentrated oil-in-water emulsions [START_REF] Ducloué | Rheological behaviour of suspensions of bubbles in yield stress fluids[END_REF]. Those emulsions are non-reactive simple yield stress fluids [START_REF] Mason | Yielding and flow of monodisperse emulsions[END_REF]Ovarlez et al., 2008): their rheological behavior does not depend on the flow history. This well-controlled behavior justifies their use as model yield stress fluids. It has been shown that the yield stress of these foamed emulsions is equal to the one of the suspending emulsion providing that capillary effects are strong enough in the system. More precisely, the rheology of the foamed simple yield stress fluid is the result of the interplay of the two kinds of stresses acting on the bubbles when the sample starts to flow. On the one hand, the just yielded suspending fluid exerts a stress on the bubbles, which is of the order of magnitude of the fluid's yield stress and tends to deform the bubbles. On the other hand, capillarity acting on the surface of the bubbles tends to minimize to bubbles' surface area by restoring a spherical shape. The competition of these two mechanisms determines the relative rigidity of the bubbles in the just yielded fluid and can be quantified by introducing a plastic capillary number:

Ca τ = τ y (0) 2σ/R b (3)
which is the ratio of the paste yield stress τ y (0) to the capillary stress 2σ/R b .

When surface tension forces largely dominate over the yield stress of the suspending paste, this capillary number is very low and the bubbles behave as rigid spherical objects in the paste: the stress in the paste is not high enough to deform them. Note however that they differ from rigid particles in the boundary condition with the fluid, which slips on the bubbles' surface. As the relative importance of surface tension is decreased, Ca τ increases and the bubbles become increasingly soft in the paste.

Experiments have shown that below a capillary number of approximately 0.1, the yield stress of the samples is not affected by the presence of the bubbles. For the model systems of monodisperse bubbles in simple yield stress fluids at moderate void volume fraction (well below the percolation threshold of 64%), it is possible to quantitatively describe and estimate the behavior of the foamed samples at zero capillary number thanks to homogenization techniques (Thy [START_REF] Linh | Overall Properties of a Soft Porous Material: Surface Tension Effects[END_REF]. The result of the estimation is in good agreement with the experimental data obtained on those model systems and is plotted with a full line in figure 6. In the case of the foamed metakaolin samples, the plastic capillary numbers are 0.01 and 0.03 for the R b = 35 µm and R b = 100 µm samples, respectively. From the results on foamed model yield stress fluids, it is then expected that the yield stress of the foamed metakaolin samples should not depend on φ in the range of volume fractions investigated. The constant yield stress of the foamed metakaolin samples shows that their behavior at yielding can be predicted in the same way as in the model systems, although the rheology of the suspending paste is in fact more complicated. Experiments on suspensions of bubbles in model yield stress fluids have also shown that for higher plastic capillary numbers, the bubbles become deformable at yielding of the suspending paste. At the scale of the samples, the softness of the bubbles in the fluid results in a drop of the yield stress with φ. For large deformations, it has also been observed experimentally that around the same critical capillary number (approximately 0.5 for the suspensions in the concentrated emulsions), bubbles rupture during mixing with the paste [START_REF] Kogan | Mixtures of foam and paste: suspensions of bubbles in yield stress fluids[END_REF]. We did not investigate this regime for metakaolin pastes but the same features can be expected as in model materials. The plastic capillary number is thus also of particular interest in any application requiring the preservation of bubble size during the flow of a foamed metakaolin paste: it gives an estimate of the maximum shear stress that can be applied to the system without breaking the bubbles.

Plastic viscosity

The plastic viscosity of the foamed samples dramatically increases with φ for both series of suspensions studied. Interestingly, data for both series of suspensions fall on the same curve, which suggests that the bubble size is not a relevant parameter in this viscosity increase. Indeed, as we have explained in the experimental procedure, the maximum stress applied during the shear stress ramp is about twice the static yield stress. We have shown in the discussion of the static yield stress measurements that τ ys (0) is much lower than the capillary stress scale, resulting in the bubbles being non-deformable in the suspending metakaolin paste. During the flow curve measurement, the scale of the capillary stress will as a consequence also be very large compared to the stress scale in the metakoalin paste, meaning that the bubbles remain rigid in the flow within the range of shear rates considered here.

We can then quantify the relative increase in the plastic viscosity for suspensions of rigid bubbles in the metakaolin paste by fitting our experimental data with a polynomial function. The result of this fit is plotted with a full line in figure 8. The increase for the metakaolin samples is quadratic (equation of the curve: k 1 (φ)/k 1 (0) = 1 + 12φ 2 ). For comparison, the relative increase in the plastic viscosity of suspensions of rigid bubbles in model yield stress fluids is plotted with a dotted line in figure 8 (best fit to the experimental data obtained on those systems). The increase is much less, but the results on the metakaolin paste and the concentrated emulsion cannot be compared directly because the flow behavior of the emulsion has a different power law than the metakaolin paste: its flow curve is well fitted to a Herschel-Bulkley model τ ( γ) = τ + k γn with n = 0.45. For this system, the measured increase in plastic viscosity is linear: k 0.45 (φ)/k 0.45 (0) = 1 + 0.49φ. The difference in the rheology of the suspending fluid is significant, and can account for the two different responses for the plastic viscosity with the volume fraction. Indeed, the global dissipation in the foamed samples is very sensitive to the way shear is localized between the bubbles during flow, and this local behavior of the suspending fluid is determined by the flow curve of the suspending fluid.

Conclusions

We have prepared model foamed metakaolin samples, in which the rheology of the suspending metakaolin paste, the bubble size and the bubble volume fraction are controlled. Two series of samples have been studied, with two different mean bubble radii. We have shown thanks to careful rheology measurements that the static and dynamic yield stress of the foamed samples are the same as the ones of the suspending paste for the bubble sizes we have investigated. This result can be understood in terms of bubble deformability, quantified by a capillary number. Our rheology experiments have also shown that the presence of the bubbles in the paste increases the dissipation during flow. A comparison with suspensions of bubbles in model yield stress fluids shows that it is also the case in those systems, and that it is likely that dissipation increases faster for suspensions in fluids of larger plastic index. We have given phenomenological quantification of this dissipation in the foamed model fluid and the foamed metakaolin samples. Those results are very promising for the understanding of the rheology of foamed metakaolin pastes, but more importantly have proven that the framework developed for model foamed yield stress fluids can also apply to more complex systems. This generality could make it relevant to the flow and processing of other foamed mineral pastes, such as foamed clay, concrete or plaster slurries.

Ackowledgments

We are thankful to AGS Minéraux (Imerys group) for kindly providing us with the metakaolin as a sample. Financial support from Saint-Gobain Recherche (Aubervilliers, France) is gratefully acknowledged.

Figure 1 :

 1 Figure 1: Micrograph of a dilute suspension of M1200S in water. A drop of the suspension has been deposited on a glass slide.

Figure 3 :

 3 Figure3: Characterization of the suspending paste: stress as a function of strain obtained for the static yield stress measurement; a shear rate of 0.01 s -1 is applied to the sample after a rest of 4 min.

Figure 4 :

 4 Figure 4: Characterization of the flow behavior of the suspending paste: (a) Stress as a function of shear rate obtained during the decreasing shear stress ramp. The dashed line is a linear fit to the data in the region of constant plastic viscosity. (b) Derivative of the flow curve with respect to the flow rate. A viscosity plateau is observed at intermediate shear rates. The onset of shear thickening at higher flow rates is clearly visible.

Figure 5 :

 5 Figure 5: Stress as a function of strain during the static yield stress measurement on suspensions of (a): R b = 35µm and (b): R b = 100µm bubbles in the metakaolin paste.A shear rate of 0.01 s -1 is applied to the samples after a 4 minute rest. The dynamic yield stress of the same samples, obtained from the flow curves, is plotted on the right axis.

Figure 6 :

 6 Figure 6: Dimensionless yield stress (static: , and dynamic:) as a function of the gas volume fraction, for the two series of suspensions studied. The full line is a micromechanical estimate for the dimensionless static yield stress of suspensions of monodisperse bubbles in model yield stress fluids[START_REF] Ducloué | Rheological behaviour of suspensions of bubbles in yield stress fluids[END_REF].

Figure 7 :

 7 Figure 7: Flow curves of the two series of suspensions studied: (a) R b = 35µm; (b) R b = 100µm.

Figure 8 :

 8 Figure 8: Dimensionless plastic viscosity of the two series of suspensions of bubbles in the metakaolin paste (n = 1), as a function of the gas volume fraction. The full line is a phenomenological best fit to the data. The dashed line is a best fit to data obtained on suspensions of bubbles in model yield stress fluids with n = 0.45.

  Barnes, H. A., 1997. Thixotropya review. Journal of Non-Newtonian fluid mechanics 70 (1), 1-33. Bell, J., Kriven, W., 2009. Preparation of ceramic foams from metakaolinbased geopolymer gels. In: Ceram Eng Sci Proc. Vol. 29. pp. 97-112. Bich, C., Ambroise, J., Péra, J., 2009. Influence of degree of dehydroxylation on the pozzolanic activity of metakaolin. Applied Clay Science 44 (3), 194-200.


Mathieu Lefort
email: mathieu.lefort@ensta-paristech.fr

Alexander Gepperth
email: alexander.gepperth@ensta-paristech.fr

Active learning of local predictable representations with artificial curiosity

, in the neighborhood of each local representation. Thus, PROPRE simultaneously learns interesting representations of the input flow (depending on their capacities to predict the output) and explores actively this input space where the learning progress is the higher. We illustrate our architecture on the learning of a direct model of an arm whose hand can only be perceived in a restricted visual space. The modulation of the projection learning leads to a better performance and the use of the curiosity mechanism provides quicker learning and even improves the final performance.

I. INTRODUCTION

Developmental robotics is a recent and active research field that targets the conception of robots that are able to learn to interact with an unknown environment in an autonomous and lifelong open-ended manner, which raises a lot of challenging and yet unsolved problems [START_REF] Lungarella | Developmental robotics: a survey[END_REF], [START_REF] Guerin | Learning like a baby: a survey of artificial intelligence approaches[END_REF], [START_REF] Weng | Developmental robotics: Theory and experiments[END_REF]. The constructivist learning of predictive representations from a sensory-motor data flow is one of them. Especially, this learning does not fit with the classical machine learning framework as large areas of the sensory-motor space are unlearnable such as, for instance, trying to predict the gustatory consequence of moving your arms. One way to deal with learning in such high dimensional sensory-motor spaces is to take inspiration from the infant's development, especially studied in developmental psychology (see [START_REF] Piaget | The origins of intelligence in children[END_REF] e.g.), by providing to the agent intrinsic motivation [START_REF] Ryan | Intrinsic and extrinsic motivations: Classic definitions and new directions[END_REF] by mean of an artificial curiosity mechanism. This curiosity mechanism will motivate the agent to explore areas of the sensory-motor space that are interesting for its own development. Various implementations of artificial curiosity were proposed, based on different measures, such as error maximization [START_REF] Barto | Intrinsically motivated learning of hierarchical collections of skills[END_REF], [START_REF] Thrun | Exploration in active learning[END_REF] (areas where the prediction error is the higher are the more interesting) or similarity-based progress maximization [START_REF] Schmidhuber | Curious model-building control systems[END_REF], [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF] (areas where the learning progress is the higher are the more interesting, thus avoiding to get stuck in stochastic areas of the input space) (see [START_REF] Oudeyer | Intrinsic motivation systems for autonomous mental development[END_REF] for a review).

Especially, IAC [START_REF] Oudeyer | Intrinsic motivation systems for autonomous mental development[END_REF] (Intelligent Adaptive Curiosity) and its derivative R-IAC [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF] (Robust IAC) propose a generic framework for implementing a curiosity mechanism upon a prediction learning model and was applied to the developmental robotics field. This framework consists of the monitoring of the prediction learning progress in various local regions paving the input space. This progress is measured in each region by the difference of accumulated errors in two equal and consecutive sliding temporal windows. In R-IAC, the next input experienced by the prediction learning model is randomly chosen either in the input space (typically 30% of times) or in one of the regions that is picked up with a probability depending on its learning progress. When some predefined number of experiments have been processed in one region, it is split in two along one axis of the input space, so that to maximize the difference between the learning progress in the two newly created regions. For more details on the algorithm, please refer to [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF].

PROPRE, that stands for PROjection-PREdiction, is a generic and modular hybrid framework that provides online learning of input data flow representations that are useful to predict another data flow. It combines an unsupervised learning module, based on a self-organizing map (SOM), that autonomously learns representations from the input flow, with a discriminative module that learns the correspondence between the SOM representations and the output by mean of a linear regression. The main originality of PROPRE consists of a predictability module that computed an interestingness measure of the current stimulus, based on the comparison between a prediction quality measure and a sliding threshold. This interestingness measure modulates the generative learning so that to favor the learning of representations that predict the target better than a local average. This predictability modulation mechanism provides a better representations learning, for instance for visual pedestrian pose classification [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF]. Moreover, it leads to the gathering of the representations where a relationship with the output is defined, thus avoiding unlearnable areas of the input space [START_REF] Lefort | Learning of local predictable representations in partially structured environments[END_REF], so that PROPRE can be well suited for the developmental robotics field.

In this article, we propose to extend the PROPRE capabilities by including a curiosity mechanism in order to simultaneously learn and actively explore an unknown sensorymotor space in a closed perception/action loop. The curiosity mechanism is inspired by the one proposed in R-IAC [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF]. For that purpose, each unit of the SOM, that provides a local representation of the input flow based on prototype learning, is associated with a learning progress measure that monitors the model performance evolution in the corresponding Voronoi cell in the input space. Thus, a SOM unit is similar to a region in R-IAC, except that the region is here not defined by a splitting mechanism but dynamically depends on the SOM learning, thus providing a fixed a priori memory usage. The next motor action performed by the system is chosen in some unit prototype neighborhood, the unit being picked up randomly with a probability depending on its learning progress, similarly to what was proposed in R-IAC.

In the next section, we introduce the PROPRE architecture including the new artificial curiosity mechanism. In section III, we show that the curiosity mechanism leads to better asymptotic performance and better performance in almost every time steps when PROPRE learns the direct model of a simple two dimensional planar arm with a limited visual field so that large areas in the motor space are unlearnable. We conclude and discuss possible perspectives of our work in section IV.

II. PROPRE

A. Architecture PROPRE is a generic and modular neural paradigm that combines projection and prediction for online learning of input/output relationship from raw data flows. It consists of the interaction between three modules (see figure 1):

• A projection module (see section II-B for details) that transforms the current input stimulus in a low dimensional representation. This module uses the selforganizing map paradigm (SOM) that provides a topological projection of the input space on the map manifold. • A prediction module (see section II-C for details) that learns to predict the current output target from the representation given by the projection module. This learning is done with a linear regression. • A predictability module that analysis the quality of the current prediction and plays two roles. First, it modulates the projection module so that to learn representations that are more efficient to predict the target flow than a local average (see section II-D1 for details). Second, it influences the choice of the next input of the model so that to favor areas of the input space leading to learning progress, thus providing an active learning of the input/output relationship (see section II-D2 for details). Thus, PROPRE is an hybrid architecture that provides online, adaptive, active learning [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF] that can also be unsupervised when using PROPRE in a multimodal context where representations learned from each flow try to predict the one of the other flow [START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF]. Besides, depending on the way to use the SOM activity to define the representations, PROPRE can be seen as an extension of a radial basis function network [START_REF] Hecht | Using self-organizing maps for regression: the importance of the output function[END_REF].

From a computational point of view, the next input stimulus of the model is the (proprioception of the) motor command that is actively determined depending on the predictability module (equation 1.d in figure 1). The value of the target corresponds to the perception of the environment resulting from the execution of this motor command. The input stimulus is processed in the model by a feed-forward evaluation of each module activity (equations 1.a-b-c). Then, the plastic connections are updated depending on the corresponding modules activities (equations 2.a-b) and so on.

B. Projection

In our previous articles [START_REF] Lefort | Learning of local predictable representations in partially structured environments[END_REF], [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF], the projection step consisted of the classical self-organizing map model proposed by Kohonen [15]. This model is related to the minimization of the quantization error -plus a topological term -so that the distribution of the prototypes tends to be similar to the one of the input data [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF], [START_REF] Cottrell | Theoretical aspects of the som algorithm[END_REF]. In this article, we introduce a curiosity mechanism to actively choose the next input (see section II-D2 for more details), thus deeply modifying the input data distribution. The coupling of this curiosity mechanism with the Kohonen learning rule creates an undesirable dynamic attractor corresponding to the convergence of all prototypes in a very tiny area of the input space. Indeed, the input are chosen around some of the learned prototypes, i.e. in a subspace of the one mapped by the SOM, leading to the mapping of this subspace by the SOM and so on and so forth.

In order to tackle this problem, in this article, we use a Dynamic Self-Organizing Map (DSOM) [START_REF] Rougier | Dynamic self-organising map[END_REF] for the prediction module. DSOM model, compared to the one of Kohonen, includes the distance between the stimulus and the best matching prototype in the computation of the learning rate and the Gaussian neighborhood (see equation 2.a) so that the model is only slightly changed if a prototype is already close to the stimulus. Thus, even if the curiosity mechanism concentrates the input in some new interesting subspace of the input space for some time, it will have few consequences on the previously learned projection as DSOM does not reduce a quantization error but tends to map the input space [START_REF] Rougier | Dynamic self-organising map[END_REF]. Thus, DSOM provides a kind of incremental learning, dealing with a fixed number of units in the map, necessary to the use of a curiosity mechanism [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF].

In practice, S is a discrete bi-dimensional square grid of neurons, each one receiving the input data flow D (see figure 1). Each prototype of a unit x in the map, denoted w SD (x, t), is updated at each time step t with the following equation, corresponding to the one of DSOM modulated with the interestingness measure I (t) provided by the predictability module (see section II-D1):

∆w SD (x, t) = d * (t)ηI(t)e -||x-x * (t)|| 2 2 2(d * (t)σ) 2 (D(t) -w SD (x, t))
(2.a) with η the learning rate and σ the variance of the Gaussian neighborhood radius which are both constant to provide a lifelong learning. || • || 2 is an euclidean distance in the map, x * (t) is the best matching unit defined as the unit whose prototype is the closest to the current input D(t), i.e. x * (t) = arg min The activity of any unit x of the map is defined as:

x |w SD (x, t) -D(t)|
S(x, t) = 1 if x = x * (t) 0 otherwise (1.a)
This classical activation function [START_REF] Hecht | Using self-organizing maps for regression: the importance of the output function[END_REF] provides a prediction that only depends on the best matching unit (see section II-C). Thus, it is consistent with the monitoring of the prediction quality, computed in each Voronoi cell of DSOM units, processed in the predictability module (see section II-D).

C. Prediction

The projection activity S(t) is used to compute a prediction P(t) of the target data flow T(t) at time t. The activity of a unit x in P is computed as a weighted sum of the S activity:

P (x, t) = y w P S (x, y, t)S(y, t) (1.b)
with w P S (x, y, t) the weight from the units y in S to x in P . The connection weights between S and P are learned with a classical stochastic gradient descent implementation of a linear regression [START_REF] Bishop | Pattern recognition and machine learning[END_REF], which minimizes the mean square error between the prediction P(t) and the current target value T(t). Thus, the weights are updated with the following equation:

∆w P S (x, y, t) = η S(y, t)(T (x, t) -P (x, t)) (2.b)
with η the constant learning rate.

D. Predictability module

The predictability module monitors the quality Q(t) of the current prediction P(t) with respect to the true target value T(t) with a simple and generic measure:

Q(t) = P (z * , t) z P (z, t) with z * = arg max z T (z, t)
In our experiments, the target value represents the visual position of the end effector in a matrix of pixels, when the hand is visible (see section III). Thus, Q(t) represents the percentage of prediction of the right position of the hand. This measure was also used for classification learning [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF] or multimodal regression [START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF].

1) Projection learning modulation: In each Voronoi cell1 associated to a unit x of DSOM, we compute online the average prediction quality θ(x, t) over a sliding window:

θ(x, t) = (1 -τ )θ(x, t -1) + τ Q(t) if x = x * (t) θ(x, t -1) otherwise (1)
with τ the smoothing factor and x * (t) the best matching unit at time t (see section II-B).

The interestingness measure of the current stimulus for predicting the target, that modulates the projection learning (equation 2.a), is defined as:

I(t) = Q(t) -θ(x * (t), t) if Q(t) >= θ(x * (t), t) 0 otherwise (1.c)
Thus, the current stimulus is only learned if it provides a prediction locally more accurate than the average and the interestingness measure depends directly on this accuracy difference to the local average.

2) Artificial curiosity: As proposed in [START_REF] Oudeyer | Intrinsic motivation systems for autonomous mental development[END_REF], [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF], the artificial curiosity mechanism is based on the monitoring of learning progress of the model on various local areas paving the input space, here the Voronoi cells associated with the DSOM units. In practice, the learning progress LP(x, t) of each unit x of the DSOM at time t is computed as the difference between two average prediction quality computed online with different smoothing factors τ and τ (τ < τ ). The learning progress itself is averaged over time with τ LP as smoothing factor:

LP(x, t) =      (1 -τ LP )LP(x, t -1) + τ LP (θ(x, t) -θ (x, t)) if x = x * (t) LP(x, t -1) otherwise θ (x, t) = (1 -τ )θ (x, t -1) + τ Q(t) if x = x * (t) θ (x, t -1) otherwise
The average prediction quality θ(x, t), computed on the shortest time window, is the one used for the interestingness measure (see previous section).

The next action to perform is uniformly chosen in the input space with some probability (25% in our experiments) for exploration, otherwise it is actively determined by the curiosity mechanism. For that purpose, the model computes for each unit x of the DSOM a probability p(x, t), depending of its learning progress, to be picked up :

p(x, t) = |LP(x, t)| x |LP(x , t)|
Let denote x(t) the effectively randomly picked up unit by the model at time t. Then, the next input D(t+1) = (D(i, t+1)) i of the model is obtained by adding to each dimension of the prototype w SD (x(t), t) = (w SD (x(t), i, t)) i , associated with the randomly chosen unit x(t), a value in [-r, +r] picked up with an uniform probability:

D(i, t + 1) = w SD (x(t), i, t) + σ i (1.d)
with σ i ∼ U(-r, r). It has to be noticed that the interestingness measure (section II-D) can be interpreted as the rectified temporal evolution of the prediction quality measure between short term (current quality Q(t)) and medium term (θ(x, t)) averages. Thus, both the projection modulation and the artificial curiosity are based on the same mechanism operating at different time scales and their intertwining leads to the emergent properties of PROPRE.

III. EXPERIMENTS AND RESULTS

A. Protocol

We tested our PROPRE architecture on the learning of the direct model of a simple simulated robotic planar arm (see figure 2). In our setup, the hand is seen by a 5 × 10 pixels matrix covering a restricted part of the reachable area so that the direct model is learnable only for around 20% of the motor space (see figure 3). This experiment, a simplified and adapted version of the hand-eye-clouds experiment proposed in [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF], illustrates one of the main problem in developmental robotic: exploration of a sensory-motor space where large areas are unlearnable.

With the curiosity mechanism, the next motor command is chosen in some neighborhood of a learned prototype (see section II-D2) so that it can be outside of the joint limits. In this case, the motor command is bounded within the joint limits and the input/output couple provided to PROPRE is the proprioception and visual perception of the effectively performed motor command.

B. Results

In order to study the influence of the predictability module on the performance, we tested three different architectures:

• no modulation of the projection learning (i.e.

∀t, I(t) = 1) and no curiosity, denoted DSOM+LR, • modulation of the projection learning and no curiosity, denoted PROPRE, • modulation of the projection learning and curiosity, denoted PROPRE ACTIVE.

When no curiosity mechanism is used, the motor actions are randomly chosen in the input space with an uniform distribution. When used, the curiosity mechanism chooses 75% of the performed actions, the others are randomly picked up in the motor space.

For each architecture, the DSOM size was 10 × 10, the prediction learning rate was set to η = 10 -3 (equation 2.b). We independently tuned, with reasonable effort, the projection parameters for DSOM+LR (σ = 3, η = 1) and PROPRE (σ = 4, η = 3) (equation 2.a). Then, we tune the smoothing factor used in the interestingness measure, τ = 10 -3 (equation 1.c). These parameters, found for PROPRE, are also used for PROPRE ACTIVE is order to fairly quantify the influence of the curiosity mechanism. Finally, we tuned the curiosity mechanism and set the long term smoothing factor τ = 10 -4 and the range r = 0.35 (see section II-D2). Moreover, the initial weights of each unit of DSOM were initialized in [(π -1)/2, (π + 1)/2]2 , i.e. in the center of the motor space but outside of the motor area providing a visible hand.

In order to evaluate the performance of the models, we defined a benchmark of 10000 motor/visual couples where the hand is visible and recorded an error each time the maximum of the prediction did not correspond to the real position of the hand. The results are presented in figure 4. By the way, similar qualitative results are obtained with the same parameters when putting a random pixel at 1 and the others at 0, instead of white noise, when the hand is not visible during learning. We can clearly see that the modulation of the projection learning provides better performance at each time step compared to the DSOM+LR system. This confirms the results we obtain in [START_REF] Lefort | Learning of local predictable representations in partially structured environments[END_REF] but when using a Kohonen SOM module for the projection. More interestingly, we can observe that the curiosity mechanism, other things being equal, has also a significant influence on the performance. Indeed, the PROPRE ACTIVE model is the one that has better performance at almost every time step including the better asymptotic performance. This improvement is obtained because the curiosity mechanism performs mainly motor actions providing a visible hand, 60% in average over the ten simulations 2 , whereas only 20% of the total motor space leads to a visible hand.

On figure 5, we illustrate an example of the temporal evolution of the repartition of the performed motor actions obtained when using the curiosity mechanism. We can observe that the motor actions seem to first mainly concentrate on the orange-red areas of figure 3 before spreading over all the motor area leading to a visible hand. These orange-red areas are the easiest ones to learn as the activation of one specific pixel can be obtained by a larger range of motor actions (compare the relative size of the colored areas in the motor space in figure 3). Thus, this seems to indicate that the curiosity mechanism leads to a developmental trajectory of the model from simple to more complicated learning as already found when using the IAC framework [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF]. A more extensive study, including statistical results over various trials, is necessary to confirm this developmental trajectory in PROPRE.

IV. CONCLUSION AND PERSPECTIVES

PROPRE is a neural paradigm for online and open-ended learning of input/output relationship from raw data flows. It combines a generative learning, by projecting the input space on a self-organizing map -here the DSOM model, with a discriminative learning, by mean of a linear regression of the output. The main originality of PROPRE is the use of a predictability module that monitors the quality of the prediction, with a simple measure, and modulates the projection learning so that to favor representations that predict the output better than a local average. Especially, when the input/output relationship in only defined in some areas of the input space, this predictability modulation leads to the gathering of representations in these learnable areas [START_REF] Lefort | Learning of local predictable representations in partially structured environments[END_REF], an important property for developmental robotics.

In this article, we propose to integrate in PROPRE an artificial curiosity mechanism, derived from the IAC paradigm [START_REF] Oudeyer | Intrinsic motivation systems for autonomous mental development[END_REF], based on the monitoring of learning progress in various regions paving the input space. For that purpose, we compute the temporal evolution of the prediction quality obtained in each Voronoi cell of the DSOM by the difference between two averages computed online with different smoothing factors. The next input is randomly chosen, with an uniform distribution, in some neighborhood of the prototype of one DSOM unit picked up with a probability depending of its associated learning progress. Thus, PROPRE simultaneously learns to represent efficiently the input space and actively explores around learned areas, where it seems the more promising (through the curiosity mechanism monitoring the learning progress), in a closed perception/action loop.

We tested this architecture on the learning of a direct model of a two degrees of freedom planar arm, whose hand position is perceived by a matrix of pixel only covering a limited subspace of the reachable space. The modulation of the projection learning by the predictability module provides a better visual prediction performance. The additional use of the curiosity mechanism provides an even better asymptotic performance. Moreover, by performing mainly motor commands providing a visible hand, the curiosity mechanism reduces the number of actions needed to obtain a defined performance.

These very promising results on a simple setup open the way to test PROPRE for the learning and exploration of more realistic sensory-motor spaces, especially high-dimensional and redundant. We also want to improve the exploration part of the curiosity mechanism, which require yet to know a priori the boundaries of the motor space, for example using social guidance. Moreover, IAC was improved by two paradigms including some features whose integration in PROPRE can lead to interesting perspectives. First, R-IAC (Robust IAC) [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF] monitors the learning progress at several scales by using a tree structure of regions. One way to include this feature in PROPRE can be to use a tree of self-organizing maps, a structure that was already studied to obtain different granularities in the mapping (see [START_REF] Astudillo | Topology-oriented self-organizing maps: a survey[END_REF] e.g.). This question can be coupled with the study of the differences between having an increasing number of fixed regions, as in IAC and its derivatives, versus a fixed number of dynamic regions, as in PROPRE ACTIVE. Second, SAGG-RIAC (Self-Adaptive Goal Generation RIAC) [START_REF] Baranes | Active learning of inverse models with intrinsically motivated goal exploration in robots[END_REF] monitors a competence progress, i.e. the ability to reach some goal in the output space, instead of a learning progress, i.e. the ability to predict the consequence of a motor action. This competence-based curiosity seems to be particularly efficient in highly redundant sensory-motor spaces. PROPRE can be use for multimodal learning, by providing as output of a data flow processing the representations learned from another data flow [START_REF] Lefort | Learning of local predictable representations in partially structured environments[END_REF]. Thus, one way to integrate competence-based curiosity in PROPRE could be to learn representations from the sensory and the motor flows and to associate the curiosity mechanism to the sensory representations instead of the motor representations as in this article. Moreover, both mechanisms could be coupled and alternatively used depending on the sensory-motor area currently explored.

Fig. 1 .

 1 Fig.1. Data flow processing in PROPRE. The system learns representations of the input data flow (D) by combining generative learning (S) and discriminative learning of the target (T ). The predictability module, that monitors the quality of the prediction (P ), modulates the generative learning so that to favor representations that are better to predict of the input/output relationship than a local average. It also chooses the next input received by the model, depending on the learning progress, closing the perception/action loop.

  2 and d * (t) is the distance between this best matching unit and the current input, i.e. d * (t) = |w SD (x * (t), t) -D(t)| 2 , with | • | 2 an euclidean distance on the input space.

Fig. 2 .

 2 Fig. 2. A robot moves its arm (d 1 = 3 and d 2 = 2.5, the units are arbitrary) in a plan depending on an input joint motor command ((θ 1 , θ 2 ) ∈ [0, π[ 2 ).It can only see its hand with a pixel matrix providing a restricted visual field. If it sees its arm (left), the corresponding pixel is set to 1, otherwise (right) the visual perception is a white noise with amplitude 1.

Fig. 3 .

 3 Fig. 3. Example of visual (top) and motor (down) couples. White area correspond to an invisible hand, whereas each colored area corresponds to some pixel activated in the visual matrix (better viewed in color).

Fig. 4 .

 4 Fig. 4. Average and variance (over 10 simulations) of the temporal evolution of performance obtained by the different models.

Fig. 5 .

 5 Fig. 5. From top left to bottom right: temporal evolution of the distribution of motor actions executed by the system using the curiosity mechanism in one of the simulations. Each figure represents, in the motor space, the 100000 actions executed by the system during the time window mentioned above the correspond figure.

In practice, as the system continuously learns, unit prototypes are continuously smoothly modified so that θ(x, t) is only a reasonable approximation of the average prediction quality in the Voronoi cell of the unit.

For exploration purpose, 25% of actions are randomly chosen in the motor space (see section II-D2), leading to around 20% of total actions performed by the system in the unlearnable area. So that half of the actions leading to an invisible hand are due to the exploration mechanism.


E Koorambas

Weak Space-Time Curvature Effects for Lepton Electric Charge Swap at High Energy Scale

Keywords: number(s): 11.30.-j, 02.20.-a, 11.25.Uv, 14.80.-j, 11.5 Group theory, Hypothetical particles, Extra dimensions, Gauge theory of gravity

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

General relativity, published by Albert Einstein in 1915 [START_REF] O'connor | General relativity[END_REF], generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time.

The curvature of space-time, in particular, is directly related to the energy and momentum of whatever matter and radiation are present. This relation is specified by a system of partial differential equations, the Einstein field equations,  E-mail:elias.koor@gmail.com In quantum field theory, an attempt to extend Yang-Mills' universal description of the fundamental interactions [2], to describe gravity, resulted in gauge gravitation theory.

The first gauge model of gravity was suggested by R. Utiyama in 1956 [START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF] and others [START_REF] Blagojević | Gauge Theories of Gravitation: A Reader with Commentaries[END_REF][START_REF] Brodsky | [END_REF]6], just a few years after the emergence of the gauge theory itself [START_REF] Blagojević | Gauge Theories of Gravitation: A Reader with Commentaries[END_REF]. In these initial attempts to construct a gauge theory of gravity by analogy with gauge models of internal symmetries, the Lorentz group is localized and the gravitational field is not represented by gauge potential, but by the metric field [7,8,9].

In 2003, N. Wu proposed a Quantum Gauge Theory of Gravity (QGTG) based on the gravitational gauge group (G) [10][11][12][13]. In Wu's theory, the gravitational interaction is considered as a fundamental interaction in a flat Minkowski space-time and not as space-time geometry. Also within the framework of QGTG and the De Broglie-Bohm approach to gravitational gauge fields, a model of interacting massive gauge gravitons and a (possible) heavy gauge graviton, resulting from shell decay of Higgs bosons, were developed recently by the author [14][START_REF] Koorambas | Recent Developments in Bosons Research[END_REF][START_REF] Koorambas | [END_REF][17][START_REF] Koorambas | Can gravitons be effectively massive due to their zitterbewegung motion? hal-00990723v1[END_REF].

Electric Charged Swap (ECS) symmetry for leptons was also proposed by the author [19]: A family of particular transformations may be continuous (e.g. the rotation of a circle) or discrete (e.g. the reflexion of a bilaterally symmetric figure, or the rotation of a regular polygon). ECS-transformation between ordinary families of leptons produces heavy neutral non-regular leptons of masses of order O (1TeV). These particles may form cold dark matter. Furthermore, certain properties of lepton families are explained from this symmetry within the framework of superstring theories [19,[20][21][22][23][24].

A-Wollmann Kleinert and F. Bulnes, based on ECS symmetry (in this case of leptons [19]), they proposed that leptons are the subtle Fermions [25].

A quark (q) and an ECS-quark (q῀) bound state (qq῀) were recently proposed by the author to explain the electrically charged charmonium Z + c [START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF]9) meson as a charm quark (c), and the charm ECS-quark (c῀) bound state (cc῀) [START_REF]Koorambas Quark-quark of swap electric charge bound state hal-00852409v2[END_REF]. This suggestion predicted that J/ψ and π + mesons are the decay products of a Z + c [START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF]9), as it was later observed at BES III [48]. Furthermore, the proposed charm ECS-quark (c῀) of mass 2,3GeV predicts two new mesons: an electrically charged charmed D¯* +(zm) and a neutral charmed D *0(zm) meson.

In the present paper, starting with the six-dimensional space time, we investigate the nature of a new kind of space-time curvature effects produced by electric charge swap (ECS) transformations between families of leptons that produce non-regular leptons (ECS-leptons) at high energy scale. We find that lepton ECS-transformations form the proper Lorentz group SO + (1,3) (ECS) . We also show that Einstein-type theory of weak space-time ECS-curvature effects can be formulated from the SL(2,C) (ECS) gauge group by using ECS-tetrad formalism. The latter yields to electrically charged massless spin-2 (ECS-graviton), which mediates the ECS exchange interaction between families of leptons. The proposed ECS-curvature effects may have far-reaching applications in astrophysics and cosmology.

FUNDAMENTALS OF THE ELECTRIC CHARGE SWAP (ECS) SYMMETRY IN SIX-DIMENSIONAL SPACE-TIME

Following Gogberashvilli et al. [27], we consider a six-dimensional space-time with signature ( , , , , , )       . Einstein's equations in this space-time have the form:

4 11 () 2 AB AB AB AB R g R g T M     , ( 1
)
where M is the six-dimensional fundamental scale,  is the cosmological constant and A,B are capital indices equal to 0,1,2,3,4,5,

To split the six-dimensional space-time into four-dimensional and two-dimensional parts, we use the metric ansatz [27]:

2 2 2 2 2 2 2 ( ) ( ) ( sin ), a ds g x dx dx d b d            (2)
where  and b are constants and ()  is the warp factor. This warp factor equals one at brane location ( 0)

  , and decreases to zero in the asymptotic region ()   at the south pole of the extra two-dimensional sphere. Here the metric of the ordinary four-dimensional () a gx  has signature ( , , , )

    , with , , 0,1,2,3

   

. The extra compact 2-manifold is parameterized by the spherical angles ,  ( 0 ,0 2 2), the extra 2-surface is exactly a 2-sphere with radius  (0.07TeV -1 ).

        ).
The ansatz for the energy-momentum tensor of the bulk matter fields is:

( ), T g E     ( ), ij ij T g P   0. i T   (3)
Lower case Latin indices in equation ( 3) correspond to the two extra coordinates. The source functions E and P depend only on the extra coordinate θ. For these ansätze, Einstein's equations (1) take the following form:

22 24 3 3 3 cot 1 [ ( ) ], E M                22 24 6 4 cot [ ( ) ], P M           22 24 4 6 [ ( ) ]. P M             (4)
where the prime denotes differentiation d/dθ.

For the four-dimensional space-time we have assumed zero cosmological constant. Einstein's equations take the form:

(4) (4) 1 0, 2 R g R    (5)
where (4) R  and (4) R are four-dimensional Ricci tensor and scalar curvature, respectively. M. Gogberashvili and D. Singleton have found a non-singular solution of (4) for boundary conditions (0) 1   , (0) 0.

  [28]. This solution is given by: 2 ( ) 1 ( 1)sin ( / 2), a

      ( 6
)
where a is the integration constant. The source terms for this solution are given by:

with the radius of the extra 2-spheroid given by 24 10 / . M   For simplicity, in this paper we take 0 a  , so that the warp factor takes the form:

       (8)
This warp factor equals one at the brane location (θ= 0) and decreases to zero in the asymptotic region θ=π, i.e., at the south pole of the extra two-dimensional spheroid. The expression for the determinant of our ansatz (2) used here is given by: (4) 2 4 ( )sin ,

gg        (9) where (4)
g  is the determinant of four-dimensional space-time.

Non-Regular Leptons in Six Dimensions

Here we assume that the zero mode corresponds to the non-regular leptons which are copies of the third family of leptons. Although uncertain, this assumption is not physically implausible: it is reasonable to expect that, when entering the six-dimensional bulk, third family leptons change their properties profoundly and lose, so to speak, their 'individuality; (e.g. their observable masses) to their bare masses, spin and magnetic moment [19].

Let us now consider spinors in the six-dimensional space-time (2), where the warp factor ()  has the form (8). The action integral for the six-dimensional massless fermions in a curved background is:

6 . BA B A S d x g i h D h c           . ( 10
)
A D is the covariant derivative and A  is the 6-dimensional flat gamma matrices.

We also introduce the sechsbein A A h following the usual definition [27],

AB AB A B AB g h h   , (11)
where , AB are local Lorenz indices.

The six-dimensional spinor is given by: ( ) .  The representation of the flat (8 × 8) gamma-matrices is given by [27] as:

0 0 1 0 , , , 0 1 0 0 i i                               (13)
where 1 denotes the four-dimensional unit matrix and   are ordinary (4×4) gamma- matrices. Representation (13) gives the correct space-time signature ( , , , , , ).

      The generalization of 5  matrix is:

5 7 5 0 . 0       (14)
The variation of action (10) yields the following six-dimensional massless Dirac equation:

( ) ( ) 0, B B B A B BB h D h D h D x              (15)
with the sechsbein for our background metric (2) given by:

1 1 1 ,, sin B B B B A h              . (16)
From the definition of spin connection we have:

11 ( ) ( ) 22 1 ( ) . 2 h MN NM N N NN M M M M N N M M N N M PM QN R P Q M QR PR h h h h h h h h h h              (17)
The non-vanishing components of the spin connection are:

sin     , sin 2
         . ( 18
)
The covariant derivatives of the spinor field take the form:

sin ( ) ( ) ( ) 4 ( ) ( ) cos ( ) ( ) ( ) 2 AA AA AA D x x D x x D x x                               . ( 19
)
The Dirac equation takes the form [29], [30]:

1 sin 1 1 cot ( ) 4 sin 2 1 1 sin cot 1 ( ). 2 sin A A x x x x                                                                                           (20)
This system of first-order partial differential equations has the following solutions:

    00 2 00 () 1 () 2 ( ) () A ax x x               , ( 21
)
where 00 ( ), ( )

xx  
are the four-dimensional Dirac spinors.

We note that since the dimensions of We are looking for four-dimensional leptonic zero modes. To this end, we consider the conditions under which equation ( 21) obeys the four-dimensional, massless Dirac equations; 00 ( )

() A x  in six dimensions is 5 / 2 m ,
( ) 0 xx              . (22)
Of course, there are very massive Kaluza Klein (KK) modes of masses n/ε. Here we assume that 1/ε ≈14TeV. Under this assumption, these massive KK modes, have a much higher mass and are distinct from the third family of leptons.

For the massless case, the 4 spinors 00 ( ), ( )

xx 
are indistinguishable from the four- dimensional point of view, and we can write 00 ( )

( ) xx   
. Insertingly, introducing (21) and ( 22) into (20) converts the bulk Dirac equation into:

0 0 () cot 0 ()                    . ( 23
)
The solutions of equation ( 23) equation are:

00 00 ( ) , ( ) sin sin AB       , (25)
where A 0 and B 0 are integration constants with the dimension of mass. The normalisable modes are those for which:

  6 (4) 4 0 0 0 0 gd x g d x          . ( 26
)
In other words, we want the integral over the extra coordinates  and  to equal 1.

Inserting (21), (25) and the determinant ( 9) into (26), the latter requirement gives:

2 * * 0 0 0 0 ( ) 1 A A B B   . (27)
Explicitly, the expressions for the three normalisable 8-spinors ( 21) that solve the sixdimensional Dirac equations ( 20) are:

0 00 2 0 1 ( ) ( ) 2 sin ( ) A A xx B          , ( 28
)
where constants A 0 and B 0 obey relations (27).

Set-Up of the ECS Symmetry in Six-Dimensional Space-Time

In the four-dimensional part of six-dimensional space-time non-regular leptons have the same mass as ordinary third family leptons. Hypothetical non-regular leptons are, a) a zerocharged version of the tau, 0  (1784MeV) and, b) a positive charged version of the tau neutrino,    (0,1eV) . Non-regular leptons may, therefore, be obtained by the swap of electric charges between tau and tau neutrino particles in the six-dimensional part. We call these proposed non-regular leptons electric charge swap (ECS) leptons [19].

ECS leptons have the same mass as ordinary third family leptons. They differ from ordinary third family leptons in lepton numbers ( 1 s L  for ordinary leptons; 1 s L  for ordinary antileptons) and in electric charge (positive or neutral for ordinary leptons; negative or neutral for ordinary antileptons). We hypothesize that ECS leptons are produced from third family leptons when these enter the six-dimensional bulk, loose their 'individuality' and swap their electric charge [19]. Το formulate the electric charge swap between ordinary leptons, we have to look for symmetry that characterizes swap processes in the framework of 2-extra dimensions with compactification scale 14 TeV [19].

We consider the 2-sphere 2 S as a quotient space 2

(2) / (1)

L S SU U  
and express the latter in terms of the new symmetry between the original lepton and the new, ECS lepton doublets.

We do this in the following steps [19]:

First, we observe that both the ordinary lepton doublet

L SU [START_REF] Halzen | Quarks and Leptons: An Introduction Course in Modern Particle Physics[END_REF]. This fundamental representation is given by:

[ , ]

j k jkl l I I i I   (40)
Τhe generators are denoted as:

1 2 ii I   . ( 41
)
where

1 2 3 1 0 0 1 0 ,, 0 1 0 0 1 i i                           (42)
are the isospin versions of Pauli matrices.

The action of the latter on the new leptons states is represented by: 0 10 , 01

L                 . ( 43
)
To link the two distinct sectors, ordinary and ECS leptons, we assume that neither ordinary L nor ECS s L lepton numbers are conserved, while the overall lepton number is conserved obligatorily.

0 overall s L L L    . ( 44
) s LL  , ( ) ( ) 1 s LL       . ( 45
) s LL  , 0 ( ) ( ) 1 s LL    . ( 46
)
The quantum numbers of the new ECS leptons of mass 1784 MeV and 0,1eV respectively, are given in Table 1 [19].

Table.1. Quantum numbers (mass M, weak isospin I, charge Q, hypercharge Y S , Lepton number L S ) of the ECS leptons

0 L  , v   New lepton M I I-z Q Y S, L S v   0,1eV ½ ½ 1 1, -1 0 L  1784MeV ½ -½ 0 1, -1
The next step is to define the group transformation that can account for the swap of electric charges between the tau and tau neutrino particles. The ECS transformation must be derived from a transformation from The quotient space SU (2)/U ( 1) is diffeomorphic to the unit 2-sphere S 2 . Consequently, the swap of the electric charges between the tau and neutrino of tau particles must be an automorphism of the 2-sphere to itself [19].

Here, since the two extra dimensions are endowed with the Fubini-Study [1] metric [START_REF] Fubini | [END_REF], [33], not all Möbius transformations (e.g. dilations and translations) are isometries. Therefore, the automorphism from the 2 (2) / (1) S SU U  to itself, which brings the electric charge swap between the tau and neutrino of tau particles, is given by the isometries that form a proper subgroup of the group of projective linear transformations (3) ECS SO [START_REF] Fubini | [END_REF], [33], [19], which is the isometric group of the unit sphere in three-dimensional real space 3 R . The automophism of the Riemann sphere Ĉ is given by: (2) ECS SU [19]. This group is also differomorphic to the unit 3-sphere S 3 .

() ( ) 2( arg ) ( ) (3)
We regard ordinary and ECS leptons as different electric charge states of the same particle -analogous, that is, to the proton-neutron isotopic pair. Finally, in terms of rotational symmetry between the original lepton and the proposed ECS leptons, the ECS two-extra dimensional sphere 2 3 ECS S  is given by: 33 [1] The round metric of the 2-extra dimensional sphere can be expressed in stereographic coordinates as 22 12 22

() 2 3 ( ) (2) / (1) s ECS ECS Y Y S SU U   [19] (48)
(1 )

dy dy g    
, where The ECS symmetry have been tested in various numbers of lepton families and spacetime dimensions [19]. Furthermore, some properties of lepton families are explained from the provided symmetry, within the framework of superstring theories [20][21][22][23][24]. This local symmetry breaks at energy scale below 14TeV. A symmetry breaking mechanism has also been proposed that can make the unobserved non-regular leptons highly massive [19]. This mechanism may 'explain' the invisibility of ECS leptons in the Large Electron Positron ring (LEP) I, II and neutrino oscillations experiments at energy scales below 14TeV [34][35][36][37], [38][39][40]. The existence of ECS leptons can be tested once the Large Hadron Collider (LHC) becomes operative at 14 TeV energy-scales [41][42][43].

INDUCED ECS LORENTZ TRANSFORMATIONS IN FOUR-DIMENSIONAL SPACE-TIME

Let us first define induced electric charge swap Lorentz transformations (ECS) in fourdimensional space-time. In the six-dimensional space-time, the ECS Mobius group of two extra dimensions Möb( 2) is the group of all orientation-preserving conformal isometries of the two extra dimensional sphere 2 (2) / (1)

L S SU U  
to itself that bring about electrical charge swap between leptons.

If we consider the two extra dimensional conformal sphere as the space of future-pointing rays of the null cone in the four dimensional Minkowski space R 1,3 , then there is an isomorphism of ECS Möb(2) group with the restricted Lorentz group SO + (1,3) of Lorentz transformations in four-dimensional space-time with positive determinant, preserving the direction of time [START_REF] Penrose | Two-spinor calculus and relativistic fields[END_REF]. Therefore, it is the ECS Mobius group of two extra dimensional sphere induced Lorentz transformations that brings about the electric charge swap between leptons in four dimensional space-time. We call this the electric charge swap (ECS) Lorentz group SO + (1,3) ECS of induced (ECS) Lorentz transformation.

Let us first consider the Lorentz group O + (1,3) ECS [START_REF] Misner | Relativity, Groups and Topology[END_REF] with infinitesimal generators J  and the associated Lie algebra given by:

  , J J i g J g J g J g J                 . ( 49
)
The ECS restricted Lorentz group is a subset of the Lorentz group O + (1,3) ECS given by: 0 0( )

(1,3) { (1,3) | det 1, 0} ECS ECS ECS ECS ECS SO O         , ( 50
)
which excludes parity and time-reversal transformations, These are thus considered as separate, discrete operations P and T.

A generic element ECS  of the ECS-Lorentz group is given by exponentiating the generators together with the parameters of the transformation, exp( / 2)

ECS iJ       . ( 51
)
The antisymmetric tensor of parameters   is given by the rotations that produce electric charge swap between leptons, ECS-rotations   52) ensures that the effect of ECS physics at high energy scales (close to the 2extra dimensions compactification scale: (M c =14TeV), on the physics at the low energy scale E in the four-dimensional part of six-dimensional space-time, is vanished.

THE ECS POINCARE GROUP IN FOUR-DIMENSIONAL SPACE-TIME

Adding to the ECS Lorentz group space-time translations that bring the electric charge swap between leptons to the set of (homogeneous) ECS-Lorentz transformations, we get:

( , ) () ECS ECS a ECS ECS x x a          , (53)
where 4 ECS a   . The Poincare group [START_REF] Misner | Relativity, Groups and Topology[END_REF] can thus be also called the inhomogeneous Lorentz group. Here the proper,or restricted Poincare group [START_REF] O'connor | General relativity[END_REF][START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF] ECS ISO  is a subgroup of the full Poincare group that contains only proper orthochronous ECS-Lorentz transformations. This Poincare group is given by a semidirect product, 4 [START_REF] O'connor | General relativity[END_REF][START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF] (1,3)

ECS ECS ISO SO   . (54)
The ECS-translation subgroup

4     0, ECS a is a normal subgroup, since equation 11 ( , ) (0, )( , ) (0,
)

ECS ECS ECS ECS ECS ECS ECS a a a a          
is still a translation for any ECS-Poincare transformation ( , )

ECS ECS a   . The Lorentz subgroup     ,0 ECS  , however, is not normal, because 1 1 1 1 ( , ) ( ,0)( , ) ( ,( ) )
ECS ECS ECS ECS ECS ECS ECS ECS ECS ECS ECS ECS a a a                         , (55)
which is not a homogeneous ECS-Lorentz transformation.

When only one of the factors in the group product is normal, one speaks of a semidirect group product  , instead of a direct group product  . To generate ECS-translations, one has to add a new generator P  to the Lorentz algebra Eq. ( 49) to form the Poincare algebra:

    ,, , 0,
,

J J i g J g J g J g J PP P J i g P g P                              . (56)
A correct relativistic quantum field theory has to be not only ECS -Lorentz covariant, but ECS-Poincare covariant as well.

ECS LEPTONS CURVATURE EFFECTS IN FOUR-DIMENSIONAL SPACE-TIME

In this section we discuss the possibility that ECS-transformations between leptons could cause space-time curvature effects and the implications of such effects for a possible ECS gravity like theory.

Let us first consider an electron and the electron neutrino in the presence of an electromagnetic field. The electron is accelerating but the electron neutrino is not influenced by the electromagnetic field. We suppose that, in an arbitrary high scale of energy, electron and electron neutrino swap their charge by ECS-Lorentz transformation. After this ECS-Lorentz transformation, the electron stops being influenced by the electromagnetic field, whereas the electron neutrino begins to accelerate as a result of the field influence.

To illustrate the implications of this, let us consider Newton's Second Law, which relates the force exerted on an object to the acceleration it undergoes by setting them proportional to each other, with the constant of proportionality being the inertial mass m in :

in ECS ECS f m a   (57)
The inertial mass clearly has a universal character, related to the resistance you feel when you try to push on the object; it is the same constant no matter what kind of force is being exerted.

We also have the law of electromagnetism, which states that the electric force exerted on leptons due to swapping of their electric charge is proportional to the gradient of a scalar field  . The constant of proportionality in this case is called the swapped electric charge () Here, the swapped electric charge ()

ECS e 
is the ordinary electric charge (e ± ) which has been swapped between the leptons.

In an arbitrary high (close to the compactification) scale of energy, we may assume that:

em ECS a    (59)
Equations ( 57), ( 58), (59) yield to the swapped electric charge and inertial mass equivalent ()

ECS in em  
. Einstein shows us that, because of the empirical fact that inertial (m in ) and gravitational mass (m g ) are equal, an experiment cannot distinguish between an accelerated laboratory or a fixed laboratory in the presence of a gravitational field. Doing experiments in a free falling laboratory in the presence of a gravitational field is, therefore, equivalent to doing them in a laboratory in the absence of gravity [START_REF] O'connor | General relativity[END_REF]. We call this the Einstein's principle of equivalence (EPE).

Based on the above, in an arbitrary high (close to the extra dimensional compactification) scale of energy in four-dimensional space-time, no possible experiment could distinguish between an accelerated laboratory or a fixed laboratory in the presence of an ECS-field. Here, an ECS-field is defined as the electrically charged field which mediates the ECS exchange interaction between families of leptons [19]. We extend the EPE into the case ECS symmetry and call this ECSEPE This ECS-Einstein's principle of equivalence (ECSEPE) can be defined in the following way:

Any experiment in four-dimensional space-time in high (close to extra dimensional compactification) energy scale, performed locally in an accelerated frame of reference in the presence of an ECS field will obtain the same results as the same experiment in fourdimensional space-time and at the same high energy scale performed in an inertial frame in the absence of ECS-field. Here, the term 'inertial' refers to the relativistic variant of Newton's inertial frame. Relativistic inertial frames are related by ECS-Poincare transformations as mentioned.

The proposed ECSEPE can also be translated into the statement that, due to electric charge swap between leptons, for any curved space-time, its local tangent space is always a flat space-time with Minkowski metric.

Of course, in low (well below the extra dimensional compactification) energy scales ECSEPE is strongly violated.

SL (2,C) ECS GAUGE THEORY OF WEAK ECS CURVATURE EFFECTS

Following K.P Sinha 1984 [START_REF] Sinha Pramana | [END_REF], in this section we propose an SL(2,C) gauge theory formulation of Einstein type theory of weak ECS curvature effects using the tetrad formalism (Sivaram and Sinha 1975[46]; Dennis and Huang 1977 [47].

SL(2,C) ECS group is homomorphic to the proper ECS-Lorentz group. The tetrad formalism is ideally suited to display the gauge aspect of the theory. The special feature of the formalism is invariance under tetrad ECS-rotation. This corresponds to invariance under the gauge group SL(2,C) ECS , and is analogous to the invariance of the fields under local ECSisotopic spin rotation in SU(2) ECS [START_REF]Koorambas Quark-quark of swap electric charge bound state hal-00852409v2[END_REF].

In tetrad formalism four reference vectors are constituted at each space-time point, in addition to the four-coordinate system. We denote the tetrad by:

a t  ,( 0,1,2,3 a
The relationship between the Lorentz metric (1, 1, 1, 1) ab diagonal      and the metric f  of curved space-time due to electric charge swap between leptons is given by:

ab ab f t t      . ( 62
)
The constant Dirac matrices (on flat space-time) satisfy the relation:

f          . (65)
The a  Dirac matrices provide a representation,by the following expression

1 ( ) ( ) a a b b ECS ECS L S L S L    , ( 66
)
where S is the space-time dependent spinor representation of the tetrad ECS-rotation L ECS . the homomorphism ()

a b ECS L S L  (67)
is thus defined.

The transformation matrix S can be chosen as    is the Lagrangian density of matter field (scalar, vector or spinor). In the action (75) the parameter a is given by the equation (52) To proceed further, we use a modified invariance principle: action (75) must be invariant under the group of general coordinate transformations and, also, under the ECS-Lorentz group transformation at each space-time point (tetrad ECS rotation). As a consequence, the fields in question will be scalars or tensors under general coordinate transformation and scalars, tensors or spinors under the ECS-Lorentz group transformation. Furthermore, we must consider C  to be an ECS-curvature tensor with second derivative of a t  . On varying the action with respect to a t  , (75) then yields the equation:

                           (79)
which can also be expressed as:

1 1 2 ECS H f H g t J       . ( 80
)
This bears resemblance to Einstein's equation. The ECS-curvature tensor and J  , nonetheless, are not necessarily symmetric. The proposed SL(2,C) ECS gauge theory of weak space-time curvature effects is mediated by an electrically charged massless spin-2 (ECS-graviton).

CONCLUSION

In the six-dimensional space-time, we propose a new kind of induced 4-dimensional space-time curvature effects, which can be produced by electric charge swap (ECS) transformations between families of leptons that produce non-regular leptons (ECS-leptons) at high energy scale. We find that lepton ECS-transformations form the proper Lorentz group SO + (1,3) (ECS) . We show that Einstein-type theory of weak space-time effects in four-dimensions can be formulated by the SL(2,C) (ECS) gauge group by using ECS-tetrad formalism. The latter yields to electric charged massless spin-2 (ECS-graviton), which mediates the ECS exchange interaction between families of leptons. This proposition may have far reaching applications in astrophysics and cosmology.

This 2 -

 2 surface is attached to the brane at point 0   . When  changes from 0 to π, therefore, the geodesic distance into the extra dimensions shifts from the north to the south pole of the 2-spheroid. For 1 b  in equation (

22 ( ) 1

 221 sin ( / 2) cos ( / 2).

  six-dimensional spinor has eight components and is equivalent to a pair of fourdimensional Dirac spinors ,.

m and 3/ 2 m

 2 , respectively.

2

 2

  ,

	Rot	PSU 	SO	ECS
	ECS	Ch e
						(47)
	2 ˆS     ,
	where ˆis the extended complex plane,	PSU	2( arg ) Ch e	is the proper subgroup of the
	projective linear transformations, and swap symmetry,	SO	() (3) ECS	is the group of rotations
	in three-dimensional vector space	3 R . The universal cover of	SO	() (3) ECS	is the special
	unitary group	()

  

				for tetrad ECS-components)	(60)
		0,1,2,3	for local coordinate indices)
	and their inverses a t  which satisfy
	a  t t a 	,   a t t   b   . b a

  .e. the derivate, is not a spinor. Hence we introduce the SL(2,C) ECS covariant derivative: Relations (74) is similar to those for Yang-Mills fields under ECS-isospin SU(2) ECS transformation[START_REF]Koorambas Quark-quark of swap electric charge bound state hal-00852409v2[END_REF]. In the ECS tetrad formalism the Einstein-Hilbert type action becomes:

		 	( ) x	(     S  		S	) 	,	(73)
	D  	( ) x	    	2	ECS g C  	.	(74)
	 A a d x f C t t 4 ( ) a b     		ab		2	g	ECS m 		,	(75)
	where
		det( ) ft  	,			(76)
							a
	S		exp	1	ig 	ab	,	(68)
							ECS ab
						4
	and
		m
	where	g				represents the ECS coupling constant,
			ECS
		ab		, a b      i	.		(69)
					2
	The gauge potential and the electrically charged ECS-gauge field for the case at hand are denoted by C  and , ECS C C C ig C C               . (70) The transformation laws for () x   , C  and C  under (68) (i.e. under SL(2,C) ECS are: 1 11 1 , ( ) , . SS C SC S i S S C SC S               (71) Furthermore, either an ECS or ordinary spinor () x   obeys the transformation: ( ) ( ) x S x   (72)
	and remains invariant under coordinate transformation.
	However,

i