Abdenbi Mohand Ousaid
email: mohandousaid@isir.upmc.fr

Sinan Haliyo
email: haliyo@isir.upmc.fr

Stephane Regnier
email: regnier@isir.upmc.fr

Vincent Hayward
email: hayward@isir.upmc.fr

Stéphane Régnier

Micro-force sensor by active control of a comb-drive

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.

I. INTRODUCTION

Interaction forces play a key role in micro-scale probing, characterization, manipulation, and assembly systems. Their range, i.e dependence on the distance between interacting bodies, and magnitude vary greatly as a function of the intended application. For example in biology, the contractile forces in the mN range from isolated ventricular heart cells (cardiac myocytes) differs in normal or diseased heart [START_REF] Lin | Miniature Heart Cell Force Transducer System Implemented in MEMS Technology[END_REF]. In nanotechnologies, measurement of adhesion forces is required for experimental validation of different models for micro objects interactions, like model of adhesion developed in [START_REF] Alvo | A van der Waals Force-Based Adhesion Model for Micromanipulation[END_REF]. In microscale manipulation, those interaction forces affect greatly gripping and assembly strategies [START_REF] Haliyo | Controlled rolling of micro objects for autonomous micro manipulation[END_REF], [START_REF] Dionnet | Autonomous micromanipulation using a new strategy of accurate release by rolling[END_REF].

Several types of micro and nano force sensors were developed to provide small force measurements. Most are based on a passive and indirect measurement, i.e through the deformation or displacement of a sensitive part with a calibrated stiffness. Most designs are based on elastic microstructures such as microcantilevers [START_REF] Sepaniak | Microcantilever transducers: A new approach in sensor technology[END_REF], piezoresistive sensors using the variation of resistance on a piezoelectric layerb [START_REF] Arai | 3d micromanipulation system under microscope[END_REF], capacitive sensors based on the variation between two armatures over distance [START_REF] Sun | Mechanical property characterisation of mouse zona pellucida[END_REF], direct piezoelectric sensors through the correlation between their deformation and produced current [START_REF] Li | Novel micro gripping, probing and sensing devices for single-cell surgery[END_REF] Nevertheless, the most widespread approach is the Atomic Force Microscopy (AFM) [START_REF] Mate | Atomicscale friction of a tungsten tip on graphite surface[END_REF] [START_REF] Xie | Enhanced accuracy of force application for AFM nanomanipulation using nonlinear calibration of the optical lever[END_REF]. The deflection of a cantilever of known characteristics is measured by a two or four quadrants photodectectors [START_REF] Rougeot | Forces analysis for micromanipulation[END_REF] or by interferometry [START_REF] Kato | Force-balancing microforce sensor with an optical-fiber interferometer[END_REF]). The applied force is then deduced using the measured deflection and the stiffness of the cantilever.

Despite their popularity, passive force measurements with cantilever-based probes have serious limitations. First, cantilevers are hard to calibrate. Geometry-dependent probe/sample interactions cause the occurrence of a combination of bending and torsion modes that makes data interpretation difficult. AFMs, because of the Bernoulli assumption, also suffer from a restricted dynamic range when high accuracy is required. Other limitations comes from several non-linear phenomena, such as hysteresis and input saturation on optical levers. Perhaps the greatest limitation of the cantilevered beam is the unavoidable trade-off that must be made between stiffness, sensitivity, and structural response characteristics.

The concept of active force measurement principle or force balancing is introduced to overcome these limitation of passive systems, such as used in some accelerometers [START_REF] Chau | An integrated force-balanced capacitive accelerometer for low-G applications[END_REF] and pressure sensors [START_REF] Gogoi | A Low-voltage Force Balanced Pressure Sensor with Hermetically Sealed Servomechanism[END_REF]. In those devices, an actuator is integrated to keep the moving part of the sensor in equilibrium position. In direct force sensing, an active sensors where the applied force is balanced using an electrostatic micro actuator is recently proposed [START_REF] Sun | Actively servoed multi-axis microforce sensors[END_REF]. Another sensor, based on cantilevered composite beam structure with embedded piezoelectric polyvinylidene fluoride (PVDF) actuating and sensing layers is mentioned [START_REF] Shen | Closed-Loop Optimal control-Enabled Piezoelectric Microforce Sensors[END_REF]. However, this design still carries the trade-off related to the cantilevered beam stiffness, mentioned in the previous paragraph. To deal with this limitation, the application of the active force measurement principle to the sensing approach seems a promising solution [START_REF] Sidobre | Calibrated measurement of the behaviour of mechanical junctions from micrometre to subnanometre scale : the friction force scanner[END_REF] .

An improvement of the design of the device proposed in [START_REF] Sidobre | Calibrated measurement of the behaviour of mechanical junctions from micrometre to subnanometre scale : the friction force scanner[END_REF], along an adapted control scheme, is proposed in this paper. The systems uses electrostatic energy, which can be modulated at electronic speeds, instead of strain energy to act against external interaction forces. The probe can be represented by a lumped parameter system, that is, by rigid body motion, to dissociate the sensor stiffness from the force measurement. The movements of the probes are constrained along one single dimension by a fiber suspension that provides nearly perfect kinematic guidance with very low stiffness and virtually no friction. Precise guidance makes it possible to measure the displacement of the moving probe carrier by an optical lever. The suspended probe carrier includes a differential electrostatic actuator, a comb-drive. This actuator is used to balance the external force through the feedback loop and hence provides direct force measurement over the entire operating range.

II. SYSTEM DESCRIPTION

The proposed force sensor is designed around a moving probe kinematically constrained to a single translation motion, through a glass fiber suspension design, and carrying the moving part of a comb-drive armature.

A. Fiber suspension normal direction

Fig. 1. Five-fiber suspension.

In order to constrain the motion of the probe to a single axis, a suspension with 5 glass fibers is designed, as depicted in Fig. 1. The length of fibers is much higher then their expected deformation. They are held at both extremities with a rocker-and-leaf mechanism allowing to manually adjust their tensions. It is safe to consider that the only possible motion for an object fixed in their middle is in a direction perpendicular to their axis, given that their axial stiffness is much higher then the bending one. Moreover, given the very low deformation-to-length ratio, this bending stiffness can be considered constant.

The 3 vertical fibers effectively constrain the motion of the carrier to planar motions. The 2 horizontal fibers add additional constrains and effectively limit the motion to a single translation. The suspension compliance is tuned by adjusting the tension of the fibers and the tension in the front fibers equalize spurious torque.

An optical lever, consisting of an external laser source, a mirror embedded on the carrier, and an external four quadrant photodiode is used to measure the displacement of the probe. The resulting system is shown in Fig. 2, including fiber fixations and the optical lever.

B. Differential bipolar comb actuator

(2)

(3) (1)

x Fig. 3. Differential bipolar comb actuator.

A bipolar differential electrostatic actuator, called a combdrive, is used in the system to balance the external force. A moving armature ( 1) is included in the carrier, and two fixed armatures ( 2) and ( 3) are mounted on the external frame. Given three voltages v 1 , v 2 , v 3 corresponding to the armatures (1), [START_REF] Alvo | A van der Waals Force-Based Adhesion Model for Micromanipulation[END_REF], and (3), the electrostatic energy is

E = 1 2 (c 11 v 2 1 +c 22 v 2 2 +c 33 v 2 3 +2c 12 v 1 v 2 +2c 13 v 1 v 3 +2c 23 v 2 v 3 ) (1
) where the c i j represent the capacities of armature pairs. Considering x the displacement of the moving armature, the actuator force is

∂ E/∂ x = 1 2 c n (-v 2 2 +v 2 3 -2v 1 v 2 +2v 1 v 3 )
, where c n depends on the size of armatures and their gap, all considered equal. Controlling v 2 , v 3 to be opposite in sign and equal in magnitude, assigning them to a control voltage v n such that v 3 = v n , v 2 = -v n , and fixing v 1 , the force developed by the actuator is

a n = 2c n v 1 v n
Thus, the force provided by the actuator is proportional to v n and v 1 can be used to adjust the actuator constant.

It is important to note that this force does not depend on the position of the probe on the normal direction, as long as the overlapping surface on mobile and fixed armatures is constant. It is also linear in respect to control voltage. This is a considerable benefit over flexible systems with intrinsic dependencies between force and displacement.

C. Mechanical behavior

The dynamic of the system is then well described by a second order differential equation involving all forces applied to the probe,

f = m ẍ -a n -kx -b ẋ ( 2
)
where f is the interaction force between probe and sample, m ẍ is the inertial term, a n is the force provided by the actuator, kx is the force due to suspension compliance, b ẋ is the viscous damping term.

D. Prototype

The meso-scale prototype illustrated in Fig 4 is comprised a frame made of polyoxymethylene plastic (180 mm×180 mm×40 mm, 10 mm thickness) supporting the fiber positioning and tensioning mechanisms and the fixed actuator armatures. The carrier is machined out of magnesium to minimize mass (1.2 g). Its dimensions are 22 mm×8.5 mm×7.4 mm and the gap between fixed and moving armatures of 0.3 mm. The springs are made out pinchbeck. Care was exercised to minimize hysteresis by using long glass fibres (180 mm longer, 0.1 mm diameter) and careful fiber attachment design. The probe is made from pulled glass tube of 1 mm in diameter (Sutter glass puller, PI1000). To measure the displacements of the carrier, a laser (Camero, 4 mW, 635 nm wavelength) and a quadrant photodiode detector (Silicon Sensor, QP5.8-6 SD) are placed behind the frame.

The electrical connection to the moving armature actuator is provided by a tungsten incandescent lamp filament (Fig. 4). The voltages applied to the fixed armatures, v n and -v n , were provided by two high voltage operational amplifiers (±200 V, APEX, PA82J). The reference for each amplifier are digitally set trough a DAC/ADC card (National Instrument, PCI-6259) and the control algorithm is implemented on a PC running RTAI real-time Linux kernel. The sampling rate is set to 1 kHz.

III. CALIBRATION OF THE SENSOR

The calibration of the system as a force sensor requires three parameters: the optical lever calibration constant, the stiffness of the suspension, and the electromechanical constant of the actuator. Moreover, dynamic properties of the sensor are investigated by experimental identification.

A. Optical lever

B. Stiffness

The suspension stiffness is found by measuring the natural frequency of the mass-spring system and the known mass of the carrier. For the particular tuning used in the experiments described in the Section V, the measured natural frequency is 9.4 Hz. Given that the natural frequency is given by f 0 = 1 2π k/m and the mass is m=1.2 g, the stiffness, k, is 3.83 N/m.

C. Actuator

At rest, the actuator force balances the elastic force of the suspension, thus a n = kk 1 k 2 U. According to this expression the actuator constant is equal to kk 1 k 2 , here k a =40.68 µN/V. The force produced by the actuator is indeed found to be proportional to the input voltage, validating its linear .

D. Identification

As shown in Section II-A, in the absence of interaction force f , the mechanical behavior of the system is described by

m ẍ -a n -kx -b ẋ = 0 (3)
that is a system of order two, with a natural pulsation, w n , and damping coefficient, ζ . The Laplace transfer function is

G(s) = k n w 2 n s 2 + 2ζ w n s + w 2 n , (4)
where k n , w n and ζ are to be defined.

The system is identified from a step response and the values k n = 0.032, ζ = 0.0047, for w n = 64 rad/s, are found (step reference is fixed to 4 volts). Figure 6 compares the identified response to the experimental response of the system, indicating a very good match around 95%.

IV. ACTIVE CONTROL FOR FORCE MEASUREMENTS

With passive sensing, the force is provided by measuring the deformation of a structural element. As the objective here is to detach the measurement from a structural stiffness, an active control approach is applied. It consists in opposing, by an artifice, the value of the force to be measured with an adjustable value of the same or different nature, in the objective of immobilizing the probe at its equilibrium position. In this case, the force measurement is not computed using the signal coming from the displacement sensor, but the value of the opposed artifice used to balance the force.

Controller Mechanical probe

Fig. 7. Active force sensing principle.

Figure 7 illustrates the general principle of this active force measurements method. The interaction force F ext appears as a perturbation to be rejected by the controller in order to maintain the sensitive part of the sensor at its equilibrium position. This compensation provides directly the measure of F ext which is proportional to the control signal u(t). This principle has many advantages :

-maintain the sensitive part of the sensor in linear range, -improve the precision of the micro-force measurements, -increase the force measurements range, -improve the robustness and stability of the sensor,

A. Control scheme for active force sensing

An H ∞ controller is used to implement this scheme. Controller output is the actuator control voltage u(t)(represents the signal v n defined above), and its input is the feedback given by the voltage U h supplied by the photodiode. The output voltage allows to compute the applied external force which is balanced by the controller using actuator calibration constant. The bloc diagram of this closed-loop controller is shown in Fig 7.

Fig. 8 shows the generalized plant of the H ∞ problem. Three weighting functions are used to get adjustable parameters for the control design. Its synthesis requires to find a rational function K(s) to stabilize the closed-loop, by considering (Re f ,e 3 ) as inputs and (e 1 ,r 2 ) as signals to supervise. More details are given in [START_REF] Cao | The Characteristics Analysis of Magnetic Bearing Based on H-infinity Controller[END_REF], [START_REF] Iwasaki | All controllers for the general h ∞ control problem : LMI existence conditions and state-space formulas[END_REF], [START_REF] Gahinet | A linear matrix inequality approach to h ∞ control[END_REF], [START_REF] Glover | All optimal hankel-norm approximations of linear multivariable systems and their l ∞ error bounds[END_REF], [START_REF] Doyle | State space solutions to standard h2 and h1 control problem[END_REF] about H ∞ problem and method used to resolve it.

The weighing functions, W 1 (s) and W 2 (s), respectively applied to the error signal ε and the control signal u(t), specify the performance that the controller should achieve [START_REF] Li | An H ∞ design for aircraft longitudinal flight control[END_REF]. The weighing function W 1 specifies a tracking error of about 1% in the low frequencies, a modulus margin of 0.6, and a bandwidth of 60 rad/s. W 2 specifies the limitations on the Fig. 8. Set up to determine a controller from H ∞ optimal minimization. control signal imposed by the saturation of the amplifiers and a high cut frequency to avoid dynamics not properly modeled. The weighing function W 3 is null.

W 1 (s) = s + 102 1.7s + 0.6 , W 2 (s) = s + 200 0.001s + 20000 (5)
The H ∞ controller, K(s), is then found by minimizing the bound the H ∞ -norm from b and Re f to e 1 and e 2 . It gives the minimum value of 1.17 and a controller K(s) of the fourth order. The resulting open-loop system response is shown in Fig. 9, providing interesting gain and phase margins, at G m =16.6 dB and P m =67. [START_REF] Dionnet | Autonomous micromanipulation using a new strategy of accurate release by rolling[END_REF] The discrete time transfer function K(z) using a Zero-order hold method with sampling period h=1 ms is given below. This function is computable in real-time with reasonable computational effort.

K(z) =

2.232 + 11.65z -29.95z 2 + 16.11z 3 0.0005445 -0.7045z + 2.371z 2 -2.667z 3 + z 4 (7)

V. EXPERIMENTAL VALIDATION OF THE ACTIVE SENSOR

The system is validated on a complex case for microscale force sensing, by measuring the time-course of the interaction of a thin glass probe (40 µm in radius) with a water droplet. Such an interaction, illustrated in Fig. 10, has four main phases, namely 'approach', 'contact', 'retract', and 'contact break'.

A schematic representation of such a theoretical curve is given in Fig. 11, coming from a classical AFM probing of a hard surface. The cycle begins in A and terminates in E, where the probe-surface interaction force varies as a function of the displacement of the probe, Fig. 11(a). The same behavior is represented in Fig. 11(b) where the force is shown as a function of time. This phenomenon highlights the type of interaction that are characteristic of the physics at microscopic scale. Two discontinuous events occur when the probe is attracted by the substrate (pull-in, point A on the plot) and when the contact is broken (pull-off, point E on the plot). Two types of forces are at play, attractive forces dominating in B and D, and repulsive forces dominating in C [START_REF] Bolopion | Analysis of stability and transparency for nanoscale force feedback in bilateral coupling[END_REF], [START_REF] Bolopion | Haptic Teleoperation for 3D Microassembly of Spherical Objects[END_REF]. Please note that this curves are theoretical representations of an interaction on a hard surface. The inherently more complex interaction on a water droplet is chosen here to depict the superior characteristics of the proposed force sensor.

As shown in Fig. 12, the experimental bench is comprised of a motorized xyz micro-positioning stage for coarse motion (Sutter MP-580), an electromagnetic transducer for fine motion, a microscope (Veho-VMS-004), the laser-based optical lever, and the active probe. The bench is placed on a antivibration table.

The drop is approached to the probe, contacted and then retracted to produce an approach-retract cycle. The interaction force between the drop and the probe, during the cycle, is shown in figure 13, as a function of time. Besides, the corresponding plot force according to the relative position of the droplet is shown in figure 14.

As shown in figures 13 and 14, the resulting curves exemplify the characteristics of the approach-retract cycle over a droplet of water. A pull-in effort of 2 µN corresponding to the instant when the drop attracts the probe is observed during the approach stage and a pull-off effort of 7 µN is also observed when the contact is broken during the retract stage. Further, attractive and repulsive forces dominate the interaction. The force observed in the experiment is similar to classical approach-retract curve presented previously. However, in this case the curve is filtered by the nature of the interaction with liquid, but also, by sensor whose bandwidth is about 60 rad/s and the hysteresis ensuing from the transducer's position. This clearly represents the effectiveness and efficiency of the developed active probe system. The tests show that the resolution of the active sensing can reach 0.4 µN. Nevertheless, due to the mass of the carrier (1.2 g) the bandwidth of the sensor is limited to 10 Hz.

VI. DISCUSSION AND CONCLUSIONS

Designed around a glass fiber suspension, a differential bipolar comb actuator and an active force measurement principle, an active micro-force sensor was proposed in this paper. It is distinct from previous designs in several aspects. Actively controlled electrostatic forces are used instead of bending stresses or structural stiffness to balance the force acting on the sample. The suspension is arranged to constrain movements accurately to a single degree of freedom in translation. An electrostatic actuator combined with an optical lever for position feedback through an especially designed controller yields a large range of measurements both in force and displacement. Based on the feedback, the actuator generates an opposite balance force to cancel the action of the interaction forces. The measurement is related to the balanced force which makes the probe remains at equilibrium position. Preliminary experimental results verify the performances of the active micro-force sensor and its effectiveness.

The future work would overcome the limitations of the present prototype with improved fabrication, while retaining the same design principle. An important improvement would be the reduction of the mass of the carrier. To this end, MEMS fabrication techniques could be applied advantageously with drastic reduction in size and mass. The bandwidth of micro-scale active sensor is then expected to reach the kHz range.

Fig. 2 .

 2 photodiode

Fig 3

 3 is a schematic representation of such an actuator.

Fig. 4 .

 4 Fig. 4. Mechanical elements.

Fig. 5

 5 Fig. 5 shows the position of the carrier as a function of the output voltage, U h , of the photodiode and U h as a function of the excitation voltage, U. Experimental data were fitted with straight lines. Calling k 1 and k 2 the calibration constants of the photodiode and of the actuator respectively, led to k 1 =387 µm/V and k 2 =0.0274. The relation between the U h and U is linear as expected.

Fig. 5 .

 5 Photodiode output (V)Excitation voltage (V) 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Fig. 6 .

 6 Fig. 6. Experimental and simulated step response of the prototype.

FrequencyFig. 9 .

 9 Fig. 9. Bode diagram of the open loop of the system.

Fig. 10 .

 10 Fig. 10. Interaction probe-droplet.

Fig. 11 .

 11 Fig. 11. Approach-retract curve. (a) force-position curve. (b) time-force curve.

Fig. 13 .

 13 Fig. 13. Measured approach-retract curve, force vs time.

Fig. 14 .

 14 Fig. 14. Measured approach-retract curve, force vs position.

Fig. 12 .

 12 Photodiode

  deg respectively. + 1.56810 8 s 2 + 7.46810 7 s + 3.86110 11 s 4 + 7516s 3 + 2.57110 6 s 2 + 3.12410 8 s + 1.09910 8

	K(s) =	78.4s 3

ACKNOWLEDGEMENTS

The authors wish to thank Sheng Chao WONG for advise and help on electronic design. This work has been supported by the French National Project ROBOTEX ANR-10-EQPX-44-01.


Daniele Funaro
email: daniele.funaro@unimore.it

Newtonian Forces Exerted by Electromagnetic Waves Traveling into Matter

Keywords: Electrodynamics, Stress-energy tensor, Einstein's equations, EM-Truster

Electromagnetic waves, developing in vacuum or into matter, produce dynamical alterations of the space-time metric. This is a consequence of Einstein's equation, that we are able to solve explicitly in some circumstances. Solutions are in fact obtained by plugging on the right-hand side of the equation some appropriate energy tensors. Hence, the passage of a wave generates both electrodynamics and 'gravitational' (local and temporary) modifications of the molecular lattice of a dielectric. If the wave or the dielectric body are asymmetric, we could theoretically obtain a distribution of Newtonian-like forces with nonzero resultant. This hypothesis suggested a laboratory experiment where an electromagnetic signal applied to a ring with a particular geometry imparts a directional thrust in apparent violation of the action-reaction principle. This test was recently realized with success. Therefore, the present theoretical approach, once appropriately refined, may constitute a crucial referring point for further developments.

Introduction

A number of relatively recent papers analyze the role of general relativity in the description of phenomena happening inside matter. For instance, relations between phonons (or even sound waves) and space-time deformations have been theoretically studied in [START_REF] Sabín | Phonon creation by gravitational waves[END_REF][START_REF] Kornreich | Light induced gravity phonons[END_REF][START_REF] Nicolis | Mutual interactions of phonons, rotons, and gravity[END_REF][START_REF] Esposito | Gravitational mass carried by sound waves[END_REF]. The findings suggest possible applications in various fields, with the aim of taking advantage of the gravitational mass component. In the present paper we disclose the results of a preliminary study concerning the space-time deformations following the passage of an electromagnetic wave traveling into vacuum or a dielectric. This analysis, in part heuristic, 1 has suggested the construction of a device able to generate asymmetric thrust [START_REF] Funaro | An efficient ring-shaped electromagnetic thruster[END_REF]. The prototype has been successfully tested in laboratory, confirming the theoretical expectations. At this point, a further validation of both theory and practice follows parallel paths.

The device tested in [START_REF] Funaro | An efficient ring-shaped electromagnetic thruster[END_REF] is an asymmetric ring supplied by radio-frequency electromagnetic signals, which undergoes a directional shift in violation of the laws of momentum conservation. The device, belonging to the family of the so called EM-thrusters (see for instance the EmDrive [START_REF] Shawyer | Digitally controlled beam former for a spacecraft[END_REF]), may represent a valid propulsion engine in particular in those applications where there is the need to generate unbalanced forces by only relying on electromagnetic sources, thus without the help of moving components, permanent magnets or some kind of fueling. Various explanations are possibly available. For example, momentum is transferred to the quantum vacuum [START_REF] Milonni | The Quantum Vacuum: An Introduction to Quantum Electrodynamics[END_REF][START_REF] Cornille | Advanced Electromagnetism and Vacuum Physics, SCCP[END_REF][START_REF] Puthoff | Advanced space propulsion based on vacuum (spacetime metric) engineering[END_REF][START_REF] Meis | Light and Vacuum[END_REF], a radiation of electromagnetic entity which pervades the universe. More sophisticated approaches rely upon the Unruh effect [START_REF] Cozzella | Proposal for observing the Unruh effect using classical electrodynamics[END_REF][START_REF] Sudhir | Unruh effect of detectors with quantized center of mass[END_REF][START_REF] Šoda | Acceleration-induced effects in stimulated light-matter interactions[END_REF] or the concept of warped space-time [START_REF] Fischer | Warped space-time for phonons moving in a perfect nonrelativistic fluid[END_REF][START_REF] Lobo | Fundamental limitations on warp drive spacetimes[END_REF][START_REF] Lentz | Breaking the warp barrier: hyper-fast solitons in Einstein-Maxwell-plasma theory[END_REF]. As mentioned above, we are trying here to provide the reader with an alternative version of the facts, which is the one that actually inspired the construction of the new working device.

Figure 1: An electromagnetic pulse travels from left to right within the molecular lattice of a dielectric, causing a local deformation at its passage. The picture shows the modified displacements at different times. The signal has enough power and travels at a speed larger than that needed for the recovery reaction of the molecular links. At the end of the process, the initial framework turns out to be slightly shifted backwards. A continuous reiteration of the signal recalls the wakes shaking the body of a shuffling caterpillar.

Action-reaction principle can be put into discussion when taking into account the delay occurring between two events. For example, if our Sun suddenly disappears and we admit that the gravitational information travels at the speed of light, it takes more than 8 minutes before the change will be detected on Earth. During that time, our planet will continue its trajectory along an elliptic orbit, disregarding some basic classical assumptions. In the same way, an electromagnetic wave traveling inside a medium may produce local and temporary variations of the molecular displacement. The inertia of the molecules to regain their initial displacement plays a role where an asymmetry is encountered. Moreover, the uncertainty principle of quantum mechanics, in the form of momentum-position or energy-time, allows for the violation of classical conservation properties under certain restrictive conditions.

The question is how to induce effects as the one outlined in Fig. 1, where, hypothetically, a backward shift of the medium is realized after the passage of a pulse. More specifically, the problem is how to convert electromagnetic strength into Newtonian-like forces. Moreover, we would like to have a non vanishing resultant, able to confer a directional thrust. Therefore, the introduction of some asymmetry in the system becomes a primary concern. Note that, at frequencies of the order of GHz, the wavelengths are measured in centimeters. Thus, there are good margins to be able to work with tools having a manipulable size. Even if the results are negligible for a single pulse, a repetition at a rate of billions times per second may end up in significant outcomes.

The above mentioned laboratory experiment is still waiting to be replicated by other researchers. By the way, its realization was suggested by the same theoretical considerations that we would like here to report. In order to provide a background for a better understanding of the discovery, and suggest how the performances could be improved upon, in the following pages we try to give a formal description of the model equations ruling the phenomenon, as well as their main properties. Due to the technicality of the subject, we are not be able to explain all facts, but we shall limit our analysis to the development of the theoretical foundations and the discussion of some relatively simple examples.

The model equations

We denote by E the electric field, by B the magnetic one, and by V a velocity vector field. In particular, V follows the evolution of the electromagnetic information, which is not necessarily carried by real massive bodies, such as charged particles. For this reason, the equations will also have validity in pure vacuum. In the simplest cases, V maintains the direction of the Poynting's vector E × B. A set of model equations coupling Maxwell's equations with Euler's equation for non viscous fluids can be obtained as follows:

∂ E ∂t = c 2 curl B -ρ V ∂ B ∂t = -curl E div B = 0 (1) ρ µ -1 D V Dt + E + V × B = --1 0 ∇p (2)
with ρ = div E and c is the speed of light. The above equations were introduced and discussed in [START_REF] Funaro | Electromagnetism and the Structure of Matter[END_REF][START_REF] Funaro | From Photons to Atoms[END_REF] and further analyzed in [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF]. Hence, we refer to those publications for comments and explanations. The setting is not dissimilar from the ones reported in [START_REF] Jackson | Classical Electrodynamics[END_REF], chapter 10, within the framework of plasma physics. Analogous formulations are found in Magneto-HydroDynamics (see, e.g., [START_REF] Jeffrey | [END_REF][START_REF] Davidson | An Introduction to Magnetohydrodynamics[END_REF][START_REF]Magnetohydrodynamics: Historical Evolution and Trends[END_REF]). The exception is that we do not introduce any mass density here. Indeed, the model corresponding to (1)-( 2) represents, compatibly with the existing ones and in the respect of the physics canons, the most general way to handle electromagnetic phenomena without involving the presence of massive particles.

The equations actually derive from a very general classical energy-stress tensor (see ( 9)). The model has been built to describe phenomena in vacuum, and it is naively applied to dielectrics. In truth, the description for the momentum of light in a medium is a serious issue that requires more attention, since it has been subject of different interpretations in the past [START_REF] Barnett | Resolution of the Abraham-Minkowski dilemma[END_REF]. We recall some useful facts. The first equation in [START_REF] Sabín | Phonon creation by gravitational waves[END_REF] is the Ampère's law and the second one the Faraday's equation. As usual, the term D V /Dt denotes the substantial derivative and 0 is the dielectric constant in vacuum. The scalar p is a potential denoting pressure density per unit of surface, that, differently from fluid dynamics, can also attain negative values. The term E + V × B recalls Lorentz's force. Finally, the constant µ is dimensionally equivalent to Coulomb/Kg. An estimation of µ is provided in [START_REF] Funaro | From Photons to Atoms[END_REF], appendix H, in a specific circumstance. The most important difference is that density of mass does not appear in the equations and the link between electromagnetic and Newtoniantype forces is represented by the unknown p.

The above set of equations extends the classical Maxwell's model in vacuum. Indeed, by imposing ρ = 0, we get:

∂ E ∂t = c 2 curl B ∂ B ∂t = -curl E div E = 0 div B = 0 (3)
Another significant case is when D V /Dt = 0 and p = 0, but ρ = 0. In such a circumstance, we get:

∂ E ∂t = c 2 curl B -ρ V ∂ B ∂t = -curl E ρ( E + B × V ) = 0 div B = 0 (4)
According to [START_REF] Funaro | Electromagnetism and the Structure of Matter[END_REF], the solutions of (4) (that include those of (3)) are named free waves and consist of electromagnetic phenomena that evolve according to the rules of geometrical optics. Indeed, as far as ( 4) is concerned, it turns out that the vector field V must be constant and orthogonal to both E and B (i.e., it is lined up with the Poynting's vector). The relation | V | = c, stating that the velocity of propagation of the signal equals that of light, also becomes the eikonal equation [START_REF] Born | Principles of Optics[END_REF], which actually rules the propagation of fronts in classical optics. As pointed out in [START_REF] Funaro | Electromagnetism and the Structure of Matter[END_REF][START_REF] Funaro | From Photons to Atoms[END_REF] there is plenty of these waves that do not satisfy (3), justifying the introduction of the extended model. By applying the divergence operator to the first equation in (4), the following continuity equation is automatically derived:

∂ρ ∂t = -div(ρ V ) (5)
Note that the term ρ V is not necessarily a classical current, but is related to a kind of density charge accompanying the evolution of a wave. The reason for assuming that ρ can attain values different from zero also in vacuum is explained for instance in [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF].

The equations ( 1)-( 2) can be expressed in relativistic form, by using 4vectors and introducing suitable tensors. As usual, in the system of coordinates (ct, x, y, z), we denote by F αβ the electromagnetic tensor (see, e.g., [START_REF] Fock | The Theory of Space, Time and Gravitation[END_REF][START_REF] Misner | [END_REF] or other typical text on general relativity). Within the space-time metric denoted by g αβ , we consider the electromagnetic stress tensor:

U αβ = -g γδ F αγ F βδ + 1 4 g αβ F γδ F γδ (6)
Afterwards, by introducing a velocity 4-vector V , we define a mass tensor M αβ that recalls that of a dust of particles for a perfect fluid:

M αβ = µ -1 ρ V α V β + -1 0 g αβ Π αβ (7)
with Π αβ = diag(E, -p, -p, -p), where the scalar E denotes an energy density per unit of volume and p our pressure density per unit of surface. The introduction of the constant µ is justified by the observation that ρ is a density of charge per unit of volume, and not a density of mass, as in standard mechanics.

In this way, we remain within a pure electromagnetic context. Inside matter, the information will not be transported by massive particles (such as electrons) but develops by means of compression and rarefaction waves, as a kind of sound wave. Similar consideration may be applied when modeling the behavior of phonons [START_REF] Brüesch | Phonons: Theory and Experiments II, Experiments and Interpretation of Experimental Results[END_REF].

By putting together the tensors so far examined, we arrive at a global stressenergy tensor:

T αβ = 0 (U αβ -M αβ ) (8)
The covariant version of the the model equations is obtained from requiring:

∇ β T αβ = 0 α = 0, 1, 2, 3 (9)
By specializing the expression in a Minkowski metric, we finally get (1)-( 2). In order to make this exposition more fluent, we omit the details of this computation. We also get the following equation:

∂E ∂t = -0 c div( E × B) + ρ E • V (10)
where

E = 1 2 0 (| E| 2 + c 2 | B| 2
). This is akin to the Poynting's theorem and says that the local time variation of energy follows the evolution of the vector product E × B and also depends on the scalar product E • V (which is zero for a free wave).

Some interesting solutions

In Cartesian coordinates (x, y, z), the electric and magnetic fields of a standard (suitably polarized) plane wave traveling in the direction of the x-axis can be represented as follows:

E = (E 1 , E 2 , E 3 ) = (0, f (ξ), 0) B = (B 1 , B 2 , B 3 ) = (0, 0, f (ξ)/c) ( 11
)
where ξ = x -ct and f is an arbitrary function. In this circumstance we have ρ = 0, so that we are in the usual Maxwell's setting in vacuum. More complex plane waves, where the function f also depends on y and z can be considered. In these cases it is necessary to use the modified model ( 4). The reader interested to this extension is addressed to [START_REF] Funaro | Electromagnetism and the Structure of Matter[END_REF][START_REF] Funaro | From Photons to Atoms[END_REF] for more clarifications. Here, for the sake of simplicity, we will not use such an option. The full set of equations ( 1)-( 2) also allows for solutions where a component of the electric field is lined up with V (that implies E • V = 0). Electromagnetic waves having a component in the direction of motion actually exist when traveling within a dielectric. In this case, a viable solution is:

E = (h(ξ), f (ξ), 0) B = (0, 0, f (ξ)/c) V = (V 1 , V 2 , V 3 ) = (c, 0, 0) (12)
where h is another arbitrary function. We now have ρ = h (ξ), with the prime denoting derivation with respect to ξ. Moreover, we get:

p = - 1 2 0 h 2 ∇p = -0 (hh , 0, 0) (13)
Of course, we return to the expression in [START_REF] Cozzella | Proposal for observing the Unruh effect using classical electrodynamics[END_REF] when h = 0. Note that the pressure has negative signature. It is worthwhile to observe that, when f = 0 (so that B = 0), the constant c in (1)-( 2) is multiplied by zero. This means that in principle a pure longitudinal wave is free to travel at any speed.

The last example teaches us that in order to create pressure we need two facts: there must be a divergence ρ different from zero (this also emerges from examining the left-hand side of (2)), there must be a component of E along the direction of propagation. We return to this issue in section 4.

We would like to plug the tensor (8) on the right-hand side of Einstein's equations. This means that we have to solve:

G αβ = R αβ -1 2 g αβ R = χT αβ ( 14
)
where R αβ is the Ricci's curvature tensor and R = g αβ R αβ the scalar curvature.

On the right-hand side a constant χ appears, that is not in relation with the gravitational constant. The dimension of χ is meters/joules. An estimate of such a constant in a very special circumstance has been given in [START_REF] Funaro | From Photons to Atoms[END_REF], appendix E. The question of providing a value to χ is delicate. We expect that χ is many orders of magnitude greater than the corresponding constant related to gravitation. Indeed, inside matter, masses are concentrated in the nuclei whose volume size is 10 -15 times smaller that of the corresponding atoms. The total mass is negligible from the classical gravitational viewpoint. The corresponding density of mass, taking into account the relevant spacing between nuclei, is also extremely small. Gravitational relativity refers to masses as a whole, without considering that the distribution is concentrated in tiny peaks. On the other hand, an enormous amount of free energy is circulating between atoms, with magnetic fields of the order of Teslas. The fields tend to cancel each other on the average, presenting a piece of matter like an almost neutral object, thus hiding the large energies that keep atoms together. This means that the role of χ in this context must be rethought. The constant will also depend on the material under consideration. We are unable in this short paper to provide a full theory, but the question is certainly of interest and should be further investigated, also in view of the observations that will be made in section 4.

Our analysis is conducted at a pure electromagnetic level. Hence, masses are not to be taken into account. When we talk about 'deformation of the space-time' we mean that this is due to some form of energy of electromagnetic nature which does not come from the presence of massive bodies or black holes. We are going to show that an electromagnetic wave will carry within itself a gravitational-type wave, i.e., a local time-dependent metric with nonzero curvature. Here the name 'gravity' should not be associated with the standard terminology. For instance, speaking about phonons, in [START_REF] Esposito | Gravitational mass carried by sound waves[END_REF] the authors observe that they actually carry mass, which is not due to the usual mass-energy conversion, since the effect survives in the nonrelativistic limit. Moreover, they remark that the effect is not of quantum type. Just to have an idea, the ionization energy of the Hydrogen atom is of the order of 10 -18 J (more exactly 13.6 eV). The gravitational interaction would be only of order 10 -56 J. Obviously, the description of the space-time metric inside matter cannot be a direct descendant of gravitational type phenomena at large scales.

Let us examine the first case, relative to the solution in [START_REF] Cozzella | Proposal for observing the Unruh effect using classical electrodynamics[END_REF]. Since E 1 = E 3 = B 1 = B 2 = 0, our tensors are rather simple. We anticipate that only U αβ will be necessary in this case. We then look for a metric of the form:

(ds) 2 = (c dt) 2 -(dx) 2 -(σ(ξ) dy) 2 -(dz) 2 (15)
where the function σ is the unknown to be determined. In other words, we have to find g αβ = diag(c 2 , -1, -σ 2 , -1). In the special case we are examining, the non-vanishing entries of the electromagnetic tensor F αβ are:

F 02 = E 2 = f , F 20 = -E 2 = -f , F 12 = -cB 3 = -f , F 21 = cB 3 = f .
Correspondingly, the nonzero entries of the electromagnetic stress tensor U αβ are:

U 00 = U 11 = (f /σ) 2 U 01 = U 10 = -(f /σ) 2 (16)
The next step is to evaluate the curvature tensor. Starting from [START_REF] Lobo | Fundamental limitations on warp drive spacetimes[END_REF], the Christoffel's symbols are:

Γ 0 22 = Γ 1 22 = -σ σ Γ 2 02 = Γ 2 20 = -σ /σ Γ 2 12 = Γ 2 21 = σ /σ (17)
Subsequently, we get:

G 00 = G 11 = -σ /σ G 01 = G 10 = σ /σ R = 0 (18)
The above outputs have been evaluated with the help of an algebraic manipulation software. Finally, substituting into ( 14) we must have:

-σ σ = χ 0 f 2 (19)
which is a simple nonlinear second-order equation (see also [START_REF] Funaro | From Photons to Atoms[END_REF], p. 248). Comments about its resolution will be given later in this section.

Let us now consider the electromagnetic displacement [START_REF] Sudhir | Unruh effect of detectors with quantized center of mass[END_REF]. For simplicity, we only treat the case when f = 0. This time we look for a metric of the following form:

(ds) 2 = (c dt) 2 -(σ(ξ) dx) 2 -(dy) 2 -(dz) 2 (20)
corresponding to g αβ = diag(c 2 , -σ 2 , -1, -1). The nonzero entries of the electromagnetic tensors are (recall that now B is zero):

F 10 = -h F 01 = h U 00 = U 22 = U 33 = 1 2 (h/σ) 2 U 11 = -1 2 h 2 (21)
The Christoffel's symbols are:

Γ 0 11 = -σ σ Γ 1 01 = Γ 1 10 = -σ /σ Γ 1 11 = σ /σ (22)
As far as the curvature is concerned, we get:

G 22 = G 33 = -σ /σ R = -2σ /σ (23)
Let us solve the Einstein's equation. It turns out that only the pressure part of the mass tensor M αβ in ( 7) is needed. We end up with the following relations:

-σ σ = χ 0 h 2 E = 1 2 0 h 2 /σ 2 p = -1 2 0 h 2 /σ 2 (24)
Again we get equation [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF]. The scalar curvature R is not zero and is proportional to p. This can be also checked by taking the trace of ( 14) and recalling that the trace of G αβ is equal to -R and that of U αβ is zero. Thus, we obtain:

R = χ(E -3p) = 2χ 0 h 2 /σ 2 = -4χp (25)
Let us now approach the solution of the first equation in [START_REF] Barnett | Resolution of the Abraham-Minkowski dilemma[END_REF]. If the right-hand side is h(ξ) = sin(ωξ), for some ω > 0, it is easy to check that: σ(ξ) = ( √ χ 0 /ω) sin(ωξ). This immediately says that the magnitude of the gravitational deformation is inversely proportional to the frequency ω, which is quite an interesting result. Moreover, we get:

p = -ω 2 /2χ R = 2ω 2 (26)
Thus, in presence of a longitudinal sinus wave, the scalar curvature is a constant only depending on the parameter ω.

Therefore, there might be conflicts with classical gravitational fields (the one relative to Earth, for instance), suggesting how to build tools for increasing or screening gravitational forces.

As far as different forcing terms h are concerned, the evaluation of σ must be done numerically. We show in Fig. 2 the plot of σ when h is a sawtooth signal. We imposed the boundary conditions σ(0) = σ(1) = 0. The corresponding σ is asymmetric, with σ and σ growing at ξ = 1 as the slope of h on the right becomes steeper. This means that, if we prolong the plot periodically, the junctions are not smooth. The alteration of the space-time is continuous, but presents asperities. We recall that ξ = x -ct. An asymmetric signal shifting at speed c (not that of light) inside a molecular lattice (see Fig. 1) produces slight deformations at a local level. These are certainly electrodynamics effects acting on chemical bonds, but they could be also interpreted as microscopic deviations due to a flow of 'immaterial masses' as a consequence of a gravitational type wave traveling inside the body. This wave does not necessarily develop at the speed of light and may have a sensible intensity. In fact, for the reasons specified at the beginning of this section, the constant χ in ( 14) can be relevantly bigger in comparison to the classical gravitational case. Note that the pressure gradient (where p is given in [START_REF] Barnett | Resolution of the Abraham-Minkowski dilemma[END_REF]) is oriented along the longitudinal direction. Note instead that p = 0 in those cases when the electric field has no longitudinal component. We are reminded that ∇p represents our link between forces of electrodynamical nature and forces of mechanical type. Here, we are not converting energy into matter in order to take advantage of gravitational forces. As it is known, this would give negligible outcomes. The idea is to convert electromagnetic energy directly into mass density, forgetting the intermediate passage. The trick is to stimulate asymmetrically the large electromagnetic energy hidden inside matter, having fields that tend to cancel each other, with the aim to extract a 'positive' resultant.

The phenomenon is certainly nonlinear and the reaction of the medium may be different if the signal imparted has a very slow growth followed by a fast damping. If this is true, we would entail that a periodic signal carrying an asymmetric time behavior could generate mechanical forces with nonzero resultant. Indeed, by sitting a wheelchair without the feet touching the ground, we can impart neat acceleration in one direction by moving our body back and forth, each way with a different velocity law. In the case under discussion, we would take advantage of a sort of internal 'friction' hidden in the interatomic gaps. Heuristically, this phenomenon may be a consequence of the fact that the molecules take a finite fraction of time to regain their relative position after the passage of a pulse.

If the above argument is not convincing, and we want to stick to the classical rules on momentum conservation, we may say that the backward recoil shown in Fig. 1 is due to the forward ejection of a 'massive object' of electromagnetic type. Such an emission is totally different from waves generated by an antenna, that, with the exception of a brief transient near the source, carry a transversal signal satisfying the laws of geometrical optics (see the solutions in [START_REF] Cozzella | Proposal for observing the Unruh effect using classical electrodynamics[END_REF]). Waves of this last type have p = 0 (so they do not carry mass). They only exert very mild mechanical forces (radiation pressure) when interacting with matter (a survey on theory and applications is provided in [START_REF] Cohen-Tannoudji | Manipulating atoms with photons[END_REF]). For this reason, we do not expect a significant recoil from a classical antenna as a result of energy emission. As previously observed, the ejected mass is not comparable to that attributed to concrete objects (made of an enormous amount of molecules) and successively converted into gravitational forces of very small magnitude. We are dealing with a new interpretation of relativistic phenomena, where the constants involved are far more large. The electric energy supplied to the system is partly transformed in mass density (without masses) moving inside the material and following well-known rules of conservation. A power of 1 Watt is easily realized in an electric circuit and is a relevant quantity in classical mechanics. Even if such an energy conversion was highly inefficient there could still be margins for exploitation.

It is not straightforward however to build a device able to expel longitudinal signals. This is not trivial, at least, along rectilinear paths. The experiment becomes feasible if we opt for circular type paths. The discussion will follow in the next section.

Rotating electromagnetic waves

The set (1)-( 2) also admits interesting solutions rotating about the axis of a cylinder. These are discussed in detail in [START_REF] Funaro | Electromagnetism and the Structure of Matter[END_REF][START_REF] Funaro | From Photons to Atoms[END_REF]. We report here a simplified version. We work in the system of coordinates (r, φ, z). For an integer k ≥ 1 and arbitrary constants ω > 0 we have for any z:

E = Qr ω + kJ k (ωr) ωr cos(cωt -kφ), J k (ωr) sin(cωt -kφ), 0 (27)
B = 1 c 0, 0, - Qr 2 k -J k (ωr) cos(cωt -kφ) ( 28
)
where Q is another constant and J k denotes the k-th Bessel function of the first kind, that satisfies:

J k (ξ) + J k (ξ) ξ - k 2 J k ξ) ξ 2 + J k (ξ) = 0 ( 29
)
We may take r between 0 and the first zero of J k (denoted by δ k ), in order to actually obtain a solution defined on a disk (for any fixed z). The radial part of the stationary electric field, i.e. (Qr/ω, 0, 0), is the one corresponding to a uniformly charged dielectric. The stationary part of B, i.e. (0, 0, -Qr 2 /ck), is the one generated inside a uniformly charged rotating cylinder (though there is no physical rotation of the body).

The displacement of the electric field for Q = 0 and k = 1 is shown in Fig. 3 at a fixed time t. The magnetic field is perpendicular to the page. As t evolves the picture rotates with constant angular velocity determined by the parameter ω. For r = δ 1 , we have B = 0, whereas E is tangential to the cylinder surface (E 1 = E 3 = 0). Always for Q = 0, the electromagnetic fields satisfy Maxwell's equations ( 3), since we have ρ = 0 in this case.

As Q = 0, we need to use the full set of equations ( 1)-( 2). In fact, we have that ρ = div E = 2Q/ω is constant and different from zero. We now define V = (0, cωr/k, 0). This velocity field describes a uniform angular rotation about the z-axis. As a consequence we get that D V /Dt = (-c 2 ω 2 r/k 2 , 0, 0) is radial and points towards the center. Moreover, we start having p = 0. Indeed, we obtain the expression (up to an additive constant):

p = 0 c 2 ωQ r 2 µk 2 - Q 2 r 2 ω 2 1 - ω 2 r 2 2k 2 - 2Q r kω J k (ωr) cos(cωt -kφ) (30)
Note that with this choice p is zero for r = 0. The check that ( 27)-( 28)-( 30) are actually solutions is straightforward. It is enough to compute partial derivatives and substitute into the corresponding equations. Note that (29) must be used several times. From [START_REF] Misner | [END_REF] we observe that E has the component E 1 transversal to the direction of rotation (as in a classical plane wave in vacuum) and also a longitudinal one, i.e., E 2 . The situation is similar to that illustrated in [START_REF] Sudhir | Unruh effect of detectors with quantized center of mass[END_REF]. Unfortunately, the solution of Einstein's equations is rather complicated now, therefore we do not have results to show. Explicit solutions are extremely rare, and those proposed in the previous section are the result of an intensive study using algebraic manipulation. Numerical simulations aimed at approximating periodic solutions of Einstein's equation are rather challenging. The problem may be approached with techniques similar to those used for the study of rotating black-holes. Some scattered references in this field are for instance [START_REF] Price | Colliding black holes: The close limit[END_REF][START_REF] Brügmann | Binary black hole mergers in 3d numerical relativity[END_REF][START_REF] Pretorius | Evolution of binary black hole spacetimes[END_REF]. The case of a rotating cylinder with anisotropic fluid has been also treated [START_REF] Bolokhov | Rotating cylinders with anisotropic fluids in general relativity[END_REF] in the framework of general relativity. The metric in the case of the electromagnetic fields in ( 27)-( 28) is not as simple as in [START_REF] Lobo | Fundamental limitations on warp drive spacetimes[END_REF] or [START_REF] Jackson | Classical Electrodynamics[END_REF]. For sure, the term g 02 = g 20 , coupling time with the rotation angle φ, is activated. Our guess is that, due to the complicated structure of the right-hand side T αβ , the term g 01 = g 10 is also going to be different from zero, making the resolution of Einstein's equation very tough. Figure 3: Electric field for k = 1 and Q = 0 on the disk of radius δ1 (left). The magnetic field is orthogonal, so that the whole displacement is defined on a cylinder of infinite length. The system rotates with constant angular velocity. For Q = 0, according to (30), we have pressure waves developing circularly inside the body (right).

We cannot rely on exact (or approximated) solutions, but we can however come out with useful qualitative considerations. By putting together the existence of field displacements like in ( 27)-( 28) together with the argumentation of the previous section, we entail that pressure waves, accompanied by suitable space-time deformations, rotate inside the cylinder. Some sort of nonhomogeneous mass is then circulating around the axis. In the experiment in [START_REF] Funaro | An efficient ring-shaped electromagnetic thruster[END_REF], the solicitations were obtained through a winding made of a conducting wire supplied with specific frequencies related to some resonance properties of the object. Thus, the rotating electromagnetic wave and the consequent gravitational one are expected to produce dynamical effects on the molecular bonds (Fig. 3, right). This should shake the entire body with oscillations that, of course, are supposed to be extremely small. On the other hand, as we mentioned in the introduction, these happen at a rate of a billion times per second, so that the final effect may be appreciable. We are however at a periodic regime inside a symmetric body. Therefore, we get a compensation of the various forces by integrating in a period of time. In other words, the resultant displays zero average.

The situation is different if the cylinder section is not a disk. In this occurrence, we may encounter dynamical displacements where the resultant of the forces at a fixed time is different from zero and, in addition, remains different from zero also after averaging over a period of time. This could generate a thrust towards a specific direction that depends on how the cylinder section has been designed.

As far as the determination of electromagnetic displacements is concerned, from the case of the cylinder it is relatively simple to pass to a ring geometry. The first computations were proposed in [START_REF] Chinosi | Rotating electromagnetic waves in toroid shaped regions[END_REF]. Numerical solutions of the full set of equations ( 1)-( 2) circulating in rounded cavities (also including toroids), are also available in [START_REF] Funaro | Electromagnetism and the Structure of Matter[END_REF][START_REF] Funaro | From Photons to Atoms[END_REF][START_REF] Funaro | Trapping electromagnetic solitons in cylinders[END_REF][START_REF] Funaro | High frequency electrical oscillations in cavities[END_REF][START_REF] Funaro | Electromagnetic waves in annular regions[END_REF]. Their evolution is very similar to that of fluid dynamic vortex rings [START_REF] Shariff | Vortex rings[END_REF]. Periodic behaviors are obtained by requiring that the eigenfunctions associated with the vector Laplace operator on the domain have multiplicity at least four. From the viewpoint of general relativity, finding the metric generated by the corresponding stress tensors requires an effort that at the moment is out of our possibilities.

From the practical viewpoint, the adoption of the ring makes the realization more easy and forces the magnetic field to follow closed loops inside the body, conferring additional stability properties. The analysis of real-life situations becomes more intricate if we take into account that the supplied signal (the one carried by the conducting winding) does not travel at the speed of light and its velocity is not equal in general to the one propagating inside the dielectric. Thus, the correct imposition of boundary conditions should not be underestimated.

Together with a dynamical behavior, the fields in ( 27)-( 28) also display a stationary component proportional to the constant Q. This option was also taken into account in the practical realization of the experimental device. In fact, the addition of a radial steady electric field emphasizes the properties of the thruster. We can provide a sort of justification by showing again some explicit results related to the solution of Einstein's equation. The cylinder and the ring turned out to be rather difficult to handle. We can say something regarding a charged dielectric sphere. The preparatory work was done in the recent paper [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF].

We are in the spherical system of coordinates (ct, r, θ, φ). As in [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF], we look for a metric of the form:

(ds) 2 = c 2 τ 2 (r)(dt) 2 -σ 2 (r)(dr) 2 -r 2 (dθ) 2 -r 2 sin 2 θ (dφ) 2 (31)
and we consider [START_REF] Fischer | Warped space-time for phonons moving in a perfect nonrelativistic fluid[END_REF] with the right-hand side [START_REF] Cornille | Advanced Electromagnetism and Vacuum Physics, SCCP[END_REF], which is built on the radial electric field E = (Qr, 0, 0), whereas B and V are zero. Here Q is a dimensional constant proportional to charge and depending on the relative dielectric constant of the medium. By solving equation [START_REF] Brüesch | Phonons: Theory and Experiments II, Experiments and Interpretation of Experimental Results[END_REF] in [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF] with the term χ 0 Q 2 r 2 on the right-hand side, we discover that:

τ (r) = 1 + χ 0 Q 2 r 4 /5 σ(r) = 1/τ (r) (32)
We also get:

E = -p = 3 2 0 Q 2 r 2 R = 6χ 0 Q 2 r 2 (33)
Moreover, we obtain ρ = 3Q either in the flat or the curved space (interesting result, since ρ is not a relativistic invariant).

The above finding tells us that the sole fact of charging a body produces a deformation of the space-time metric. According to [START_REF] Pretorius | Evolution of binary black hole spacetimes[END_REF] the effect is minimal near the center (r = 0), where τ and σ are approximately equal to 1.

Quantitatively, the final effect depends on the magnitude of the parameter χ (see also the comments at the beginning of section 2), which is expected to depend on the atomic properties of the dielectric material. We can try to provide a rough estimate for χ as follows. A relativistic charged body can be described by the Reissner-Nordström metric. For the electric field external to a charged sphere E = (q/4π 0 r 2 , 0, 0), the metric is obtained by choosing an appropriate τ and σ in [START_REF] Brügmann | Binary black hole mergers in 3d numerical relativity[END_REF]. In [START_REF] Funaro | The space-time outside a pulsating charged sphere[END_REF], working with the same tensor [START_REF] Cornille | Advanced Electromagnetism and Vacuum Physics, SCCP[END_REF] for the study of the electric field external to a pulsating charged sphere, the following functions were deduced in the stationary case:

τ (r) = 1 - 2mG c 2 r + χq 2 16π 2 0 r 2 σ(r) = 1/τ (r) ( 34
)
where G is the gravitational constant and m is the mass of the body.

Let us suppose that we are in the situation in which the massive and the electric components are approximately of the same magnitude, i.e.:

2mG c 2 r ≈ χq 2 16π 2 0 r 2 ⇒ χ ≈ 32π 2 0 mG q 2 c 2 r (35)
We now compare χ with the classical constant G/c 4 of Einstein's equation:

χ = γ G c 4 where γ ≈ 32π 2 mc 2 q 2 0 r (36)
If, for instance, our body is a proton, we can evaluate γ for r equal to its radius (about .85 × 10 -15 meters), so obtaining the multiplicative factor:

γ ≈ 13.93 (37)
Therefore, thanks to [START_REF] Funaro | Electromagnetic waves in annular regions[END_REF], χ has a an order of magnitude not too far from the standard constant. Moreover, we can observe that γ defined in ( 36) is adimensional, being the ratio between two energies. The situation changes drastically when we consider an entire hydrogen atom. In this case, the diameter is of the order of 10 -10 meters, but the mass is only concentrated on the nucleus and the corresponding gravitational field rapidly decays outside. On the other hand, the electric field mildly reduces its strenght. Hence, γ changes its meaning if we intepret it as a ratio of energy densities, so that the new G must be multiplied by a factor of order 10 15 , at least. Somebody certainly finds these considerations unconventional. Nevertheless, they help to better understand what actually happens inside matter. It is enough to think about the bending of light due to diffraction, where angles change within very small spatial ranges. The guess is that the rays are following geodesics of a modified geometry induced by the presence, not only of the nuclear lattice, but of the full electromagnetic internal environment. This approach supports the existence of solutions as those in Fig. 3 (see also ( 27)-( 28)), pertaining to rotating electromagnetic waves. An interpretation in terms of general relativity can provide a more robust description of optics phenomena, beyond the abstract application of geometrical rules. In this framework, also the attraction or repulsion of charges due to Coulomb's law ceases to be a mere phenomenological rule and acquires a deeper meaning in terms of modifications of the space-time (note that the potential p may take either positive or negative values). The challenge would be to show that between two neutral object, the resultant of the nonlinear mutual interactions of their internal charges still amounts to a residual weak action, which is what we call gravitational force. If this was true, a theory unifying electromagnetic and gravitational forces would become a reality. A more thoroughgoing discussion on these issues has been put forth in [START_REF] Funaro | From Photons to Atoms[END_REF], section 2.6, with more examples and remarks, though again at the level of an interrogative analysis .

Going back to the experiment, the role of the stationary electric field was proven to be significant Indeed, a second independent wiring was added to the primary one. The purpose of this circuitry was to apply a constant voltage difference of the order of KVolts between the secondary coil and the center of the dielectric (represented by a further wire running inside the ring). In this fashion, the ring itself becomes a charged capacitor.

With a mixture of rigorous and heuristic considerations, a ring with an asymmetric section was built as described in [START_REF] Funaro | An efficient ring-shaped electromagnetic thruster[END_REF] (see Fig. 4). The design of its shape and the electric circuit benefit from the above theoretical hints. As a combination of stationary and dynamical signals is applied to the device, a sensible thrust is observed in the direction of the bottom part (the rounded one). In absence of the high frequency signal, the sole presence of the high voltage does not produce any measurable effect, and this is a confirmation of the theoretical predictions. In principle, performances may vary depending on many parameters, such as: the frequency injected in the system, the magnitude of the stationary electric field (see Q in [START_REF] Misner | [END_REF]), the geometry of the ring, the properties of the dielectric, the conductivity and the number of spires in the wiring, etc. Quantitative information will be available as more tests will be conducted.

Figure 4: Section of the asymmetric ring used for the experiments in [START_REF] Funaro | An efficient ring-shaped electromagnetic thruster[END_REF]. The geometry has been suggested by heuristic arguments. The idea is to allow for a smooth rotation of the signal in the lower part of the ring and emphasize instead the acceleration term D V /Dt on the upper side. This last is relatively large at the corner line on top and points downward. As the angular velocity of the signal is constant, by neglecting the term E + V × B in (2), we expect that the integral of the pressure gradient ∇p shows a neat resultant pointing downwards. The shape can be certainly ameliorated in order to increase performances. This should be done in conjunction with numerical experiments conducted on the model equations.

What really happens is difficult to explain without a good dose of numerical simulations that are expected to be carried out in the future. These will be confronted with the experimental outcomes and ameliorated when necessary.

According to the results of this paper, an 'immaterial' mass is circulating inside the ring, following trajectories as specified by the arrows in figure. There are two kinds of asymmetries. The first one is the section shape. The sharp form of the upper side was actually designed to emphasize diversity. Secondly, we note that the semi-path passing through the ring hole is different (in terms of surface density) from the one running outside. Our guess is that the orbits are not closed, but are spiraling towards the bottom. As we specified, this may be due to some nonlinear properties of the dielectric material, or because part of the signal is ejected from the top (or both the occurrences). In this framework, the violation of the action-reaction principle is not due to the interaction with the surrounding vacuum (that would be negligible). The asymmetric Newtonian forces should come instead from the induced repositioning of some 'virtual masses' partly already present inside matter due to the strong chemical bonds. The density of such 'alternative' mass is certainly small, but it is spread all over the volume of the body, contrary to what happens to standard masses that are only concentrated at the nuclei. The rotatory process is ignited by the electromagnetic component enforced externally. Unfortunately, we are not in the position to provide more details at the moment.

Conclusions

Electromagnetic waves traveling into matter bring along a space-time deformation that can be assimilated to a kind of mass flow, with the consequence of generating longitudinal mechanical pressure variations. Analogous theoretical results have been obtained by other authors with the purpose of finding feasible applications. The results discussed above actually suggested the construction of an asymmetric thruster, as experimented in [START_REF] Funaro | An efficient ring-shaped electromagnetic thruster[END_REF]. Though the device still needs to be replicated and tested by independent research groups, the material here collected may a key point for improvements, as well as a source for further new ideas. In addition, the theoretical contributions here collected may open a new page for the study of relativistic phenomena inside matter.

Figure 2 :

 2 Figure 2: Plot of σ (blue line) solution of the first equation in (24), when h (orange line) is a sawtooth.