Waves

Field of the invention

The present invention relates to the generation and the detection of gravitational waves using purely electromagnetic sources.

Background of the invention

Out of the four fundamental interactions, gravitation remains the only one not to be under technological control. Controlling gravity requires not only to generate, but also to detect, gravitational fields.

Generating gravitational fields is a possibility offered by Einstein's Equivalence Principle. The universality of free fall teaches us that all types of energy, associated to any of the four fundamental forces, undergo an external gravitational field in the same way. But this also implies that all types of energies produce gravity in the same way. Since we cannot switch off the binding energies of matter sources, one should rely on electromagnetic energies as a source of human-made controllable gravitational fields.

A mechanism to convert electromagnetic fields into gravitational waves has been disclosed by Gertsenshtein in“Wave resonance of light and gravitational wave” of Soviet Physics JETP, volume 14, number 1 , in January 1962. This is purely electromagnetic mechanism of gravitational wave emission based on wave resonance: a static magnetic field perpendicular to an incident electromagnetic wave produces gravitational waves since the equivalence principle couples both Einstein and Maxwell wave systems.

Nevertheless, this description is purely theoretical and the extreme weakness of the conversion process of electromagnetic waves into gravitational waves and the reverse process constitutes a serious obstacle to any possible practical realization of the Gertsenshtein’s mechanism.

The difficulties for designing a strong continuous source of emission of gravitational waves that could be further detected postponed the challenge of gravity control with the joint emission and reception of gravitational waves. The strategy to develop gravitational wave physics was to use the much stronger astrophysical sources and to attempt detecting them in ground-based specific facilities. Nevertheless, detection of astrophysical gravitational waves with laser interferometers such as LIGO in the USA and Virgo in Europe allows exploring rather low frequency ranges while electromagnetic gravitational wave detecting devices would allow exploring higher frequency range. Moreover, astrophysical sources of gravitational waves are cataclysmic cosmic events occurring at random rate and only produce gravitational waves for some limited period of time.

Summary of the invention

An object of the invention is to provide a device for generating a gravitational wave underlying a practical and efficient realization of Gertsenshtein’s mechanism, allowing to emit continuously gravitational waves with free predetermined frequency and sufficiently high amplitude, for a sufficiently long duration, so that the emitted gravitational waves can be detected with accessible technology consisting of gravitational wave detecting device developed hereafter as part of the invention.

For the sake of clarity, four aspects of the invention presented in the following summary are enumerated as follows and respect the order of the claims:

- first aspect: a gravitational wave generating device;

- second aspect: a gravitational wave detecting device; - third aspect: a method for generating gravitational waves;

- fourth aspect: a method for detecting gravitational waves.

According to a first aspect, the invention provides a gravitational wave generating device comprising:

- an elongated cavity extending along a longitudinal axis, said cavity being configured for confining electromagnetic waves thereto;

- means for generating, with said cavity, at least one electromagnetic standing wave in said cavity, coaxially to said longitudinal axis, with a predetermined frequency and a predetermined amplitude; and

- means for generating within at least a portion of said cavity and in a second predetermined direction a static magnetic field;

characterized in that

- said cavity is an elongated transverse electromagnetic (TEM) waveguide comprising a central core, said device comprising:

• a first elongated electrically conducting element defining said cavity, and

• a second elongated electrically conducting element, internal to said first elongated electrically conducting element and coaxial therewith, defining said central core;

- said at least one electromagnetic standing wave consists of a transverse electromagnetic (TEM) standing wave;

- said cavity and said means for generating at least one electromagnetic standing wave are configured for confining said TEM standing wave in a complementary space of said central core, said complementary space being comprised in said cavity;

- said second predetermined direction is substantially perpendicular to said longitudinal axis;

said gravitational wave generating device being configured for generating at least one gravitational wave within said cavity through coupling between said at least one electromagnetic standing wave and said static magnetic field, said at least one gravitational wave propagating substantially in a solid angle centered on a first predetermined direction with said predetermined frequency.

The present gravitational wave generating device allows to emit continuously gravitational waves with predetermined frequency and sufficiently high amplitude, for a sufficiently long duration, so that the emitted gravitational waves can be detected with accessible technology consisting of gravitational wave detecting device developed hereafter as part of the invention.

In fact, as we will show, the invention allows to emit at least one gravitational wave having a free predetermined frequency in a free predetermined direction. In particular, this generating device allows exploring both low and high frequency ranges. The predetermined frequency of the emitted gravitational wave is inherited from the at least one electromagnetic standing wave. Preferably, the free predetermined frequency is lying between 10 ³ Hz (radio-frequencies) and 300 GHz (microwaves) for conventional electromagnetic cavities.

The generation of a gravitational wave with given free predetermined frequency and free predetermined direction can be realized by choosing the at least one electromagnetic standing wave with the predetermined frequency and by orienting appropriately the static magnetic field. In particular, the second direction of the static magnetic field is preferably carefully determined from the first predetermined direction and the longitudinal axis.

The cavity comprises a central core. It is well known for a man skilled in the art that a complementary space of the central core with respect to the cavity is a space defined by the set of points belonging to the cavity that does not belong to the central core. The at least one transverse electromagnetic standing wave is not able to take place within its central core. The generating device according the invention is configured for generating standing gravitational waves in the above-mentioned complementary space, and progressive gravitational waves toward the outside of the cavity.

The static magnetic field according to this method allows to boost the gravitational waves emission when it is properly oriented with respect to the cavity. In this case, specific polarizations of the emitted gravitational waves can reach an amplitude of order

GB ₀ E ₀ L ²

Qz := - 5m o

where G is the Newton constant, c is the speed of light, mo is the magnetic vacuum permeability, L is the characteristic size of the generator (e.g., its extent along the longitudinal direction), E ₀ is the characteristic amplitude of the electric field carried in the cavity and B ₀ is the amplitude of the outer static magnetic field. This number g _z is called the Gertsenshtein- Zeldovich number. It measures the efficiency of the cavity as gravitational waves generating device as the instantaneous rate of conversion of electromagnetic waves into gravitational ones. This amplitude for specific polarizations of the emitted gravitational waves is very high thanks to the presence of the static magnetic field. Preferably, it immerses the whole cavity therein and/or it allows to boost up the gravitational wave emission with a factor at least 1000.

Advantageously, the generating device according to the first aspect of the invention is able to generate continuously at least one gravitational wave for a long period of time as the static magnetic field and the at least one electromagnetic standing wave can be maintained for long duration thanks to available electrical technology.

Preferably, the generating device according to the first aspect of the invention allows to generate continuously standing and progressive gravitational waves during at least several months. The gravitational wave generating device according to the first aspect of the invention comes with thorough rigorous mathematical modeling, based on the standard physics of general relativity and electromagnetism on curved space-time. A practical application of the method according to the invention can lead to a very high instantaneous rate of conversion of electromagnetic waves into gravitational ones.

In fact, the Gertsenshtein-Zeldovich number of this device according to embodiments of the invention is greater than 5 x 10 ³⁹ . This number measures the efficiency of such gravitational wave generating device as the instantaneous rate of conversion of electromagnetic waves into gravitational ones. None of the previous gravitational wave generating devices according to the prior art were able to reach such a high Gertsenshtein-Zeldovich number. In this sense, the cavity as generating device according invention currently constitutes the most efficient practical realization of the Gertsenshtein’s mechanism to date.

The generating device according of the invention, features other advantages. In particular, it requires limited electromagnetic energy density in the cavity, due to considerably larger amount of energy stored in the static magnetic field, and allows an anisotropic emission of gravitational waves as a function of the orientation of the magnetic field and the cavity geometry.

Gravitational waves emitted by the gravitational wave generating device according to the invention are fully characterized in terms of spatial distribution, propagation, amplitude, wavelength, polarization and energy- momentum carried away.

Moreover, advantageously, the generating device according to the invention constitutes a tool for performing new tests on Einstein’s Equivalence Principle, since no current test combines purely relativistic sources of gravitation in the weak-field limit. The gravitational waves generated by the gravitational wave generating device according to the invention behaves in the far distance as a spherical wave with an amplitude decreasing with the inverse of the distance to the center of the generating device, and the same predetermined frequency as the at least one transverse electromagnetic standing wave.

Moreover, the gravitational wave generating device according to the invention is configured for generating at least one gravitational wave having a predetermined polarization and a free predetermined frequency and propagating along a first predetermined direction. The at least one gravitational wave is generated within the elongated transverse electromagnetic waveguide through coupling between the at least one transverse electromagnetic standing wave and the static magnetic field, said coupling underlying a practical and particularly efficient realization of the Gertsenshtein’s mechanism.

The predetermined polarization of the at least one gravitational wave generated by the generating device depends on the second predetermined direction of the static magnetic field in a plane perpendicular to the longitudinal axis.

The predetermined polarization can then be selected by a specific choice of orientation of elongated transverse electromagnetic waveguide with respect to the second predetermined direction of the static magnetic field. In particular, the predetermined polarization of the emitted gravitational wave according to further described embodiments of the invention is a free predetermined polarization. The gravitational wave generating device according to the invention generates both gravitational wave transverse polarizations h _xx and h _xy in the vacuum outside of the generating device. The first and second elongated electrically conducting elements are preferably electrically separated and not short-circuited conducting edges of the complementary space.

The elongated transverse electromagnetic waveguide according to invention has a particularly simple design. The interior part of the first elongated electrically conducting element defines the cavity which is an open cavity. The interior part of the second elongated electrically conducting element defines the central core.

Advantageously, a predetermined polarization and the first predetermined direction of any gravitational wave generated by the generating device according to the invention can be selected through the choice of the second predetermined direction in a plane perpendicular to the longitudinal axis. The possible choices of predetermined polarization depend on the geometry and orientation of the first and second elongated electrically conducting elements.

According to an embodiment of the invention, the longitudinal axis is disposed along a horizontal direction.

According to an embodiment of the invention, the length of the cavity measured along the longitudinal axis lies between 10 ² meters and 10 ² meter, preferably between 10 meters and 10 ¹ meter, more preferably between 2 meters and 2 ¹ meter.

According to an embodiment of the invention, the volume of the cavity lies between 10 ¹ m ³ and 10 m ³ .

According to an embodiment of the invention, the static magnetic field is generated by at least one high field electromagnet. Preferably, the high field electromagnet is an intense DC electromagnet. Preferably, the high field electromagnet is a solenoid. Preferably, the high field electromagnet surrounds the cavity.

According to an embodiment of the invention, the strength of the static magnetic field lies between 10 ¹ Tesla and 40 Tesla, preferably between 1 Tesla and 20 Tesla, more preferably between 5 Tesla and 15 Tesla.

According to an embodiment of the invention, the amplitude of the electric field in the at least one electromagnetic standing wave lies between 10 ³ Volt per meter and 10 ¹⁵ Volt per meter, preferably between 10 ⁴ Volt per meter and 10 ¹ ° Volt per meter, more preferably between 10 ⁵ Volt per meter and 10 ⁷ Volt per meter.

Preferably, the at least one electromagnetic standing wave is a monochromatic electromagnetic standing wave.

According to a preferred embodiment of the invention, the first and second elongated electrically conducting elements are arranged symmetrically along the longitudinal axis, each of these elongated electrically conducting elements defining a constant cross-section in any plane perpendicular to the longitudinal axis.

For example, the constant cross-section can either be an ellipse or a polygon.

On one hand, the invention can further be characterized in that the first elongated electrically conducting element defines a first rectangular cross-section in a plane perpendicular to the longitudinal axis, and the second elongated electrically conducting elements defines a second rectangular cross-section in a plane perpendicular to the longitudinal axis.

Advantageously, the first and second elongated electrically conducting elements according to this embodiment of the invention have a very simple geometry and are easy to design.

Advantageously, this embodiment of the invention allows to generate at least one gravitational wave with a free predetermined polarization which depends on the orientation of the static magnetic field with respect to the said conducting elements, and on the dimension of the first and second rectangular cross-sections. In particular, the gravitational wave generating device according to some embodiments of the invention generates both gravitational wave polarizations h _xx and h _xy .

Advantageously, this embodiment of the invention allows to generate at least one gravitational wave propagating along a free first predetermined direction which depends on the longitudinal axis. More precisely, the first predetermined direction according to this embodiment is substantially aligned to the longitudinal axis. The at least one gravitational wave generated by the gravitational wave generating device according to this embodiment of the invention behaves in the far distance as a spherical wave with an amplitude decreasing with the inverse of the distance to the center of the generating device.

Such an embodiment can further be characterized in that the area of the first outer rectangular cross-section lies between 10 ³ m ² and 10 ³ m ² , preferably between 10 ¹ m ² and 10 m ² , more preferably between 2 ¹ m ² and 2 m ² .

Such an embodiment can further be characterized in that the ratio of the area of the first rectangular cross-section with the area of the second rectangular cross-section lies between 1 and 10 ⁴ , preferably between 2 and 10 ² , more preferably between 5 and 20.

Such an embodiment can further be characterized in that each of the first and second rectangular cross-sections consists of a square.

Such an embodiment can further be characterized in that each of the first and second elongated electrically conducting elements consists of a cuboid.

On the other hand, the invention can further be characterized in that the first elongated electrically conducting element defines a first elliptic cross-section in a plane perpendicular to the longitudinal axis, and the second elongated electrically conducting elements defines a second elliptic cross-section in a plane perpendicular to the longitudinal axis. Advantageously, the first and second elongated electrically conducting elements according to this embodiment of the invention have a very simple geometry and are easy to design.

Advantageously, this embodiment of the invention allows to generate at least one gravitational wave propagating along a first predetermined direction with one polarization substantially parallel to the second predetermined direction.

An outgoing gravitational wave generated by the gravitational wave generating device according to this embodiment of the invention behaves in the far distance as a spherical wave with an amplitude decreasing with the inverse of the distance to the center of the generating device.

Such an embodiment can further be characterized in that the area of the first and second elliptic cross-sections lies between 10 ³ m ² and 10 ³ m ² , preferably between 10 ¹ m ² and 10 m ² , more preferably between 5 ¹ m ² and 5 m ² .

Such an embodiment can further be characterized in that each of the first and second elliptic cross-sections consists of a circle.

Such an embodiment can further be characterized in that each of the first and second elongated electrically conducting elements consists of a cylinder arranged symmetrically along the longitudinal axis.

Any of the above mentioned embodiments of the invention can further be characterized in that each of the first and second elongated conducting elements comprises a superconducting material.

Superconducting material comprised in the first and second elongated conducting elements according to this preferred embodiment of the invention improves the continuity of the emission of the at least one gravitational wave for a long period of time.

Preferably, the means for generating the static magnetic field comprises a superconducting solenoid surrounding said cavity. The cavity is preferably rotatable around the longitudinal axis. This allows to select arbitrary a direction of maximum gravitational wave energy emitted as well as an emitted polarization of gravitational waves by rotating the cavity with respect to the static magnetic field.

Any of the above mentioned embodiments of the invention can further be characterized in that it comprises means for varying the second predetermined direction in a plane perpendicular to the longitudinal axis. This allows to select an emitted polarization of gravitational wave by rotating the static magnetic field with respect to the longitudinal axis.

Any of the above mentioned embodiments of the invention can further be characterized in that it comprises means for varying the predetermined frequency and the predetermined amplitude of the at least one electromagnetic standing wave.

Since there is no restriction on the wavenumber for electromagnetic propagation modes generating the gravitational waves, means for varying the predetermined frequency and the predetermined amplitude of the at least one electromagnetic standing wave according to this preferred embodiment of the invention allows to choose freely the predetermined frequency of the at least one gravitational wave.

According to a second aspect, the invention provides a gravitational wave detecting device configured for detecting gravitational waves generated by the gravitational wave generating device according to the invention,

the gravitational wave detecting device comprising:

- a detection region, preferably through which a propagation axis of said gravitational waves is passing ;

means for generating a static magnetic field in a direction transverse to a propagation axis of gravitational waves to be detected and in at least a portion of said detection region; and means for measuring accumulated magnetic energy gain inside said detection region.

The magnetic energy gain inside the detection region is due to an accumulation of electromagnetic radiation induced by an interaction between the gravitational waves and the static magnetic field. In particular, the gravitational wave detecting device according to the invention comes with thorough rigorous mathematical modeling constituting a practical realization of the reverse of the Gertsenshtein’s mechanism.

The gravitational wave detecting device according to the invention is adapted to the gravitational waves emitted by the gravitational wave generating device according to the invention. The embodiments of the gravitational wave generating device according to the first aspect invention and advantages of this device apply to the present detecting device.

Preferably, gravitational waves having an axis of propagation passing through the detection region are continuously emitted by the gravitational wave generating device according to the first aspect of this invention. The gravitational wave detecting device according to this embodiment of the invention is preferably disposed just beside a generating device according to the invention.

Preferably, the detection region is bounded by an electromagnet to accumulate the induced electromagnetic radiation inside the detection region which can be measured through a magnetic energy variation inside the detection region.

Advantageously, the gravitational wave detecting device according to the invention is also adapted to the gravitational waves emitted by astrophysical sources.

Advantageously, information encoded into the gravitational waves can be retrieved through signal analysis of the time evolution of the magnetic energy stored inside the detection region. As a consequence, the gravitational wave generating and detecting devices according to the present invention constitutes a major step toward a technological control of the gravity and a tool for performing new tests on Einstein’s Equivalence Principle. It also constitutes a basis for the development of new technological telecommunication, remote sensing or geologic devices, non-invasive remote investigation methods, propulsive applications of gravitational radiation recoil using high sensitive gravitational wave detecting devices and high-power gravitational wave generating devices.

According to an embodiment of the invention, the volume of the detection region lies between 10 ⁵ m ³ and 10 ² m ³ , preferably between 10 ⁴ m ³ and 10 ¹ m ³ , more preferably between 10 ³ m ³ and 1 m ³ .

Preferably, means for generating the static magnetic field allows to generated are at least one continuous static magnetic field which is external to the detection region.

Preferably, said static magnetic field immerses the whole detection region therein. More preferably, the realization of this method involves an electromagnet generating the static magnetic field and defining the detection region.

According to an embodiment of the invention, the static magnetic field is generated by at least one high field electromagnet. Preferably, the high field electromagnet is an intense DC electromagnet. Preferably, the high field electromagnet is a solenoid.

Preferably, the high field electromagnet surrounds the detection region. The detection region is preferably internal to a superconducting solenoid and the means for generating the static magnetic field comprise this superconducting solenoid.

According to an embodiment of the invention, the strength of said at least one continuous static magnetic field lies between 10 ¹ Tesla and 40 Tesla, preferably between 1 Tesla and 20 Tesla. Preferably, magnetic energy variation is measured either through a current variation, a magnetization and/or a photon generation.

The monitor of magnetic energy in the detection region requires high-sensitive and high-precision physics. In particular, according to an embodiment of the invention, the means for measuring magnetic energy gain comprises technical tools such as those that have been used in experiments of dark matter detection (e.g., ADMX) for nearly two decades.

The invention also provides a system (or kit) comprising a gravitational wave generating device according to the first aspect of the invention and a gravitational wave detecting device according to the second aspect of the invention, the second predetermined direction of the static magnetic filed of the generating device being substantially parallel to the direction of the static magnetic filed of the detecting device.

The whole set of embodiments of the gravitational waves generating and detecting devices according to the invention and the whole set of advantages of these devices use apply mutatis mutandis to the present system. Moreover, the specific alignments of the static magnetic fields in these devices allows advantageously to maximize the variation of the magnetic energy stored in the detection region that is induced by the gravitational waves to be detected.

The invention also provides a telecommunication device comprising the above-mentioned system. The invention also provides a remote sensing device comprising the above-mentioned system. The invention also provides a geologic device comprising the above-mentioned system (or kit).

According to a third aspect, the invention provides a method for emitting, in a first predetermined direction, a gravitational wave having predetermined frequency, the method comprising the steps:

providing a gravitational wave generating device according to the invention; determining the second predetermined direction from said first predetermined direction;

using the means for generating the static magnetic field in said second predetermined direction determined in previous step; - immersing at least partially the cavity in the static magnetic field generated in previous step;

using the means for generating the at least one electromagnetic standing wave in said cavity with said predetermined frequency; generating said gravitational wave within the cavity through coupling between the at least one electromagnetic standing wave and the static magnetic field.

The whole set of embodiments of the gravitational waves generating device according to the above mentioned first aspect of the invention and the whole set of advantages of this device use apply mutatis mutandis to the present generation method.

In particular, this gravitational wave generation method allows to emit continuously gravitational waves with predetermined frequency and sufficiently high amplitude, for a sufficiently long duration, so that the emitted gravitational waves can be detected with accessible technology consisting of the gravitational wave detecting developed as part of the second aspect of the invention. The gravitational wave generation method according to the invention comes with thorough rigorous mathematical modeling and constitutes a very efficient practical realization of the Gertsenshtein’s mechanism.

The steps of the method according to this third aspect of the invention can be executed according to various sequences.

The generation method according to this third aspect of the invention can be reformulated as a gravitational wave generation method for emitting, in a first predetermined direction, a gravitational wave having a free predetermined polarization and a free predetermined frequency, the method comprising the steps:

providing an elongated waveguide made of two coaxial conductors having a longitudinal axis extending therethrough;

generating at least one transverse electromagnetic (TEM) standing wave within the complementary space between the waveguide conductors, said at least one transverse electromagnetic standing wave propagating coaxially to said longitudinal axis with said predetermined frequency;

generating at least one static magnetic field in a predetermined direction perpendicular to said longitudinal axis, said predetermined direction being related to said predetermined polarization;

immersing at least partially said elongated electromagnetic waveguide in said at least one static magnetic field; and

generating at least one gravitational wave having said predetermined polarization and said predetermined frequency, said least one gravitational wave being generated within said elongated electromagnetic waveguide through coupling between said at least one transverse electromagnetic standing wave and said at least one static magnetic field, said at least one gravitational wave propagating substantially in a solid angle centered on the said longitudinal axis.

Preferably, the generation method according to the third aspect of the invention further comprises the step: measuring accumulated power loss inside the cavity.

This power loss corresponds to the energy carried by the fleeing gravitational waves. As a consequence, measuring accumulated power loss inside the cavity according to such a preferred embodiment of the invention allows to detect indirectly the emitted gravitational waves.

The monitor of power loss requires high-sensitive and high-precision physics. In particular, such a preferred embodiment of the invention can be further characterized in that the power loss is measured with techniques such as those that have been used in experiments of dark matter detection (e.g., ADMX and CDMS) for nearly two decades.

The generation method according to the present invention can be applied in various practical applications. For example, this generation method constitutes a very simple and practical tool for the realization of new technological telecommunication methods when it is combined with an appropriate method for detecting gravitational waves such as the detection method according to a fourth aspect of the invention. Current telecommunication devices suffer from absorption and scattering by the medium or materials in which the physical support of the signal travels or passes through. Global telecommunications via electromagnetic radiation also require satellite relays. As gravitational waves almost do not interact with matter, an application of some embodiments of the generation method according to the invention constitutes a core of a future telecommunication methods allowing to communicate even through the Earth crust without relays. Moreover, such telecommunication methods will overcome any confidentiality and reliability problem on the transferred signal. In fact, a signal emitted with the current technology can be intercepted by some conventional detecting device or the communication line can be physically broken or jammed. Remote messages have therefore to be protected through cryptography and the physical support of the signal must be protected to avoid interruption or amplified to ensure correct reception. Gravitational waves flux cannot be interrupted and reading a message send by gravitational waves with a method based on the present invention will require a very specific technology of high-frequency gravitational wave detecting devices which are not easy to develop and to master, the current laser interferometer technology currently used for astrophysical applications being inappropriate for high-frequency gravitational waves. According to a fourth aspect, the invention provides a method for detecting gravitational waves generated by the gravitational wave generating device according to the invention, the method comprising the steps:

- providing a detection region;

disposing the detection region through a propagation axis of said gravitational waves;

generating a static magnetic field in a direction transverse to said propagation axis, preferably substantially parallel to the second predetermined direction along which the static magnetic field of the gravitational wave generating device is generated; immersing at least partially the detection region in the static magnetic field generated in previous step;

measuring a magnetic energy gain inside the detection region due to an accumulation of electromagnetic radiation induced by an interaction between said gravitational waves and the static magnetic field.

The detection region provided in the first step of this detection method consists preferably of a detection region of a gravitation wave detecting device according to the second aspect of the invention.

In particular, the whole set of embodiments of the gravitational waves detecting device according to the above mentioned second aspect of the invention and the whole set of advantages of this device use apply mutatis mutandis to the present detection method.

The static magnetic field is preferably generated in the detection method by a superconducting solenoid surrounding the detection region.

As described above for the detecting device, the step of measuring a magnetic energy gain preferably comprises a measure of at least one among: a current variation, a magnetization and a photon generation. The gravitational wave detection method according to the invention comes with thorough rigorous mathematical modeling, based on the standard physics of general relativity and electromagnetism on curved space-time. In particular, its application constitutes a practical realization of the reverse of the Gertsenshtein’s mechanism.

The gravitational wave generation and detection methods according to the invention are preferably applied simultaneously in order to detect emitted gravitational waves, therefore realizing a practical application of gravity control. Brief description of the figures

For a better understanding of the present invention, reference will now be made, by way of example, to the accompanying drawings in which:

- Figure 1 illustrates a schematic three-dimensional view of a gravitational wave generating device according to an embodiment of the invention;

- Figures 2A-C illustrate longitudinal and transversal two-dimensional slices of the amplitude of two gravitational wave polarizations around a waveguide of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 ;

- Figure 3 illustrates four graphs describing the propagation of metric perturbations h _xy in a direction parallel to the longitudinal axis for a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 ;

- Figure 4 illustrates a schematic three-dimensional view of a gravitational wave generating device according to an embodiment of the invention;

- Figure 5 illustrates a distribution of the transverse components of the electric field in a two-dimensional transversal section of a waveguide of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4; Figures 6A-R illustrate longitudinal and transversal two-dimensional slices of metric field distribution around a waveguide of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4;

Figure 7 illustrates a schematic three-dimensional view of a gravitational wave generating device alternatively to the invention; Figures 8A-P illustrate longitudinal and transversal two-dimensional slices of metric fields distribution around the cavity of the gravitational wave generating device illustrated in Figure 7;

Figure 9 illustrates a schematic three-dimensional view of a gravitational wave generating device and a gravitational wave detecting device according to an embodiment of the invention;

Figure 10 illustrates a schematic three-dimensional view of a gravitational wave generating device and a gravitational wave detecting device according to an embodiment of the invention;

Figure 1 1 illustrates two graphs describing the time evolution of the magnetic energy density remotely induced in the detection region of a gravitational wave detecting device according to an embodiment of the invention;

Figure 12A-D illustrate three-dimensional views of gravitational radiation power patterns around gravitational wave generating devices according to gravitational wave generating devices.

Description of the invention

The present invention will be described with respect to particular embodiments and with reference to certain drawings but the invention is not limited thereto. The drawings described are only schematic and are non limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. On the figures, identical or analogous elements may be referred by a same number. Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequential or chronological order. The terms are interchangeable under appropriate circumstances and the embodiments of the invention can operate in other sequences than described or illustrated herein.

Furthermore, the various embodiments, although referred to as “preferred” are to be construed as exemplary manners in which the invention may be implemented rather than as limiting the scope of the invention.

In the following description, G will denote the Newton constant, c will denote the speed of light and mo will denote the magnetic vacuum permeability.

In the following description, we will refer to the Cartesian harmonic coordinates

(x°, x ¹ , x ² , x ³ ) = ( ct, x, y, z )

with regard to the spatial coordinate axis X, Y and Z defining respectively three axis of a spatial orthonormal coordinate system. These axes are illustrated in most of the above mentioned figures.

In the following description, we will denote by Z the longitudinal axis along which is extended the waveguide or the cavity. In particular, in the following description and the following figures, this axis will coincide with the axis Z of said spatial orthonormal coordinate system. This choice of longitudinal axis is non limitative and is pure matter of convention. Both the representation of the longitudinal axis and the spatial orthonormal coordinate system are arbitrary.

In the following description, we will use equivalently the terms gravitational wave generating device, generating device, generator and emitter. In the following description, we will use equivalently the terms gravitational wave detecting device, detecting device, detector and receiver. In the following description, we will use equivalently the term gravitational wave and the abbreviation GW. In the following description, we will use equivalently the term electromagnetic and the abbreviation EM. In the following description, we will use equivalently the term transverse electromagnetic and the abbreviation TEM. In the following description, we will use equivalently the term transverse magnetic and the abbreviation TM. In the following description, we will use equivalently the term transverse electric and the abbreviation TE.

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[25] S. J. Asztalos et al., Phys. Rev. Lett. 104, 041301 (2010); S. J. Asztalos et al., Nuclear Inst and Methods in Physics Research, A 656 (2011 ) pp. 39-44;

[26] W. Buckles & W.V. Hassenzahl, IEEE Power Engineering Review 20 (5), 16-20 (2000);

[27] C. A. Balanis, "Antenna Theory: Analysis and Design”, fourth edition, Wiley (2016). The Gertsenshtein effect [2] is a wave resonance mechanism in which light passing through a region of uniform magnetic field, perpendicular to the direction of light propagation, produces GWs. We will apply this effect to design GW generators whose working principle will be an EM standing wave in some resonant cavity or waveguide coupling with an external static magnetic field.

Let us therefore work in the linearized approach of Einstein- Maxwell equations, for perturbations of a background Minkowski space:

where h _mn = diag(+1 ,-1 ,-1 ,-1 ) is the Minkowski metric and where e _{mn Mmn mn} « 1 represent the metric perturbations respectively due to:

(i) the external static magnetic field generated by some coil (o _mn ),

(ii) the EM wave (in _mn ), and

(iii) the coupling between the external magnetic field and the EM wave (/i _mn ).

Case (i) has been studied in [20] but does not give rise to GW since the outer magnetic field is considered static. Case (ii) has been studied for light propagating pulses in [21 , 22] and in [4, 5, 17] for EM waves in resonant hollow cavities [29]. Case (iii) actually corresponds to the resonance between the EM waves and the outer magnetic field applied to light in [2, 3] and will keep our interest here because of the amplification of the outcoming GWs it provides.

Assuming Lorenz gauge condition d _m 1i ^mn = 0, the linearized Einstein equations can be written down (in International System of Units): is the stress-energy tensor for the resonance, which is given by where (X= c, w, tot) are the Maxwell stress-energy tensors in Minkowski space for the EM fields of the coil, the EM wave and their superposition, respectively: with the total (antisymmetric) Faraday tensor F^ ^ot) = being due to the superposition of the EM fields of the coil and of the wave. 7^ ^es) is traceless (from the basic properties of Maxwell stress-energy tensor) and so does 1i _mn through Eq.(3) in Lorenz gauge only. In this gauge, the algebraic structure of the metric perturbation tensor H _mn is identical to the one of the stress-energy tensor T _mn since Einstein equation Eq.(3) in Lorenz gauge are decoupled. This will allow to directly relate the excited polarizations of GWs to the polarizations present in the EM wave acting as a source. Furthermore, Lorenz gauge condition is compatible with Eq.(3) provided the source r^ ^es) verifies 9 _m t ⁿ _b ^ = 0 or, equivalently, that the zeroth order components F^ of the Faraday tensor are solutions of the Maxwell equations in the vacuum of Minkowski space:

¾¾ = °- ( ⁶ )

Equations (3-6) describe the direct Gertsenshtein effect: the conversion of an EM wave into a GW in the presence of an external magnetic field. We can now apply these equations to the modeling of GWs generators where EM standing waves of different polarizations will couple to an external magnetic field to produce specific metric perturbations.

The EM standing waves will be produced by excitation of a waveguide or a cavity whose axis of symmetry is considered lying along the z direction. Choosing the length of the waveguide/cavity L _z as the characteristic scale of the problem, the solution of linearized Einstein equations in Lorenz gauge Eq.(3) is given by the retarded potentials where R is the dimensionless spatial position vector

( R, T ) = C C, U, Z, T ) = ( x, y, z, ct)/L _z and where g _z is a dimensionless constant we will introduce below. In Eq.(7), the triple integral is performed over the generator, which is the volume of the waveguide/cavity entirely immersed into the external magnetic field vanishes outside of the waveguide/cavity). In Eq.(7), f _mV = m ₀ )/ B ₀ E ₀ ) T _m x ^e is the dimensionless stress-energy tensor with E ₀ , Bo standing for the amplitude of the EM standing wave and the external magnetic field respectively. It is important to notice that, due to the relativistic nature of the electromagnetic source, one cannot make use of the usual compact source approximation for a finite inertial mass source and therefore cannot rely on the quadrupole formula for gravitational radiation.

We dub the dimensionless number g _z in Eq.(7),

the Gertsenshtein-Zeldovich number (GZ) associated to a given GWs generator (see also [2, 3] for dimensional considerations on the direct Gertsenshtein effect). This GZ number measures the efficiency of the generator as the instantaneous rate of conversion of EM waves into gravitational ones. The amplitude of the generated metric perturbations are roughly of order of magnitude g _z , up to some geometrical factor related to the shape of the generator.

If we take L _z = 1 m, B ₀ = 10T and E ₀ = 1 MV/m, we find g _z ~10 ³⁹ .

As a matter of comparison, the EM standing GWs generator in [5] only consists of a simple hollow toroidal TM or TE cavity, such that the amplitude of the generated GWs can be obtained from Eq.(8) with B ₀ = E ₀ /c. For an electric field of amplitude E ₀ ~ 1 MV/m and a characteristic length L _z of one meter, we find that /i _mn ~1(G ⁴² . Therefore, in a GW generator relying solely on toroidal EM waves, an electric field of about 10 ¹⁰ V/m is required to beat the efficiency of the proposed model of generator with an external magnetic field of 10T.

One could turn to extreme power laser pulses, with electric fields as high as 10 ¹⁵ V/m over a surface of 1 cm ² , to generate gravitational perturbations. However, EM pulses alone do not excite the transverse GW modes but only longitudinal ones (see also [21 , 22] and Eq.(12) below). Another possible GW generator could be to shoot extremely powerful laser pulses into a transverse (pulsed) magnetic field. However, since the size of the wave packet is only a few wavelengths long, g _z ~10 ⁴¹ for E ₀ = 10 ¹⁵ V/m, B ₀ = 100T and L _z ~ 10 ⁶ m. Therefore, using a waveguide or a resonant cavity inside a DC high-field magnet seems to us the best choice for the electromagnetic generation of GWs, with higher efficiency g _z but also offers the possibility to have a continuous emission for long durations.

Of course, metric perturbations are not directly observables, but any gain in their amplitude will increase physical effects like the energy and momentum carried away by the emitted GWs. Indeed, the emitted GWs can be in principle indirectly detected through the energy loss in the generator. The loss corresponds to the energy carried by the fleeing GWs (see for instance [23, 24]) and is of order of magnitude

For generators using petawatt laser pulses of E ₀ ~ 10 ¹⁵ MV/m into intense pulsed magnetic fields of BO ~ 100T, one gets that the power loss is of order 10 ³¹ W per laser shot. However, due to the very short time scales of the GW emission duration, this might be very hard to detect. For this reason, one may wish to adopt another strategy by using a continuous emission of GW, which can be carried out in a generator constituted of a standing wave immersed into a static intense magnetic field.

A generator of size L = 10m containing a standing wave of peak electric field E ₀ = 10 ⁶ V/m put into an external magnetic field B ₀ of 10T will produce GWs of amplitude of order 10 ³⁹ and a power loss of 10 ²³ W due to the fleeing GWs. It is worth noticing that the somewhat similar experimental design of a microwave cavity in a magnetic field used in the ADMX experiment searching for axions has a sensitivity of 10 ²⁴ W over a time integration of 10 ³ s [25]. Future work will particularly focus on modeling the ADMX experiment with the results developed here to explore its energy loss through GW emission.

We can now particularize this discussion to different types of EM standing waves in waveguides and hollow cavity and find the associated gravitational radiation polarizations and propagation.

We consider a TEM standing wave arising from the superposition of two progressive waves counter-propagating along the z-direction:

This TEM standing wave is assumed here immersed into a purely transverse static magnetic field (B _z = 0):

p( ^c ) (°) __ D

— y

p (c),(0) _ G)

JL

This gives rise to the following source of the direct Gertsenshtein effect Eq.(3)

Given the above structure of , the resulting metric perturbations must have the form:

We can now apply the above developments to two specific TEM waveguides:

(i) two coaxial cylinders and

(ii) two embedded conducting boxes,

both in an external transverse magnetic field. We assume the edges of the waveguides and cavities to be perfect conductors. We also consider the TEM waveguides are aligned along the z-axis, of length L _z , and located between z = -L _z /2 and z = +L _z /2.

Following [18, 19], we have the following field configurations for a pure monochromatic mode of the EM fields

with (E, B) 1 (x, y) stands for the transverse components of the electric (magnetic) field, k the wavenumber, w = 2 pn the pulsation of the wave with frequency v and a +(-) sign in the exponential indicates a wave propagating forward (backward) the z-axis. For TEM waves, the dispersion relation is as in vacuum, k = w/c, and the transverse magnetic field inside the waveguide is given by

In this case, Maxwell equations reduce to the two dimensional Laplace equations:

so that both E _± and B _± derive from a scalar potential i/>(x, y) also satisfying 2D Laplace equation The boundary conditions for perfectly conducting surfaces, E _h = B _± = 0 are fulfilled in the TEM mode when y = cst in both the inner and outer conductors.

For the simple case of a TEM waveguide made of two embedded cylinders of inner (outer) radius R _in (R _out ), the solution of Maxwell equations with the above boundary conditions for the standing TEM waves is given by, in cylindrical coordinates (r, Q, z), (see also [19]):

where E ₀ is the electric field amplitude on the inner conductor at r = R _in and K = 2p/l is the wavenumber of the TEM standing wave. The above standing waves are formed from the superposition of two sinusoidal pure modes counter-propagating in the z direction. We can now consider the direction of the external static magnetic field as our x-axis so that B _x = B ₀ , B _y = 0 and the source of GWs Eqs.(11 ) now becomes

TOO =

T _l 1 722

712 = 721 (15) and E _x,y are obtained from Eq.(14). This generator therefore produces GWs with polarizations h _^ + h _® = -h _xx .

A schematic view of this GW generator is given in Figure 1. This figure illustrates a schematic three-dimensional view of a gravitational wave generating device 1 according to an embodiment of the invention. This device 1 comprises an open cavity 4 consisting of an elongated transverse electromagnetic waveguide extending along a longitudinal axis Z. The open cavity 4 is made of two coaxial embedded elongated perfectly conducting elements of length L _z : an outer conducting cylinder 5 of inner radius R _out and an inner conducting cylinder 6 of inner radius R _in . The cavity 4 is immersed into an external static magnetic field B which is generated in a predetermined direction 2 perpendicular to the longitudinal axis Z, aligned with axis X. The open cavity 4 is configured for confining transverse electromagnetic standing waves that are propagating within the open cavity 4, coaxially to the longitudinal axis Z, with a predetermined frequency and a predetermined amplitude. The open cavity 4 is considered with constant circular transverse cross-sections 8. A transverse slice of the electromagnetic energy density inside the cavity 4 is represented on a circular transverse cross-section 8 for a specific transverse electromagnetic mode. It is shown that said electromagnetic energy density is maximal at the inner conducting cylinder 6.

We now present in more details the gravitational field perturbations generated by such a TEM waveguide inside a DC high-field magnet.

Figure 2A illustrates the longitudinal two-dimensional slice y = 0 of metric perturbations h ₁₂ = h _xy (in unit of g _z ) at a given time, around the open cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 with parameters Rin = 0.2, Rou _t = 1 and the outgoing gravitational radiation wavelength A _GW = 2L _Z . A circle indicates the position of the maximum of the wave pattern.

Figure 2B illustrates the transversal two-dimensional slice z = 1 of metric perturbations h ₁₂ = h _xy (in unit of g _z ) at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 with parameters Rin = 0.2, Rou _t = 1 and the outgoing gravitational radiation wavelength AGW = 2L _Z . A circle indicates the position of the maximum of the wave pattern.

Figure 2C illustrates the transversal two-dimensional slice z = 1 of metric perturbations hn = hxx (in unit of g _z ) at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 with parameters Rin = 0.2, Rou _t = 1 and the outgoing gravitational radiation wavelength AGW = 2L _Z . A circle indicates the position of the maximum of the wave pattern.

Spatial dimensions are given in units of L _z , the length of the waveguide. The GW emission appears anisotropic and a privileged direction is given by the direction of the external magnetic field B : the extrema of h _xy = h ₁₂ points in the direction of the external field B while those of h _xx = h-p lie in direction perpendicular to B (the x-axis by assumption here). It should be also noticed that the cone of emission is rather large around the direction of the external magnetic field B.

Figure 3 illustrates the propagation of metric perturbations h ₁₂ = h _xy along the direction of the generator at different times in the interval of the wave period. More precisely, figure 3 illustrates four one dimensional graphs 12 describing the propagation of metric perturbations h ₁₂ = h _xy measured in unit of Gertsenshtein-Zeldovich number on the axis 1 1 along the longitudinal direction defined by x = 1 .5, y = 0, for a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 with parameters R _in = 0.2, R _out = 0.5 and the outgoing gravitational radiation wavelength A _G w = 2L _Z . The lines 10 illustrate the position of said generating device along said longitudinal direction.

The wavelength and frequency of the continuously emitted GWs are the same as the one of the EM standing wave inside the generator. This is due to the resonance mechanism, since the coupling between the variable EM field and the external magnetic field is linear (see the source Eq.(11 )). The GWs which are due to EM standing wave alone (see Eq.(2)) have a period which is double the one of the EM wave, because their source term Eq.(5) is quadratic in the EM field.

While the central wave pattern immediately in front of the generator is that of a stationary wave (as already found in [5]), the outgoing GWs quickly fade away with inverse power of the distance as the progressive waves propagate.

However, due to rotational continuous symmetry of the waveguide around its axis, it is not possible to completely disentangle all the emitted polarizations as can be seen from Eq.(15). Since choosing both the direction of emission and the emitted polarizations is important for detection and other practical applications, we can propose the following modified design.

We now consider a TEM waveguide with rectangular transverse sections aligned along the x and y-axis and of outer (inner) length(es) L _{x y} (lx, _y ). The transverse external magnetic field B now have two independent components B _{x y} . This simple set-up allows more freedom in the emitted polarizations, according to Eqs.(11 ). Therefore, using such a rectangular cross-section has two advantages:

(i) it will become possible to excite the different GW polarizations more independently and

(ii) directing the GW emission can be made by simply changing the relative orientation of the external magnetic field and the waveguide.

In the previous design with a circular cross-section, it was necessary to rotate the outer DC magnet if one wants to change the direction of emission, which can be quite inconvenient in practice due to the size and weight of such electromagnets. A schematic view of the generator made of a TEM rectangular waveguide coupled to an external magnetic field is given in Figure 4. This figure illustrates a schematic three-dimensional view of a gravitational wave generating device 1 according to an embodiment of the invention. This device 1 comprises a cavity 4 consisting of an elongated transverse electromagnetic waveguide extending along a longitudinal axis Z. The cavity 4 is made of two coaxial embedded elongated perfectly conducting elements of length L _z : an outer conducting rectangular box 5 and an inner conducting rectangular box 6. The cavity 4 is immersed into an external static magnetic field B which is generated in a predetermined direction 2 perpendicular to the longitudinal axis Z. The cavity 4 is configured for confining transverse electromagnetic standing waves that are propagating within the cavity 4, coaxially to the longitudinal axis Z, with a predetermined frequency and a predetermined amplitude. The cavity 4 is considered with constant rectangular transverse cross-sections 8 of outer length L _x and L _y , and inner length l _x and l _y . A transverse slice of the electromagnetic energy density inside the cavity 4 is represented on a rectangular transverse cross-section 8 for a specific transverse electromagnetic mode. It shows edge effects around the inner conducting rectangular box 6.

Figure 5 illustrates a distribution of a transverse electric field E _± 7 A from a numerical resolution of the Laplace equation in a two-dimensional transverse cross-section 8 of the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with parameters L _x = L _y = L _z = 1 and L _x /I _x = L _y /I _y = 3. The contour levels 7B of the norm of the transverse electric field E _± 7A are also represented.

The above mentioned distribution can be put into Eqs.(7) and (1 1 ) together with the assumption of a static uniform transverse magnetic field B to find the metric perturbations around this generator. Figures 6A, 6D and 6G illustrate the transversal two-dimensional slice z = 0 of emitted gravitational waves h ₀ o, h-n and h ₁₂ (in unit of g _z ) respectively, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with parameters L _x = L _y = L _z = 1 and L _x /I _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to 0.45 and 0.89 respectively. Those parameters have been given for the sake of illustration.

Figures 6B, 6E and 6H illustrate the transversal two-dimensional slice z = 0 of emitted gravitational waves h ₀ o, h-n and h ₁₂ (in unit of g _z ) respectively, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with L _x = 3, L _y = L _z = 1 and L _x /I _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to 0.45 and 0.89 respectively.

Figures 6C, 6F and 6I illustrate the transversal two-dimensional slice z = 0 of emitted gravitational waves h ₀ o, h-n and hi ₂ (in unit of g _z ) respectively, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with L _x = 6, L _y = L _z = 1 and L _x / l _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to 0.45 and 0.89 respectively.

Figures 6J, 6M and 6P illustrate the longitudinal two-dimensional slice y = 0 of emitted gravitational waves h ₀ o, h-n and h ₁₂ (in unit of g _z ) respectively, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with parameters L _x = L _y = L _z = 1 and L _x /I _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to 0.45 and 0.89 respectively. Figures 6K, 6N and 6Q illustrate the longitudinal two-dimensional slice y = 0 of emitted gravitational waves h ₀ o, h-n and h ₁₂ (in unit of g _z ) respectively, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with L _x = 3, L _y = L _z = 1 and L _x /I _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to 0.45 and 0.89 respectively.

Figures 6L, 60 and 6R illustrate longitudinal two-dimensional slice y = 0 of emitted gravitational waves h ₀ o, hn and h-i ₂ (in unit of g _z ) respectively, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with L _x = 6, L _y = L _z = 1 and L _x /I _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to 0.45 and 0.89 respectively.

We now comment additionally figures 6A-R. For L _x = 1 , the wave pattern looks like the one generated with a cylindrical waveguide (see figure 2), except that the GW polarizations are now excited differently. The direction of the GW emission now depends on the relative orientation of the external magnetic field and the waveguide transverse axis. It is therefore possible to direct the emission of the different GW polarizations by simply rotating the waveguide inside the external magnetic field during the functioning of the generator. In addition, increasing the size of the generator not only modify the shape of the wave patterns (see figures 6A- I) but also increases the amplitude of the emitted GWs, since the energy stored in the waveguide is more important (the integration in Eq.(7) is then performed over a larger volume). For instance, in figures 6A-R, the amplitude of metric perturbations has increased by roughly a factor 5 when the transverse length of the waveguide L _x has been increased from 1 to 6. Let us now consider another GW generator alternatively to the invention made of a hollow resonant cavity immersed into an external static magnetic field B. The cavity is assumed filled with TM modes, with EM fields

where I, j, k = 1 , 2, 3 and e _ijk is the Levi-Civita symbol. The source terms for the wave resonance Eq.(4) are given by

In the Lorenz gauge, the linearized Einstein equations decouple so that the metric perturbation tensor H _mn must have the same algebraic structure as the source tensor Eq.(16). Consequently, 8 different GW polarizations, including longitudinal ones, can be excited in the chosen gauge with this generator design.

Let us now consider a special case with a hollow cavity made of two connected open cylinders, namely the interior of the cavity is the region defined in cylindrical coordinates by r e[R _in , R _out ], 0 e[-0/2, Q/2] and z e [— L _z /2, L _z /2] For the sake of simplicity, we will restrict ourselves to a external magnetic field oriented along the longitudinal axis of the cavity: B ^T = (0,0, B _z ). A schematic view of this GW generator using TM wave resonance is given in figure 7 for R _in = 0.2, R _out = 1 and Q = TT/3. In particular, this figure 7 illustrates a schematic three-dimensional view of a gravitational wave generating device 1 alternatively to the invention. We use the same reference as the one for the invention given that this GW generator has also the same general structure described in the preamble of the further explicit claim 1 . This device 1 comprises a cavity 4 consisting of an asymmetric elongated electromagnetically resonant cavity extending along a longitudinal axis Z. The cavity 4 is made of an outer elongated perfectly conducting element 5 of length L _z . This outer element 5 comprises two portions of cylinders 9A and 9B. The cavity 4 is immersed into an external static magnetic field B which is generated in a predetermined direction 2 parallel to the longitudinal axis Z. The cavity 4 is configured for confining transverse magnetic standing waves that are propagating within the cavity 4, coaxially to the longitudinal axis Z, with a predetermined frequency and a predetermined amplitude. A transverse slice of the electromagnetic energy density inside the cavity 4 for a particular field mode is represented on a transverse cross-section 8 for a specific transverse magnetic mode.

For hollow cavities with a uniform cross-section along their longitudinal axis z, TM/TE monochromatic modes are decomposed along transverse and longitudinal components as in Eq.(13) and the transverse components

(£, 33)JL

must satisfy the following two-dimensional Helmholtz equation (see [18, 19])

with boundary conditions for perfectly conducting surfaces: in = ¾ _j _ = 0

The EM fields are therefore eigenfunctions of the two-dimensional laplacian and the dispersion relation w = co(k) can be obtained from the resolution of the related eigenvalue problem. Pure monochromatic TM standing waves in a hollow cavity made of two connected open cylinders can be obtained from the longitudinal component:

with I > 0 some integer and where J _n (r), Y _n (r) are Bessel functions of the first and second kind, respectively. In the above equation, the constants C _mn provide normalization, for instance they are such that

in the cavity while the constants A _mn and a _mn are given by the boundary condition

From those constants and the Helmholtz equation above, one has the following dispersion relation

where the wavenumber k has been replaced by the value 2pI/I_ _z . The other components of the EM fields in the TM polarization can be obtained as with k = 2pI/I_ _z and where a subscript l still indicates transverse components (see also [18]). It can be verified that Eqs.(17,19) are solutions of Maxwell equations with the appropriate boundary conditions on the surface of the hollow cavity. Eqs.(17,19) can be combined with the assumption of uniform static external magnetic field B _z = cst to assemble the source terms Eqs.(16). Under the assumptions of external longitudinal magnetic field B _± = 0, the GW polarizations that are excited by the TM cavity have the form, in the Lorenz gauge,

Figures 8A-D illustrate the transversal two-dimensional slice z = 1 of metric field perturbations h _0i , h ₀₂ , h ₁₃ and h ₂₃ respectively, at a given time, around the cavity of a gravitational wave generating device illustrated in Figure 7 with parameters R _in = 0.2, R _out = 1 , Q = TT/2 and n = I = 1 , where the components of the vectors B along X and Y are substantially equal to zero.

Figures 8E-H illustrate the transversal two-dimensional slice z = 1 of metric field perturbations h _0i , h ₀ 2, hi ₃ and h ₂₃ respectively, at a given time, around the cavity of a gravitational wave generating device illustrated in Figure 7 with parameters Ri _n = 0.2, R _out = 1 , 0 = p/3 and n = I = 1 , where the components of the vectors B along X and Y are substantially equal to zero.

Figures 8I-L illustrate the longitudinal two-dimensional slice y = 1 of metric field perturbations h _0i , h ₀ 2, h ₁₃ and h ₂₃ respectively, at a given time, around the cavity of a gravitational wave generating device illustrated in Figure 7 with parameters Ri _n = 0.2, R _out = 1 , Q = p/2 and n = I = 1 , where the components of the vectors B along X and Y are substantially equal to zero.

Figures 8M-P illustrate the longitudinal two-dimensional slice y = 1 of metric field perturbations h _0i , h ₀₂ , h ₁₃ and h ₂₃ respectively, at a given time, around the cavity of a gravitational wave generating device illustrated in Figure 7 with parameters R _in = 0.2, R _out = 1 , 0 = p/3 and n = I = 1 , where the components of the vectors B along X and Y are substantially equal to zero.

Besides of exciting different GW polarizations, the generator presented here produces different radiation patterns and more focused directions of emission compared to the generators presented before. This establishes that directed emission of GWs can be obtained with asymmetrical resonant cavities like the one in figure 7 with 0 < 2p. The amplitude of the emitted GW are of order of the Gertsenshtein-Zeldovich number Eq.(8) up to some geometrical factor depending on the transverse size of the GW generator. Other GW polarizations can be excited by introducing a transverse component of the external magnetic field (B _x ,y ¹ 0). This allows emitting specific GW polarizations to be detected by the technique developed hereafter as part of the invention, which is sensitive to the incoming GW polarizations.

Metric quantities in general relativity are not observables, since they are not gauge invariant. This is why the quantities presented before were given for the sake of illustration. Flowever, observable quantities can be built upon these metric perturbations and their derivatives. Of course, metric perturbations are not observable quantities, although any gain in their emitted amplitude will increase detectable physical effects like the energy and momentum carried away by the emitted GWs. This can be estimated from the energy-momentum tensor of the gravitational field ΐ _mu which, for traceless metric perturbations verifying Lorenz gauge condition such as those considered here, takes the form

where <...> denotes some average in space-time around the observing point located outside of the GWs source (see for instance [23]).

Let us first estimate the energy loss in the generator due to the emission of GWs. The loss corresponds to the energy carried away by the fleeing GWs (see also [24]) which we estimate by computing the radial energy flux of these waves over a sphere large enough to encompass the GW generator [ > with r denoting the radius of the encompassing sphere. If we consider r ~ L _z and

(with L _z the characteristic length of the generator), we find the following order of magnitude for the power loss by emission of GWs:

If we consider a generator using petawatt laser pulses of intensity E ₀ ~10 ¹⁵ V/m and length L _z ~10 ⁶ m into intense pulsed magnetic fields of B ₀ ~100T , one gets that the power loss is of order 10 ³¹ W per laser shot. In addition, the duration of the GW emission, typically the traveling time of the pulse into the external magnetic field, is very short. Therefore, this seems very hard to detect. It seems better to adopt another strategy by using a continuous emission of GW, which can be carried out in a generator constituted of a standing wave immersed into a static intense magnetic field. A generator of size L=10m containing a standing wave of peak electric field E ₀ ~10 ⁶ V/m put into an external magnetic field B ₀ of 10T will produce GWs of amplitude of order G _z = 10 ³⁹ and will undergo an instantaneous power loss of 10 ²³ W due to the fleeing GWs.

It is worth noticing that the close experimental design of a microwave cavity in an external static magnetic field used in the ADMX experiment searching for axions has a sensitivity of 10 ²⁴ W over a time integration of 10 ³ s [25]. Future work will particularly focus on modeling the ADMX experiment with the results developed here to characterize its energy loss through GW emission.

Then, the energy is obviously not radiated isotropically from our generators and we can use equations ( ^* ) and ( ^** ) to compute the gravitational radiation power pattern, in the same way this is done for a true electromagnetic antenna [27] To do this, we compute the gravitational radiation intensity Eq.( ^** ) on a sphere of large radius to obtain the angular distribution of gravitational energy radiation. The energy fluxes to _o in Eq( ^* ) are obtained by averaging the outgoing GWs over one time period, since the EM standing waves used in the generators are pure harmonic modes of pulsation co that subsequently produce monochromatic GWs of same frequency. The gravitational radiation power patterns in the region z>0 for different generators are given in figures 12A-D. These power patterns are symmetrical in the z<0 direction (not illustrated).

Figure 12A illustrates a three-dimensional view of a gravitational radiation power pattern, at a given time, around the open cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 1 with parameters Ri _n = 0.1 , R _out = 1 , and where the components of the vectors B along Y and Z are substantially equal to zero.

Figure 12B (respectively, 12C) illustrates a three-dimensional view of a gravitational radiation power pattern, at a given time, around the cavity of a gravitational wave generating device according to the embodiment of the invention which is illustrated in Figure 4 with L _x = 1 (respectively, L _x = 3), L _y = l_z = 1 and L _x /I _x = L _y /I _y = 3, where the components of the vectors B along X and Y are substantially equal to V2/2.

Figure 12D illustrates a three-dimensional view of a gravitational radiation power pattern, at a given time, around the cavity of a gravitational wave generating device illustrated in Figure 7 with parameters Ri _n = 0.5, Rou _t = 1 , Q = TT, and where the components of the vectors B along X and Y are substantially equal to zero.

The GW emission from our EM devices is clearly directional. For TEM wave resonance generators, the directions of maximal GWs emission are orthogonal to the orientation 2 of the external magnetic field and directed toward the axis of the TEM waveguide. In order to change the direction of maximum GW emission, one has to rotate the transverse external static magnetic when using a generator with cylindrical TEM waveguide. For a generator with a TEM waveguide with rectangular cross- sections, one has simply to rotate the rectangular waveguide inside the external magnetic field to change the direction of maximal GW emission. This gives a clear practical advantage for the latter design. Increasing the transverse length of the TEM waveguide, as is done in Figure 12C, deforms the lobes of emission, with an increase of the radiation intensity in the direction of the longer transverse dimension. The gravitational radiation power pattern of a generator with a TM half-cylinder hollow cavity in a longitudinal static magnetic field is illustrated in Figure 12D: it shows four major emission lobes and four minor lobes around the direction of the external longitudinal magnetic field. The number of lobes, their orientation and their thickness all depend on the aperture Q of the open-cylindrical TM cavity.

We now focus on the question of the detectability of the GWs produced by the EM generators according to the present invention. We now propose a novel detection method based on magnetic energy storage.

The principle is as follows. Some GW passing through a static magnetic field B ⁽⁰⁾ will locally modify the geometry, which results in a locally varying magnetic flux and the emission of EM waves A^ This is sometimes dubbed the inverse Gertsenshtein effect, although unfortunately discarded as being "hardly of interest” by Gerstenshtein himself in his original paper. This is the conversion of GWs into EM ones we will review below. The energy stored in the region covered by the magnetic field B ⁽⁰⁾ is therefore modified by the passing GW. Indeed, the total magnetic field is the superposition of the static one B ⁽⁰⁾ and the induced one B ⁽¹⁾ and the total EM energy density is quadratic in the norm of the total magnetic field B ⁽⁰⁾ + B ⁽¹⁾ . When a GW induces an additional magnetic field in the magnet, the total energy stored by the magnetic field therefore varies by roughly an amount

with V _det the volume of the detector. Therefore, some large region of strong magnetic field can be used as a GW detector through monitoring precisely the amount of energy stored in the magnet. This could be done for instance using superconducting magnetic energy storage technology [26], with special attention paid on the high precision measurement of the discharge current when the unit is emptied.

We derive the inverse Gertsenshtein effect starting with the second group of covariant Maxwell equations n _m = 0 perturbed at first order in the metric:

If we now set R _mn and if we focus solely on the first order corrections , the previous equation simply reduces to neglecting higher order terms, given that /i _mn is traceless and satisfies the Lorenz gauge condition, and considering that zeroth order Fadaray tensor components verifies Maxwell equation on Minkowski background Equation (22) shows us that the GWs /i^ can combine with background EM fields to constitute an effective 4-current density ; ^v _eft that generates weak EM fields . We will solve Eq.(22) under its wave

= d _m A ^ - d _v A ^ and assuming Lorenz gauge

□ ² A » = n„]f· (23)

We will consider that the incoming GWs are passing by some region filled with a static uniform magnetic field B ⁽⁰⁾ , and we will use the induced variation of the magnetic energy as a way to detect GWs.

Let us now assume that the incoming GWs are under the polarizations Eq.(12) such as those generated by the TEM wave resonance mechanism. The effective current

densities in Eq.(22) are given by

From Eqs.(24), one can therefore decompose the induced vector potential as following and choose an appropriate electromagnetic gauge in which A^ both vanish. This shows that only transverse components A^ are remotely induced by the TEM wave resonance mechanism.

In the case where the incoming GWs are under the polarizations Eq.(20) as generated by the TM wave resonance mechanism, the effective current densities in Eq.(22) are given by

From Eq.(23), we obtain that the induced EM fields are given by where the factor a!, ¹ ' is dimensionless and is solution of the wave equation

The source term in the above equation ϋ _m is of order unity from Eqs. (24,25), and so does The wave equation Eq.(27) will be solved for some volume filled with a uniform magnetic field B by using a spectral method based on Fourier Transform. More precisely, if we expand over a basis of plane waves as aW(B. T) « ; v _¾ (T) cxp

J?

with ά _{m c} (T) the Fourier transform of the field (which can be approximated for efficiency by a Fast Fourier Transform), then the wave equation Eq.(27) becomes a set of ordinary differential equations describing each Fourier coefficient as a forced harmonic oscillator:

where the Fourier transform of the source term ϋ _m · Those source terms are related to the gradients of the metric perturbations (see Eqs. (24,24)) and can be obtained by differentiating Eq.(7). As a matter of boundary conditions, we will assume a simple case where the magnetic field B used for detection is switched on at time T = 0, so that

With all these elements in hand, we can now propose an experimental concept for emission and reception of GWs in a laboratory. The emitter will be given by one of the designs presented previously either using TEM or TM wave resonance. For the receiver, we consider some separate region filled by strong static magnetic field B, like the interior of a solenoid, in which the GWs generated by the emitter will induce EM fields A^ We assume the magnetic field B ⁽⁰⁾ is constant over the volume of the receiver for the sake of simplicity and aligned so that a specific GW polarization is detected, as we show below. Figures 9 and 10 illustrate two experimental concepts, with specific alignments of the magnetic fields in the emitter and the receiver.

Figure 9 illustrates a schematic three-dimensional view of a gravitational wave generating device 1 and a gravitational wave detecting device 21 according to an embodiment of the invention. The gravitational wave generating device 1 is similar to the one which is represented on figure 1 where the cavity 4 is lying inside a solenoid 3, said solenoid 3 generating the external static magnetic field B in the predetermined direction 2, perpendicularly to the longitudinal axis Z. The gravitational wave detecting device 21 comprises a superconducting solenoid 23 defining a detection region 24 configured for detecting gravitational waves emitted by the generating device 1 with propagation axis passing through the detection region 24. The solenoid 23 produces a strong magnetic field B in a direction 22, transversally to the propagation axis of the emitted gravitational waves, the strong magnetic field B immersing the detection region 24. As a consequence, the strong magnetic field B interacts with the incoming gravitational waves to produce electromagnetic waves A ^{(1 )} which is accumulated with time in the detection region 24 given that remotely induced electromagnetic waves can be produced continuously into the detecting device 21 as long as the strong magnetic field B and the gravitational waves source are maintained. This can be detected through the variation of magnetic energy stored in the detection region 24.

Figure 10 illustrates a schematic three-dimensional view of a gravitational wave generating device 1 and a gravitational wave detecting device 21 according to an embodiment of the invention. The gravitational wave generating device 1 is similar to the one which is represented on figure 7 where the cavity 4 is lying inside a solenoid 3, said solenoid 3 generating the external static magnetic field B in the predetermined direction 2, parallel to the longitudinal axis Z. The gravitational wave detecting device 21 comprises a superconducting solenoid 23 defining a detection region 24 configured for detecting gravitational waves emitted by the generating device 1 with propagation axis passing through the detection region 24. The solenoid 23 produces a strong magnetic field B in a direction 22, transversally to the propagation axis of the emitted gravitational waves, the strong magnetic field B immersing the detection region 24. As a consequence, the strong magnetic field B interacts with the incoming gravitational waves to produce electromagnetic waves A ^{(1 )} which is accumulated with time in the detection region 24 given that remotely induced electromagnetic waves can be produced continuously into the detecting device 21 as long as the strong magnetic field B and the gravitational waves source are maintained. This can be detected through the variation of magnetic energy stored in the detection region 24.

In particular, the GW detector of the invention is also able to detect GWs from the GW generator as described with regard to Figure 7.

We propose to detect the incoming GWs through the variation of the magnetic energy stored in the receiver. Neglecting the contribution of the induced electric field E ^{(1 )} , the magnetic energy stored in the volume of detection V _det is given by where the total magnetic field in the detector B _tot is the superposition of B ⁽⁰⁾ and the magnetic field B ^{(1 )} induced by the passing GWs:

The absolute variation of magnetic energy in the receiver when the GWs is passing is given by

at first order in || B ⁽¹⁾ ||. The absolute variation of stored magnetic energy is therefore proportional to the average of the induced magnetic field II B ⁽¹⁾ II over the volume of detection. Since

and considering Eqs. (24-26), we have that

(up to the factor a _m ~ 0 1)) We finally find that the absolute variation in the stored magnetic energy is of order of magnitude: However, while the detector magnetic field B ⁽⁰⁾ is constant, the induced magnetic field B ⁽¹⁾ is variable, and so does the energy variation AE. Since the induced EM fields have the same frequency w _ann as the passing GWs (see Eqs. (23-25)), the number of induced photons in the receiver is giv

since w = ^-~ 2nc/L _z . If we assume V _det * L |, the number of induced

^GW

photons N _g in the receiver is of order 1 for L _z = 10m, E ₀ = 10 ⁶ V/m, B ₀ = SW = 14T by using Eq.(8).

The reasoning detailed just above supposes the average of B ⁽¹⁾ is constant over the volume of the receiver and of order of magnitude B ⁽⁰⁾ V¾. In practice, this average varies with time and the receiver is located some distance d away from the emitter, so that the local amplitude of the passing GWs is actually smaller, of order g _z L _z /d.

To be more precise, let us now present the results of a full numerical computation that uses the GWs generated by the emitters presented above and the numerical resolution of the 3D wave equation Eq.(27) for the induced EM field in a distant receiver. We consider two possible experimental realizations for these simulations:

(a) a first situation (illustrated in figure 9) involving a generator made of a cylindrical TEM waveguide in a transverse magnetic field, with parameters R _in = 0.2, R _out = 1 and l = L _z /2;

(b) a second situation (illustrated in figure 10) involving the open cylinder TM cavity in a longitudinal magnetic field with parameters Ri _n = 0.2, R _out = 1 , Q = p/2 and m = n = I = 1.

In both cases (a-b), the receiver is constituted by some volume filled with a constant magnetic field S aligned along the x-axis and put at some distance from the emitters (see figures 9 and 10). The receiver will therefore amplify the component S _¾ ⁽¹⁾ of the induced magnetic field, given by the potentials A _y . which are generated by J _y,z . For the TEM wave resonance generator (a), we therefore only need to compute d _z h _xx from Eq.(24) while for the TM wave resonance generator (b), we need to compute d _{y z} h ₂₃ from Eq.(25). Finally, in experiment (a) we place the receiver at some distance along the axis y and in experiment (b) at some distance along axis z (see also figures 9 and 10). The relative positions of the emitter and the receiver influence the shape of the gravitational wavefront that enters the receiver, hence impacting the volume integral of

The numerical resolution of the 3D wave equation Eq.(27) is made with Fourier method so that the gradients of A^ can easily be computed and then averaged to evaluate the energy variation DE from Eq.(29). This numerical computation allows us to compute the average of II B ⁽¹⁾ II over the volume of the receiver:

such that the absolute variation of magnetic energy stored in the receiver is given by

The numerical results of the time evolution of (II B ⁽¹⁾ II ) measure on axis 32 for both experiments (a, b) are given in figure 1 1. In other words, the graph 33 describes the time evolution, measured in unit of ct/Lz on the axis 31 , of the magnetic energy density induced in the detection region of a gravitational wave detecting device according to the embodiment of the invention which is illustrated in Figure 9. The graph 34 describing the time evolution, measured in unit of ct/L _z on the axis 31 , of the magnetic energy density induced in the detection region of a gravitational wave detecting device according to the embodiment of the invention which is illustrated in Figure 10. These illustrative numerical results depend on the relative position of the emitter and the receiver.

In Figure 1 1 , one can see that the absolute variation of magnetic energy stored in the receiver undergoes oscillations with increasing amplitude as the energy induced by the passing GWs is accumulated in the receiver. The estimation Eq.(30) therefore gives a good estimation for the order of magnitude of the energy variation in the receiver.

In other words, the present invention relates to electromagnetic devices 1 , 21 and associated methods for the directional emission and the detection of gravitational waves. The gravitational wave generating device 1 consists of a cavity 4 carrying electromagnetic waves, which is immersed into an external static magnetic field appropriately oriented for boosting the emission of gravitational waves. It is possible to detect the generated gravitational waves, first through the related energy loss in the generating device 1 , and second through the electromagnetic fields that are remotely induced in a gravitational wave detecting device 21. This last device 21 comprises an electromagnet 23 producing a magnetic field which interacts with the incoming gravitational waves in such a way that a magnetic energy is accumulating in a detection region 24 defined by the electromagnet 23.

Advantageously, this invention allows:

to determine the shape of the gravitational radiation pattern, i.e. the angular distribution of the energy emitted under the form of GWs; to modify said gravitational radiation pattern;

to fix and change a particular direction of GW emission;

to fix and change the polarizations of the outgoing GWs;

to build a purely magnetic detection device that is sensitive to both direction and polarization of a passing GW.

In this detailed description, two specific GW generating devices were introduced: the one of the invention, i.e. a coaxial TEM waveguide inside an external magnetic field transverse to the axis of the waveguide, and an asymmetric TM/TE resonant cavity inside an external magnetic field longitudinal to the axis of the waveguide. In these two devices, the direction of maximum gravitational energy emitted as well as the outgoing GW polarization can be arbitrarily chosen, for example, by rotating the inner waveguide or the inner cavity with respect to the external magnetic field. The GW detector according to the invention works on the same principle. Indeed, by using an appropriate relative orientation between the incoming GWs with given polarizations and the static magnetic field of some magnetic energy storage device, it is possible to maximize the variation of the magnetic energy stored that is induced by the passing GW. Furthermore, the detection method is more sensitive than simple resonant cavities: while the energy variation in the first one is directly proportional to the GW amplitude, the energy variation in the second depends only on the square of the GW strain.

Although the present invention has been described above with respect to particular embodiments, it will readily be appreciated that other embodiments are also possible.

The term“comprising”, used in the claims, should not be interpreted as being restricted to the elements or steps listed thereafter; it does not exclude other elements or steps. It needs to be interpreted as specifying the presence of the stated features, integers, steps or components as referred to, but does not preclude the presence or addition of one or more other features, integers, steps or components, or groups thereof. Thus, the scope of the expression “a device comprising A and B” should not be limited to devices consisting only of components A and B, rather with respect to the present invention, the only enumerated components of the device are A and B, and further the claim should be interpreted as including equivalents of those components.