On the Theoretical Possibility to Generate Gravitational Waves Using Electromagnetic Waves

This paper demonstrates that according to a consequence of General Theory of Relativity, a high amount of electromagnetic waves, such as ultraviolet light, injected between two or more parallel reflective surfaces must generate through multiple reflections gravitational waves having a significant power which are radiated into space. The emitted gravitational waves have the same frequency as the incident electromagnetic waves and are directed along the normal to the reflective surfaces. This papers derives, an equation connecting the radiated gravitational power with the electromagnetic energy injected between reflective plates. The radiated gravitational power can be enhanced and focused on a point using multiple reflective calottes. Indexing terms/


INTRODUCTION
Based on the acceleration involved, the reflection of electromagnetic waves on a reflective wall (mirror) can be considered the most violent process in the Universe, which in addition involves a quadrupole-type oscillation of matter. Such type of oscillations must cause emission of gravitational waves.
In this article, the generation of gravitational waves during reflection of electromagnetic waves is deduced as a direct consequence of Einstein's General Theory of Relativity (GTR).
In addition, an intuitive demonstration of the possibility to generate gravitational radiation using electromagnetic waves is presented in appendix 1.

METHODS
For avoiding any ambiguity, Einstein's field equations and some important known demonstrations are summarized in the next section.

Einstein's field equations
Einstein field equations [1] are: where the indices of coordinates in the 4D space-time continuum, i and k take values from 0 to 3. The values 1, 2 and 3 indicate space coordinates (for example x1=x, x2=y, x3=z when using Cartesian coordinates and x1=r, i.e., radius, x2=θ, x3=φ when using spherical coordinates). The index value of zero is reserved for the temporal coordinate, which takes the form t c x 0   , where 'c' is the speed of light in vacuum and 't' is time, Rik, gik and ik T are tensors; ik R is the curvature tensor of second order or Riemann's tensor, the tensor that defines the space metric, ik g , is also called the metric or fundamental tensor, R is Ricci's scalar curvature of space and G is the universal constant from Newton's Law of gravitation.
It is known that 'contraction', (i.e., the multiplication of covariant field Eq. (1) with the metric tensor expressed in countervariant form, which is labeled g ik , followed by a summation), provides a new form of Einstein's field equations which is given below. [4] T c G 8 R 4   (4) where R is the invariant of curvature Rik (also known as Ricci's scalar curvature).
Notes: Equation (4) was derived taking into account that: 1) The definition of the counter-variant metric tensor g ik is based on the covariant metric tensor: As it can be seen from Eq. (3), the invariant T of the energy-momentum tensor Tik is null for a free electromagnetic field. Based on Eq. (4), the scalar curvature R is null, i.e., according to GTR, electromagnetic waves propagating freely through space do not generate gravitational fields. [8] The fact that electromagnetic waves, which propagate freely in space do not produce gravitational fields is a profound property of matter.

Demonstration
The possibility of generating gravitational waves using electromagnetic waves is demonstrated below.
Assume a packet of electromagnetic waves which freely propagate through space where there are no gravitational fields (see Figure 1). Assume this pack enters a massless sphere (see Figure 2). When the sphere's cover is closed, the waves are confined inside (see Figure 3). If the inner surface of sphere is perfectly reflective, an isotropic electromagnetic field is established inside the sphere in a short time (see Figure 4).  If the total energy of electromagnetic waves confined inside is E, then according to Einstein's formula, the relativistic mass of photons is: According to Newton, such a mass must produce a gravitational field with potential: where G is Newton's gravitational constant and r is the distance from sphere's center (r > r0, r0=sphere's radius).
On the other hand, according to Einstein's GTR, a concentrated mass M should modify the continuum space-time curvature; the space element being given by Schwarzschild's metric given below. [9] (7) where r, θ, φ are spherical coordinates, t is time, . In the Schwarzschild metric, the fundamental metric tensor is therefore: (8) Note: for a concentrated mass, the energy-momentum tensor takes the following form: (9) As it can be seen, all tensor components are null, except, T00 , which is equal to the energy density   2 c w (ρ is the mass density). We have obviously an unacceptable contradiction: On one hand, according to Eq.4 the electromagnetic waves confined inside the sphere should produce no space-time curvature but according to Eq. 7, the mass of electromagnetic waves must produce gravitational effects. The above contradiction can be eliminated only by admitting that electromagnetic waves confined inside the sphere produce gravitational fields only during reflection on the reflective wall of sphere. In fact, this contradiction was observed as early as 1913 during debates between Nordstrom and Einstein [10] The single explanation given at that time was that "the electromagnetic radiation enclosed within a massless box having reflective walls would not acquire a gravitational mass although that radiation would exert a pressure on the walls of the box. These walls would become stressed and, simply because of this stress, the walls would acquire a gravitational mass". Today, it is very clear for everybody that such a massless wall cannot be stressed simply because in the absence of mass, such a stress can have no physical sense. So, while a sphere of condensed matter generates gravitational field in every point, a massless sphere full with electromagnetic waves generates a gravitational field only on its surface i.e., obviously during reflection of electromagnetic waves by the sphere surface. For understanding further why a photon is able to generate during reflection a graviton, one should be aware that in a way, the reflection of a photon on a reflective surface can be considered the most violent process in Universe even more violent than the head on collision of neutron stars because the acceleration of matter during normal reflection has a huge value due to the high speed of light: where ΔV is speed variation, Δt is the duration of photon-surface interaction, c is the speed of light in vacuum, T the oscillation period of wave and  is the frequency of electromagnetic wave. During reflection, the matter of photon obviously oscillates as a quadrupole (see Figure 5) and for this reason it must generate gravitational waves (it was demonstrated that only a quadrupole and not a dipole can generate gravitational waves. [11] The direction of the electric field is perpendicular to the photon's travel direction. During reflection, the sense of the electric field is reversed and obviously the photon's matter also oscillates longitudinally due to the reflection process. A fundamental question arises: how to determine the direction of a graviton emitted during normal reflection of a photon by a reflective wall? Intuition would indicate the graviton is emitted in the direction of the incident wave. But the opposite is true. The inertia must oppose the cause, therefore the emitted graviton and incident photon must have opposite directions, see Figure 6. In this case, the graviton-photon interaction would tend to counter the deceleration process produced by reflection i.e., the graviton 'attempts to accelerate' the photon along the initial direction.
This conclusion is supported even by the laws of energy and momentum conservation. Assume the mass of the emitted graviton is mg. The energy balance in the reflection process is given by: where Eif is the energy of incident photon, Erf is the energy of reflected photon and Eg is the energy of graviton.
Based on Eq. (11), reflection causes a red shift i.e., the frequency of the reflected wave is smaller than the frequency of This red shift is obviously extremely low and cannot be observed visually. On the other hand, according to the law of momentum conservation: where if  is the frequency of incident photon, rf  is the frequency of reflected photon and mg is the mass of graviton.
Equation (12) shows that the direction of radiated gravitational wave is opposite to the incident electromagnetic wave and that the propagation speed of gravitational wave is c, fact, which was already demonstrated for weak gravitational fields. [12] The energy of a graviton emitted in this manner can be easily evaluated using dimensional analysis. The graviton energy should depend on Planck's constant, h, frequencies of incident and reflected electromagnetic wave, frequency of gravitational wave), c-speed of light and G -gravitational constant: where *  is a constant. One can easily find that  = 1,  = 2, i.e., The value Eq. (15) reveals a connection between radiated gravitational energy and electromagnetic energy: The radiated gravitational energy is proportional with the cube of electromagnetic energy, and the proportionality constant is which obviously can also be considered a universal constant.

The enunciation of conversion theorem and equation of the radiated gravitational power
According to the above demonstrations, the following theorem can be stated: 'During normal reflection of an electromagnetic wave on a reflective surface, electromagnetic energy is partially converted into gravitational energy. The frequency of the emitted gravitational wave is equal to the frequency of incident wave. The energy of emitted gravitational wave is proportional with the cube of frequency of electromagnetic wave. The direction of gravitational wave is opposite to that of the incident electromagnetic wave'.
Assume two packs of electromagnetic waves 1, 2 having the same frequency and continuously reflected by two parallel reflective surfaces (mirrors A, B) separated by a gap,   d (see Figure 7). Assume the incidence angle is zero, each pack has the same number of photons, n, and total energy E. Obviously such equipment continuously generates gravitational power because reflection of pack 2 on mirror A and of pack 1 on mirror B is instantaneously followed by reflection of pack 1 on mirror A and of pack 2 on mirror B.
The radiated gravitational power Pg is calculated dividing the right hand side of Eq. 14 by wave period c / T   and multiplying by n, the number of photons in a pack and by 2 because both packs are reflected simultaneously.
Eq. (16) shows that the radiated gravitational power generated by electromagnetic waves reflecting between two parallel mirrors separated by a gap  is proportional to the square of incident wave energy and its frequency. In addition, as Figure  7 shows, gravitational energy is directed along the normal direction pointing away from the mirror surface. If (q+1) reflective surfaces are placed at distances   d (see Figure 8), and 2 packs of electromagnetic waves with energy E are inserted between each pair of reflective surfaces, the total power of gravitational radiation increases by a factor of q.

The magnitude of radiated gravitational power produced in this way
Assume the reflectivity coefficient R=1 for all the (q+1) reflective layers from Figure 8. During reflection, the electromagnetic energy E is consumed. While the number of photons remains the same, the wave length must increase, i.e. the light enters the visible spectrum and then infrared spectrum, microwave and lower frequencies. For this reason, in practical applications, the distance d must exceed the wavelength of injected light which usually is ultraviolet light. When the wave length of electromagnetic waves reaches the maximum possible length allowed by the gap between mirrors, , the wave can no longer exist between the reflective surfaces and is absorbed by the electrons from the surface of mirror. Thus, the energy of remaining electromagnetic waves is converted into heat. This is the reason for which the equipment must be intensively cooled.
Eq. (17) shows that although (G/c 5 ) is an extremely small number, the quantity ( ) and for high amounts of electromagnetic energy (E >1kJ or more) injected between every pair of plates, the radiated gravitational power becomes significant (Table 1). Obviously the gravitational radiation becomes very intense if it is generated and focused by the equipment presented in Figure 9.
Intense research is necessary in solid physics and in the field of nanotechnology for finding reflective materials having high reflectivity coefficients. An intense technological research is also necessary for developing ultraviolet sources with very high power. The values of radiated power given in the table 1 show that in the future, gravitational radiation produced in this way could be used for practical applications.

Design of equipment which further concentrates the artificial gravitational radiation
Using the above results it is possible to further increase the intensity of radiated gravitational waves if the waves are focused in one point (see Figure 9). The equipment is made of (q+1) reflective calottes separated by a gap   d .The highest intensity of radiated gravitational field is obtained in the calottes center, C.
The existence of the gravitational force in Newton's sense can be tested using a Cavendish balance.

CONCLUSIONS
1-This paper demonstrates the theoretical possibility of generating gravitational waves using electromagnetic waves. To eliminate any possible mistake, this demonstration is strictly based on the accepted GTR paradigm.
2-To summarize , electromagnetic waves with high energy, which are reflecting between two or more reflective parallel surfaces (mirrors) can generate gravitational waves (radiation).
3-The radiated gravitational waves are propagating in the direction normal to the reflective surfaces.
4-The propagation sense of radiated gravitational waves is opposite to that of incident electromagnetic waves.

5-
The radiated gravitational power is significant if the gap between reflective parallel surfaces is comparable to the wave length of photons.
6-Multiple reflective calottes having a common center can be used for increasing the intensity of gravitational radiation. 7-Cavendish balances can be used for proving the emission of artificial gravitational waves produced by as discussed above.
8-During multiple reflections, the frequency of ultraviolet radiation, which is injected between plates, decreases progressively until becoming visible light, infrared or microwave radiation. Infrared or microwave radiation is absorbed by electrons from the reflective surfaces and transformed into heat. 9-The mastery of artificial gravitation generated in this way will be important for industrial, medical and transportation application.
10-Fundamental research in solid physics will be necessary for improving the reflectivity of materials. At the same time, intense technological research is necessary for developing sources of ultraviolet light with high power. 11-Generation of gravitational radiation for practical applications is a difficult objective and needs sustained research and important resources.

APPENDIX 1-A SIMPLIFIED DEMONSTRATION OF THE POSSIBILITY OF GENERATING GRAVITATIONAL WAVES USING ELECTROMAGNETIC WAVES
The starting point of the following demonstration is a GTR prediction validated by astronomical observations. According to GTR a system formed by two planets generates gravitational waves.
The approximate power of gravitational radiation generated by two planets having masses m1 and m2, respectively (see where G is the universal constant of gravitation and c is the speed of light. There are more accurate expressions for the radiated gravitational power [14,15], but simplicity makes Eq. (18) preferable here. Planets having identical masses equal to m, create gravitational waves, which are uniform and propagate through the space-time continuum as symbolically illustrated in Figure 11. For this case, Eq. (18) takes the following form: The minus sign indicates that the system losses energy through gravitational radiation. As a result of this continuous emission of energy in gravitational form, the distance d between planets decreases and as a result the pair of bodies collapses. [13] (Note: Due to the low amount of radiated power, the collapse process takes billions of years).
According to Einstein's formula, the relativistic energy of a body having mass m is 2 c m E   . Therefore, the Eq. (19) is equivalent to: If the two bodies of condensed matter moving on circular trajectories with velocity V are replaced by two photon packs, i.e., energy pulses having the same relativistic energy as the planets and moving with the speed of light on a circular trajectory having the same diameter, d (see Figure 12), the gravitational power radiated by these packs of photons must be far greater than the power radiated by the pair of planets because the frequency of rotation would obviously increase.
Assume that the electromagnetic waves from the two packs have the same frequency,  , and each pack has energy E.
Also assume the two photon packs are constrained to move on circular trajectories inside a fiber optic coil or wave-guide having diameter d, and the two packs always remain diametrically opposed, see Figure 13. It is important to note that a greater amount of electromagnetic energy E and a smaller diameter d boosts the power of gravitational radiation, |dE/dt|. Assume the coil of optic fiber is made from a perfectly flexible, transparent and nondispersive material, which therefore does not absorb electromagnetic energy and does not scatter the photon packs. Also assume photon packs with frequency i  and total energy Ei are periodically injected into this coil, which has an enormous number of spires ns having diameter d. If the injection period, T* is given by: