In the framework of the theory of informatons [1] - [5] , a gravitational field is a dual entity, having a field- and an induction component ( $E_g$ and $B_g$) simultaneously created by its common sources: time-variable masses and mass flows. It is the manifestation at the macroscopic level of a cloud of informatons: mass and energy less granular entities that—relative to an inertial reference frame—are moving with the speed of light and that are carriers of information referring to the position (“g-information”) and the velocity (“β-information”) of their emitter.

The Maxwell-Heaviside [1] equations, that describe how a gravitational field ( $E_g,B_g$) is generated and how it evolves in space and in time, imply fluctuations in its intensity that are propagating at the speed of light as waves outward from the source of that field.

More specifically, from the postulate of the emission of informatons [2] - [5] , it can be deduced that an accelerated mass particle is the source of a gravitational field of which the time dependent components of $E_g$ and $B_g$ represent waves that are traveling with the speed of light. It follows that a harmonically oscillating mass particle is the source of an harmonically gravitational wave transporting energy in the form of “gravitons”: quanta of energy carried by informatons.

A change of the rest mass of an object—what occurs during radioactive decay— is another source of a gravitational wave. Indeed, the change of the rest mass of an object immediately results in a change of the rate at which it emits informatons which gives rise to a disturbance of its gravitational field that propagates with the speed of light.

Let’s also mention that gravitational waves were first proposed by Oliver Heaviside [6] in 1893 and then later by Henri Poincaré [7] in 1905 as the gravitational equivalent of electromagnetic waves. More recently, attention was also paid to the phenomenon in the work of Oleg Jefimenko [8] .

In free space—where ${\rho }_G=J_G=0$—the Maxwell-Heaviside [1] equations are:

$div{E}_g=0$ (1)

$div{B}_g=0$ (2)

$rot{E}_g=-\frac{\partial B_g}{\partial t}$ (3)

$rot{B}_g=\frac{1}{c^2}\frac{\partial E_g}{\partial t}$ (4)

To attempt a solution of a group of simultaneous equations, it is usually a good plan to separate the various functions of space to arrive at equations that give the distributions of each.

It follows from (3):

$rot\left(rot{E}_g\right)=-rot\left(\frac{\partial B_g}{\partial t}\right)$ (3’)

Because [9]

$rot\left(rotF\right)=grad\left(divF\right)-{\nabla }^{2}F$, where ${\nabla }^2$ is the Laplacian,

(3’) leads to:

$grad\left(div{E}_g\right)-{\nabla }^2{E}_g=-rot\left(\frac{\partial B_g}{\partial t}\right)=-\frac{\partial }{\partial t}\left(rot{B}_g\right)$

And taking into account (1) and (4):

${\nabla }^2{E}_g=\frac{1}{c^2}\cdot \frac{{\partial }^2{E}_g}{\partial t^2}$ (5)

This is the general form of the wave equation. This form applies as well to the g-induction, as is readily shown by taking first the rotor of (4) and then substituting (2) and (3):

${\nabla }^2{B}_g=\frac{1}{c^2}\cdot \frac{{\partial }^2{B}_g}{\partial t^2}$ (5’)

Solutions of this equation describe how disturbances of the gravitational field propagate as waves with speed c.

To illustrate this, we consider the special case of space variation in one dimension only. If we take the x-component of (5) and have space variations only in the z-direction, the equation becomes simply:

$\frac{{\partial }^2{E}_{gx}}{\partial z^2}=\frac{1}{c^2}\cdot \frac{{\partial }^2{E}_{gx}}{\partial t^2}$

This equation has a general solution of the form

$E_{gx}=f_1\left(t-\frac{z}{c}\right)+f_2\left(t+\frac{z}{c}\right)$ (6)

The first term of (6) represents the wave or function f₁ traveling with velocity c and unchanged form in the positive z-direction, the second term represents the wave or function f₂ traveling with velocity c and unchanging form in the negative z-direction.

In Figure 1 , we consider a mass particle with rest mass m₀ that, during a finite time interval, moves with constant acceleration $a=a\cdot e_z$ relative to the IRF O. At the moment t = 0, m₀ starts from rest at the origin O, and at t = t it passes at the point P₁. Its velocity is there defined by $v=v\cdot e_z=a\cdot t\cdot e_z$, and its position by

$z=\frac{1}{2}\cdot a\cdot t^2=\frac{1}{2}\cdot v\cdot t$

We limit our considerations to the situation where the speed of the particle remains much smaller than the speed of light: $\frac{v}{c}\ll 1$.

The informatons that during the infinitesimal time interval (t, t + dt) pass near the fixed point P (whose position relative to the moving particle m₀ is defined by the time dependent position vector $r$) have been emitted at the moment $t_0=t-\Delta t$, when m₀ passed at P₀ with velocity $v_0=v_0\cdot e_z=v\left(t-\Delta t\right)\cdot e_z$. The position of P relative to P₀ is defined by the time dependent position vector $r_0=r\left(t-\Delta t\right)$. Δt, the time interval during which m₀ moves from P₀ to P₁ is the time that the informatons need to move—with the speed of light—from P₀ to P

from which we can conclude that $\Delta t=\frac{r_0}{c}$, and that

$v_0=v\left(t-\Delta t\right)=v\left(t-\frac{r_0}{c}\right)=v-a\cdot \frac{r_0}{c}$

Between the moments $t=t_0$ and $t=t_0+\Delta t$, m₀ is moving from P₀ to P₁. That movement can be considered as the resultant (the superposition) of a uniform movement with constant speed $v_0=v\left(t-\Delta t\right)$ and a uniformly accelerated movement with constant acceleration a.

1) In the below figure, we consider the case of the particle m₀ moving with constant speed $v_0$ along the Z-axis. At the moment $t_0=t-\Delta t$ m₀ passes at P₀ and at the moment t at ${P^{\prime }}_1$: $P_0{P^{\prime }}_1=v_0\cdot \Delta t$. The informatons that, during the infinitesimal time interval (t, t + dt), pass near the point P—whose position relative to the uniformly moving particle m₀ at the moment t is defined by the position vector $r^{\prime }$—have been emitted at the moment t₀ when m₀ passed at P₀.

$z^{\prime }=tg\alpha \cdot y^{\prime }-\frac{1}{2}\cdot \frac{a}{c^2\cdot {\cos }^2\alpha }\cdot {y^{\prime }}^2$

At the moment $t=t_0+\Delta t$, when they pass at P, the tangent line to that trajectory cuts the Z’-axis at the point M, that is defined by:

${z^{\prime }}_M=\frac{1}{2}\cdot a\cdot {\left(\Delta t\right)}^2=\frac{1}{2}\cdot a\cdot \frac{r_0^2}{c^2}$

That means that the g-indices of the informatons that at the moment t pass at P, point to a point M on the Z-axis that has a lead of

${P^{″}}_1{P^{″}}_2=P_{0}M=\frac{1}{2}\cdot a\cdot {\left(\Delta t\right)}^2=\frac{1}{2}\cdot a\cdot \frac{r_0^2}{c^2}$

on ${P^{″}}_1$, the actual position of the mass particle. And since $P_0{P^{″}}_1=P_0{P^{″}}_1+{P^{″}}_1{P^{″}}_2$, we conclude that:

$P_0{P^{″}}_2=a\cdot \frac{r_0^2}{c^2}$

In the inertial reference frame O ( Figure 1 ) $s_g$ points to the point P₂ on the Z-axis determined by the superposition of the effect of the velocity (1) and the effect of the acceleration (2):

$P_0{P}_2=P_0{P^{\prime }}_1+P_0{P^{″}}_2=\frac{v_0}{c}\cdot r_0+\frac{a}{c^2}\cdot r_0^2$

The carrier line of the g-index $s_g$ of an informaton that—relative to the inertial frame O—at the moment t passes near P forms a “characteristic angle” $\Delta \theta$ with the carrier line of its velocity vector $c$, that can be deduced by application of the sine-rule in triangle $P_0{P}_{2}P$ ( Figure 1 ):

$\frac{\sin \left(\Delta \theta \right)}{P_0{P}_2}=\frac{\sin \left({\theta }_0+\Delta \theta \right)}{r_0}$

From which it follows that

$\sin \left(\Delta \theta \right)=\frac{v_0}{c}\cdot \sin \left({\theta }_0+\Delta \theta \right)+\frac{a}{c^2}\cdot r_0\cdot \sin \left({\theta }_0+\Delta \theta \right)$

From the fact that P₀P₁—the distance travelled by m₀ during the time interval Δt—can be neglected relative to P₀P—the distance travelled by light during the same period—it follows that ${\theta }_0\approx \theta$ and that $r_0\approx r$. So:

$\sin \left(\Delta \theta \right)\approx \frac{v_0}{c}\cdot \sin \theta +\frac{a}{c^2}\cdot r\cdot \sin \theta$

We can conclude that the g-index $s_g$ of an informaton that at the moment t passes near P, has a longitudinal component, this is a component in the direction of $c$ (its velocity vector) and a transversal component, this is a component perpendicular to that direction. It is evident that:

$\begin{array}{c}{s}_g=-s_g\cdot \cos \left(\Delta \theta \right)\cdot e_c-s_g\cdot \sin \left(\Delta \theta \right)\cdot e_{\perp c}\\ \approx -s_g\cdot e_c-s_g\cdot \left(\frac{v_0}{c}\cdot \sin \theta +\frac{a}{c^2}\cdot r\cdot \sin \theta \right)\cdot e_{\perp c}\end{array}$

The informatons that, at the moment t, are passing near the fixed point P—defined by the time dependent position vector $r$—are emitted when m₀ was at P₀ ( Figure 1 ). Their velocity $c$ is on the same carrier line as $r_0=P_{0}P$. Their g-index is on the carrier line P₂P. According to §3, the characteristic angle $\Delta \theta$— this is the angle between the carrier lines of $s_g$ and $c$—has two components:

1) a component $\Delta {\theta }^{\prime }$ related to the velocity of m₀ at the moment ( $t-\frac{r_0}{c}$) when the considered informatons were emitted. In the framework of our assumptions, this component is determined by:

$\sin \left(\Delta {\theta }^{\prime }\right)=\frac{v\left(t-\frac{r}{c}\right)}{c}\cdot \sin \theta$

2) a component $\Delta {\theta }^{″}$ related to the acceleration of m₀ at the moment when they were emitted. This component is, in the framework of our assumptions, determined by:

$\sin \left(\Delta {\theta }^{″}\right)=\frac{a\left(t-\frac{r}{c}\right)\cdot r}{c^2}\cdot \sin \theta$

The macroscopic effect of the emission of g-information by the accelerated particle m₀ is a gravitational field ( $E_g,B_g$). We introduce the reference system ( $e_c,e_{\perp c},e_{\phi }$) ( Figure 1 ).

1) With N the density of the flow of informatons at P (the rate per unit area at which the informatons cross an elementary surface perpendicular to the direction of their movement), the g-field at that point is

$E_g=N\cdot s_g$

According to the postulate of the emission of informatons, the magnitude of $s_g$ is the elementary g-information quantity:

$s_g=\frac{1}{K\cdot {\eta }_0}=6.18\times {10}^{-60}{\text{m}}^3\cdot {\text{s}}^{-1}$

and the density of the flow of informatons at P is:

$N=\frac{\stackrel{\dot{}}{N}}{4\pi \cdot r_0^2}\approx \frac{\stackrel{\dot{}}{N}}{4\pi \cdot r^2}=\frac{K\cdot m_0}{4\pi \cdot r^2}$

Taking into account that $\frac{1}{{\eta }_0\cdot c^2}={\nu }_0$, we obtain:

$\begin{array}{l}{E}_g=-\frac{m_0}{4\pi \cdot {\eta }_0\cdot r^2}\cdot e_c-[\frac{m_0}{4\pi \cdot {\eta }_0\cdot c\cdot r^2}\cdot v\left(t-\frac{r}{c}\right)\cdot \sin \theta \\ +\frac{{\nu }_0\cdot m_0}{4\pi \cdot r}\cdot a\left(t-\frac{r}{c}\right)\cdot \sin \theta ]\cdot e_{\perp c}\end{array}$

2) $B_g$, the g-induction at P, is defined as the density of the cloud of $\beta$ -information at that point. That density is the product of n, the density of the cloud of informations at P (number per unit volume) with $s_{\beta }$, their $\beta$-index:

$B_g=n\cdot s_{\beta }$

The $\beta$-index of an informaton refers to the information it carries regarding the state of motion of its emitter; it is defined as:

$s_{\beta }=\frac{c\times s_g}{c}$

And n, the density of the cloud of informatons at P, is related to N, the density of the flow of informatons at that point by: $n=\frac{N}{c}$.

So:

$B_g=n\cdot s_{\beta }=\frac{N}{c}\cdot \frac{c\times s_g}{c}=\frac{c\times \left(N\cdot s_g\right)}{c^2}=\frac{c\times E_g}{c^2}$

With the expression of that we have derived above under 1 we finally obtain:

$B_g=-\left[\frac{{\nu }_0\cdot m_0}{4\cdot \pi \cdot r^2}\cdot v\left(t-\frac{r}{c}\right)\cdot \sin \theta +\frac{{\nu }_0\cdot m_0}{4\cdot \pi \cdot c\cdot r}\cdot a\left(t-\frac{r}{c}\right)\cdot \sin \theta \right]\cdot e_{\phi }$

The time dependent components of $E_g$ and of $B_g$ represent waves traveling with the speed c in the direction of $c$ [5] We say that an accelerated mass particle is the source of a “gravitational wave” $\left\{E_g,B_g\right\}$. Its components are both transverse to $c$ and mutually perpendicular.

From the mathematical expressions derived above it can be concluded that at a point P, sufficient far from the accelerated particle m₀, the components of the

wave under consideration are proportional to $\frac{1}{r}$ and they are determined by the

acceleration of the source at the time the involved informatons were emitted:

$E_g=-\frac{{\nu }_0\cdot m_0}{4\pi \cdot r}\cdot a\left(t-\frac{r}{c}\right)\cdot \sin \theta \cdot e_{\perp c}$

$B_g=-\frac{{\nu }_0\cdot m_0}{4\cdot \pi \cdot c\cdot r}\cdot a\left(t-\frac{r}{c}\right)\cdot \sin \theta \cdot e_{\phi }$

In Figure 2 , we consider a mass particle with rest mass m₀ that harmonically oscillates around the origin of the inertial reference frame O with frequency

$\nu =\frac{\omega }{2\cdot \pi }$. Now t it passes at P₁. We suppose that the speed of the particle described by:

$v\left(t\right)=V\cdot \cos \omega t$

is always much smaller than the speed of light.

The elongation z(t) and the acceleration a(t) are then expressed as:

$z\left(t\right)=\frac{V}{\omega }\cdot \cos \left(\omega t-\frac{\pi }{2}\right)$ and $a\left(t\right)=\omega \cdot V\cdot \cos \left(\omega t+\frac{\pi }{2}\right)$

We restrict our considerations about the gravitational field of m₀ to points P that are “far away” from the origin O: we assume that the amplitude of the oscillation is very small relative to the distances between the origin and the points P on which we focus.

Thus, relative to O, $B_{g\phi }$ and $E_{g\perp c}$ are, according the conclusions of the preceding paragraph, in the “far field” expressed as functions of the space and time coordinates as:

$\begin{array}{c}{B}_{g\phi }=\frac{E_{g\perp c}}{c}=\frac{{\nu }_0\cdot k\cdot m_0\cdot V\cdot \sin \theta }{4\pi r}\cdot \sin \left(\omega t-kr\right)\\ =\frac{{\nu }_0\cdot m_0\cdot \omega \cdot V\cdot \sin \theta }{4\pi cr}\cdot \sin \left(\omega t-kr\right)\\ =-\frac{{\nu }_0\cdot m_0\cdot a\left(t-\frac{r}{c}\right)\cdot \sin \theta }{4\pi cr}\end{array}$

With $k=\frac{c}{\omega }$.

So, an harmonically oscillating particle emits a transversal “gravitomagnetic” wave that propagates out of the mass with the speed of light and that in the “far field” is defined by the previous equations.

The intensity of the “far gravitational field” is inversely proportional to r, and is determined by the component of the acceleration of m₀, that is perpendicular to the direction of $e_c$.

Another phenomenon that is the source of a gravitational wave is the conversion of rest mass into energy (what per example happens in the case of radioactive processes). To illustrate this, let us—relative to an inertial reference frame—consider a particle with rest mass m₀ that—due to intern instability—during the period (0, Δt) emits EM radiation.

This implies that that particle during that time interval is emitting electromagnetic energy U_EM carried by photons (and gravitational energy U_GEM² carried by gravitons) that propagate with the speed of light. Between the moment t = 0 and the moment t = Δt, the rest mass of the particle is, because of this event, decreasing

with an amount $\frac{U_{EM}\left(+U_{GEM}\right)}{c^2}$ from the value m₀ to the value ${m^{\prime }}_0$. Because the

gravitational field is determined by the rest mass, this implies that if t < 0 the source of the gravitational field of the particle is m₀ and if t > Δt it is ${m^{\prime }}_0$. It follows that at the moment t the gravitational field at a point P at a distance r > c.t is proportional to m₀, and at a point at a distance $r<c\cdot \left(t-\Delta t\right)$ to ${m^{\prime }}_0$.

During the period (t, t + Δt) the gravitational field at a point at a distance r = c.t changes from the situation where it is determined by m₀ to the situation where it is determined by ${m^{\prime }}_0$. So, the conversion of rest mass of an object into radiation is the cause of a kink in the gravitational field of that object, a kink that with the speed of light—together with the emitted radiation—propagates out of the object.

We can conclude that the conversion of (a part of) the rest mass of an object into radiation goes along with the emission by that object of a gravitational wave.

The effect of the decrease—during the time interval (0, Δt)—of the rest mass of a point mass on the magnitude of its g-field E_g at the point P at a distance r is shown in the plot of Figure 3 .

1) Until the moment $t=\frac{r}{c}$, the effect of the conversion of rest mass into radiation has not yet reached P. So, during the period ( $0,\frac{r}{c}$) the quantity of

mass-energy enclosed by an hypothetical sphere with radius r centered on the particle is still m₀ (the remaining part of the rest mass + all the radiation that during the mentioned period has arisen from the conversion of rest mass). From the first GEM equation it follows:

$E_g=\frac{m_0}{4\pi {\eta }_0\cdot r^2}$

2) From the moment $t=\frac{r}{c}+\Delta t$, the radiation generated by the conversion of

rest mass has left the space enclosed by the hypothetical sphere with radius r, that from that moment only contains the remaining rest mass ${m^{\prime }}_0$. From the first GEM equation it follows:

$E_g=\frac{{m^{\prime }}_0}{4\pi {\eta }_0\cdot r^2}$

3) During the time interval ( $\frac{r}{c},\frac{r}{c}+\Delta t$), the mass-energy enclosed by the

hypothetical sphere with radius r is decreasing (not necessary linearly) because mass-energy flows out in the form of radiation. So, during that period $E_g$ at P is decreasing.

The existence of gravitational or gravitomagnetic waves is embedded in the GEM description of gravity. According to the theory of informatons a gravitational wave propagating outward from an oscillating mass particle at the speed of light is the macroscopic manifestation of the fact that the “train” of informatons emitted by that source is a spatial sequence of informatons whose characteristic angle is harmonically fluctuating along the “train” what implies that the component of their g-index perpendicular to their velocity $c$ and their β-index fluctuate in space.

¹Both terms refer to the same phenomenon.

²Negligible in first approximation.