New Approach to the General Theory of Relativity
by Jorge A. Franco Rodríguez
Part 8. Gravitational Waves?
Effect of the Electric, Magnetic and Gravitational fields on a pure (neutral) mass m.
Let's start saying that a moving mass m at velocity v under the effect of a varying gravitational and central field, produced by another moving mass M, the acting force F (according to obtained results in Parts 4 and 5) is given by:
F = -(G.M.mo/r²).[v²/(w².r²)]Ur
But, if the mass m is also under a magnetic field B and also under an electric field E, both varying or time-dependent, in which the mass m is a neutral particle (or that each atom has the charges +q and -q, equillibrated), the total acting force over the total charges +Q and -Q becomes:
F = -(G.M.mo/r²).[v²/(w².r²)]Ur + Q.(E + v´
B) - Q.(E + v´
B) = -(G.M.mo/r²).[v²/(w².r²)]Ur
Say, the external magnetic and electric fields have a null effect on the neutral mass (let´s discard the electric and magnetic dipole effect, because it is irrelevant for this analysis).
This means that the total effect on the mass m is only given by the gravitational field G = -(mo/m).(G.M/r²).[v²/(w².r²)]Ur
Effect of the Electric, Magnetic and Gravitationals fields on a pure charge +q.
As we can observe, this case of a pure charge +q moving at speed v is not similar to the previous one. On the contrary, as it is known, the force of these three fields on such charge +q, is only given by:
F = +q.(E + v´
B)
Say, there is a null effect of the gravitational field on the charge +q.
Thus, we can say that the total electromagnetic field acting on the charge +q, EM is given by:
EM = F/q = E + v´
B
With this example, we only tried to establish the different behavior between a mass m and a charge +q when they are under the effect of these three fields.
Field produced by a photon.
Nevertheless, the previous observation is not completely general, specially when we analize the field produced by a moving particle of mass. For instance, in the case of a photon, which is a neutral particle, with a mass given by m = p/c, all its kinetic energy K = m.c², can also be represented by the Planck's constant times the frequency of an electromagnetic wave, K = h.n. From here, we can realize and observe that the mass has associated an electromagnetic wave, or, time-dependent electric and magnetic fields, when it moves. But, how a time-varying electromagnetic field can be produced by a neutral particle of mass?. That is a good question to start with. By now, I don't know how this works, but this trascendent fact actually leads us to suspect or think about the inherent unification of the effects of the fields around this fact. Let's try to analize this subject to see what can we obtain.
In the case of photon, its neutral mass character (Ñ.E = 0) leads to that the electric and magnetic fields of its associated electromagnetic wave, meet the wave equation through its speed c:
¶²E/¶t² - c².ѲE = 0
¶²B/¶t² - c².ѲB = 0
Which implies that Faraday's Law holds:
Ñ´E = - ¶B/¶t
and also its counterpart equation, in the absence of a current:
c².Ñ´B = ¶E/¶t
Field produced by a moving mass m.
We had observed in the case of photon its dual character of particle and wave when it is in movement, in which the kinetic energy could be expressed either as in its relativistic presentation K = m.c² (particle character), or as K = h.n (wave character). We could "extrapolate" and infere that any pure mass (Ñ.E = 0) has associated an electromagnetic wave, in which, Kinetic energy can be expressed also in both ways, K = h.n = m.(2.v² - c²) + Mo.c², and the magnetic and electric fields of such wave would meet the wave equation through its speed v:
¶²E/¶t² - v².ѲE = 0
¶²B/¶t² - v².ѲB = 0
Similarly meeting:
Ñ´E = - ¶B/¶t
and
v².Ñ´B = ¶E/¶t
This way of reasoning would lead us to establish the following two statements:
1) Kinetic energy of a moving mass could be also expressed through the Planck's constant, as K = h.n, where n is the frequency of the associated electromagnetic wave.
2) Then, this would imply that the constant relationship between the linear moment p and the wavelenght l would not longer be true (in other words, p.l would not be always equal to h). The contrary is what has been assumed until now in Physics. If what we had concluded were not so, we would obtain that K = h.n = p.l.v/l = m.v², which is not true!.
(In fact, as we will see later, h = p.l is only valid for Mo = 0 and v = c).
In sum, the relationship between the relativistic kinetic energy and the electromagnetic wave, according to this development, could be written as:
K = h.n = m.(2.v² - c²) + Mo.c²
The actual relationship between Planck's constant and momentum-wavelenth can be obtain as:
h.n = m.v² + m(v² - c²) + Mo.c² = p.v + m.(v² - c²) + Mo.c² = p.l.n + m.(v² - c²) + Mo.c²
h = p.l + [m(v² - c²) + Mo.c²]/n
Where, as it can be observed, h = p.l, is only valid for Mo = 0 and v = c (say, for the case of photon).
In this way, we could obtain that the frequency of the electromagnetical wave is given by:
n = [m.(2.v² - c²) + Mo.c²]/h
And, its wavelength by:
l = (v.h)/[m.(2.v² - c²) + Mo.c²]; or by: l = v/n
These results would oblige us to review Quantum Mechanics, because this discipline strongly depends on the momentum-wavelenght product constancy. Moreover, this relationship is extremely important for deriving Schrödinger's equation. If it no longer holds, as it was previously concluded, then the results obtained from such equation would not be the correct ones. If this development is correct, Quantum Mechanics need to be completely rebuilt. In Part 7 we proposed a way to start with this review.
Curl of the Gravitational Field produced by a moving mass M.
On the other hand, we had said, and shown in Parts 3, 4 and 5, that the gravitational field G, time-varying (produced by a moving mass M over a mass m which also moves at speed v relative to M), is central and has an almost general expression, with no other terms: G = -(mo/m).(G.M/r²).[v²/(w².r²)]Ur. Remember, we had observed in Parts 4 and 5, that in a radial movement of the two masses the Gravitational Field becomes G = -(G.M/r²)Ur. These concepts had lead us to obtain consistent results (See Parts 4 and 5). As it can be shown, the gravitational field G, in general (static or time-varying), can be represented by the gradient of a potential function: ÑV, where V = Ep/m is the gravitational potential and Ep the potential energy.
This would imply, by vector identities, that:
Ñ´G = Ñ´ÑV = 0
But, we had observed that there is an associated electromagnetical wave to the moving mass. And, it is apparent that the Gravitational field doesn't depend on other variables such as B or E. It would seem that the mathematical consideration of a fact like this could be expressed through an equation, dimensionally consistent, in the following manner:
Ñ´G = (q/m).[Ñ´E + ¶B/¶t] = 0
Through this equation, we could establish that a moving mass originates time-varying electric and magnetic fields in such a way that their effects lead to that the curl of its gravitational field be null. But in where, the relationship between these Electric and Magnetic fields (Faraday's law) preserves intact, with no other additional terms that modify its expression. The existence of these time-varying Electric and Magnetic fields with such features determine completely an electromagnetic wave; This way of interpreting such equation, preserves the classic and accepted way of deriving the wave equation for the photon through its speed c, as we know it. Also, the same way of derivation of the wave equation could be applied to the case of the moving mass through its speed v, as it was previously suggested.
Gravitational waves do not exist!.
The result, that the curl of the time-varying gravitational field, produced by a moving mass m, be numerically null but implicitly related to the existing time-varying electric and magnetic fields, as the unique contribuitors to the total kinetic energy, leads us to establish a third statement (as a consecuence of such result):
3) Gravitational waves do not exist!..
Comment 1: A fact that also had lead us to suspect the consistence of the previous result, is that the total energy of the photon is completely and exactly determined by the energy of the electromagnetic wave, K = h.n = m.c², which is only associated with the energy conveyed by the electric and by the magnetic fields. As it is known, half of this energy is conveyed by each field. If the gravitational waves there would have existed they had to be present (in some way) in the total kinetic energy expression, as the energy conveyed by the gravitational field, say, as part of that total energy, but this is not the case.
Comment 2: At that speed c, the maximum speed that any particle can develope, the contribution of the gravitational field to the total energy of the photon must be noticeable, if such gravitational energy contribution had existed. Then, the correctness of the third statement seems to be ensured by the result Ñ´G = 0.
Comment 3: If the Curl of the Gravitational field hadn't been null, it should have been related to the Electric or Magnetic field in some way. Thus, it would imply that, either it is separately related to the Electric field, but because of the existing relationship between the magnetic and Electric fields, it also will be related separately to the magnetic field, implying that the kinetic energy of the moving mass can be equally represented by the energy conveyed also by the Gravitational field alone, a thing that has not been observed nor experimentally measured or demonstrated; OR it is related simultaneously to the magnetic and electric fields, introducing new constraints to the shape of the magnetic and electric fields (which are already determined by their curls and divergences), things that lead to inconsistent results. In sum, it is apparent that the only way that Faraday's Law holds and half of the kinetic energy be conveyed by the Electric field and the other half by the Magnetic field, as it is known in the case of photon, is that Ñ´G = 0. So, all previous arguments arrive at the same conclusion: Curl of the Gravitational field must be null.
Divergence of the Gravitational Field produced by a moving mass M.
Additionally, let's establish, as a logical statement, that the Divergence of the Gravitational field is the mass density d divided by a constant k ("Gravitational permeability"?), in order to allow the consistence of units (similar to the case of the electric charge). Then, if this is so, we would have defined completely the vector G, because we also had previously defined its Curl, and a vector is completely defined if its Curl and its Divergence are defined. So, the expression of the Divergence of the Gravitational field should become:
Ñ.G = d/k
Which is the same as stating the Gauss' Law for the mass. Say, that the total flux of the Gravitational field times the proportionality constant k from a volume V is equal to the net mass contained within V. If d represents the mass density in Kg per cubic meter, Gauss' Law applied to mass may be wrtten as:
òS kG.dS = òV ddV = m
Because V is arbitrary.
Two more Maxwell Equations.
In this way we would have added to the Maxwell Equations two more equations, becoming six in total, that would govern the Magnetic, Electric and Gravitational fields, unifying in this way, the field of Materia. So, such six equations completely defining the three fields, and perfectly matching, would be written as:
Ñ.B = 0
Ñ´H = ¶D/¶t + J
Ñ.D = r
Ñ´E = - ¶B/¶t
Ñ.G = d/k
Ñ´G = (q/m).[Ñ´E + ¶B/¶t] = 0
In which, the last two obtained equations, as we have to expect and it can be shown in the same way as we did with other equations in Part 2, are also invariant to the Lorentz transformations. Or, in other words, they also holds in any reference system.
Conclusion.
If this work is correct, it would imply that looking for Gravitational Waves is useless!. Also, it would imply that Quantum Mechanics needs to be rebuilt in order to obtain exact results in the properties of the electromagnetic wave for a moving mass at speed v, through the Wave Equation, in which Planck's constant is not always equal to the product of the linear momentum times the wavelength (as it was concluded in this Part 8 and in Part 7).
See you later!
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