Conceptual error in the Maxwell Stress Tensor equation as formulated by the D J Griffiths book “Introduction to Electrodynamics” and applications

Goran Dakov, Independent, <goran.dakov@gmail.com>

Journal of Physics A

Journal of Physics B

Keywords: Math Derivation Errors, Electrodynamics, Radion Field, Faster that Light

# Summary

A conceptual error is found in the derivation of Maxwell Stress Tensor equation. This is the derivation in the book “Introduction to Electrodynamics” by D J Griffiths. The misuse is due to wrong “type” applied to the electric field in the force equation. When the misuse of mathematics is corrected for, the new equation is non-conservative. An extra force term appears. A correction is not needed to Maxwell’s Equations so that an extra Power term also exists to make it consistent with the principle of relativity. Waves are propagating independently of reference frame, exactly like to Albert Einstein’s theory of special relativity. However, conservation laws appear to be false. Note that it is not a tautology that the laws of nature must be consistent with an inevitable apocalypse.

A Lagrangian-sourced light speed modifying field is proposed, which would make the universe conservative, when the field is included, based on an observation in a Bulgarian village. The observation is an installation that appears to be solar panels but has visual disturbances in the surrounding air of size roughly 1-2 cm. A perpetual motion machine is described that could explain the visual disturbances and be in use at these installations and utilised the non conservative extra impulse in Maxwell’s Equations. The conclusion is, the expansion of the universe can potentially be explained by this field without the use of dark energy. The dis-balance of matter and anti matter can also be explained in addition. A steady-sate theory is proposed as a conclusion, that, if anti matter generates anti gravity, due to some reason written in the paper, and under some circumstances, a “Great Furnace” exists at the centre of the universe, made up of anti matter, that continuously spits out new matter and anti matter, the anti matter collapsing into primordial black holes more frequently than matter. An explanation is offered for dark matter based on 6 dimensional space. An attempt is made in the conclusion to guess a conserved Lagrangian that is the same for both radiation and force, which is possible but it doesn’t disagree with the differentiation between matter and anti matter. If this conserved Lagrangian is used. Then, an assessment is made whether some form of the second law of thermodynamics still applies. It is concluded that it will if the Lagrangian sourced field is quantized. In that case, entropy will be carried away by the field, assuming that it can only be absorbed in quantized amounts or worse.

# Math Error

In the “proof” of Maxwell Stress Tensor equation, the following[1] steps are taken:

$F=\rho E+J×B$
(1)

$F={\epsilon }_{0}\left(\nabla .E\right)E+\left({\mu }_{0}^{-1}\nabla ×B-{ϵ}_{0}\frac{\partial }{\partial t}E\right)×B$
(2)

It then proceeds to derive the Maxwell Stress Tensor, assuming that in the term

${\epsilon }_{0}\left(\nabla .E\right)E$

the E on the right is the electric field. However, the E on the right is the electric field at the given point, and is not a field over space. This is because (1) is derived form (3), and in (3) the force should not be a field over space. Also, the same applies to B. This is so, because otherwise it would be possible to split the particle’s probability density by applying forces in varying directions at two ends.

$F=qE+v×B$
(3)

Therefore, the correct Maxwell Stress Tensor(2) is:

${T}_{\mathit{ba}}={\epsilon }_{0}\mathit{VAL}\left({E}_{a}\right){\partial }_{b}{E}_{B}-\frac{1}{2}{\epsilon }_{0}{\partial }_{a}\left({E}^{2}\right)+{\mu }_{0}^{-1}\mathit{VAL}\left({B}_{a}\right){\partial }_{b}{B}_{b}-\frac{1}{2}{\mu }_{0}^{-1}{\partial }_{a}\left({B}^{2}\right)$
(3.2)

Where the VAL function takes a field over space and reduces it to a value at a given point.

The force density equation becomes:

$F=\nabla .T-\frac{\partial S}{{c}^{2}\partial t}+{\epsilon }_{0}\left(\mathit{VAL}\left(E\right).\nabla \right)E+{\mu }_{0}^{-1}\left(\mathit{VAL}\left(B\right).\nabla \right)B$

The Pointing theorem remains so far unchanged due to energy being a field over space.

However it only describes the electric energy power. If the work done by the non-conservative impulse change is added, it is consistent with the principle of relativity.

$P=-\nabla .S-\frac{\partial u}{\partial t}+v.{F}^{\text{*}}$
(5)

However, certain components of the extra momentum can be further eliminated, at least for the static field approximation.

# “Near” field momentum violations

For near field, all electric and current charges can be represented as polarisations. Thus it can be eliminated at the static approximation. However, the magnetic field component cannot always be represented by magnetisation. Current densities with a divergence are not equivalent to magnetization bound currents. Thus the force between a ferromagnetic slab and current with a divergence will be non-conservative since one of them is of the form:

${\mu }_{0}^{-1}\left(B.\nabla \right)B$

While the other force is not of that form.

Thus the extra momentum change per unit time is:

${F}^{\text{*}}={\epsilon }_{0}\left[\frac{\partial }{\partial t}\left(\nabla V\right)\right]×B$
(*6)

V is the scalar potential. It is not clear which gauge should be used, since we are using static approximation. The force is seated at the source of the scalar electric potential.

# Perpetual motion machine design

## For propulsion

Basically a C-shaped super-conductive resonator, consisting of adjacent and parallel C-shaped wires. The C-shaped wires are put in a cavity resonator to be fed energy and to prevent radiation from escaping. The static potential will build up at the opposite arms of the C shaped wires and is time derivative will be in anti phase to the magnetic field.

## For energy

In the Bulgarian village that I visited, there was an installation that looks like solar panels, about 20 sq m. However, in the air surrounding it there were visible disturbances of size 1-2 cm similar to those seen on a road in a hot summer day. This could be explained by the solar panels being actually perpetual motion machines, emitting Lagrangian compensation field. This field would cause the visual disturbances as it will modify the speed of light, compensating the change in Lagrangian.

$\left({\nabla }^{2}-\frac{\partial }{{c}_{G}^{2}\partial t}\right)\psi =-L$
(7*)

${c}_{G}=c+{k}_{C}\nabla {\psi }^{2}$
(8*)

Note that the constant term might be due to a constant field, but this would depend on modelling the universe as a whole.

The kinetic energy would be proportional to the first power of the modified speed of light.

$T=m\gamma c{c}_{G}$
(9)

Where  the kinetic energy might be a vector-like functional quantity.

The Lagrangian compensation field will carry kinetic energy when there is time derivative of the Lagrangian. Thus if it were all absorbed, the universe as a whole would remain mith no change of the Lagrangian.

Now about the device I suggest they are using. One thin bar magnet, with “conical” polarisation, made of non conductive material. On top of it, a solenoidal coil aligned with its axis. The electric field will polarise the sides of the bar magnet, creating static potential in the other direction. (Itself being vector potential, mostly). This will be perpendicular to the inward facing field, creating a force aligned with the axis of the coil.

${F}^{\text{*}}\approx {\epsilon }_{0}{B}_{1}B\frac{\pi r{\omega }^{2}}{2}$
(10)

The axial polarisation of the magnet will create a force, which will in turn produce velocity, in the same direction.

$F\approx {\mathit{BB}}_{2}{\mu }_{0}^{-1}{r}^{-1}$
(11)

Given the resistivity of copper, these two will produce power at wavelength of approximately 1 cm, the magnet sizes being of the order of a millimetre or two.

# Dark energy substitute

The Lagrangian sourced field, together with the global light speed it generates, can be used to explain the expansion of the universe.

$\nabla {\psi }^{2}=\frac{-\stackrel{˙}{L}\stackrel{^}{R}}{4\pi {R}^{2}c}$
(12)

Turning it to a volume integral:

$\nabla {\psi }^{2}=\int \frac{\stackrel{˙}{\psi }\stackrel{^}{R}}{4\pi {R}^{2}c}d\tau$
(13)

$\nabla {\psi }^{2}=H\psi R{c}^{-1}$
(14)

And then we can find the energy in space, although there still is a free constant.

${k}_{C}{\psi }_{\mathit{BIAS}}^{2}=c$
(15)

In addition, the positive[2] charge on stars will cause the light emitted by stars to “boost” the star formations, causing the more rapid expansion of denser areas of the universe due to the increased “global” light speed[3]. This is due to the surface charge wobbling inwards when photons are emitted, which triggers destruction of kinetic energy.

# The dis-balance of matter and anti-matter

The positive charge on stars will cause them to move apart more. However, anti-matter stars would have negative charge. Thus anti-mater star clusters are more likely to collapse into black holes, at least if they are very dense. If matter and anti-matter and energy is coming from a central point in the universe, compensated by the Lagrangian sourced field, then, some way along, the amount of anti-matter would be reduced by more frequent collapse into black holes, and then, the remainder would have been annihilated with matter. Thus, an explanation for the lack of anti matter. The source of matter and energy, could be very hot plasma on the surface of an anti-matter sphere. The extra light speed would pull it inwards, while the extra energy from self-increasing kinetic energy would keep it hot, as it traverses it. The sphere surface would have to be negatively charged, like the anti-stars. The negative energy of the Lagrangian sourced field will presumably cancel out the gravity of the energy generator aka “great furnace”.

# Conclusion

It is not a tautology that any physics law must be compatible with an inevitable apocalypse. This would only be so if the goal of physicist was to convince people that there is no God or gods. Alternatively, it could be so if its goal was to preserve religious dogma as certain religious beliefs are predicated on the world ending. The long standing consensus on conservation laws can be explained by alpha male physicists trying to outrun each other into proving the universe is conservative. This explanation explains away dark energy and the dis-balance between matter and anti-matter is accounted for. In addition, the global velocity doesn’t transform as a four vector. This means that the extra 3 time-like dimensions would shift when changing between inertial reference frames. Stars in shifted extra 3 dimensions could be the candidate for dark matter. In addition, the disappearance[4] of some stars from the sky could be explained by changing galactic rotational velocity. I also attempt to guess an almost single Lagrangian for fields and forces, approximately.

$L=\frac{-1}{2}{\epsilon }_{0}{E}^{2}+{\epsilon }_{0}\mathit{VAL}\left(E\right)\otimes E-\frac{1}{2{\mu }_{0}}{B}^{2}+\frac{1}{{\mu }_{0}}\mathit{VAL}\left(B\right)\otimes B+\frac{\stackrel{⃗}{J}\otimes \stackrel{⃗}{{c}_{g}}}{{c}^{2}}-\mathit{Vq}+mc\stackrel{⃗}{{c}_{G}}\otimes \text{{}1,e\text{}}$
(16)

Equations of motion:

${ϵ}_{0}\left(\mathit{VAL}\left(E\right).\nabla \right)E+\frac{1}{{\mu }_{0}}\left(\mathit{VAL}\left(B\right).\nabla \right)B+\frac{1}{2}{\epsilon }_{0}\nabla {E}^{2}+\frac{1}{2\mu }{B}^{2}-\nabla .\left(\rho {c}_{G}{c}^{-1}\right)+\frac{\partial }{{c}^{2}\mathit{dt}}\left(\rho e.{c}_{G}\right)$
(17)

${\partial }_{\left(a\right)}\left(\rho {c}_{G\left(a\right)}\right)+\frac{\partial }{c\partial t}\left(\rho {c}_{G\left(a\right)}\right)=?$
(18)

Where the unit vector, points to a preferable direction, comprising the extra 3 dimensions. If the angle between these 3 dimensions is too large, two observers presumably can’t see each other.

(18) is equal to the force on the kinetic energy side. On the radiation side,

${\partial }_{\left(a\right)}\left(\rho {c}_{G\left(a\right)}\right)+\frac{\partial }{c\partial t}\left(\rho {c}_{G\left(a\right)}\right)=\frac{1}{{\mu }_{0}}\nabla {B}^{2}-\frac{1}{{\mu }_{0}}\left(B.\nabla \right)B-{ϵ}_{0}\left(E.\nabla \right)E$
(19)

${\partial }_{\left(a\right)}\left(\rho {c}_{G\left(a\right)}\right)+\frac{\partial }{c\partial t}\left(\rho {c}_{G\left(a\right)}\right)=-\frac{1}{{\mu }_{0}}B×\left(\nabla ×B\right)-{ϵ}_{0}\left(E.\nabla \right)E$
(20)

Removing the near field internalised forces due to embedded E field (EM mass) and magnetic force, we get:

${\partial }_{\left(a\right)}\left(\rho {c}_{G\left(a\right)}\right)+\frac{\partial }{c\partial t}\left(\rho {c}_{G\left(a\right)}\right)={\epsilon }_{0}\left(\frac{\partial }{\partial t}E\right)×B$
(21)

${\partial }_{\left(a\right)}\rho {c}_{G\left(a\right)}=-{\epsilon }_{0}\left(\frac{\partial }{\partial t}\nabla V\right)×B$
(22)

$\frac{d{c}_{G}}{dt}-\frac{\partial {c}_{G}}{dt}={k}_{C}\frac{\stackrel{˙}{\rho }v.\stackrel{^}{R}}{4\pi {q}_{e}{R}^{2}}×B$
(23.)

Now lets asses whether some form of compliance with the second law of thermodynamics could apply to the energy gathering device. When atoms bounce, they temporarily convert some of their kinetic energy to potential and back. Thus for the purpose of quantisation, only the kinetic energy of the electrons counts. Thus,

${E}_{T}\approx \frac{1}{2}{k}_{B}\frac{T×1}{4000}=\mathrm{\hslash }\omega$
(24)

The negative kinetic energy quantum will have a higher temperature and the positive kinetic energy quantum will have lower temperature according to (10) and (11). It seems that it operates similarly to a Carnot engine in that respect. The frequency of ~20 GHz will be safe from boosting the energy of the fires and or sparks. With respect to plasma, the factor no longer applies and sun surface like temperatures are safe, however it might still boost and cool nuclear explosions. At this frequency, it will only boost a nuclear explosion. However it would cool in at some distance from the epicentre thus restricting its damage.

Note that v is offset by ${c}_{G}$
for the moving magnet, while the base will simply feel some tidal forces due to being attached to the ground. This means that the amount of field needs to be closely matched to ${k}_{C}$ to get high power density, otherwise it will be negligible power output.

## Acknowledgments

I acknowledge the efforts of those physicists who spent 8-9 years studying for a PhD, and apologise for publishing a paper without having done so. I thank Sir Roger Penrose for the cyclic cosmology theory. Even though this is a different take on physics, there are some similar ideas in it.

## References

1. Daved J. Griffiths, Introduction to Electrodynamics, (Prentice Hall inc. 1981)

2.  MOTZ, L. Existence of Net Electric Charges on Stars. Nature 189, 994-995 (1961).

3. B. Joachimi, C.-A. Lin, M. Asgari, T. Tröster, C. Heymans, H. Hildebrandt, F. Köhlinger, A. G. Sánchez, A. H. Wright, M. Bilicki, C. Blake, J. L. van den Busch, M. Crocce, 8, A. Dvornik, T. Erben, F. Getman, B. Giblin, H. Hoekstra, A. Kannawadi, K. Kuijken, N. R. Napolitano, P. Schneider, R. Scoccimarro, E. Sellentin, H. Y. Shan, M. von Wietersheim-Kramsta, and J. Zuntz, submitted to Astronomy and Astrophysics arXiv:2007.01844

4.Beatriz Villarroel, Johan Soodla, Sébastien Comerón, Lars Mattsson, Kristiaan Pelckmans, Martín López-Corredoira, Kevin Krisciunas, Eduardo Guerras, Oleg Kochukhov, Josefine Bergstedt, Bart Buelens, Rudolf E. Bär, Rubén Cubo, J. Emilio Enriquez, Alok C. Gupta, Iñigo Imaz, Torgny Karlsson, M. Almudena Prieto, Aleksey A. Shlyapnikov, Rafael S. de Souza, Irina B. Vavilova, Martin J. Ward, The Vanishing & Appearing Sources during a Century of Observations project: I. USNO objects missing in modern sky surveys and follow-up observations of a "missing star", Earth and Planetary Astrophysics (2019)