squaring Lorentz force

[AntonioLao]

The electromagnetic force is sometimes called the Lorentz force (L) given by L=eE+ev´B where e is the unit of electric charge, E is the electric field, v is the velocity of the electric charge, and B is the magnetic field.

In vector analysis, the cross outer vector product of v´B is not commutative. That is to say that v´B= -B´v. Therefore the Lorentz force can be written in two valid forms. L(a)=eE(a)+ev(a)´B(a) and L(b)=eE(b)-eB(b)´v(b). Their scalar inner dot product is L(a)·L(b)=e²[ E(a)·E(b)-v(a)´B(a)·B(b)´v(b)]. If L(a)·L(b) is now multiplied by the space-time interval of Einstein’s special relativity then the product is equivalent to the square of energy E² given by E²=ds²L(a)·L(b) where the space-time interval ds² is given by ds²=dx²+dy²+dz²-c²dt². Furthermore by a principle of hidden symmetry |L(a)|=|L(b)| such that [L(g)]²=|L(a)|²=|L(b)|² and E²=ds²[L(g)]². If ds² is light-like then ds² vanishes such that E²=0. However, the Lorentz force remains positive definite. On the other hand if [L(g)]² is multiplied by the absolute square of the probability wave function |y(g)|² then E²=|y(g)|²[L(g)]². These two forms for the square of energy imply the wave-particle duality of E²=ħ²n² and E²=m²c where n is its frequency and m is the rest mass of the particle.

[Guille]

Very interesting. I wonder, what is the relationship between [∂F(α)×∂r(α)]▪[∂F(β)×∂r(β)] and |L(a)|, |L(b)|??

And would |L(a)|+|L(b)|=|L(a)+L(b)|?? I remember the thread were we discussed that dealing with frequencies of E and B, the length of one plus the length of the other isn't the same as the length of the sum of them. This is obvious. Unless of course, the two vectors are paralell. What would this mean in terms of E (or E^2)? would it have value 0?

[AntonioLao]

what is the relationship between

∂F(a)=L(a) and ∂F(b)=L(b), these are all local infinitesimal orthogonal forces.

And would |L(a)|+|L(b)|=|L(a)+L(b)|??

|L(a)|+|L(b)|º|L(a)+L(b)|. They are identical (note the 3 bars symbol).

Unless of course, the two vectors are paralell.

The two vectors are perpendicular.

[Guille]

Thanks Antonio for your answers. Is there any difference between equal and identical (in mathematics)?

One thing I don't see is the relationship between gravity and the Lorentz force?

[AntonioLao]

Is there any difference between equal and identical

All primary forces are equal but two given primary forces are identical if and only if they are equal in magnitude and in direction. Both gravity and Lorentz force are secondary forces but the Lorentz forces sometimes also become primary forces due to the non-commutativity of the vector product of velocity and magnetic field.

[Guille]

All primary forces are equal but two given primary forces are identical if and only if they are equal in magnitude and in direction. Both gravity and Lorentz force are secondary forces but the Lorentz forces sometimes also become primary forces due to the non-commutativity of the vector product of velocity and magnetic field.

What if we look at gravity from the 'field' point of view, and state that it's magnitude and direction are identical to those of the Lorentz force. What would that mean? (keep in mind that here the lorentz force is secondary. also, what would 'direction' mean to the field of gravity, would it be always to the centre of mass?)

[AntonioLao]

that it's magnitude and direction are identical to those of the Lorentz force.

If they are identical then they all repel each other. Therefore all of them become primary forces and are all oriented away from the center. Secondary forces can also become primary when equalized with other secondary forces.

[Guille]

If they are identical then they all repel each other. Therefore all of them become primary forces and are all oriented away from the center. Secondary forces can also become primary when equalized with other secondary forces.

And can secondar forces be equal/identical to each other? If they can, what would be the reaction?

Also, if we consider the case of gravity being identical to the lorentz force, and it would impply them to go away from the centre, would light be emitted away of the centre of mass?

In addition, is it that when a primary force approaches identicity or equality with a secondary force then they get attracted less and less and finally when they are equal they repel, or is it another mechanism?

Have you written any paper about secondary and primary forces?

[AntonioLao]

Have you written any paper about secondary and primary forces?

Nope. But if I ever restore the latex typesetting for my personal computer this paper about primary and secondary forces will be my priority.

If secondary forces become equal then by definition they are primary forces in a relative sense and they repel.

Light is always radiated away from its source. Light never go back toward its source or any other sources of light. In QED, light as photons are destroyed the moment they interacted with matter. Light is created when matter and antimatter interact.

In addition, is it that when a primary force approaches identicity or equality with a secondary force then they get attracted less and less and finally when they are equal they repel, or is it another mechanism?

This is a reasonable conjecture. I could use it in my paper with your permission of course.

[Guille]

Nope. But if I ever restore the latex typesetting for my personal computer this paper about primary and secondary forces will be my priority.

If secondary forces become equal then by definition they are primary forces in a relative sense and they repel.

Light is always radiated away from its source. Light never go back toward its source or any other sources of light. In QED, light as photons are destroyed the moment they interacted with matter. Light is created when matter and antimatter interact.

This is a reasonable conjecture. I could use it in my paper with your permission of course.

The idea belongs to you for it came from your theory, us it wjhenever and however you want. One thing, if photons in QED are destroyed when they interact with matter, how can the leave out from matter from a start?