Maxwell’s equations in four *spatial* dimensions

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We have shown throughout this blog there are many theoretical advantage to defining the universe in terms of the field properties of four *spatial* dimensions instead of four-dimensional space-time.

One is that it would allow one to define a physical link between the quantum mechanical properties of electromagnetic energy, Maxwell’s equations and gravity by extrapolating the observable properties of our three-dimensional environment to a fourth *spatial* dimension.

For example the article “Electromagnetism in four *spatial* dimensions” Sept 27, 2007 defined the propagation of electromagnetic energy in terms of a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
While the article “Why is energy/mass quantized?” Oct. 4, 2007 derived the quantum mechanical properties of energy/mass and a photon in terms of the discrete energies associated with a resonant “system” formed in space by a matter wave on the continuous or field properties of that manifold.  However, it also showed why all energy must be propagated through space in these quantized resonant systems.

Briefly it showed the four conditions required for resonance to occur in a classical three-dimensional environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in one consisting of four.

The existence of four *spatial* dimensions would give a matter wave the ability to oscillate spatially on a “surface” between a third and fourth *spatial* dimensions thereby fulfilling one of the requirements for classical resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital.  This would force the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

However, the oscillations caused by such an event would serve as forcing function allowing a resonant system or “structure” to be established in a continuous field of energy/mass.

Classical mechanics tells us the energy of a resonant system can only take on the quantized values associated with its resonant or a harmonic of its resonant frequency.

It was shown these resonant systems are responsible for the quantum mechanical properties energy/mass.

Therefore they define a common mechanism responsible for the quantum mechanical and wave properties of electromagnetic radiation because it defines one in terms of the other.

The article “Gravity in four spatial dimensions” Dec. 15, 2007 derived the propagation of gravity in terms of a displacement caused by a curvature in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension as well as one in a four dimensional space-time manifold.

(Our apologizes for the many referrals to other chapters.  However, even thought the concepts we are trying to present are relatively straight forward they have many interconnecting parts that have been discussed earlier.  Therefore, to steamily the presentation we will give a brief explanation of them and allow the reader the option of reviewing them if he or she desires.)

The link between gravity and electromagnetism was established in the article “Electromagnetism in four *spatial* dimensions” Sept.27, 2007 where it was shown the electromagnetic properties of a matter wave share a common property with gravity in that their energy is propagated by displacements in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly it showed one can derive the causality of the electrical component of an electromagnetic wave in terms of the displacements associated with the “peaks” and “troughs” of a matter wave that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement.

Classical Mechanics tells us a horizontal force will be developed by a perpendicular or vertical displacement which will always be 90 degrees out of phase with it.  This force when extrapolated to a fourth *spatial* dimension is called magnetism.

This is analogous to how the vertical force pushing up of on mountain also generates a horizontal force, which pulls matter horizontally towards the apex of that displacement.

However it also defines a physical reason for the attractive and repulsive properties of charges.

This can be demonstrated by comparing a matter wave on a “surface” of three-dimensional space responsible for both the quantum mechanical and wave properties of electromagnetic radiation to a wave on water.

In a classical world, the trough of a wave on water displaces the vertical volume of water below its surface with air while its peak displaces the air above it with water.

Additionally, classical wave mechanics tells us a force will be developed by the differential displacement of its surfaces, which will result in the elevated and depressed portions of the water moving towards or become “attracted” to each other and the surface of the water.

Similarly, the “trough” of a matter wave on a “surface” of a three-dimensional space manifold would displace a volume of three-dimensional space below its “surface” with a component of four-dimensional space.  While its “peak” would displace a “volume” of four-dimensional space above a “surface” of a three-dimensional space manifold with three-dimensional space.

However, as mentioned earlier classical wave mechanics when extrapolated to a fourth *spatial* dimension tells us that a force would be developed by the differential displacements of the “surfaces” of three-dimensional space, which will result in the elevated and depressed portions “moving” towards or becoming “attracted” to each other.

Therefore, by extrapolating the laws of classical wave mechanics to four *spatial* dimensions one can derive the causality of the attractive forces of unlike electromagnetic charges in terms of a force developed by the differential displacement of a point on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by a matter wave on that “surface”.

However, it also provides a classical mechanism for understanding why similar charges repel each other because observations of water show that there is a direct relationship between the magnitude of a displacement in its surface to the magnitude of the force resisting that displacement.

Similarly, the magnitude of a displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two similar charges will be greater than that caused by a single one.  Therefore, similar charges will repel each other because the magnitude of the force resisting the displacement will be greater for two similar charges than it would be for one.

Therefore, deriving the electromagnetism in terms of a displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension provides a link between it, gravity as shown above each can be derived in terms of the field or continuous properties of that manifold.

Additionally the reason an electromagnetic wave is observed to be made up of discrete quantized units called photons and a continuous wave is because, as mentioned earlier the article “Why is energy/mass quantized?” Oct. 4, 2007 showed the energy of a matter wave forms resonant “system” formed in four-dimensional space.  Therefore, its energy will propagated in quantized resonant systems called photons.

However, if, as we are suggesting the electromagnetic properties of an electromagnetic wave are the results of the displacements associated with the “peaks” and “troughs” of a matter wave on a “surface” of a three-dimensional space manifold one should be able to physically derive Maxwell’s equations in terms of those displacements.

First the mechanism responsible for Gauss’s law of Electricity, the total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity will be derived in terms of the geometric properties of four *spatial* dimensions.

Next Gauss’s law of Magnetism, the net magnetic flux out of any closed surface is zero and that there are no magnetic monopoles will be defined in terms of the alignment of the axis of a fourth *spatial* dimension with respect to the axes of three-dimensional space.

Following that the physical mechanism responsible for Ampere’s Law, the magnetic field in space around an electric current is proportional to the electric current that serves as its source, will be derived in terms of an energy gradient in a three-dimensional space manifold caused by the “peaks” and “troughs” of a matter wave on its surface.

Then the fourth of Maxwell’s equations or Faraday’s law of induction that any change in a magnetic environment of a coil will cause a voltage to be “induced” in the coil that is equal to the negative rate of change of the magnetic flux times the number of turns of the coil will be derived in terms of the displacements in a “surface” of three-dimensional space associated with a charge and the inertial properties of four *spatial* dimensions defined in the article “What is space?” Sept. 17, 2007

Gauss’s law of electricity states the total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity or “integral of E*dA =Q / permittivity”.

However, one can easily explain the causality of Gauss’s law of electricity using the matter wave concept for electromagnetic energy developed in the article “What is electromagnetism?” by extrapolating laws of Classical Wave Mechanics in a three-dimensional environment to a fourth *spatial* dimension.

The peaks and troughs of wave on the two-dimensional surface of water radiates outward in a circular manner from its origin.

Similarly the “peaks” and “troughs” of a matter wave associated with electromagnetic flux would radiate outward form a point on a “surface” of three-dimensional space.

However, this means the electric flux out of a closed surface must be equal to the charge enclose because as mentioned earlier electromagnetic flux is a result of the “peaks” and “troughs” of a matter wave moving on a “surface” of three-dimensional space.  Therefore, all of the energy associated with it must pass through the closed three-dimensional “surface” encompassing its origin.

The reason why the flux is equal to the quantity of charge particles enclosed is because as was shown in the article “Why is energy/mass quantized?” flux or energy is contained in quantized resonant systems.

Therefore, the total electric flux or energy out of a closed “surface” of a three-dimensional space surrounding a charge will be equal, as Gauss’s Law of Electricity states, to number of charge particles enclosed by that surface.

The fact that the total electric flux out of a closed surface must be divided by the permittivity of the volume of space an electric flux is transversing is related to the mechanism responsible for its propagation.

As was shown in the article “What is electromagnetism?” electromagnetic flux is propagated thought space by a matter wave in a continuous non-quantized field of energy/mass.

(Note the reason why the velocity of light is constant despite the fact that it is propagated on a continuous non-quantized form of energy/mass is due as explained “The “Relativity” of four spatial dimensions” Dec. 01, 2007 to a relativistic  property of four *spatial* dimensions.)

However, as was shown in that article there would be direct relationship between the density or permittivity of space and the density of a continuous non-quantized field of energy/mass in that space.

Therefore, the electric flux out of a closed surface must be divided by the permittivity of a volume because according the concepts developed in the article “What is electromagnetism?” the attenuation of the “peaks” and “troughs) of a matter wave associated with electromagnetic flux would be directly related to the density of the continuous non-quantized field of energy/mass that it was moving through.

This completes the derivation of the physical mechanism responsible for Gauss’s law of electricity in terms of the existence of four *spatial* dimensions and a continuous non-quantized field of energy/mass.

We will now derive the mechanism responsible for Gauss’s law for magnetism, that the net magnetic flux out of any closed surface is zero and that magnetic monopoles do not exist in terms of the existence of four *spatial* dimensions.

The mechanism responsible for net magnetic flux out of a closed surface being equal to zero is related to the “alignment” or angle the “W” axis of the fourth *spatial* dimension has with respect to coordinate planes of a three-dimensional space.

(The “W” axis of the fourth *spatial* dimension was defined in the article  “The reality of the fourth spatial dimension”.

The article “What is electromagnetism?” showed magnetic flux associated with electrical energy was a result of the misalignment of the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension by the “peaks” and “troughs” of a matter wave on that “surface”.

However, the “angle” or “alignment” the axis of the fourth *spatial* dimension makes with a coordinate plain in a three-dimensional space manifold would be analogous to the angle a line makes when drawn through a plain on a coordinate graph.

A line will always make symmetrically equal and opposite angles above and below the coordinate plain it transverses.

Similarly, the axis of a fourth *spatial* dimension will always make symmetrically equal and opposite angles “above” and “below” the coordinate plain in three-dimensional space it transverses.

Therefore, the net magnetic flux or force out of any closed surface will always be equal to zero because the magnitude and “direction” of the magnetic flux associated with the W axis of the fourth *spatial* dimension will always be equal and opposite “above” and “below” a three-dimensional coordinate plain it is transversing.

This also defines the physical reason why magnetic monopoles cannot exist.

Magnetic monopoles cannot exist because the axis of a fourth *spatial* dimension is infinite in length.

This is analogous to how the each axis in three-dimensional space is infinite in length.

Therefore, magnetic monopoles cannot exist because the axis of a fourth *spatial* dimension is infinite in length it must generate two magnetic fields of equal and opposite polarity “above” and “below” any coordinate plain in three-dimensional space it transverses.

This completes the derivation of the physical mechanism responsible for Gauss’s law of Magnetism in terms of the orientation of the axis of a fourth *spatial* dimension with respect to the axes of three-dimensional space.

We will now move on and derive a physical mechanism responsible for Ampere’s law that the magnetic field in space around an electric current is proportional to the current which serves as its source or the integral of B*ds equals the permittivity of space times the current or “I”.

The mechanism responsible for Ampere’s law can be derived from the “alignment” of the axis of a fourth *spatial* dimension with respect to the tangential forces associated with the electromagnetic component of a matter wave.

Earlier it was mentioned one can derive the causality of its electrical component in terms of the energy associated with the “peaks” and “troughs” of a matter wave that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement.

Classical Mechanics tells us a horizontal force will be developed by that perpendicular or vertical displacement which will always be 90 degrees out of phase with it.  This force when extrapolated to a fourth *spatial* dimension is called magnetism.

In an electric current, the tangential forces associated with magnetic flux will have a common perpendicular alignment along the velocity vector associated with charged particles.

Therefore, this “tangential” force, which defines a magnetic field associated with the individual particles in an electric current, will be aligned and therefore each will contribute to the total “tangential” force on a “surface” of a three-dimensional space.

This results in the magnetic field in space around a current to be directly proportional to the electric current.

This completes the derivation of the mechanism responsible for Ampere’s law in terms of the existence and “alignment” of the “W” of the fourth *spatial* dimension.

Lastly we shall derive the mechanism responsible for Faraday’s Law of Induction, that any change in a magnetic environment of a coil will cause a voltage to be “induced” in it that is equal to the negative rate of change of the magnetic flux times the number of turns of the coil or the integral of E*ds= -dB/dt.

The mechanism responsible for Faraday’s Law of Induction is related to the inertial properties of space defined in the article “What is space?”

In the derivation of Gauss’s law of Electricity it was shown the existence of a continuous non-quantized field of energy/mass was partially responsible for why the total electric flux out of a closed surface being inversely proportional to the permittivity of space.

However as was shown in the deviation of amperes law each additional turn of a coil will add charges to the space effected the movement of them through it.  This will increase the total “tangential” or force generated on a continuous non-quantized field of energy/mass in the space affected by those charges.

Later in the derivation of Ampere’s law, it was shown the “tangential” force generated on a “surface” of three-dimensional space by an electric current is directly related to the number of charged particles it contains.

This means each additional turn of a coil will make a proportional increase the magnitude of the magnetic flux or force associated with the current in it.

This defines the mechanism responsible for why the magnitude of the magnetic flux around a coil that is carrying an electric current is directly proportional to the number of turns in it.

One aspect of electrical energy that has not yet been mentioned in this chapter is the mechanism responsible for the force or voltage associated with the unit electric charge.

The article “Defining potential and kinetic energy?” Nov. 26, 2007 showed one can derive the energy associated with the voltage and magnetic flux in terms of energy gradients, which as mentioned earlier, would be caused by the “peaks” and “troughs” of a matter wave in a continuous non-quantized field of energy/mass.

This suggests the change in the magnetic environment of a coil, which causes a voltage to be “induced” in that coil, may be related to the inertial properties of space associated with continuous non-quantized field of energy/mass component of space.

This is because Newton’s third law of motion involving an interaction between masses which “For every action there is an equal and opposite reaction” may be applied to the energy/mass component of three-dimensional space.

In other words, the voltage is induced in a coil is equal to the negative rate of change of the magnetic flux because a change the dimensional characteristics of space associated with a change in voltage could be considered to be an equal and opposite reaction to that associated with a magnetic flux based on Newton’s third law of motion.

This shows that one can derive a “physical” link between the quantum properties of electromagnetic energy, Maxwell’s equations and gravity by extrapolating the observable properties of our three-dimensional environment  to a fourth *spatial* dimension.

Later Jeff

Copyright Jeffrey O’Callaghan 2010

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