Electromagnetism in four *spatial* dimensions

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As the article This Month in Physics APS News documents Albert Einstein spent the last thirty years of his life on a fruitless quest for a way to combine gravity and electromagnetism into a single elegant theory.

He along with several other scientists, including Theodor Kaluza showed that by extending space-time to five dimensions, one could produce the Einstein equations in four dimensions, plus an extra set of equations that is equivalent to electromagnetism.

Einstein liked this idea saying "The idea of achieving unification by means of a five-dimensional cylinder world would never have dawned on me…At first glance I like your idea enormously." even though he was not successful in incorporating it into his space-time theories.

However his theory already contained the elements of the fifth dimension required to integrate it into it because he qualitatively and quantitatively defined the geometric properties of a space-time universe in terms of the constant velocity of light.  This allows one to redefine a unit of time in his space-time universe to unit of space in a one consisting of only four *spatial* dimensions. 

In other words by defining the geometric properties of a space-time universe in terms of constant velocity of light he provided qualitative and quantitative means of redefining it in terms of the geometry of four *spatial* dimensions which allows one to view electromagnetism and gravity in terms of its spatial instead of its time properties thereby adding a fifth dimensional perspective to his theories. 

This provides the basis, as was done in the article Defining energy in four spatial dimensions for assuming that all forces and be derived in terms of a spatial displacement in a "surface" of a three dimensional space manifold with respect to a fourth spatial dimension.

In other words the mathematics Einstein used to define his theories allows one to derive gravity in terms of a curvature in three-dimensional space with respect to both a time and a spatial dimension.

However it also allows one to define the forces associated with electromagnetism in those same terms. In other words one can explain the classical mathematical properties of Maxwell’s wave equations in terms of a wave moving on the "surface" of a three dimensions space with respect to a fourth spatial dimension.

For example a wave on the two-dimensional surface of water causes a point on that surface to be become displaced or rise above or below the equilibrium point that existed before the wave was present.  A force will be developed by the differential displacement of the 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 a matter wave on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would cause a point on that "surface" to become displaced or rise above and below the equilibrium point that existed before the wave was present.

Therefore, classical wave mechanics, if extrapolated  to four *spatial* dimensions tells us the force developed by the differential displacements caused by a matter wave moving on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension will result in its elevated and depressed portions moving towards or become "attracted" to each other.

This defines the causality of the attractive forces of unlike charges associated with the electromagnetic wave component of a photon in terms of a force developed by a differential displacement of a point on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

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 magnitudes 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 charges than it would be for a single charge.

One can define the causality of electrical component of electromagnetic radiation in terms of the energy associated with its "peaks" and "troughs" that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement.

However, 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 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.

This shows how one can explain and predict the electrical and magnetic properties of an electromagnetic wave by extrapolate the laws of classical wave mechanics in a three dimensional environment to a matter wave moving on a "surface" of a three dimensional space manifold with respect to a fourth *spatial* dimension.

Additionally, the article "Photon: a matter wave?" showed it would give physicists the ability to derive the quantum mechanical properties of electromagnetic energy in terms of a classically resonating system generated by that same matter wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension that define its electromagnetic properties.

There are four conditions required for resonance to occur in a classical Newtonian 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.

The existence of four *spatial* dimensions would give a continuous non-quantized field of energy/mass (the substance) 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.

Therefore, these oscillations in a continuous non-quantized field of energy/mass, would meet the requirements mentioned above for the formation of a resonant system or "structure" in space. 

Observations of a three-dimensional environment show the energy associated with resonant system can only take on the incremental or discreet values associated with a fundamental or a harmonic of the fundamental frequency of its environment.

Similarly the energy associated with resonant systems in four *spatial* dimensions could only take on the incremental or discreet values associated a fundamental or a harmonic of the fundamental frequency of its environment.

These resonant systems in four *spatial* dimensions are responsible for the incremental or discreet energy associated with quantum mechanical systems.

(In a latter article "The geometry of quarks" it will be shown how and why quarks join together to form these resonant systems in terms of the geometry of four *spatial* dimensions.)

The only way to dampen the frequency of a classically resonating system is to add or remove energy from it, which results in changing the characteristics of that system on an incremental basis. 

Additionally the energy in a classically resonating system is, as mentioned earlier is discontinuous and can only take on the discrete values associated with its fundamental or harmonic of its fundamental frequency.

However, these properties of a classically resonating system are the same as those found in a particle in that they are made up of discreet or discontinuous packets of energy/mass and when energy is either added or removed from it, its characteristics are changed incrementally.

This shows that it is not necessary to extend Einstein’s space-time to five dimensions or even add extra set of equations to incorporate electromagnetism in his General Theory of Relativity to combine gravity and electromagnetism because as was mentioned he had already done so when qualitatively and quantitatively defined the geometric properties of a space-time universe in terms of the constant velocity of light.  This is because it allows one to redefine a unit of time in his space-time universe to unit of space in a one consisting of only four *spatial* dimensions. 

It should be remember Einstein’s genius allows us to choose to define a all forces in either a space-time environment or one consisting of four *spatial* dimension when he defined the geometry of space-time in terms of the constant velocity of light. This interchangeability broadens the environment encompassed by his theories thereby giving us a new perspective on how forces physically interact with our three-dimensional environment.

Later Jeff

Copyright 2007 Jeffrey O’Callaghan

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