How many dimensions do we need

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to completely define the reality of our universe?

Einstein was able to define the relativistic properties of motion and the equivalence of gravitational and accelerated reference frames in terms of the geometry of space-time.  However, as Lee Smolin mentions in Chapter Three (page 49) of his book “The Trouble with Physics” he was unable to physically connect them to        quantized electromagnetic properties of light.

Einstein himself acknowledge this when he said, “I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism.”

One of the reasons may be because electromagnetism is observed to have the spatial properties of a wave while Einstein defined gravity in terms the non-spatial properties of a time or a space-time dimension.

But as Lee Smolin also mentions on page 39 of his book Gunnar Nordstrom discovered electromagnetism “pops out” of Einstein’s theory if one increases the dimensions of space by one.  In other words, just by adding an extra dimension of space you get a unification of gravity with electromagnetism that was consistent with Einstein’s Special Theory of Relativity.”

However, Einstein’s theories predicted that light would be bent by gravity while Nordstrom’s did not. 

In 1919, Arthur Eddinngton observed that light was bent by gravity thereby verifying that Einstein’s theory was more consistent with observations and that gravity was not the result of an extra dimension as Nordstrom had postulated.

Since then there have been many attempts to unite gravity with electromagnetism simply by adding dimensions to Einstein’s space-time manifold.

The most promising of these is called string theory, which attempts to define all of the observed properties of our universe in as many as ten dimensions.

However, as is pointed out on page 51 of “The Trouble with Physics” all attempts at unifying physics through extra dimensions suffer from the same problem.  There are a few solutions that lead to the world we observe but there are many which do not.  One has to set the initial conditions, which are found by observing our world to determine which solutions define what we observe.  The use of the circular methodology means its validity is not based on its theoretical structure but on its flexibility.

But it may be possible to develop a theoretical connection between electromagnetism, gravity and its equivalence to accelerated reference frames if instead of adding a spatial dimension as Nordstrom did to Einstein’s space-time we replace its time component with a spatial one. 

Einstein gave us the ability to do this when he use the equation E=mc^2 and the constant velocity of light to define the geometric properties of space-time because that provided a method of converting a unit of space-time associated with energy to unit of space associated with position.  Additionally because the velocity of light is constant he also defined a one to one quantitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words by deriving the geometric properties of space-time in terms of the equation E=mc^2 and the constant velocity of light he provided a way of replacing the time dimension with a spatial one without changing any of its predictive value.

However as the article “The Photon: a matter wave?” Oct. 1 2007 showed one can also derive the quantum mechanical properties of an electromagnetic field in terms of resonant system or structure formed in space by a matter wave moving on a “surface” of a three-dimensional space manifold with respect to fourth *spatial* dimension.

Briefly it showed they can be derived in terms of a resonant system by extrapolating the laws of classical wave mechanics in a three-dimension environment to a matter wave on a surface of a fourth *spatial* dimension.

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

However, 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 four spatial dimensions.

Classical mechanics tells us the energy of a resonant system can only take on the discrete “quantized” energies associated with their fundamental or a harmonic of their fundamental frequency.

In other words one can derive the discrete quantum mechanical properties of an electromagnetic field by extrapolating the laws of classical mechanics in a three-dimensional environment to a resonant system in four “spatial” dimensions.

However, as mentioned earlier one can also derive its electromagnetic properties by extrapolating the laws of classical wave mechanics to a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

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 a fourth *spatial* dimension tells us a force will be 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 which 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 also define the causality of electrical component of an electromagnetic wave 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 from the apex of that displacement.

However as mentioned earlier the article “The Photon: a matter wave?” showed the quantum mechanical properties of a photon can also be derived by extrapolating the laws of classical resonance in a three-dimensional environment to a matter wave on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension

Therefore assuming the universe is composed of four *spatial* dimensions instead of four dimensional space-time allows one to theoretically bridge the gap between gravitation and both the electromagnetic and quantum mechanical properties of an electromagnetic field because one can define their causality in terms of a matter wave moving on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Additionally, because as mentioned earlier the matter wave associated with quantum mechanical properties of electromagnetism is propagated on a “surface” of three-dimensional space manifold with respect to a fourth *spatial* dimension the curvature in that “surface” cause by gravity will “bend” it by the same amount as is predicted by Einstein’s theories.

This means one may not have to increase number of spatial dimensions “in order to bridge the gap between gravity and electromagnetism” as Nordstrom did because, as is shown in this blog it is possible define a theory that makes predictions identical to those of Einstein’s and “bridge the gap between gravitation and electromagnetism” not by adding a dimension but by replacing his time dimension with a spatial one.

Therefore, the answer to the question as to “How many dimensions do we need to completely define the reality of our universe?” may be four.

It should be remember that Einstein’s genius allows us to choose whether to define the properties of our universe” 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.

Later Jeff

Copyright Jeffrey O’Callaghan 2009

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