The Theory of Orthogonality, or the 4th dimension of space

[Kharakov]

Revision 1.9e(sfversion) October 14, 2008

Original version: September 21, 2008

The Theory of Orthogonality, or the 4th dimension of space

Matthew Benesi

 

This theory is a paradigm shift in classical relativistic physics. It shows that there is a 4th dimension of space which photons travel in. It shows that the classical idea that matter is a different form of energy is not exactly correct: it is simply light traveling with an energy vector in the 4th dimension of space (as well as the other 3). It shows that energy is a vector rather than a scalar quantity (which eliminates the discrepancy of energy being a vector when you divide it by the scalar c to get a photon’s momentum). It explains what rest mass really is and why matter cannot be accelerated to the speed of light. Why photons do not travel away from 3dspace is covered at the bottom of this webpage in the section titled: The 4th dimension and expansion of 3dspace.

 

There are certain implications for spin ½ particles: they are photons (spin 1) with components in both dimensions (therefore having ½ spin in 3dspace, and you guessed it, the orthogonal dimension as well (spin is scalar in this case)). For the spin 1 W+, W-, and Z symmetry breaking bosons I believe there is full spin in both orthogonal and 3dspace dimensions or they are some weird form of composite particles. The charge of particles is due to the direction of travel in the orthogonal dimension (for neutrinos, the direction of travel is not on the y axis of the orthogonal dimension, rather on the x, therefore no electric charge (compared to other particles) due to direction of travel, however antineutrinos would have an opposite non-electric “charge” value). More to be added to this section later.... Which happens to be now: I may be a fool for proposing this, but I believe that the orthogonal dimension has 3 directions as does "our own" (3dspace).

 

I don’t think the idea of a 4th dimension of space is new: I’ve read about them in science fiction, watched shows on String Theory, etc. It is however a paradigm shift in the way we think about matter: matter is simply light with a velocity component orthogonal to 3dspace. I suppose some people have thought this before, but I found mathematical proof for it, it agrees with classical relativistic physics and the equations thereof (except where it improves upon them by addressing the 4th dimension of space). Looking at space as 4 dimensions instead of 3 is like looking at the earth as revolving around the sun instead of the sun revolving around the earth.

 

It’s slightly counterintuitive, but I hope you see it in the math.

 

As the theory fits right in with existing theory, it really just explains what we already know in a more exact way. The only deficiency in current relativistic physics is not acknowledging the 4th dimension of space. Perhaps we will find that particles and antiparticles are simply light with opposite directions in the 4th dimension (I haven’t gotten that far yet, and that may be for sharper minds then mine). I just think this will allow theoretical physicists to take the next step. Each insight is one step closer, and I hope this helps all of us understand the universe a little better. This doesn’t debunk special relativity or the classical equations, it simply gives us new insight into what they describe.

 

The results that support this theory are all old discoveries by the greats: special relativity(confirmed) ^[5,6], de Broglie hypothesis (confirmed) ^[3,4], Planck constant ^[1,2], the Lorentz transformation ^[7,8], the Lorentz factor ^[7,8], all of which are based on the knowledge of the great philosophers of the past.

 

3dspace: The normal 3 dimensional space we perceive as we go through our day to day activities.

Orthogonal (or 4th) dimension: The 4th dimension of space in which photons travel. When photons “travel” in this direction, they increase the volume of space in 3dspace rather than “leaving” 3dspace.

 

The rest mass of a particle is simply the energy component of a photon orthogonal to 3dspace. Since we can’t directly access this energy yet (until we figure out how to reflect the photon’s orthogonal component to make it into a positron) we can only change it relative to other particles in 3dspace (add energy to its 3dspace component). If we add energy to the photons 3dspace component the total energy of the photon increases but the orthogonal component does not change at all. As the velocity of the photon does not change, we only alter its direction relative to 3dspace by adding energy to it (and increase its overall energy).

 

Let’s start out with a demonstration of the theory, with an electron. The electron is a photon that has a velocity component orthogonal to 3dspace (if it’s moving through 3dspace, it has a component in both 3dspace and the orthogonal (4th) dimension).

 

Electron mass: 510,998.910 eV/c2 ^[1,2]

Electron energy: 510,998.910 eV ^[1,2]

Planck constant in eV s: 4.13566733 * 10-15 eV s ^[1,2]

Speed of light in vacuum ©: 299792458 m/s ^[1,2]

 

Let’s find the frequency and wavelength of a photon with this energy.

[math]f=\frac{E}{h} = \frac{510998.910 \ eV}{4.135667733 \ \times \ 10^{-15} \ eV \cdot \ s} = 1.23558998 \times \ 10^{20} \ Hz[/math]

 

 

[math]\lambda=\frac{c}{f} = \frac {299792458 _{m/s}}{1.23558998 \times 10^20 \ Hz} = 2.42631021 \ \times \ 10^{-12} m[/math]

 

Now we are going to accelerate the electron to .5 c [math](\cos{60} \ \times \ c)[/math] and calculate its de Broglie wavelength.

The equation for the de Broglie wavelength is:

[math]\lambda = \frac{h}{mv} \sqrt{1-\frac {v^2}{c^2}}[/math]

 

[math]\lambda=4.20249256 \ \times \ 10^{-12} \ m[/math]

 

Wavelength, frequency, and energy are all vectors (they have direction) in 4d space. We are used to looking at them as scalar quantities, but that is not correct. The de Broglie wavelength is the wavelength component of the photon (or photons) on the standard 3 dimensional space axis (3dspace). The wavelength we calculated for the rest mass of an electron earlier is the wavelength component of the photon (electron) on the orthogonal dimension axis (orthogonal to 3dspace).

 

Let’s determine what the total wavelength of the photon (the electron) is.

To find the total wavelength, we have to invert the wavelengths, square them, add them together, take the square root of the result, and invert the result of this. We invert the wavelengths because the total wavelength is shorter than either of the components. We could just convert the wavelength to frequencies as well, square them, add them, and then take the square root of the results.

 

Here are the 2 equations we can use (depending on whether we have wavelength or frequency vectors to calculate the totals of):

 

[math]\lambda_{total} = \frac{1}{\sqrt{\frac{1}{\lambda_{3dspace}}^2+\frac{1}{\lambda_{orthogonal}}^2}}[/math]

 

Note: There is actually only one true frequency for the photon (the total frequency component). The frequency components are those of equivalent wavelength photons to the wavelength of the 3dspace and orthogonal components. We have to use the scalar c (instead of actual velocities) to arrive at the correct total frequency (which is c/ total wavelength).

 

[math]f_{total}=\sqrt{f^2_{3dspace}+f^2_{orthogonal}}[/math]

 

We can also calculate the relativistic energy if we have both energy components (3dspace and orthogonal).

 

[math]E_{total}=\sqrt{e^2_{3dspace}+{e}^2_{orthogonal}}[/math]

 

Anyways, let’s keep going with our previous example and calculate the total wavelength of the photon (electron).

 

[math]\lambda_{total}=\frac{1}{\sqrt{{\frac{1}{4.2024925 \ \times10^{-12}m}}^2+{\frac{1}{2.42631021 \ \times10^{-12}m}}^2}}[/math]

 

[math]\lambda=2.10124628 \ \times \ 10^{-12} \ m[/math]

 

Let’s calculate our total energy from the wavelength:

 

[math]E= \frac{hc}{\lambda}[/math]

 

[math]E_{total}=590050.717 \ eV[/math]

 

Let’s go ahead and check this against the result from the standard relativity energy equation^[5,6,7,8]:

 

 

[math]E= \frac{e}{\sqrt{1-\frac {v^2}{c^2}}}[/math]

 

 

[math]E= \frac{510998.910 eV}{\sqrt{1-\frac {(.5c)^2}{c^2}}}[/math]

 

[math]E=590050.716 \ eV[/math]

 

It’s the same (except for slight error due to rounding).

 

We can also determine the angle of the photon from the 3dspace axis and all of its component vector magnitudes in a much easier manner, simple trigonometric functions. I wanted to show you the classical method first.

 

If we know the velocity of the photon along the 3dspace axis, we can do something really simple.

Since:

 

[math]v_{total}=\sqrt{v^2_{3dspace}+v^2_{orthogonal}}=c[/math]

 

We can take the inverse cosine of the velocity unit vector component along the 3dspace axis divided by the total velocity of the photon. In other words, the inverse cosine of (.5c)/c (the scalar © is taken out) gives us the angle away from 3dspace. If we know the total relativistic energy of the photon (particle) we can use the angle to generate its 3dspace and orthogonal component vectors.

 

[math]\theta=\arccos{.5}=60[/math]

 

Say we know the relativistic energy of this particle, how do we find its rest mass energy (the orthogonal component of the photons energy)? Simple trigonometric calculation in 5d physics:

 

[math]e_{rest\ mass}=\sin\theta \times E_{total}[/math]

 

[math]e_{rest\ mass}=\sin 60 \ \times \ \590050.717 \ eV=510998.910 \ eV[/math]

 

Likewise if we know the photons rest mass energy component and its velocity, we can take the inverse cosine of the vector, get the angle, divide its orthogonal (rest mass) energy by the sin of the angle and get its relativistic energy.

 

[math]v=\sqrt{.96}c[/math]

 

[math]e_{orthogonal}=510998.910 \ eV[/math]

 

[math]\theta = \arccos{\sqrt{.96}}=11.5369590[/math]

 

[math]E_{total}=\frac{510998.910}{\sin{\theta}}= 2554994.55 \ eV[/math]

 

Go ahead and check this result against the standard relativistic energy equation.... you'll see that the results are the same.

 

Energy is a vector in space with 4 dimensions, rather than a scalar quantity. It has direction.

 

Anything with mass is comprised of photons with a velocity component orthogonal (in a 4th dimension) to 3dspace. From this we can discover new ways of interacting with … photons. To see why the photons don’t leave 3dspace… scroll down (past this set of citations which applies to the above section of the paper only):

 

1. www<dot>physicstoday<dot>org/guide/fundconst.pdf CODATA Recommended Values of the Fundamental Physical Constants – 2006

 

2. "Planck constant in eV s". 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.

 

3. en<dot>wikipedia<dot>org/wiki/De_Broglie_hypothesis]de Broglie hypothesis - Wikipedia, the free encyclopedia (The equation is not cited on this page, so must not be copyrighted: it is de Broglie’s equation, however)

 

4. L. de Broglie, Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925). Reprinted in Ann. Found. Louis de Broglie 17 (1992) p. 22.

 

5. Albert Einstein in letter to L Barnett (quote from L. B. Okun, “The Concept of Mass,” Phys. Today 42, 31, June 1989.)

 

6. Tolman, R. C. (1934). Relativity, Thermodynamics, and Cosmology. Oxford: Clarendon Press. LCCN 340-32023. Reissued (1987) New York: Dover ISBN 0-486-65383-8.

 

7. Larmor, J. (1897), "A dynamical theory of the electric and luminiferous medium — Part III: Relations with material media", Philosophical Transactions of the Royal Society 190: 205–300, doi:10.1098/rsta.1897.0020

 

8. Lorentz, Hendrik Antoon (1899), "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam I: 427–443; and Lorentz, Hendrik Antoon (1904), "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam IV: 669–678

 

 

 

The 4th dimension and expansion of 3dspace

Matthew Benesi

In the Theory of Orthogonality, photons are described moving in a 4th dimension of space, but what wasn’t described is why they do not appear to recede from 3dspace.

 

It’s simple. Conservation of energy (in 3dspace) cannot be violated. Therefore, the photon “pulls” space into 3dspace from the 4th dimension as it “travels” in that direction. In the beginning, the first photons traveling this way produced tons of space (compared to the volume of the universe). Now, the volume is soooooo big that we don’t see as much expansion (but it still is going on, it’s just a smaller ratio to the total volume).

 

 

Now this really makes it look like 3dspace has an absolute reference frame. It just happens to be that for every particle with momentum in 3dspace has a corresponding momentum (maybe in a group of particles, maybe one particle) in the opposite direction as well (conservation of energy). The velocity of all particles in 3dspace (related to one another) cancels out to produce an average orthogonal component into the 4th dimension (which may be related to the Gravitational Constant ^[1]). Of course, there are some non-symmetrical theories out there, so other stuff could be happening as well.

 

As to conservation of energy, photon velocity is being absorbed by space expansion (at cosmological distances). As to conservation of space, well, the universe is expanding due to all those orthogonal photons. Maybe we can put a stop to it. :D

 

 

Citation for this section of paper only:

 

1. "University of Washington Big G Measurement". Astrophysics Science Division. Goddard Space Flight Center (2002-12-23). "Since Cavendish first measured Newton's Gravitational constant 200 years ago, "Big G" remains one of the most elusive constants in physics."

 

Edited to add the following notes section to help people understand the mathematics of the theory:

 

5d Physics notes on the Theory of Orthogonality

 

3dspace: standard 3 dimensional space (x,y,z)

Orthogonal: 4th dimension of space (w)

Theory of Orthogonality: a theory describing space as having greater than 3 dimensions in which photons move. Photons moving orthogonal to 3dspace are perceived as matter in 3dspace.

 

1. If we have the velocity of an object in 3dspace, how do we calculate the angle of the photon from the 3dspace axis?

 

The velocity component of the photon along the 3dspace axis can be used to calculate the photons angle ([math]\theta[/math]) from the 3dspace axis.

 

[math]v_{3dspace}=\cos \theta \ \times \ c[/math]

 

[math]\cos \theta = \frac {v_{3dspace}}{c}[/math]

 

[math]\theta= \arccos (\frac{v_{3dspace}}{c})[/math]

 

2. If we have the orthogonal (rest mass) energy and total (relativistic) energy of a particle, how do we calculate the velocity of the photon along the 3dspace axis?

First, we need to realize that the orthogonal (rest mass) component of a photon’s energy is:

[math]e_{orthogonal}= \sin \theta \ \times \ E_{total relativistic)}[/math]

 

From this we can deduce the angle of the photons travel relating it to the 3dspace axis:

 

[math]\sin \theta= \frac {e_{orthogonal(rest mass)}}{E_{total(relativistic)}}[/math]

 

[math]\theta= \arcsin (\frac{e_{orthogonal}}{E_{total}})[/math]

 

[math]v_{3dspace}= \cos \theta \ \times \ c[/math]

 

3. How do we find the de Broglie wavelength or 3dspace component of a photon’s wavelength with only the photon’s total energy (relativistic) and angle of travel from the 3dspace axis?

 

[math]\lambda_{total}=\frac{hc}{E_{total (relativistic)}}[/math]

 

[math]\lambda_{3dspace (de Broglie)}=\frac{\lambda_{total}}{\cos \theta}[/math]

 

4. Likewise, how do we find the photons orthogonal wavelength component?

[math]\lambda_{orthogonal}=\frac{\lambda_{total}}{\sin \theta}[/math]

 

5. From this, how do we find the photons orthogonal (rest mass) energy component?

[math]e_{orthogonal}=\frac {hc}{\lambda_{orthogonal}}[/math]

 

That's it for now.... perhaps more will be forthcoming in a bit.

For a nicer looking PDF version of the theory (and "lecture notes/questions" for when it is taught... eventually) check out (replace <dot>s with . (can't post links yet)):

 

www<dot>personal<dot>psu<dot>edu/kfv100/boyfriends%20theory/The%204th%20dimension%20of%20space.pdf