art theory 101~ The LUFE Matrix: E=mc^{2} ...~ and then some ~ 

The LUFE Matrix: E=mc^{2}
(~ and then some ~)
A supplement to "The LUFE Matrix". The focus is on energy. From Einstein's initial insight into the transformation between mass and energy, we now have a large number of similar transformations between energy and the other fundamental parameters of matter...i.e., charge and spin...and their activities within their spacetime environments such as velocity and acceleration, frequency and temperature, momentum, pressure and force, potential difference, heat and work, etc. The original PreviewReview Slide Show "Can you say 'Energy' 10 different ways?" (see "The LUFE Matrix link below), included here, is now further enhanced with a showcase of select LUFE Matrix Supplement examples all focused on the various forms that energy can take.
Copyright© 2005, Reginald Brooks. All rights reserved
NOTE: This graphic supplement presupposes that the reader has prepared her/himself by reviewing the earlier works upon which this work is based...especially "DarkDarkLight: Dark Matter = Dark Energy (the inverse of)." Works to review include:
"Naughty Physics (a.k.a. The LUFE Matrix )"
"The History of the Universe in Scalar Graphics"
"The LUFE Matrix Supplement: Examples and Proofs"
"The LUFE Matrix Supplement: References"
"The LUFE Matrix: Infinite Dimensions"
The original PreviewReview Slide Show "Can you say 'Energy' 10 different ways?" is presented here as a simple, graphic introduction to the workings of The LUFE Matrix. It walks you through the layout and basic operation of the matrix as it solves "E" for energy in 10 different ways.
It is based on both the full introduction to "The LUFE Matrix: The distillation of SI units into more fundamentally base units of SpaceTime (ST) dimensions" and has been presented in depth in "DarkDarkLight: Dark Matter = Dark Energy (the inverse of): The Conservation of SpaceTime by the Conservation of Force". This paper will enhance and advance that simple presentation with actual examples of the working matrix from "The LUFE Matrix Supplement: Examples and Proofs" with a focus on the numerous other transformations between and the matter and fields which encapsulate, store and conserve its essence.
PreviewReview Slide Show "Can you say 'Energy' 10 different ways?"
...and then some
Click on each of the "Energy" links...the [equation numbers in brackets]...below to reveal its LUFE Matrix presentation.
Energy in the fundamental identifying parameters of any fundamental particle (made of leptons or quarks) must be related to its:
*massa scalar parameter
*chargea vector parameter
*spina vector parameter
*Vector in the sense that they have both magnitude and some sort of directional pointer such that opposites will subtract or cancel and likes will add to the "net" expression of that parameter. Whereas, mass effectively appears to be directionless...if you include or aggregate it, it only increases. Thus far, the intrinsic (rest) energies of fundamental particles like electrons and neutrinos, and, protons and neutrons (and mesons), as well as photons and gravitons, have not, in large part undoubtedly due to the conservation laws, been exploited to release this intrinsic energy... outside of transformations resulting from the collisions of them against each other at velocities approaching the speed of light.
Energy relating to the actions on or of any single or aggregate particle(s) (atoms, molecules, compounds, ...) invokes a wide range of transformational issues. The possibilities of expressing that energy in a different form are greatly expanded. This energy is typically in addition to the fundamental energy of any single particle. Thus:
E=^{1}/_{2}mv^{2} relates to the energy of moving a mass [1]
while
E=mc^{2} relates to the energy of aggregating or splitting masses. [2]
The former relates to moving any particle or clump of particles at sublight velocity. Eintein's famous equation relates to the clumping or splitting smaller clumps of aggregated particles from or onto larger ones, brokered by the amount of available energy and limited by the square of the velocity of light as m = E/c^{2}. Implicit in this expression, mass is slowed down energy and may be completely created or absorbed by it. The LUFE Matrix relates to both relationships equally in terms of spacetime interactional dimensions (STIDS).
Similarly, consider charge. Moving an electric charge, q_{e} (measured in coulombs, C, in the mks system and as statcoulombs, STC, in the cgs system), generates a current, I (measured in amperes, A), which in the presence of a potential difference, V (measured in volts, V), the product of which is Power, P(measured in watts, W) as;
P = IV or watts = amps x volts. [3]
Power is simply the energy per time unit or;
P = E/t = joules/sec = watts [4].
Now consider the energy in another form, the form expressing the potential energy, P.E., expressed as a negative value...the potential electrostatic and the potential gravitational energy...we get;
P.E. = Φ_{e} = q_{1}q_{2}/r [5]
which is very similar to the gravitational potential energy of one mass, m_{1} to another m_{2}, where r= radius of separation;
P.E. = Φ_{g} = Gm_{1}m_{2}/r [6]
Both of which are directly derived from Newton's Universal Law of Gravitation:
F_{g} = Gm_{1}m_{2}/r^{2} [7]
and, Coulomb's Law for Electrostatic Forces:
F_{e} = q_{1}q_{2}/r^{2} [8]
Since Force, F, is but Energy/spatial displacement, Δs;
F = E/Δs [9]
solve for E;
E = FΔs [10]
Substituting [9] into [7] and [8], respectively and solving for E we get the expression for the potential energy as above in [6] and [5]:
P.E. = FΔs = Gm_{1}m_{2}Δs/r^{2} = Gm_{1}m_{2}/r [11]
and;
P.E. = FΔs = q_{1}q_{2}Δs/r^{2} = q_{1}q_{2}/r [12]
and from Newton's Second Law, force equals mass x acceleration, a, as
F = ma [13]
we see that acceleration due to gravity, a_{1} and a_{2} is;
a_{1} = Gm_{1}/r^{2} and a_{2} = Gm_{2}/r^{2} [14],[15]
and acceleration due to electrostatic attraction, are found by substituting [13] into [7] and [8];
a_{1} = q_{1}q_{2}/m_{1}r^{2} and a_{2} = q_{1}q_{2}/m_{2}r^{2} [16],[17]
Solving the two acceleration expressions for G, Newton's Universal Gravitational Constant (a fudge factor to relate disparate SI units...it has zero ST units) gives;
Gm_{1}/r^{2} = q_{1}q_{2}/m_{1}r^{2} [18]
and,
G = q_{1}q_{2}/m_{1}m_{2} = 0 ST units [19]
More importantly, though, in front of us all along, is what is found by substituting [13] into [10] as;
E = FΔs = maΔs = mc^{2} [20]
a relationship that is crystal clear in The LUFE Matrix right from the beginning.
What is also clear in The LUFE Matrix is that mass and charge occupy the same ST interactional dimension space, and therefore suggest that indeed,
E = qc^{2}. [21]
Similarly, and intimately related to the mass, charge is also a form of slowed down energy and may be completely created or absorbed by it.
q = E/c^{2} [22]
Intimately related in that while there are particles with zero net charge, there are no such particles with charge (and spin) and no mass. The true conservation of charge points to charge as being a kind of strict accountant for all the sloweddownlight masses (even a chargeless particle, one with a net charge expression of zero has an intact, internal charge accounting system which reaches the net zero sum by the opposing orientations of its charge vectors, i.e. fractionally charged quarks). When there is a net expression of charge, it is always a whole integral of one...positive or negative and the total charge of the universe...via the conservation of charge...remains the same, equal and opposite.
That conservation of mass is not strictly true, as the energy broker demands a fee of pure energy for every transformation, it is curiously tied close to the fate of the charge. And yet one acts as a scalar expression (mass), the other a vector (charge)...and the vector seems to be in charge (no pun intended!) while the mass plugs along.
The question is, what portion of the energy involved in the mass to energy and energy to mass transformation is contributed by this charge expression? Clearly, some part of a particle must be able to reproducibly generate a universally standard expression of a net, integral charge, yet different particle types have mass expressions which can vary from each other by several thousand fold or more. Does that different part, that part that expresses charge, also carry a lump of energy unique to itself and separate from the contribution of its mass component? This is a subtle point and not to be confused with the common technique of converting all particles to their energy equivalents in electron volts, eV. The energy per second, or power, that moving electrons generate as watts is of course well documented and part of the world's practical use of energy. What we are addressing here is the energy involved in the transformation of particles into others or to pure energy itself. We know about the mass. What about the charge? It seems all forms of expression require energy, so it certainly would seem to be true that the expression of charge accounts for some, albeit a standardized amount, of the energy that we have traditionally assigned exclusively to mass during the E = mc^{2} transformations. What are the E = qc^{2} transformations?
While these relationships are quite clear on The LUFE Matrix, we shall substitute in The LUFE Matrix Constant L_{c}^{2} for one of the velocity of light elements, c,
E=qcc = qcL_{c}^{2} [23]
as a fudge factor to reconcile disparate SI units (similar in function to Newton's Gravitational Constant, G) as;
L_{c}^{2} = 1.36 x 10^{5} g cm/s STC [24]
built from the parameters of the Planck mass, γ2.18 x 10^{5}g, thus;
E=eV=joules=mc^{2}=g(cm/s)^{2}=qc^{2}=qcc=qcL_{c}^{2}=STC(cm/s)(g cm/s STC) [25]
now...can you say Energy 40 different ways?
The examples shown below have been selected from The LUFE Matrix Supplement: Examples and Proof and individual links to the specific page is provided. The mouseover activation of the text links to generate the new graphics requires that you allow for the small graphics to download completely for the first time before you begin. Use your "up" and "down" arrows if needed.
~hover your cursor over the text on the right for images~
~click image for full matrix blowup~ ~click the number to go to the page~ 



~hover your cursor over the text on the right for images~
~click image for full matrix blowup~ ~click the number to go to the page~ 



~hover your cursor over the text on the right for images~
~click image for full matrix blowup~ ~click the number to go to the page~ 



~hover your cursor over the text on the right for images~
~click image for full matrix blowup~ ~click the number to go to the page~ 



The large number of energy expressions that have unfolded over time in various guises and units can readily and clearly be seen on The LUFE Matrix. The distillation of SI units into fundamentally more base units of space and time dimensions, as spacetime interactional dimensions, has the great advantage of presenting a visual graphic of the physical parameters and their complex interrelationship in an easy to read, easy to learn colorful matrix. In our discussions and presentations of examples it is always important to keep in mind the differences in the identifying parameters intrinsic to fundamental particles...like mass, charge and spin angular momentum...and the additional energy transformations and expressions resulting from the manipulation of these particles in to and out of aggregates of matter and in the presence of fields. The conservation laws are the interface between these two fronts and the spacetime dimensions of which they are composed.
As originally described in "DarkDarkLight: Dark Matter = Dark Energy (the inverse of): ...," the Universe is formed, ruled and propagated by the Conservation of Energy...which is itself the result and manifestation of the Conservation of SpaceTime. This is clearly presented in The LUFE Matrix... the new visual language combining mathematics, physics and a colorful geometry of squares (rectangles) within a dynamic matrix.
Energy, especially as energy/sec (Power, S_{V}/T_{V}) occupies the apex of the matrix at five dimensions of space (S_{V}) over five dimensions of time (T_{V})... and forming a diagonal line back to the origin, passing through its first S/T entry, c, the velocity of light (S_{I}/T_{I}), also a constant conserved parameter. In a sense, everything between these two points are details. Important details for sure! Conservations of Momentum, Angular Momentum, Force, Charge and Mass (100% only as massenergy), are but restatements of energy reduced in time and/or space dimensions. This is easily seen on The LUFE Matrix. They are like the children of a parentchild relationship. Starting with the parents...a fixed, constant and conserved amount...offspring are derived which may be slower, fatter and not nearly as full of promise as their energetic parents, they nevertheless never forget...or lose accountability to...the source from which they came and ultimately may be transformed back to. To enjoy the comfort and safety of the Universe we must embrace and act responsibly to the parentchild relationship...as we are but a part of it.
This paper and all its contents © 2005, Reginald Brooks. All rights reserved. Permission is hereby granted for single copies to be made for personal, noncommercial use for students and teachers of schools, colleges and universities provided that: either the entire paper, including figures and tables, is kept intact; or, any extracts of the text, or figures or tables (in part or whole), be properly and visibly cited as to authorship and source.
NEXT: On to Page 5a DarkDarkLight: Dark Matter = Dark Energy  Page 5b The History of the Universe in Scalar Graphics
Back to Naughty Physics (a.k.a. The LUFE Matrix )
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Page 5a DarkDarkLight: Dark Matter = Dark Energy
Page 5b The History of the Universe in Scalar Graphics
Page 5c The History of the Universe_update: The Big Void
Page 6a Geometry Layout
Page 6b Geometry Space Or Time Area (SOTA)
Page 6c Geometry SpaceTime Interactional Dimensions(STID)
Page 6d Distillation of SI units into ST dimensions
Page 6e Distillation of SI quantities into ST dimensions
Page 7 The LUFE Matrix Supplement: Examples and Proofs: IntroductionLayout & Rules
Page 7c The LUFE Matrix Supplement: References
Page 8a The LUFE Matrix: Infinite Dimensions
Page 9 The LUFE Matrix:E=mc^{2}
Page 10 Quantum Gravity ...by the book
Page 11Conservation of SpaceTime
Page 12LUFE: The Layman's Unified Field Expose`