The Larson Research Center

From another post on the BAUT forum series: 

Last night I watched a PBS special on E=mc², called “Einstein’s Big Idea: The Ancestors of E=mc².” I learned several things from the show.  For instance, I learned that the letter c, as a symbol for the speed of light, comes from the latin word for swiftness, “celeritas.”  I never knew that, I don’t think.

However, I learned something else as well, something much more significant.  I’ve written before that I can easily understand how energy and matter are related in the RST, since it’s so straightforward in the new system, but I really have trouble understanding how Einstein knew this.  Now, after seeing the show, the mystery has been cleared up for me.  Einstein accepted, as most others did by then, that energy could be defined by the mass of an object times its velocity squared, because Emilie du Châtelet, following Leibniz lead, had done it, and she demonstrated the veracity of the definition with the help of Willem ‘sGravesande’s clay penetration experiments.

So, when Einstein realized that the velocity of mass was limited to the value of c, he realized, at the same time, that its energy must also be limited by c.  In other words, what was being asserted was that the [i]maximum[/i] energy that a moving object could obtain, given the accepted definition of energy, was

E = mv²,

which, given the newly discovered speed limit of v at c, naturally leads to

E = mc².

However, it’s not clear to me that Einstein understood that this means that the mass of matter could be converted into energy by some process other than putting the energy into it, by accelerating it to c-speed (or as close as possible). In other words, it wasn’t necessarily understood right away that mass and energy are equated by the equation.  At least this seems to be the logical conclusion; especially, since the dimensions of mass, kg, are obviously not the same as the dimensions of energy, it would to be the most logical conslusion. Energy is measured in units of joules in the SI system of units (probably ergs in the cgs system of his day).  Therefore, the dimensions of the equation must have initially been interpreted as

ergs = mass x velocity x velocity,

but then what is velocity times velocity?  I don’t think Leibniz, or du Châtelet, or Einstein knew the answer to this question. I wish I could find out WHY Leibniz thought it was necessary to square the velocity in the energy equation, but all I can find so far is that he did think so, even though he couldn’t prove it.  Therefore, as far as I can tell, there is no theoretical explanation of the definition of kinetic energy as mv²/2.

On the other hand, using the RST dimensions of motion, energy clearly has space/time dimensions,

t¹/s¹,

and momentum has space/time dimensions,

t²/s²,

while velocity has space/time dimensions,

s¹/t¹.

However, the dimensions of the energy equation balance, only if mass has dimensions,

t³/s³,

as can be seen in Einstein’s equation of energy,

t¹/s¹  = (t³/s³) * (s²/t²).

But Einstein did not suspect that mass has these space/time dimensions. Indeed, LST scientists do not understand it this way, even today. Kilograms, the units of mass, like the units of temperature. or the units of apples, or bananas, are understood as scalar units, as units of an amount, or a quantity, of a substance. Energy is also a scalar quantity; that is, it has magnitude only, with no direction in space, but the mathematical dimensions of scalar units are zero, not three, as required by the dimensional analysis of Einstein’s energy equation show above.

Yet, as we have discussed previously, while the dimensions of a scalar quantity are zero, the dimensions of the pseudoscalar octonion, are three, so this admits the possibility that a substance can also have three dimensions and still be a “scalar.”  In this case, the units of the amount of substance would be units of volume.

However, as it turns out, even though energy is a scalar quantity, its definition in LST physics, where its divided into two categories, potential and kinetic, has the dimensions of work. In Wikipedia’s article on energy, we read:

The most common definition of energy in the context of mainstream science is “the ability to do work”. Thus in physics, energy is mathematically defined as a mechanical work done by a force (gravitational, electromagnetic, etc) and has many different forms that can be broken down into two main forms: kinetic energy and potential energy. According to this definition, energy has the same units as work; a force applied through a distance. The SI unit of energy, the joule, equals one newton applied through one meter, for example. (emphasis added)

Hence, in LST physics, the scalar value of energy is transformed, by virtue of its definition, into the scalar value of “work done by a force,” which is the scalar product of force and a so-called “displacement vector,” and, therefore, can have one of three values: positive, negative, or zero, depending upon the degree of orthogonality of the force vector.  In other words, energy is a scalar defined as work, but work is a scalar that is defined as something only manifest as a vector product, the inner vector product that can be either positive, negative, or zero. From the Wikipedia article on work, we read:

Mechanical work is a force applied through a distance, defined mathematically as the line integral of a scalar product of force and displacement vectors. Work is a scalar quantity which can be positive or negative.

As seen from the above definition, force can do positive, negative, or zero work. For instance, a centripetal force in uniform circular motion does zero work (because the scalar product of force and displacement vector is zero as they are orthogonal to each other). Another example is Lorentz magnetic force on moving electric charge which always does zero work because it is always orthogonal to the direction of motion of the charge.

Thus, energy, in LST physics, is defined as the projection of one vector upon another.  When the vectors are perpendicular, no shadow is cast by the vertical upon the horizontal, but, if the angle between them is greater than, or less than 90 degrees, the shadow (projection) takes on a definite positive or negative value - just like a scalar.  Maybe we could call this a pseudoscalar too, but not in the same sense.  It’s a physical pseudoscalar, we might say, not a mathematical one.

The interesting aspect of all this, in the present context, is the insight that it gives us into the relation of numbers and magnitudes.  LST physics deals only with vectorial motion magnitudes, as we have been explaining, so to define scalar motion magnitudes (in this case the motion of inverse scalar velocity, or energy), a way has been found to describe it in terms of vector magnitudes.  It would be considered ingenious were it the work of one individual, but really it’s a concept that has evolved unconciously, as it were, over centuries.  The driving evolutionary force, success in physics, like biological success in living and reproducing, is powerful, if not all that efficient.

Nevertheless, with this much understood, we are left to ponder the meaning of scalar.  If energy is a scalar quantity, and a scalar has no direction, then how can it have the mathematical dimensions of 1 in Einstein’s energy equation? As scalars in a three-dimensional physical system, one would think that mass and energy should have the mathematical dimensions of the octonion scalars (2⁰), or pseudoscalars (2³), but, while we can see that, in the RST at least, mass has the three dimensions of the fourth line in the tetraktys, the dimensions of the pseudoscalars, energy has the dimensions of the second line in the tetraktys, the 2¹, or one-dimensional, line, not the first line, the 2⁰ line.  What’s up with that?

I’ll try to answer that question in the next post.