The Square Root of the Speed of Light in a Vacuum

and Planck Units

Charles William Johnson

Introduction

          What is the square root of the speed of light in a vacuum (c)? In a sense, the answer is, it depends on how you write it out in numbers. In another sense, one could say that there are two answers to that question, and in still another sense there are an infinite number of answers. This question is easier to answer than how does the speed of light in a vacuum affect the natural units, or Planck Units.

          Briefly, since some of the Planck Units have formulae which are expressed with square roots, and specifically with the square root of the speed of light, its effect is directly related to the values derived in Planck Units.

          There are different ways in which one may write out the numerical value of the speed of light. Let me exemplify this with the metric system as that is the one used mainly in expressing Planck Units. Depending upon where one places the decimal, there are many different ways to write out c in the metric system.

          Generally, in the CODATA, the expression of the speed of light, c, is given as:

          299 792 458 meters / second

In scientific notation c appears as

2.99792458 x 108 m/s

thus, indicating that the decimal place is to be moved eight places to the right of the initial number 2.

And, one of the more common expressions, which is probably the most impressive for everyone except a scientist, is to say,

          299792.458 kilometers / second

which can be conceived easily as being real fast.

          Obviously, according to the metric system, one could translate those numbers into any particular decimal expression desired, just by moving the decimal place to the left or right of the significant number 2 that begins the expression in scientific notation: 0.299792458, 2.99792458, 29.9792458, etc.].

          From the perspective of a floating decimal place and how numbers are written in the metric system, the apprehension about the speed of light in a vacuum is quite straight forward. And, we learn that all of the above answers are correct as variations of the same measurement. The floating decimal within the metric system is one of its most potent weapons in mathematical analysis.

The level of the fractal manner in which the speed of light is given above, causes us to drop our guard when analyzing formulae such as Planck Units. At the fractal level of variations in the expression of the speed of light in the metric system, one generally considers that by employing any one of the cited expressions in a formula that the same answer will result for all of the variations.

          Well, that is true if one is not dealing with the roots of numbers. But, as soon as the formulae have square roots, cube roots or higher within their terms of expression, as do some of the Planck Units, then a single equation has multiple answers resulting from the computations.

          The reason for that is simple. In one case, I stated above that the square root of the speed of light has two answers. These are:

          Fractal 1.731451582 and,

          Fractal 5.475330657

If you square the first answer, you obtain the value for c as 2.997924581, similarly to the scientific notation of the speed of light. And, if you square the second answer, you obtain the value for c as 29.9792458, which is not commonly found in the literature, but is one of its numerical expression.

          Besides these two possible answers [1.731451582 and 5.475330657],  another answer is to state that there exist an infinite number of square roots to c.  In fact, a different square root value exists for every decimally different expression of the speed of light in the metric system. Consider, then the square roots of the speed of light in a vacuum in the metric system as of its different decimal expressions.

Value of  Speed of Light in a Vacuum

Square Root Value

2.99792458

1.731451582

29.9792458

5.475330657

299.792458

17.31451582

2997.92458

54.75330657

29979.2458

173.1451582

299792.458

547.5330657

2997924.58

1731.451582

29979245.8

5475.330657

299792458.0

17314.51582

And so on, infinitely so in either direction of decimal placement…

          From this perspective one realizes that the answer to the question about what is the square root of the speed of light is infinite in itself. One could move the decimal place in either direction to the left or to the right and obtain an infinite number of answers.

          Similarly, the same occurs with regard to other root expressions for the cube root and above. For cube roots, the number of significant variations are three, for roots to the fourth power, the number is four, and for roots to the fifth power, there are five variations, and so on for as many powers as one may wish to derive.

The cube root of 2.99792458

equals

1.441916908

Cube root of 29.9792458

=

3.106515806

Cube root of 299.792458

=

6.692785417

Cube root of 2997.92458

=

14.41916908

Cube root of 29979.2458

=

31.06515806

Cube root of 299792.458

=

66.92785417

and so on, infinitely so in either direction of the decimal placement.

The fourth root of 0.0299792458

=

0.416107147

Fourth root of 0.299792458

=

0.739954772

Fourth root of 2.99792458

=

1.315846327

Fourth root of 29.9792458

=

2.339942447

Fourth root of 299.792458

=

4.161071475

Fourth root of 2997.92458

=

7.399547727

Fourth root of 29979.2458

=

13.15846327

Fourth root of 299792.458

=

23.39942447

Fourth root of 2997924.58

=

41.61071475

Fourth root of 29979245.8

=

73.99547727

Fourth root of 299792458.0

=

131.5846327

and so on, infinitely so in either direction of the decimal placement.

          When one expands the question to what are the roots of the speed of light in a vacuum, then one really begins to realize the infinite nature of the answer. So, depending upon whether employs nanometers, millimeters, centimeters, meters, kilometers, etc., in the terms to be derived into square roots, one will be employ variations on one of two fractal root values [either fractal 1.731451582 or fractal 5.475330657].

          Given the fact that, minimally, two fractal roots exist for every number there exist the possibility of multiple answers for equations that contain square root expressions, as do some of the formulae for the Planck Units.

Planck Units

Planck length

Length (L)

l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}

1.616 252(81) × 10−35 m

Planck mass

Mass (M)

m_\text{P} = \sqrt{\frac{\hbar c}{G}}

2.176 44(11) × 10−8 kg

Planck time

Time (T)

t_\text{P} = \frac{l_\text{P}}{c} = \frac{\hbar}{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}}

5.391 24(27) × 10−44 s

Planck charge

Electric charge (Q)

q_\text{P} = m_\text{P} 2 \pi \sqrt{G \varepsilon_0} = \sqrt{\hbar c 4 \pi \varepsilon_0}

1.875 545 870(47) × 10−18 C

Planck temperature

Temperature (Θ)

T_\text{P} = \frac{m_\text{P} c^2}{k} = \sqrt{\frac{\hbar c^5}{G k^2}}

1.416 785(71) × 1032 K

Source: Wikipedia.com

          So, how does all of this affect the scientists who search for natural units, or what have become to be known as Planck Units. These natural units are supposedly the minimum unit measure of length, mass, time, electric charge and temperature [among other derived units such as area, volume, momentum, energy, force, power, density, frequency, pressure, electric current, voltage and resistance]. And, supposedly, each Planck Unit offers a single correct answer expressed generally in scientific notation ---meaning the decimal place floats.

          Some of the Planck Units contain a square root expression within their formulae and therefore are relevant to the previous observations about square roots. These are as follows, each with its fractal value.

Planck length:

1.616252 fractal

Planck mass:

2.17644 fractal

Planck time:

5.39124 fractal

Planck charge:

1.875545870 fractal

Planck temperature:

1.416785 fractal

Planck momentum:

6.52485 fractal

Planck energy:

1.9561 fractal

Planck angular frequency:

1.85487 fractal

Planck current:

3.4789 fractal

Planck voltage:

1.04295 fractal

          Consult any list of the Planck Units and basically that is what one will see, each unit with its own fractal value in significant numbers, generally expressed in scientific notation according to each particular unit of measurement [meter, kilogram, kelvin, second, etc.].

          The impression is that there is a single correct answer for each Planck Unit. But in the previously cited Planck Units, their formulae contain square root expressions and they generally contain root expressions of the speed of light in a vacuum. But, as we have seen above, the square root of c has different fractal numerical values: minimally two, 1.731451582 and 5.475330657.

Immediately, then, one asks how can a Planck Unit that has the square root of c within its equation produce a single, correct result or constant value. The answer is that mathematically it cannot produce a single answer. Necessarily, it must produce minimally two answers, and the expanded answer would be that an infinite number of results are possible as of the infinite expression of the square root of the fractal value of c. 

          Just to offer one example, consider the Planck Length generally given as:

Planck length:                  1.616252(81) x 10- 35 meter

In order to derive the particular answer fractal 1.616252 a fractal root value for c of 5.475330657 needs to be employed in the equation for Planck length. If one employs a fractal root value of 1.73451582 for c in the equation for Planck length, then the result would be a fractal 5.111039039 constant value.                         

          The two minimally possible results for Planck length are then:

Planck length

1.616252 fractal

5.111039039 fractal

Every Planck Unit that contains an expression of square roots in its formula will produce minimally two distinct possible fractal answers as shown. Each result depends upon the fractal root expression for c, or upon other fractal root terms within its equation [such as, the vacuum constant]. In other words, any one of the terms within the square root expression will produce two distinct minimal roots and therefore modify the constant values coming out of the equations as illustrated.

The number of possible results for equations with root expressions that have various terms within those equations is therefore multiple. The Planck Units that have possible multiple answers, then, are Planck length, mass, time, charge, temperature, which are the main ones. Then, among the derived Planck Units, those that have square roots in their formulae are: Planck momentum, energy, frequency, electric current, and voltage. In other words, ten of the seventeen cited Planck Units have square root expressions in their formulae and are susceptible to rendering multiple answers for their constant values.

Nonetheless, in the literature only a single answer or numerical result is offered for these Planck Units, as though only one answer were available or was the correct answer.

If one employs millimeters [299792458000.0] or kilometers [299792.458] as the value for c in the Planck Units with square roots, then the root fractal value employed relates to the 5.475330657 fractal number.

If one employs meters [299792458.0] or the scientific notation [2.99792458], then the 1.731451582 fractal number prevails in the computations of square roots.

In either case, the constant values coming out of the computations will be different from the values coming out of the other for each Planck Unit with a square root based formula. There is no single answer to a Planck Unit equation/formula, there are two minimally, multiple maximally.

          The final question concerns how does one choose between one constant value and another for a Planck Unit. The answer is that one does not choose. The problem is the initial proposition that a constant value for units of measurement may be derived from formulae with square root expressions in their equations. The problem lies with the interpretation regarding the meaning of the results coming out of those assigned formulae. It is not so much a question that the Planck Units are incorrect, but that they are partial in their responses.

***

Link to the complete analysis in a pdf file.
"Planck Units: Natural Units and the Key Equations in Physics"

Extract

The author examines the Planck Units whose formulae are based upon expressions of square roots. He considers the performance of those units as of the roots of the speed of light in a vacuum, fractal 1.731451582 and 5.475330657. He illustrates how different constant values appear for the Planck Units as of those root expressions. The Planck Units that have square root expressions in their equations by design cannot derive a single fractal numerical constant, when in fact square roots for any number produce two significant root numbers, just as cube roots produce three significant root numbers. The author explores the math behind the equations of the Planck Units in order to discern which fractal numerical constants are employed in each one. The author shows that those Planck Units based on square root expressions forward a single constant in their results, when mathematically there exist two variations due to the square root procedure. The deficiency in the Planck Units come from conceptualizing the metric system of a floating decimal as uniform for all equations, when the root variations produce seemingly contradictory fractal numerical results. The results or constants are not contradictory, but merely two options to the same mathematical procedure.

©2010-2014 Copyrighted by Charles William Johnson

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