Natural Units vs Planck Units
by
Mircea Bidian
14/oct/2024
1.0 Natural Units
When I discovered natural units, I was still deeply anchored in Planck units. I was strongly influenced by the constant that bears his name, and it seemed natural to align myself with the vision of the senior physicists who laid the foundations of modern science. Thanks to their contributions, we have all the science and technology we rely on today. However, the day I identified the natural units of mass \(M\), length \(L\), and time \(T\)—which I consider to be the fundamental building blocks of the universe—and realized that I could use these units to redefine Planck's constant \(h\), the gravitational constant \(G\), force \(F\), the speed of light \(c\), and other fundamental constants, I understood that my approach needed to diverge from the tradition of classical physics.
This decision did not imply a complete departure from tradition, as classical physics holds many valuable aspects. For example, the speed of light constant, \( c = 299,792,458 \, \text{m/s} \), has been established as a fixed value since 1983, and this reference remains essential. In 2019, Planck's constant, with the value \( h = 6.62607015 \times 10^{-34} \, \text{J·s} \), was also adopted, but the unit \( \text{J·s} \) does not fully capture the true nature of angular momentum. In the table below, you can see the true value of angular momentum \( S \). Along with the speed of light, I accepted the meter and the unit of time, as at that moment, there was no better solution available, and the adoption of the kilogram was a natural consequence. The adoption of the values for the speed of light \( c \) and Planck's constant \( h \) by the International System (SI) was the result of thousands of experiments confirming these values, making it a justified decision.
However, since I was looking for a physics model that would allow me to calculate my non-inertial propulsion engine, not just a generally accepted physics model, I did not hesitate to adopt a personalized value for the gravitational constant, where \( G = \frac{1}{50 \cdot c} \). I can say that this was a turning point, without which the book æther - The First Element and, evidently, this blog would not exist. Thus, I arrived at the natural proportions of mass, length, and time, which became universal in all my calculations. In the table below, you can see the proportions for each unit.
With the MLT values from the table above, all the constants in physics can be calculated. Similarly, any black hole will have equal MLT units, meaning the number of masses \(M\) will be equal to the number of radii \(L\), and implicitly to the number of times \(T\). Knowing this, it becomes extremely simple to establish the units of measurement we desire, because we start from unitary values. We can get an idea of the values of these new units of measurement in the lines below:
1M77 = \(7.37249732381270789292 \times 10^{26} \, [\text{kg}]\)
1L77 = \(0.54724589239635650329 \, [\text{m}] \) (54.72 cm), an approximate measure of the ancient cubit.
1T77 = \(1.82541580948095933517 \times 10^{-9} \, [\text{s}]\)
Where 1M77 means that for a mass of \(7.372 \times 10^{26} \, [\text{kg}]\), we have \(1 \times 10^{77} [M]\) quantum units of mass. Similarly, 1L77 means that at \(0.547 \, [\text{m}] \) (54.72 cm), we have \(1 \times 10^{77} [L]\) quantum units of length, and similarly for time, where we have \(1 \times 10^{77} [T]\) quantum units of time for \(1.825 \times 10^{-9} \, [\text{s}]\).
Abandoning the kg-m-s units of measurement would imply adopting natural units, but due to the large discrepancy between mass and length, which we are accustomed to, this might not be practical. We could adopt intermediate but proportional measures, though we would be replacing one imperfect system with another, which would come with its own problems. For example, the wavelength of green light, with a value of 547.2 nm, could be expressed as 1G71, meaning '1 green with \(10^{71} [L]\)' quantum radii lengths. But let's leave this task to the metrologists.
To get an idea of the simplicity of calculations with natural units, let's look at the following example.
We will take as an example a black hole with a radius of 0.5 meters, which in natural units is composed of:
- \(9.13666063\times 10^{76}\, [\text{L}]\) (quantum lengths)
- \(9.13666063\times 10^{76}\, [\text{M}]\)(quantum masses)
- \(9.13666063\times 10^{76}\, [\text{T}]\)(quantum times)
At a distance of 1,000,000 meters from the center of the black hole, there is a mass of \[m=75[\text{kg}]\]. We are asked to find the gravitational force, FMm. The first step is to transform the distance of 1,000,000 meters into natural units, using the ratio d/L, resulting in:
\[\frac{d}{L}=1.82733213\times 10^{83}[\text{L}]\]
We convert the mass of 75 kg into quantum values using the ratio m/M, resulting in:
\[\frac{m}{M}=1.01729437\times 10^{52}[\text{M}]\]
The gravitational constant is 1. Thus, we can calculate the gravitational force in natural units:
\[F_{Mm}=\frac{G M m}{r^2}[MLT^{-2}]\]
\[ F_{Mm}=\frac{1\cdot 9.13666\times 10^{76}\cdot 1.01729\times 10^{52}}{(1.82733\times 10^{83})^2}=2.78355\times 10^{-38}\,[MLT^{-2}]\]
Using the force constant \[F=50 c^5\], we obtain the force in metric units:
\[F_N=F\cdot F_{Mm}=50 c^5\cdot 2.78355081\times 10^{-38}\\=3.37033191\times 10^{6}\,[N]\]
To verify the result, we perform the calculations in metric units:
\[F_{Mm}=\frac{G M m}{r^2}[MLT^{-2}]\]
\[F_{Mm}=\frac{6.67128190\times 10^{-11}\cdot 6.73600060\times 10^{26}\cdot 75}{(1000000)^2}\\=3.37033191\times 10^{6}\,[N]\]
The results are identical, validating both methods of calculation.
Calculating Fundamental Constants:
The value of fundamental constants of physics can be determined from natural units using precise definitions. For example:
- Planck's constant, h:
\[h=\frac{M L^2}{T^2}\]
\[h=\frac{7.37249732\times 10^{-51}\cdot(5.47245892\times 10^{-78})^2}{(1.82541580\times 10^{-86})^2}\\=6.62607015\times 10^{-34}\,[J]\]
- The speed of light, c:
\[c=\frac{L}{T}=\frac{5.472458923963565\times 10^{-78}}{1.825415809480959\times 10^{-86}}\\=299792458\,[\text{m/s}]\]
- The gravitational constant, G:
\[G=\frac{L^3}{M T^2}\]
\[G=\frac{(5.4724589\times 10^{-78})^3}{7.372497\times 10^{-51}\cdot(1.82541\times 10^{-86})^2}\\=6.671281903\times 10^{-11}\,[\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}]\]
For other fundamental constants, the calculation follows the same principle presented in the above examples.
2.0 Planck Units
In his original works, Max Planck used the Planck constant \(h\) in his formulas for quantum energy, and the units he proposed in 1899 were based on this constant. Here are the original formulas using \(h\):
1. Planck Mass:
\[m_P = \sqrt{\frac{h c}{2 \pi G}}\]
2. Planck Length:
\[ℓ_P = \sqrt{\frac{h G}{2 \pi c^3}}\]
3. Planck Time:
\[t_P = \sqrt{\frac{h G}{2 \pi c^5}}\]
4. Planck Energy:
\[E_P = m_P c^2 = \sqrt{\frac{h c^5}{2 \pi G}}\]
The symbol \(\hbar\) (the reduced Planck constant) was introduced later, in 1926, by Paul Dirac to simplify notation in quantum mechanics. Each Planck unit can be expressed in several ways, which we will display and analyze below:
a. Planck Mass:
\[m_P = \sqrt{\frac{h c}{2 \pi G}} = \frac{E_P}{c^2} = \frac{50 G E_P}{c} = 2.176926597 \times 10^{-8} \, \text{kg}\]
b. Planck Length:
\[ℓ_P = \sqrt{\frac{h G}{2 \pi c^3}} = \frac{G m_P}{c^2} = \frac{G E_P}{c^4} = 1.61588955 \times 10^{-35} \, \text{m}\]
c. Planck Time:
\[t_P = \sqrt{\frac{h G}{2 \pi c^5}} = \frac{E_P}{50 c^6} = \frac{G E_P}{c^5} = 5.390027358 \times 10^{-44} \, \text{s}\]
d. Planck Energy:
\[E_P = m_P c^2 = \sqrt{\frac{h c^5}{2 \pi G}} = 5 \sqrt{\frac{h c^6}{\pi}} = \frac{m_P l_P^2}{t_P^2} = 1.956524053 \times 10^9 \, \text{J}\]
Since all Planck units can be calculated without the involvement of \({2}\pi\), we are left to question what exactly \({2}\pi\) represents and, more importantly, what purpose it serves. To explore this, we redefine \({2}\pi\) as \(n\) and examine its mathematical expression for each Planck unit.
For Planck mass, we have:
\[m_P = \sqrt{\frac{h c}{n G}}\]
From which we derive \(n\) using the following equation:
\[n = \frac{c h}{G m_P^2} = \frac{L T^{-1} M L^2 T^{-2}}{L^3 M^{-1} T^{-2} m_P^2} = \frac{M^2}{T m_P^2}\]
In this equation, we recognize the definitions of natural units for the speed of light:
\[c = \frac{L}{T}\]
Planck's constant:
\[h = \frac{M L^2}{T^2}\]
The gravitational constant:
\[G = \frac{L^3}{M T^2}\]
And Planck mass \(m_P\).
By replacing \({2}\pi\) with \(n\) in each equation, we directly arrive at the solution for \(n\) in the Planck length formula:
\[n = \frac{L^3 M^{-1} T^{-2} M L^2 T^{-2}}{l_P^2 L^3 T^{-3}} = \frac{L^2}{T l_P^2}\]
And for Planck time:
\[n = \frac{L^3 M^{-1} T^{-2} M L^2 T^{-2}}{l_P^2 L^5 T^{-5}} = \frac{T^2}{T t_P^2}\]
As we can observe, the final solutions for \(n\) in each case represent the ratio of the square of the natural unit to the square of the Planck unit, multiplied by the frequency \(T^{-1}\).
\[n = \frac{N_U^2}{P_U^2 T} = 6.283185307\]
Thus, we can now conclude that \({2}\pi\) is simply a number that can take any value derived from the ratio of two units.
And to be sure of what has been stated, let's see what the units of measure are:
\[m_P=\frac{50 G E_P}{c}=\frac{M T^{3}L^{-4}L^3 M^{-1}T^{-2}M L^{2}T^{-2}}{L T^{-1}}=M\]
\[ℓ_P=\frac{G E_P}{c^{4}}=\frac{L^{3}M^{-1}T^{-2}M L^{2}T^{-2}}{L^{4}T^{-4}}=L\]
\[t_P=\frac{G E_P}{c^{5}}=\frac{L^{3}M^{-1}T^{-2}M L^{2}T^{-2}}{L^{5}T^{-5}}=T\]
To analyze Planck units in more detail, we need to consider the real values. Since the gravitational constant value used to calculate Planck units is based on our planet, which is not a black hole, the following calculations will use the value for black holes, specifically \( G = \frac{1}{50 c} \).
Using the Planck units recommended by CODATA and comparing the ratio of \( ℓ_P \) to \( t_P \), we observe a result very close to the speed of light, which suggests that the dimensions of Planck units correspond to the proportions of a black hole, but which do not respect the Schwarzschild radius.
\[\frac{ℓ_P}{t_P}=\frac{1.616255\times 10^{-35}}{5.391247\times 10^{-44}}={299792422.791}[m]\]
Now that we know the calculation relationships, we can calculate the values of \( m_P \), \( ℓ_P \), and \( t_P \).
\[m_P=\sqrt{\frac{\hbar c}{G}}=2.17692659\times 10^{-8}\]
\[ℓ_P=\sqrt{\frac{\hbar G}{c^{3}}}=1.61588955\times 10^{-35}\]
\[ ℓ_P=\frac{G m_P}{c^2}={1.61588955\times 10^{-35}}\]
\[t_P=\sqrt{\frac{\hbar G}{c^{5}}}=5.39002735\times 10^{-44}\]
\[ t_P=\frac{G m_P}{c^3}={5.39002735\times 10^{-44}}\]
The ratio between \( l_P \) and \( t_P \) should equal the speed of light.
\[\frac{ℓ_P}{t_P}=\frac{1.61588955\times 10^{-35}}{5.39002735\times 10^{-44}}={299792458}[m]\]
The ratios \( ℓ_P/L \), \( t_P/T \), and \( m_P/M \) must also be equal.
\[\frac{m_P}{M}={2.95276689\times 10^{42}}\]
\[\frac{ℓ_P}{L}={2.95276689\times 10^{42}}\]
\[\frac{t_P}{T}={2.95276689\times 10^{42}}\]
Looking at the three equalities, we can say that the Planck black hole does not have a Schwarzschild radius, which should be 2 times larger than the Planck length. Because the mathematical relationships are identical to the quantum, Planck black hole or M87, the Schwarzschild radius will not be found anywhere in this or any other Universe.
Based on experience with natural units, the first thing we notice is that Planck units reference a black hole with a mass \({2.95276688\times 10^{42}}\) times greater than that of a quantum, meaning that all mathematical relationships and constants are equivalent. Naturally, the units for mass, length, time, energy, and angular momentum differ, as they reflect different characteristics for each black hole.
Planck units have long been viewed as a natural set of units that describe the fundamental limits of the universe. These units, defined in terms of universal constants like the speed of light \(c\), the gravitational constant \(G\), and Planck's constant \(h\), provide a reference scale for phenomena like quantum gravity and black holes. However, a more detailed analysis shows that Planck units are not as fundamental as they might seem; they depend on a more basic set of units, known as natural units.
Natural units are distinct because they exist independently of Planck units. These units, such as length, mass, and time, are the "building blocks" of the universe, and the relationships between them are essential for understanding the fundamental forces. In contrast, Planck units, though useful in many contexts, are derived from natural units, effectively representing a larger scale. When viewed through the lens of natural units, Planck units are simply \({2.95276688\times 10^{42}}\) times larger.
One interesting aspect is that the foundation of Planck units is actually tied to black hole concepts. Essentially, these units are closely linked to the mass and dimensions of such an object, and this becomes clear when we derive fundamental constants like the gravitational constant \(G\) using Planck units. However, when we reverse the process and use natural units to derive the same constants, we get the same results, but without needing to rely on Planck units.
It's important to note that while Planck units have a well-established place in modern theoretical physics, they cannot exist without natural units. On the other hand, natural units remain valid even without Planck units. This is not a criticism of Planck units, but rather a recognition that natural units offer a more fundamental framework from which all other units are derived.
In other words, Planck units are useful in specific contexts—black holes and high-energy phenomena—but they do not represent the ultimate foundation of physical reality. In this sense, they depend on natural units for their justification. This is an important relationship that deserves better understanding and recognition within the scientific community.
Conclusion:
Planck units and natural units are not in competition but complement each other. While Planck units provide us with a framework for exploring the extremes of the universe, natural units are the roots from which these understandings grow. By accepting both sets of units and understanding their relationship, we can build a more coherent model of physical reality.
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