The removal of the conventional gravitational constant “G” from its constancy has an effect throughout physics, as there are many formulas in which G plays a role. This article uses the Planck units to show the effects of the variation of the gravitational constant on the physical system of units. The uncertainty in determining of the “gravitational constant” and the assessment of some phenomena, which are still regarded as anomalies today, becomes easier as a result. The changes in people’s everyday lives are marginal. For all processes and calculations that relate to distances from a few millimetres to distances such as Sun-Earth, the deviations are smaller than those resulting from the tolerance of the gravitational constant that applies today. In this range, it is therefore still possible to work with the conventional gravitational constant. However, for the evaluation of measurement results and for theoretical considerations relating to distances that exceed the size of our solar system, it makes sense to use the gravitational variable. It is just as useful for distances that lie below the Planck horizon.

It is no secret that the law of gravitation is disputed in physics. If the gravitational lines of force were to extend parallel to infinity, then Newton would be right with his law of gravity. However, it is no secret in physics that the law of gravity is controversial. There is plenty of evidence that gravity does not behave quite as simply as Newton’s law suggests [2] . At very large distances, the observable effects of gravity indicate a reduction in the force of attraction. In contrast, the force of attraction is greater at very small distances. This would fit exactly into the picture of an exponential curve or a power function. However, today such deviations are calculated relativistically, i.e. according to the rules of the General Theory of Relativity (GRT). There are a number of gravitational anomalies. The most spectacular is the perihelion aberration of the planets. This provides us with a wonderful template, because the perihelion deviation of the planets is closely related to the variable gravitation. This allows us to draw direct conclusions about the functional relationship. The planetary orbits would have to describe an exact ellipse if Newton’s law were correct. But this is not the case. The measured deviations are very small and only become apparent when you add up the measurements over several years. They are most obvious for the planet Mercury [1] [3] , which led to the formulation “the perihelion precession of Mercury”.

In his article from 2023, Weber [1] on the subject of the “gravitational constant”, the results of a calculation were presented, which showed the dependence of the perihelion precession of the planets on the distance between two bodies (in this case the planet and the Sun). A formula was thus proposed that modifies the gravitational constant and uses it to calculate gravity from zero to infinity. The values of the perihelion shift merely serve as diagram points of a function that can be used to calculate the deviation of the variable gravitational constant (G_var) from Newton’s gravitational constant for the distance. of two points.

Why not use this perihelion precession indicator? The course of the variable gravitational force deviates only very slightly from a straight line, so that it was previously interpreted as a straight line, or, therefore, as the gravitational constant.

The proposed formula for a variable gravitational in this article is:

$G_{var}=\frac{k_1}{r^{k_2}}$(1)

where $k_1=6.674296\times {10}^{-11}$ (conventional gravitational constant); $k_2=\frac{1}{R}=\frac{\text{1}}{\text{Rydberg Constant}}=\frac{1}{10973731.57}$.

The formula for the force of gravity becomes then:

$F=\left(\frac{k_1}{r^{k_2}}\right)\left(\frac{m_1\cdot m_2}{r^2}\right)$(2)

If you modify a formula that contains a G, so that the result of the formula = 1 by adjusting the G accordingly, you get a series of values that characterize the gravitational constant and form a scale. The Planck units are essentially formed by combining the three fundamental constants [4] - [8] :

c = vacuum light velocity = 199,792,458 m/s;

G = gravitational constant = 6.674296E−11 m³/kgs²;

$\bar{h}$ = quantum of action = 1.054571817E−34.

It is assumed that the vacuum speed of light and the quantum of action are fixed quantities, so that it is only worth varying the gravitational constant. This variation makes the gravitational constant a variable and leads to a scale of the gravitational constant. The variation of the Planck units and the units derived from the electron was carried out by means of iteration (successive approximation) by two separate steps. First, however, an attempt was made to bring the natural units related to the electron into agreement with the Planck units by changing G. In other words, G was changed so that the electron mass became equal to the electron mass (em). G thus takes the value:

$G_{var}em=3.80994675\times {10}^{+34}$

If we set the Planck energy equal to the electron energy (ee), then we get:

$G_{var}ee=3.80994675\times {10}^{+34}$

And if you set the Planck length equal to the Compton wavelength (comp) of the electron, then you also get:

$G_{var}comp=3.80994675\times {10}^{+34}$

So, there is a constant quantity that can be inserted into the formulas in place of the conventional G, making the Planck units the natural units of the electron. However, the same procedure carried out with the electron radius (er) for the Planck length gives:

$G_{var}er=2.0288479\times {10}^{+30}$.

And equating the Planck time with the electron time (et) (this is the time it takes for a photon to fly through an electron) gives:

$G_{var}et=2.0288479\times {10}^{+30}$

Obviously, there are only two positions on a G-scale where the natural units derived from the electron are hidden. For example, they could be named:

$G_{va{r}^1}$ and $G_{va{r}^2}$.

α is defined as the ratio of the squares of the electron mass to the Planck mass. The point $G_{va{r}^1}$ fulfills the condition that the Planck mass is equal to the electron mass. It was found under this condition. But this also means that at this point, the gravitational coupling constant $\alpha G=1$, because the numerator and the denominator now have the same values.

Thus, the point $G_{va{r}^1}=3.80994675\times {10}^{+34}$ is the point where gravity is equal to the strong force.

The point $G_{va{r}^1}$ is different from the point $G_{va{r}^2}$ by the factor α². Thus, the gravitational coupling constant α in the point $G_{va{r}^1}$ becomes the square of the coupling constant of the electro-magnetic interaction α.

At this point, the gravitational force is equal to the electro-magnetic force.

It turned out that there are only these two points on the G_var scale that are relevant for the representation of the “natural” units derived from the electron. They were named $G_{va{r}^1}$ and $G_{va{r}^2}$. They differ by the factor α. Just as you get from $G_{va{r}^1}$ to $G_{va{r}^2}$, you can multiply further and get $G_{va{r}^3}$, $G_{va{r}^4}$, $G_{va{r}^5}$ and so on. This also works in the other direction from $G_{va{r}^1}$. If you divide $G_{va{r}^1}$ by α, you get $G_{va{r}^1}$, $G_{va{r}^2}$, $G_{va{r}^3}$ and so on. This gives you a scale of points from which you can assume that you will find further natural units. However, this has so far proved to be a mistake. All natural units relating to the electron are concentrated on the points $G_{va{r}^1}$ and $G_{va{r}^2}$

However, it is interesting for further research to determine whether the free places on the scale of natural units are occupied by other units.

The ratio of these two G_var values results in:

$\frac{G_{va{r}^2}}{G_{va{r}^1}}=0.000053251345$ or $\frac{G_{va{r}^1}}{G_{va{r}^2}}=137,03601133$

This is the famous magic number whose origin was unknown for decades. Apparently, it represents the equal distance between the natural units. This is the square of the coupling constant of the electro-magnetic interaction α, which is also known as Sommerfeld’s fine structure constant.

It is $\sqrt{\frac{G_{va{r}^2}}{G_{va{r}^1}}}=0.007297351923=\alpha$

From this point of view, it was interesting to find out whether the G_var scale of units might contain other values, for example, those related to the Planck units. For this purpose, 17 Planck units were adjusted by changing G, so that the result of the Planck formula became equal to 1.

In contrast to the units related to the electron, the resulting G_var values of the Planck units were well distributed over the entire scale generated using α. But they did not hit any of the points exactly. In Table 1 , the values in column Gα are all hypothetical except for the start values. However, it could be seen that they followed their own direction. They had their own factor, which differed slightly from α. While α has the numerical value 0.0072973525664, the new factor, say α', has the value 0.00759967158. That is 4.143% more than α. This second scale is related to the speed of light “c”. The reciprocal of 0.00759967158 is 137.035999 and if you square this twice, i.e. to the power of 4, you get 299,792,457 = “c”—the speed of light.

So, each of the two scales has a famous constant as its “godfather”. It can be seen here that the distance between the two scales Gα. and Gα' corresponds in percentage terms almost exactly to the difference found when determining the proton radius with electrons on the one hand and with muons on the other, for which there is no plausible explanation to date.

Table 1 shows the two “scales” for the range in which the derived Planck units are located. On the left is the scale generated using α, on the right and in the center is the scale generated using α'. The latter contains values marked in red. These are the G values for which the respective Planck unit assumes the value 1. The middle column has calculated values. It is merely intended to show that the “values marked in red” correspond to the calculated values to at least the sixth decimal place. The values marked in bold are the starting values ( $G_{va{r}^1}$ and $G_{va{r}^2}$) from which the development of the scales started. The Gα' values have two different starting values. If the mean value is calculated from these points, the result is the value G(mean) = 8.07760872970777E+33. This in turn is the value that results when the Planck force equals 1.

The third scale listed in Table 1 corresponds to scale α'. Here, the values marked in red were generated by iteration. These are the G values for which the respective Planck unit assumes the value 1. The centre column was calculated in full. It is merely intended to show that the “values marked in red” correspond to the calculated values to at least the sixth decimal place. The values marked in bold are the starting values ( $G_{va{r}^1}$ and $G_{va{r}^2}$) from which the development of the scales started.

In this work, a result is generated in two different ways. One is with the Planck formulas and the other is by trial and error. It has to be emphasized that the “twin formulas” were guessed until both formulas produced the exact same result in one field each. The amazing thing is that the guessed formulas can all be represented solely by the speed of light and another constant. The constant, which was given the working name K, arose by chance in the run-up to processing. The correspondence between the two formulas per unit is so clear that there were no problems in the respective search for the formula.

When working with the Gα' scale, the constant 1.173927 kept coming to mind. As it did not yet exist in physics, it had to be given a name. It was given the working name α' constant or Kα'. This constant is found several times in the Gα' scale. For example, it is contained in the ratio of the starting value “−” (minus) to the mean value and in the ratio of the mean value to the starting value “+” (plus).

Table 1 is to be regarded as an intermediate step. The final relationships are shown in Table 2 and Table 3 . Looking at Table 1 in its current form, it is noticeable that it says nothing about how the other units would behave if the gravity variable were to change. The units must therefore be linked to this variable with their own formulas. Each unit receives a column with formulas for the units that want to use the variable “G_var”. This is conveniently placed in the column header. The result is a matrix of units, as shown in Table 2 .

Table 2 is a matrix of the Planck Units with values set equal to 1. The calculation method in the matrices is as follows. In a spreadsheet, all the Planck formulae are written one below the other and the gravitational constant G is placed in the header of this table (column). In the formulae, the G is replaced with a reference to the G in the column header. Then, there is the original Planck value in each row. The G value in the column header is now changed for as long as necessary—until a Planck formula in the body of the column equals 1. This is best done with a spreadsheet iteration programme. This is how the relationships are created. This process is repeated once for each Planck unit. This creates a matrix. As many columns are needed as there are rows. The variable G values that arise in the column header form the G_var scale of the Planck units. This matrix is unique. It was not calculated, but developed from the Planck units.

Header line 2 in the table body contains the original mathematically programmed Planck formulas. This is done in such a way that the Planck formulas access the respective “G” in the header. This creates the original Planck units on each “line 2” in the body of the table. Then, column by column, the G is set using the Excel function “Target value search” so that a Planck unit of the column assumes the value 1.00000... (iteration). It starts with “mass” and ends with “acceleration”. This produces Table 2 with its conversion factors. The former “G” in the column header has thus become “G_var determined”.

If the Planck units in Table 2 are set to 1, all other Planck units that want to

In Table 3 , the conversion factors are presented similar to Table 2 . A five-line field was set up for each factor, with the top “line 1” showing the “pairing” as the heading in each case.

The table was reduced from 17 Planck units to 11 units in order to be able to print the table. The “line 2” is reserved for the factor itself as it results from the Planck formula.

The units are connected by conversion factors, which also reflect the equivalence of relations. The condition is that the “gravitational constant” is not accepted as constant, but as variable. The application of this system will be interesting, especially for the measuring and calibration technology by limiting oneself to the provision, maintenance and care of an original measure and deriving the other measures from it. The application of this system will be of particular interest to metrology and calibration technology by limiting itself to the provision, maintenance and care of one original measure and deriving the others from it. In addition, the system presented here may stimulate further research in the field of physical units. For example, we were able to show that at least, the constant $\bar{h}$, similar to the Planck units, can be traced back to c and K.

$\bar{h}=\frac{1}{c^{4}K}$

Note: If interested, a functional Table 2 or Table 3 can be requested by email from the author as a spreadsheet. These spreadsheets with descriptions can be used to perform extensive calculations.