1. Backround

1.1. The Planck scale and the Compton scale

Max Planck [17,18] assumed in 1899 and 1906 that there were three important universal constants: the speed of light c, the Planck constant1ℏ, and Newton’s gravitational constant G. By combining these constants with dimensional analysis, he found a unique length: $l_p=\sqrt{\frac{G\hslash }{c^3}}$, time: $t_p=\sqrt{\frac{G\hslash }{c^5}}$, mass: $m_p=\sqrt{\frac{\hslash c}{G}}$, and temperature: $T_p=\frac{1}{k_b}\sqrt{\frac{\hslash c^5}{G}}$. To calculate temperature, one must also use the Boltzmann constant $k_b$, which is simply a constant used to convert from one unit scale to another, namely from joules to temperature.

Eddington [19] was likely the first to suggest in 1918 that a quantum gravity theory should be connected to the Planck length. This viewpoint is still widely held among most experts in the field of quantum gravity [20,21,22], but despite extensive efforts, no breakthrough has yet been achieved and there is no consensus on any quantum gravity theory. In this paper, we propose a new and simple method for incorporating the Planck scale and achieving quantum gravity. It is so straightforward that it may be easy to dismiss it based on preconceptions, but we encourage the gravity community to examine it closely before rejecting it. Sometimes, the simplest solution is the best.

In 1984, Cahill [23,24] suggested that the Planck units might be more fundamental than the gravitational constant and proposed that the gravitational constant could be expressed as a composite constant of the form $G=\frac{\hslash c}{m_p^2}$, which is simply the Planck mass formula solved with respect to G. Cohen [25] derived the same formula for G in 1987, but correctly pointed out that since no one had been able to find the Planck mass or some of the other Planck units independently of G, expressing G in terms of the Planck units would lead to a circular and unsolvable problem. This has been a view held at least to 2016, see [26]

In 2017, Haug [27] showed for the first time that the Planck length can be determined independently of any knowledge of G. He used a Cavendish apparatus and derived the following equation:

$$l_p=\sqrt{\frac{\hslash 2{\pi }^{2}L{R}^2\theta }{M{T}^2{c}^3}},$$

(6)

where ℏ is the reduced Planck constant, L is the distance between the two small balls in the apparatus, R is the distance between the center of the large and small balls in the apparatus, $\theta$ is the measured angle, T is the oscillation period, c is the speed of light, and M is the mass of the large ball in kilograms. The mass of the large ball can be determined using a standard balance weight and does not require knowledge of G. Generally, we only need to know G when we need to determine the kilogram mass of astronomical objects.

Further investigations [28] have shown that the Planck length can also be determined in a Cavendish apparatus using the following equation:

$$l_p=\sqrt{\frac{\overline{\lambda }2{\pi }^{2}L{R}^2\theta }{T^2{c}^2}},$$

(7)

In this case, we are no longer dependent on knowing either the kilogram mass of the large ball in the Cavendish apparatus or the Planck constant. However, we now do need to know the reduced Compton wavelength of the large ball in the apparatus. This we have demonstrated in a series of papers [28,29,30] can be done for any mass independent on knowing both the Planck constant or the kilogram mass of any object and also without knowing G.

The fact that we can find the Planck length independent of G means that we can express the gravitational constant in terms of Planck units. We can solve the Planck length formula for G, which gives us the equation:

$$G=\frac{l_p^2{c}^3}{\hslash }$$

(8)

There are many ways to express the gravitational constant using Planck units. For example, we could have solved the Planck time formula for G, which would give us $G=\frac{t_p^2{c}^5}{\hslash }$. Other suggestions can be found in the literature, as discussed in [31], which provides a review of the composite view of the gravitational constant.

Now that we have the composite gravity constant, we can input it into the modified 1873 Newton formula:

$$F=G\frac{Mm}{R^2}=\frac{l_p^2{c}^3}{\hslash }\frac{Mm}{R^2}$$

(9)

However, this does not provide much new or deeper insight. While one could argue that Newton’s force of gravity is now a function of both the Planck constant and the Planck length, as well as the speed of light, we need to look more closely at the mass.

As discussed in our review paper [31], there is more to the story than just the Composite constant of G and the Planck units. We need to consider the role of mass in relation to the gravitational constant and how it affects our understanding of gravity. So while the composite view of the gravitational constant is an important step forward, it is just one of two steps to get to a quantum gravity theory.

Compton [32] formulated the Compton wavelength formula in 1923 as $\lambda =\frac{h}{mc}$. We can solve this formula for the mass m in kilograms, giving us:

$$m=\frac{h}{\lambda }\frac{1}{c}=\frac{\hslash }{\overline{\lambda }}\frac{1}{c}$$

(10)

In fact, any kilogram mass can be expressed in this form. Even if a composite mass does not have a single physical reduced Compton wavelength, it has what we can call an aggregated reduced Compton wavelength see [29,33], given by:

$$\overline{\lambda }=\frac{1}{{\sum }_i^n\frac{1}{{\overline{\lambda }}_i}}=\frac{1}{\frac{1}{{\overline{\lambda }}_1}+\frac{1}{{\overline{\lambda }}_2}+\frac{1}{{\overline{\lambda }}_3}+\frac{1}{{\overline{\lambda }}_4}\cdots \frac{1}{{\overline{\lambda }}_n}}$$

(11)

Even energy, when looked at as mass, can be described with a Compton wavelength, because:

$$\begin{array}{ccc}\hfill \frac{E}{c^2}& =& m\hfill \\ \hfill \frac{h\frac{c}{{\lambda }_{\gamma }}}{c^2}& =& \frac{\hslash }{\overline{\lambda }}\frac{1}{c}\hfill \\ \hfill \frac{\hslash \frac{c}{\overline{\lambda }\gamma }}{c^2}& =& \frac{\hslash }{\overline{\lambda }}\frac{1}{c}\hfill \\ \hfill {\overline{\lambda }}_{\gamma }& =& \overline{\lambda }\hfill \\ \hfill {\lambda }_{\gamma }& =& \lambda \hfill \end{array}$$

(12)

Here, ${\overline{\lambda }}_{\gamma }$ is what we can call the reduced photon wavelength, which must be equal to the reduced Compton wavelength for the mass equivalent of the energy, for $\frac{E}{c^2}=m$ to be true. It simply means the photon wavelength can be seen as the Compton wavelength of the photon. Therefore even such things as binding energy [34] can be treated within this framework of aggregating Compton wavelengths to get the Compton wavelength of the mass in question. So assume for example we have a mass consisting of three elementary particles and two types of binding energy, then we have

$$\begin{array}{ccc}\hfill m& =& m_1+m_2+m_3+\frac{E_1}{c^2}+\frac{E_2}{c^2}\hfill \\ \hfill \frac{\hslash }{\overline{\lambda }}\frac{1}{c}& =& \frac{\hslash }{{\overline{\lambda }}_1}\frac{1}{c}+\frac{\hslash }{{\overline{\lambda }}_2}\frac{1}{c}+\frac{\hslash }{{\overline{\lambda }}_3}\frac{1}{c}+\frac{\hslash \frac{c}{{\overline{\lambda }}_4}}{c^2}+\frac{\hslash \frac{c}{{\overline{\lambda }}_5}}{c^2}\hfill \\ \hfill \overline{\lambda }& =& \frac{1}{\frac{1}{{\overline{\lambda }}_1}+\frac{1}{{\overline{\lambda }}_2}+\frac{1}{{\overline{\lambda }}_3}+\frac{1}{{\overline{\lambda }}_4}+\frac{1}{{\overline{\lambda }}_5}}\hfill \end{array}$$

(13)

The important point here is simply that we indeed can express any kilogram mass in the form of $m=\frac{\hslash }{\overline{\lambda }}\frac{1}{c}$. Next, let’s get back to gravity.

If we examine the modern Newtonian gravity force formula, we can observe that any formula used to predict observable gravitational phenomena includes $GM$ and not $GMm$. In two-body problems where m is not much smaller than M (i.e., m is also significantly large and important for gravitational predictions), the gravitational parameter is $\mu =G(M+m)=GM+Gm$. Even in such cases, we do not encounter $GMm$ in any formula that predicts observable gravitational phenomena.

However, when $m\ll M$, the smaller mass m is necessary only during derivations, as it cancels out before we arrive at a formula that can be utilized to predict gravity phenomena that can be verified through observations. The gravitational force itself is not directly observable.

If we substitute G with $G=\frac{l_p^2{c}^3}{\hslash }$ and M with $M=\frac{\hslash }{{\overline{\lambda }}_M}\frac{1}{c}$, we obtain:

$$GM=\frac{l_p^2{c}^3}{\hslash }\frac{\hslash }{{\overline{\lambda }}_M}\frac{1}{c}=c^3\frac{l_p}{c}\frac{l_p}{{\overline{\lambda }}_M}$$

(14)

Here, ${\overline{\lambda }}_M$ represents the reduced Compton wavelength of the larger mass M in the Newton formula, while $l_p$ denotes the Planck length and c represents the speed of light. Several important points related to Eq. (14) deserve attention. The composite gravitational constant contains the embedded Planck constant, which cancels out with the Planck constant present in the kilogram mass. Of the components $l_p$, ${\overline{\lambda }}_M$, c, and ℏ, only the Planck constant has kilogram units embedded in it. Therefore, everything related to kilogram units cancels out when predicting gravitational phenomena.

This outcome should not surprise those familiar with pre-1873 Newtonian theory, which did not include anything about kilograms. In our view, multiplying G with the kilogram mass M replaces the Planck constant with the Planck length, converting an incomplete kilogram mass into a complete gravitational mass. We have in multiple papers [35,36] claimed that the last part of Eq. (14):

$$\frac{l_p}{c}\frac{l_p}{{\overline{\lambda }}_M}=t_p\frac{l_p}{{\overline{\lambda }}_M}$$

(15)

can be described as the real gravitational mass, or what we have also called collision-time mass ($M_t)$, as it has dimension time $\left[T\right]$. If we multiply the collision-time mass by $c^3$ and call that mass, we arrive back at the original Newton mass that also Maxwell described with units $[L^3·T^{-2}]$. Therefore, from a deeper perspective, the Newton mass is equal to

$$M_n=GM=c^3{M}_t=c^3{t}_p\frac{l_p}{{\overline{\lambda }}_M}$$

(16)

One can therefore argue the Newton mass as a mass where G is embedded. However, it is important to be aware that finding $GM$ requires less information than finding G and M separately. This is because $GM$ contains no Planck constant, i.e., no information about the kilogram, while both G and M separately contain information about the kilogram. For example we can find $GM$ from measuring the gravitational acceleration at the surface on Earth. We have $GM=R^{2}g$ and we can find g by dropping a ball from hight H and measure how long time it took before it hit the ground, we have $g=\frac{2H}{T_d^2}$ where $T_d$ is the time it took for the ball from it was dropped to it hit the ground. This is why the mass of the Earth ($M_n$) can be directly calculated from gravitational acceleration or any other gravitational observation, and $GM$ can also be directly calculated from a gravitational acceleration or another gravitational observation. However, finding the kilogram mass (M) of the Earth requires finding the gravitational constant G, for example, from a Cavendish apparatus.

The collision-time mass can be derived by dividing the Newton mass by $c^3$, which in this model acts as a gravitational constant. Thus, we have $GM=M_n=c^3{M}_t$. However, a deeper understanding of these values reveals that they are all linked to the Planck scale through the expression $c^3{t}_p\frac{l_p}{{\overline{\lambda }}_M}$. This implies that even Newton gravity is linked to the Planck scale since the Planck length can be extracted from a Cavendish apparatus without knowing G. Newton naturally did not assume this, however the Planck scale get embedded in the mass by measuring the mass from gravitational phenomena.

Quantum gravity theory assumes that gravity is quantized, and since quantization in quantum mechanics is associated with the Planck constant, it is reasonable to expect that the Planck constant would also play a role in quantum gravity. Some experiments involving gravity have even claimed to be linked to the Planck constant, but as we will see in Section 4, this is not correct.

The Planck constant is not present in $M_n$ or $c^3{M}_t$ and even cancels out when multiplying G with M. Nevertheless, we assert that quantization still exists within the expression $\frac{l_p}{\overline{\lambda }}$, which is equivalent to the reduced Compton frequency per Planck time. The reduced Compton frequency per second is $f=\frac{c}{\overline{\lambda }}$, but the reduced Compton frequency for the Planck time must be $f=\frac{c}{\overline{\lambda }}{t}_p=\frac{l_p}{\overline{\lambda }}$. This can also be understood by considering how far light travels per Planck time. In other words, we can measure the speed of light in meters per Planck time instead of per second. Since light can travel the Planck length per Planck time, we can conclude that the reduced Compton frequency per Planck time is as described above.

This indicates a connection between quantization in gravity, the reduced Compton wavelength, and the Planck length. For a particle with mass equal to the Planck mass, the reduced Compton frequency per Planck time is equal to one $f=\frac{l_p}{\overline{\lambda }}=\frac{l_p}{l_p}=1$, as the reduced Compton wavelength of the Planck mass particle is the Planck length: $\frac{\hslash }{m_{p}c}=l_p$. It is natural to assume that the minimum frequency that can be observed in an observational time window is one. If the shortest possible time window is the Planck time, then the minimum observation in terms of reduced Compton frequency in such a time window is also one. Particles with mass smaller than the Planck mass will have a frequency of less than one in the Planck time window. For instance, an electron will have a reduced Compton frequency per Planck time of only $f_e=\frac{l_p}{{\overline{\lambda }}_e}\approx 4.19\times {10}^{-23}$. We have suggested in multiple papers [35,37] that this can also be interpreted as a frequency probability when it is below one, which we will discuss further later.

In summary, we make the bold claim that even Newton gravity is quantized and linked to the Planck scale and the Compton frequency when understood from a deeper perspective. We are naturally not suggesting that Newton himself hid quantization in his formula since the Planck length and the Compton wavelength were unknown during his time. Rather, we assert that this is what even Newton gravity represents when the gravitational mass is understood at a deeper level. Moreover, it should be possible to determine the reduced Compton frequency per Planck time from gravitational observations. For instance, by measuring the gravitational acceleration on Earth, we can determine the reduced Compton frequency of the Earth per Planck time. Equation 17 shows that the reduced Compton frequency per Planck time of the Earth can be determined by knowing the gravitational acceleration, the Planck length, the speed of light, and the radius of the Earth.

$$\frac{l_p}{\overline{\lambda }}=\frac{g{R}^2}{l_p{c}^2}$$

(17)

By comparing the value obtained from this formula with traditional methods for finding the Compton wavelength and frequency, we can verify its accuracy. Additionally, we can determine the reduced Compton wavelength of the Earth by knowing its kilogram mass, which requires knowledge of the gravitational constant in standard theory. However, once we have the Earth’s mass, we can use ${\overline{\lambda }}_E=\frac{\hslash }{Mc}$ to find the reduced Compton frequency per Planck time, given by $f=\frac{c}{{\overline{\lambda }}_e}{t}_p=\frac{l_p}{{\overline{\lambda }}_e}$. Equation 17 works for any astronomical object and provides the correct prediction for the reduced Compton frequency per Planck time.

The key point is to understand so far is that, at a deeper level, Newton’s gravity exhibit quantization of gravity and a link to the Planck scale and what we could call the Compton scale or Compton frequency. Next we will try to see how this fit in with Einstein’s general relativity theory.

Table 1 shows that both the original Newtonian gravitational force formula with mass $M_n$ and no gravitational constant, as well as the modern 1873 Newtonian gravitational force formula, predict the same results. From a deeper perspective, they are exactly the same formulas.

Table 1. The table shows a series of gravity predictions that typically are linked to Newton gravity theory which is the weak field approximation of general relativity theory.

Table 1. The table shows a series of gravity predictions that typically are linked to Newton gravity theory which is the weak field approximation of general relativity theory.

Prediction	Formula:
Gravity acceleration	$g=\frac{GM}{R^2}=\frac{M_n}{R^2}=\frac{c^2{l}_p}{R^2}\frac{l_p}{{\overline{\lambda }}_M}$
Orbital velocity	$v_o=\sqrt{\frac{GM}{R}}=\sqrt{\frac{M_n}{R}}=c\sqrt{\frac{l_p}{R}\frac{l_p}{{\overline{\lambda }}_M}}$
Orbital time	$T=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi R}{\sqrt{\frac{M_n}{R}}}=\frac{2\pi R}{c\sqrt{\frac{l_p}{R}\frac{l_p}{{\overline{\lambda }}_M}}}$
Velocity ball Newton cradle	$v_{out}=\sqrt{2\frac{GM}{R^2}H}=\sqrt{2\frac{M_n}{R^2}H}=\frac{c}{R}\sqrt{2H{l}_p\frac{l_p}{{\overline{\lambda }}_M}}$
Frequency Newton spring	$f=\frac{1}{2\pi R}\sqrt{\frac{GM}{x}}=\frac{1}{2\pi R}\sqrt{\frac{M_n}{x}}=\frac{c}{2\pi R}\sqrt{\frac{l_p}{x}\frac{l_p}{{\overline{\lambda }}_M}}$

2. General relativity theory

Table 2. The table shows a series of gravity predictions given by general relativity theory, but not only on their traditional form, but also from their deeper level when G is replaced by its composite form and the mass is understood from a deeper level. All the predicted gravity phenomena contains the factor $\frac{l_p}{\overline{\lambda }}$ which is the reduced Compton frequency of the mass M per Planck time. We will boldely claimn that his gives both the link to the Planck scale and the quantization of gravity without altering general realtivity theory.

3. Possible interpretation

4. The claims that the Planck constant plays role in some observed gravity phenomena is likely wrong when understood from a deeper perspective

“First of all, they demonstrate the validity of quantum theory in a gravitational potential on the 1% level. The gravity-induced interference is purely quantum mechanical, because the phase shift is proportional to the wavelength λ and depends explicitly on Planck’s constant ℏ , but the interference pattern disappears as $\hslash \to 0$. The effect depends on ${(m_n/\hslash )}^2$ and the experiments test the equivalence principle."

5. Conclusion

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