Politics, Power, and Science: Newton’s Gravitational Law in Planck Units and the Role of SI Constants

Thursday, May 15, 2025

Newton’s Gravitational Law in Planck Units and the Role of SI Constants

J. Rogers, SE Ohio, 15 May 2025, 1647

Natural (Planck) units vs. SI units:
In natural-unit systems, fundamental constants are set to 1 so that equations lose explicit conversion factors. For example, in Planck units, the speed of light $c$ , Newton’s gravitational constant $G$ , Planck’s constant $h$ , etc., are all normalized to unity. In such a system, all dimensions “collapse” and physical laws involve only dimensionless ratios. As one source explains, a “purely natural system of units has all of its dimensions collapsed, such that the physical constants completely define the system of units and the relevant physical laws contain no conversion constants.” In contrast, the SI (meter-kilogram-second) system employs fixed numerical values of these constants to define its base units. For instance, the meter is defined by fixing $c = 299 792 458$  m/s, and the kilogram by fixing $h = 6.62607015 \times 10^{- 34}$  J·s. In effect, each SI constant acts as a scaling factor between human-defined units and the underlying dimensionless physics.

Planck units definitions:
Planck units set $c = h = G = k_{B} = 1$ . In particular, the Planck length $L_{P}$ , mass $M_{P}$ , and time $T_{P}$ are defined by combinations of $c, h, G$ as:

$L_{P} = \sqrt{\frac{h G}{c^{3}}}, M_{P} = \sqrt{\frac{h c}{G}}, T_{P} = \sqrt{\frac{h G}{c^{5}}}$

Numerically, $L_{P} \approx 1.616 \times 10^{- 35}$  m and $M_{P} \approx 2.176 \times 10^{- 8}$  kg. In Planck units, these base quantities all equal 1, so any physical quantity $Q$ is expressed as $Q = Q^{'} U_{P}$ , where $Q^{'}$ is dimensionless and $U_{P}$ is the Planck unit. For example, a mass $m = m^{'} M_{P}$ , distance $r = r^{'} L_{P}$ , and force $F = F^{'} F_{P}$ , where $F_{P}$ (Planck force) is defined below. With these definitions, $c = h = G = 1$ means that the dimensionful constant $G$ no longer appears explicitly, and physical laws (written in the primed dimensionless variables) take their simplest form.

Newton’s law in SI vs. Planck form:
In SI units, Newton’s gravitational law reads

$F_{S I} = G_{S I} \frac{m_{1} m_{2}}{r^{2}}$

where $G_{S I} \approx 6.674 \times 10^{- 11}$  m³/(kg·s²). Dimensional analysis shows $[G_{S I}] = L^{3} / (M T^{2})$ , ensuring $[F] = M L / T^{2}$ as usual.

Now express each quantity in Planck units:

$m_{i} = m_{i}^{'} M_{P}, r = r^{'} L_{P}, F_{S I} = F^{'} F_{P}$

Substituting into Newton’s law gives:

$F^{'} F_{P} = G_{S I} \frac{(m_{1}^{'} M_{P}) (m_{2}^{'} M_{P})}{(r^{'} L_{P})^{2}} = G_{S I} \frac{M_{P}^{2}}{L_{P}^{2}} \frac{m_{1}^{'} m_{2}^{'}}{r^{' 2}}$

By construction, the Planck force $F_{P}$ is chosen so that $G_{S I} M_{P}^{2} / (L_{P}^{2} F_{P}) = 1$ . Indeed, one finds

$F_{P} = \frac{c^{4}}{G_{S I}}$

which exactly cancels the prefactor. Thus, the equation reduces to

$F^{'} = \frac{m_{1}^{'} m_{2}^{'}}{r^{' 2}}$

with no explicit $G$ . This matches the dimensionless form quoted in references: “if by a shorthand convention each physical quantity is the corresponding ratio with a coherent Planck unit, the ratios may be expressed simply with symbols… $F^{'} = m_{1}^{'} m_{2}^{'} / r^{' 2}$ .” This last equation (without $G$ ) is valid with $F^{'}, m_{1}^{'}, m_{2}^{'}, r^{'}$ being the dimensionless ratio quantities. In other words, Newton’s law itself becomes a pure statement about the geometry and masses, free of units, once expressed in Planck units.

Dimensional consistency check:
The algebra above is a dimensional check of the transformation. Starting from $[G_{S I}] = L^{3} / (M T^{2})$ , $[M_{P}] = M^{1 / 2} L^{1 / 2} T^{- 1}$ (from $M_{P} = \sqrt{h c / G}$ ), $[L_{P}] = L^{1 / 2} M^{1 / 2} T^{0}$ (from $L_{P} = \sqrt{h G / c^{3}}$ ), and $[F_{P}] = [F] = M L / T^{2}$ , one verifies

$G_{S I} M_{P}^{2} / (L_{P}^{2} F_{P}) = 1$

Hence the prefactor cancels. Equivalently, one uses $F_{P} = M_{P} c / T_{P} = c^{4} / G_{S I}$ to show $G_{S I} M_{P}^{2} = F_{P} L_{P}^{2}$ . This validates that $F^{'} = m_{1}^{'} m_{2}^{'} / r^{' 2}$ is dimensionally consistent and correct. (A related fact: the product $m_{1}^{'} m_{2}^{'} / r^{' 2}$ is truly dimensionless since it’s a ratio of like-dimensioned quantities.)

Interpretation: $G_{S I}$ as a conversion factor:
The above derivation shows that $G_{S I}$ is solely the factor that compensates for our choice of SI units. When we “set $G = 1$ ” we are effectively measuring everything in natural (Planck) units. Any time one uses $G_{S I}$ in an equation, one is implicitly converting between SI and natural scales. In this sense, $G_{S I}$ acts as a structured projection scaler: it projects the dimensionless gravitational law into the meter–kilogram–second framework. It fixes the numerical output of a computation in SI units given the same underlying physics in natural units. The same is true of other constants: for example, setting $c = 1$ in relativity equates mass and energy (so $E = m$ dimensionlessly), and reintroducing $c$ rescales back to SI units. In all cases, the fixed constants (like $G_{S I}, c, h, \dots$ ) serve as built-in conversion factors. As one author notes, using “ $c = G = 1$ ” is mathematically a convenient trick but “physically it represents a loss of information and can lead to confusion.” That “information” is exactly which units we are using; the information is still present but hidden by these constants when we stick to SI. The natural-law statement $F^{'} = m_{1}^{'} m_{2}^{'} / r^{' 2}$ is fully valid in any unit system once one understands it refers to dimensionless ratios. Reintroducing SI units demands putting back the combination $G_{S I} M_{P}^{2} / (L_{P}^{2} F_{P})$ (which equals 1)-equivalently multiplying by $G_{S I}$ with the appropriate unit factors-to recover $F = G_{S I} m_{1} m_{2} / r^{2}$ .

Dimensional Transformation Steps

Define Planck-unit ratios:
Write each SI quantity as a Planck ratio times a Planck unit:
$m_{i} = m_{i}^{'} M_{P}$ , $r = r^{'} L_{P}$ , $F = F^{'} F_{P}$ . Here $m_{i}^{'}, r^{'}, F^{'}$ are dimensionless numbers and $M_{P}, L_{P}, F_{P}$ carry dimensions.
By definition, $M_{P} = \sqrt{h c / G_{S I}}$ and $L_{P} = \sqrt{h G_{S I} / c^{3}}$ .
Express Newton’s law:
Plug these into $F_{S I} = G_{S I} m_{1} m_{2} / r^{2}$ :
$F^{'} F_{P} = G_{S I} \frac{(m_{1}^{'} M_{P}) (m_{2}^{'} M_{P})}{(r^{'} L_{P})^{2}} = G_{S I} \frac{M_{P}^{2}}{L_{P}^{2}} \frac{m_{1}^{'} m_{2}^{'}}{r^{' 2}}$
All SI units ( $m, M_{P}, L_{P}$ , etc.) are explicit here.
Use Planck definitions:
Recall that $F_{P} = M_{P} c / T_{P} = c^{4} / G_{S I}$ and $T_{P} = L_{P} / c$ , giving $F_{P} = M_{P} c^{2} / L_{P} = c^{4} / G_{S I}$ . Equivalently, one can check $G_{S I} M_{P}^{2} / (L_{P}^{2} F_{P}) = 1$ by substituting $M_{P}, L_{P}$ from above. This cancels the prefactor.
Dimensionless law:
The result is
$F^{'} = \frac{m_{1}^{'} m_{2}^{'}}{r^{' 2}}$
with no $G$ . This equation involves only the dimensionless ratios $m_{i}^{'}, r^{'}, F^{'}$ . In Planck (natural) units, the gravitational force between two mass-1 bodies separated by distance 1 is simply 1, by construction. This is “universally valid” in the sense that it is independent of human units-it is the underlying geometry of inverse-square attraction.
Reintroducing SI units:
To convert back to SI, one multiplies by the product of Planck units and/or re-inserts $G_{S I}$ . In practice, this means rewriting $m_{i}^{'} = m_{i} / M_{P}$ , $r^{'} = r / L_{P}$ , $F^{'} = F / F_{P}$ in the formula. One obtains
$\frac{F}{F_{P}} = \frac{m_{1}}{M_{P}} \frac{m_{2}}{M_{P}} \frac{1}{(r / L_{P})^{2}} ⟹ F = \frac{F_{P} M_{P}^{2}}{L_{P}^{2}} \frac{m_{1} m_{2}}{r^{2}} = G_{S I} \frac{m_{1} m_{2}}{r^{2}}$
since $F_{P} L_{P}^{2} / M_{P}^{2} = G_{S I}$ . Thus, the SI form is recovered exactly, showing that every appearance of $G_{S I}$ is just compensating the choice of units.

Summary and Implications

This derivation verifies the dimensional arithmetic and shows no algebraic inconsistency. It highlights that the only reason $G_{S I}$ is present in the SI-version of Newton’s law is because of our unit definitions. In a system where lengths, masses, and times are measured in fundamental (Planck) units, Newton’s law has no extra constant.

SI Base Units as Fixed Constants: A Layered Natural-Unit View
Modern metrology explicitly ties SI units to fundamental constants. As the BIPM states, “The SI is defined in terms of a set of seven defining constants. The complete system of units can be derived from the fixed values of these defining constants, expressed in the units of the SI.” These seven constants (e.g. cesium hyperfine frequency, $c, h, e, k_{B}, N_{A},$ etc.) are given exact numerical values, which define the units. For example, fixing $c = 299 792 458$  m/s defines the meter, and fixing $h = 6.62607015 \times 10^{- 34}$  J·s defines the kilogram. In this sense, the SI behaves like a layered natural-unit system: each unit is “anchored” by a fundamental constant. A NIST resource explains that the revised SI “rests on a foundation of seven values, known as the constants… these constants completely define the seven base SI units.” In other words, SI units are not arbitrary prototypes but derived from Nature’s constants. Because of this, one can regard SI as effectively choosing a particular set of natural units (those set by $c, h, k_{B}, \dots$ ) at the first layer. All derived units then follow through algebraic relations of those constants. BIPM emphasizes: “All units of the SI can be written either through a defining constant itself, or through products or quotients of the defining constants.” For instance, the newton (SI unit of force) is $k g \cdot m / s^{2} = (h / c^{2}) \cdot (c^{2}) / s^{2}$ (up to fixed numbers), ultimately tied back to $h$ and $c$ . Specifically for gravity, one can say the SI definition of force implicitly involves $c$ and $h$ , and Newton’s gravitational constant $G_{S I}$ is itself determined by the SI definitions of kg, m, s. In this way, $G_{S I}$ is not a deep fundamental quantity of Nature so much as a conversion factor imposed by our unit choices.

Implications:
The fact that $G_{S I}, c, h,$ etc. serve as fixed bridges between units and natural scale suggests that the SI system is essentially an “overlay” on top of dimensionless physics. One can view it as if the SI constants project the elegant, constant-free equations of Nature (in Planck or other natural units) into the layer of human measurements. This layered perspective is supported by metrology literature: the SI Brochure and BIPM documents make clear that specifying constants fixes the unit system. While SI is immensely useful for communication and engineering, it has this structure of encoding the underlying natural units into fixed numbers.

Conclusion:
In conclusion, we have shown that Newton’s gravitational law in SI units is literally a rescaled form of a simpler dimensionless law. By adopting Planck units (a natural-unit system), the formula becomes

$F^{'} = \frac{m_{1}^{'} m_{2}^{'}}{r^{' 2}}$

with no explicit constants. Reintroducing SI units re-attaches the scaling factor $G_{S I}$ , which we have seen is exactly the conversion factor relating those systems. Every appearance of $G_{S I}$ in an SI equation is thus implicitly an instance of converting between natural scales and human scales. Moreover, the SI’s modern definition indeed uses fixed constants (like $c, h, \dots$ ) as foundations, making it effectively a unit system “layered” upon Nature’s constants. This analysis underscores that the “mystery” of Newton’s constant is largely an artifact of unit choice: at heart, gravity follows a simple inverse-square law once one factors out the unit conventions.

References:

Planck and natural unit definitions
Dimensional analysis of Newton’s law
Concept of natural units and constant=1
SI base-unit redefinitions by constants
Metrology discussions of defining constants

Politics, Power, and Science

Thursday, May 15, 2025

Newton’s Gravitational Law in Planck Units and the Role of SI Constants

J. Rogers, SE Ohio, 15 May 2025, 1647

Dimensional Transformation Steps

Summary and Implications

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