Exploring h , h c , and G

 

$G$

This table presents the values of fundamental constants under different scalings of the meter ($s_{m}s\_m$). It explores the behavior of Planck’s constant ($hh$), momentum ($pp$), mass ($mm$), the product of $hh$ and $cc$ ($hchc$), and Newton’s gravitational constant ($GG$).

Key Features:

1. Constancy of Mass ($mm$):

   • Despite changing the scaling of the meter ($s_{m}s\_m$), the mass ($mm$) remains constant at $7.3724973238\times 10^{-51}7.3724973238\; \backslash times\; 10^\{-51\}$.

   • This consistency highlights that mass is unaffected by the redefinition of the meter.

2. Relationship Between $hh$, $pp$, $mm$, and $hchc$:

   • At $s_m=1s\_m\; =\; 1$ (where $c=1c\; =\; 1$), the constants $hh$, $pp$, $mm$, and $hchc$ become equal, indicating a simplified relationship where the inverse of $cc$ in $hh$ becomes 1.

   • They are all spread apart from mass by powers of c so as c comes closer and closer to unity they all get closer and closer together.  I was shocked that they all converged to mass. 

   • This emphasizes the interconnectedness of these quantities and the normalization of their values in natural units.

3. Scaling of Planck’s Constant ($hh$):

   • The values of $hh$ change with different scalings of the meter, showing how the unit definitions affect the numerical representation of physical constants.

   • Despite these changes, the intrinsic relationships between $hh$, $pp$, $mm$, and $hchc$ are preserved.

4. Behavior of Newton’s Gravitational Constant ($GG$):

   • The values of $GG$ become smaller and smaller  as the meter scalings approach c=1, reflecting how the meter scalings become smaller and smaller over the consistent mass scaling.

   • This variation underscores the importance of unit definitions in understanding gravitational forces.

Summary:

This analysis reveals the profound relationships between fundamental constants and the role of unit systems in defining and understanding these relationships. By examining how $hh$, $pp$, $mm$, $hchc$, and $GG$ transform under different scalings, we gain new insights into the intrinsic properties of these quantities and their interconnectedness. The constancy of mass and the simplification of relationships at $c=1c\; =\; 1$ highlight the fundamental nature of these constants and their underlying symmetries.