Is there any relation between Planck constant and Gravitational constant?

6,166

Solution 1

$G$ is not exactly larger than $h$ by a factor of $10^{23}$ in SI units, as you are probably aware (just making sure). There is also no expected numerical relationship between the two that has a physical interpretation. You have to understand that these constants are mostly just due to our (to some extent) arbitrary choices of units. These are, of course, motivated by everyday convenience. But this doesn't mean that the commonly used SI units have any physical significance. In fact, there are several other unit systems. One particularly interesting one that is quite popular among physicists doing fundamental research is known as the Planck unit system.

In terms of Planck units, both $G$ and $\hbar$ are equal to $1$, as well as $c$, $k_B$ and $4\pi\epsilon_0$, the speed of light, Boltzmann's constant and the inverse of the Coulomb constant respectively. The Planck unit system attempts to eliminate the arbitrary choices due to the perspective of humans, which just so happen to live on certain energy, length, etc. scales. This is done by defining the units of measurement only in terms of fundamental constants of nature. The idea is that these constants are really what 'nature measures in', so setting their numerical value to $1$ makes sense. Related is the concept of a natural unit system, of which several exist. These all attempt to formulate things in a 'natural' sense (which, among other things, depends on the field of study).

Solution 2

Planck's constant divided by the gravitational constant is 9.93x10^-24 kg s^2/m. This is not exactly 10^-23, and even if it was there is nothing special about 10^-23 kg s^2/m, since the number is dependent in the conventions used to define the units.

Solution 3

Express the Planck constant as $2.34\times 5^{-48}$ and the gravitational constant as $2.01\times 5^{-15}$ and the coincidence goes away. Physics doesn't care that we have ten fingers.

Solution 4

There is nothing special about the number 10 (nor about the number 23, though not everybody agrees about that)

Solution 5

The $G$ constant can be related to Planck's various constants,

$$G=\frac{\ell_P^3}{m_P\cdot t_P^2}$$

where $\ell_P$ is the Planck length, $m_P$ is the Planck mass, and $t_P$ the Planck time.

G = 6.673x10^-8 cm^3/g sec^2

volume per mass acceleration

Its an interesting correlation......

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Updated on August 01, 2022

Comments

  • user48710
    user48710 over 1 year

    Why is the Gravitational constant about $10^{23}$ times of the Planck constant in SI-units? Is there any relation between them? I mean Planck constant is about $6.6\times 10^{-34}$ $Js$ and Gravitational constant is about $6.6×10^{−11} \frac{N·m^2}{kg^2}$.

    • Qmechanic
      Qmechanic over 9 years
      Comment to the question (v2): If one only uses two significant figures for $G$, then one should round up to $6.7 \times 10^{−11} \frac{N·m^2}{kg^2}$.
    • Harry Johnston
      Harry Johnston over 9 years
      Short answer: it's a coincidence.
  • Admin
    Admin over 9 years
    His emphasis was on 6.6 not 23.
  • Double AA
    Double AA over 9 years
    Setting the Coulomb constant itself to $1$ is equivalent.
  • Danu
    Danu over 9 years
    @DoubleAA I talked to Jamal about this as well ;) his point was just that it's called the inverse Coulomb constant. Mathematically, it's irrelevant, and he realizes that too :)
  • Double AA
    Double AA over 9 years
    Ahh I hadn't seen the edit history. Why not just list it as $c, k_B$ and $\frac{1}{4 \pi \epsilon _0}$ and call it the Coulomb constant?
  • Danu
    Danu over 9 years
    @DoubleAA edit your heart out ;)
  • Zane
    Zane over 9 years
    As both are constants, the relation between them is their ratio.
  • Danu
    Danu over 9 years
    @Zane ...which is still unit-dependent.
  • user43470
    user43470 over 9 years
    Doesn't the very fact that in the planck system both h bar and G have the same value which is 1 suggest that they have some relationship with each other?
  • eqb
    eqb over 9 years
    Let $c = \hbar = G = \pi = 1$.
  • Danu
    Danu over 9 years
    @eqb setting $\pi$ to 1 is not 'ok'. It is a dimensionless constant already.
  • Kyle Kanos
    Kyle Kanos about 9 years
    That's not an interesting correlation at all, given that the Planck units are defined by using $G$, $c$, $\hbar$, $k_e$, and $k_B$.