# Newton's gravitational constant $G$, the reduced Planck constant $\hbar$, the speed of light $c$: the Dream Team of moderators?

The three great constants of Nature are well known:

• the speed of light $c$ (special relativity),
• the reduced Planck constant $\hbar$ (quantum mechanics),
• Newton's gravitational constant $G$ (general relativity).

However, is it possible that they play, in fact, a similar role, a moderator role, in the precise sense of decreasing energy?

Consider the following baby model. We consider a particle of radius $r$ and mass $m$. We admit an auto-gravitational potential, and a pseudo auto-quantum potential.

The energy of the particle is given by

$$E(m,r) = m c^2 - \frac{\hbar^2}{m r^2} - \frac{G m^2}{r}.$$

(I do not care about possible coefficients of order unity.)

We demand the physical condition $$E(m,r) \ge0.$$

We may easily observe that, putting $\hbar = 0$, we recover the gravitational limit for the radius of particle when it becomes a black hole, and, by putting $G=0$, we recover the "quantum limit" for the radius (a taste of Heisenberg inequalities).

What is the effect of choosing $c^{-1} > 0$ ($c$ not infinite), or choosing $\hbar > 0$, or choosing $G > 0$?

This is the same effect, that is, it decreases the energy $E(m,r)$.

So the team $(c^{-1}, \hbar, G\,)$, when they are not zero, decreases the energy, it appears as a (dream) team of moderators of energy.

Of course, we are working with a baby (particle) model.

But do you think that these conclusions could be extended to the real word?

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Hi, you may calculate various "Planck units" such as Planck mass, Planck time etc. by minimizing similar sums. But the purpose of the "teamwork" you're describing isn't too clear. The fact that you get the Planck units is guaranteed by the dimensional analysis. Do you actually conjecture that such random formulae for energy apply to something in Nature, or are they just a way to get some Planck units? If it's the latter, the potentials aren't really needed, and the form of the calculation doesn't have to match yours. –  Luboš Motl Oct 9 '12 at 11:39
Well, even without considering this baby model, could we consider quantum mechanics, special relativity, and gravitation, as, "in fine", decresasing energy (v.s. a non-quantum, non-relativist and non-gravitational world)?. –  Trimok Oct 9 '12 at 12:12
I don't understand what you mean by decreasing energy. Decreasing in time? Or decreasing as a function of what? The energy is conserved (in time-translational invariant theories) it is not decreasing. As function of arguments, it's decreasing in one direction and increasing in the opposite direction. What the hell are you talking about? –  Luboš Motl Oct 9 '12 at 14:30
@LubošMotl: I meant the following thing. Speak about gravitation. Without gravitation, you could put any energy $E$ in any sphere of radius R, whatever R is. So there is no limit for $E/R, E/R^2/ E/R^3$, etc.... With gravitation, there is a limit for $E/R$. So the presence of gravitation (relatively to the absence of gravitation) has a direct effect of limiting energy/by length. And I suspect there should have a deep reason for this. –  Trimok Oct 31 '12 at 17:41
@Trimok - any particular reason why you included a minus sign in the quantum uncertainty ($\hbar^2$) term in your baby model? –  Johannes Aug 15 '13 at 18:59