Numerology and Gravity

Richard Borcherds wrote a post a little while back in which he remarked that we shouldn’t take the Planck units very seriously, since Newton’s constant $G$ doesn’t have quite the same stature as $c$ and $\hbar$.

His argument went more or less as follows: The latter two seem to be quite fundamental, but from a modern point of view $G$ (actually $\frac{G^{-1}}{8\pi}$) is just one of the coupling constants that appears in the effective Lagrangian for gravitational fields

$L = \lambda \sqrt{|g|} + \frac{G^{-1}}{16\pi}R\sqrt{|g|} + \mbox{sub-sub-leading terms}$

where the coupling constant $\lambda =\frac{\Lambda_c G^{-1}}{16\pi}$ is basically the cosmological constant, normalized by $16\pi G$.

So, $G$ isn’t privileged in any way. It isn’t the coupling constant of the leading order term; a cursory analysis of the scaling dimensions of $\lambda$ and $G$ would lead us to believe that the cosmological constant term should be dominant. It isn’t even normalized nicely, what with that $8\pi \simeq 25$. And while it seems quite sensible to work in units where $\hbar = 1$ and $c = 1$, we should be a little more cautious about about the meaning we assign to units where $G = 1$. What we’re actually doing is identifying the length scale where our non-renormalizable effective field theory description of gravity should break down.

[Updated: The computation that previously appeared here involved one of the classic blunders: not checking your units.  $\lambda \simeq 10^{-86} \mbox{GeV}^{\bf{+}6}$; that’s a +6, not a -6.  Thanks to Thomas Larsson for pointing that I’m an idiot.]

5 thoughts on “Numerology and Gravity”

1. I hadn’t considered this point before. Although it may not carry any real authority, I’m convinced. One way to say it is that c and h-bar are ultimately conversion units, as cut and dried as Boltzmann’s constant, or even the constant 2.54 in converting inches to centimeters. But G is not known to be a fundamental conversion unit at the length scale that it defines. Rather, it’s just a length scale.

(Although ironically conversion units are as shallow as they are fundamental. They have no logical content.)

2. Thomas Larsson says:

The cc scale is 10^30 times the Planck scale, which to the fourth power gives you the infamous 120 orders of magnitude. Many people have of course noted that at 0.01 eV, it agrees more or less with neutrino masses. The geometric mean between the cc and Planck scales are at 10 TeV, which could be good news for the LHC.

3. Other point for c and h is that they seem more real: c is an speed, and h is an angular momentum. What G is what? Moreover, c is the maximum speed, and h is the minimum angular momentum. G does not have a similar role. It is not even a length scale: when combined with h and c, it is only a unit for area, not for length.

4. A.J. Tolland says:

Alejandro,

One could argue that G is a unit for converting area and entropy; it does play that role in Hawking’s formula, albeit in combination with Boltzmann’s k.

5. Hmm A.J, never thought of it. It does not sound as intuitive as speed or angular momentum (for which every science museum has demostrations) but at least it fits with the idea of being a maximum, a physical bound.

A different view for G is that it has some role in the dimensionality of space time. Consider a force law F= G M m / r^n, so that the units of G depend of n. Thus the definitions of Planck length and Planck time also depend of n, and it is easy to find requeriments that only fit for n=2. For instance, what I call the “kepler length”: the radius for a classic orbit to sweep one unit (or some multiple) of Planck area in one unit of Planck time. For n=2, this radius does not depend of G. I collected some random speculations around this in http://arxiv.org/abs/gr-qc/0404086
Again, in this kind of views G does not seem a fundamental object, but a tool to keep information of some other entity.

Also, note http://arxiv.org/abs/physics/0110060, the trialogue of Duff, Okun and Veneziano.