# I Dimensionless coupling constant of gravity

Tags:
1. Jun 6, 2017

### gerald V

In analogy to the fine structure constant, the dimensionless coupling constant of gravity is defined as some reference mass divided by the Planck mass, squared.

But what is the reference mass?? I have read thread https://www.physicsforums.com/threa...-fine-structure-constant.428622/#post-2878965 and other stuff, and conclude that currently the most popular suggestions are: The electron mass, the molecular mass of some gas a star is made from, the QCD scale. None of them appears as fundamental.

I can see the following three fundamental possibilities:

- the smallest mass in the universe (whose deBroglie wavelength is as large as the entire universe); in this case the coupling constant would in fact be tiny
- the Planck mass; the coupling constant is unity identically
- the entire mass-equivalent of the universe; in this case the coupling constant would be huge.

I tend towards the point of view that the coupling constant is unity identically. Has anyone a view on that? Furthermore, Wilczek argues that the coupling constant (for the electron mass inserted?) is tiny at our energy scales but runs towards unity at higher scales. I know the story about running coupling constants, but do not think that this is any answer to the question about their values (which are those at low energies like 1/137).

Thank you in advance for any comment.

2. Jun 6, 2017

### Staff: Mentor

The electron mass would be as fundamental as for the fine structure constant.

I think that what you may be more hinting at is that, unlike charge which appears only in integer multiples of 1/3 e, there isn't such an elementary mass.

Last edited: Jun 6, 2017
3. Jun 6, 2017

### Staff: Mentor

There isn't one, at least not in the sense that the fine structure constant is a dimensionless coupling constant for electromagnetism. The corresponding coupling constant for gravity has dimensions of inverse mass squared (it's the inverse Planck mass squared). That's what makes the simplest quantum field theory of gravity (massless spin-2 field) nonrenormalizable.

The information in that thread is not reliable, at least not for the use you are trying to put it to. The "gravitational coupling constant" referred to in the Wikipedia page linked to in that thread is not the analogue of the fine structure constant in electromagnetism. It's just one possible heuristic measure of the "strength of gravity".

This is true, and it's probably part of the reason why the QFT coupling constant for gravity, as I said above, is not dimensionless. We won't really understand all this until we have a good theory of quantum gravity (which the QFT I referred to above is not, it's just the first and simplest thing to try and has already been tried, and its limitations are why further efforts are still being pursued).

4. Jun 6, 2017

### Staff: Mentor

Can you give a reference?

5. Jun 6, 2017

### gerald V

Dear Dale, dear PeterDonis, thank you for the answers. I think I got the essential message. Here is the reference to the Wilczek et.al. article, which still puzzles me somewhat: http://frankwilczek.com/Wilczek_Easy_Pieces/172_Unification_of_Couplings.pdf The important figure is on page 30. This article appeared more than 25 years ago, but Wilczek still argues along this line (there are talks on the web).

6. Jun 6, 2017

### Staff: Mentor

Yes, and note also the key text on p. 32:

"Because it is characterized by a dimensional coupling--Newton's constant--rather than the dimensionless couplings that characterize the other interactions..."

Newton's constant in "natural" QFT units is just the inverse Planck mass squared. So the figure on p. 30 is a little misleading, since the "coupling strength" is dimensionless for the other three interactions but not for gravity, so the scale of the vertical axis is not well-defined. In a schematic diagram in a pop science article, that's OK, but it illustrates why you have to be very careful when reading pop science articles. In a peer-reviewed paper such a figure would never have passed muster.