# Have we derived the Gravitational constant via General Rel?

1. May 2, 2015

### Tweej

My friend and I were discussing this, and google didn't seem to reveal any result.

Much like how Bohr's theory managed to derive the Rydberg constant without much effort, has the gravitational constant been found in terms of other quantities yet?

It just seems bizarre that Newton proposed a constant hundreds of years ago, and despite General relativity tearing down the walls of why gravity works, we still haven't derived G.

2. May 2, 2015

### PAllen

The value of G is determined by units of measurement. By choosing appropriate units, you can make it exactly 1. To get a fundamental quantity from it, you need to form a meaningful dimensionless constant from it. The one commonly used is the gravitational coupling constant, which is independent of units:

http://en.wikipedia.org/wiki/Gravitational_coupling_constant

However, recasting your question as why does this have the value it does? Nobody knows, it remains a mystery, especially as it is so much smaller than the fine structure constant, which can be viewed as the corresponding coupling constant for electromagnetism.

[edit: The idea that this constant corresponds to fine structure constant is a bit weak. The fine structure constant uses the charge of the electron, but this is a fundamental charge unit shared by all fundamental, free, particles. For mass, you wouldn't want to use a quark mass because it can only exist bound, but there is nothing to favor an electron over a tauon, for example - they are both free fundamental particles. You could say, only the electron is stable. Obviousely, you would get a much more reasonable size gravitational coupling constant if you used the tau mass.]

Last edited: May 2, 2015
3. May 2, 2015

### bcrowell

Staff Emeritus
The Rydberg constant isn't logically equivalent to the gravitational constant. The Rydberg constant is a complicated property of a system that contains a complicated mix of particles (three quarks and an electron). The gravitational constant is a universal coupling constant whose value is what it is simply because of our choice of units.

The gravitational analogue of the Rydberg constant would probably be something at the Planck scale that we have no idea how to predict, since we don't have a theory of quantum gravity.

The field equations of general relativity do not pick out any specific scales of mass, time, or distance, so if the only fundamental building blocks that existed in our universe were massless things such as gluons, photons, and gravitational waves, then we would live in a universe where there was no system of measurement whatsoever. GR can never predict any number that has units, unless we input something that has units, e.g., the mass of the sun. And even if we did that, we'd be getting some specific property of a system such as our solar system, not something universal like G. In this sense GR is totally different from a theory like the standard model, which has dozens of input parameters, many of them dimensionful (e.g., the mass of the electron).

Last edited: May 2, 2015
4. May 2, 2015

### PAllen

Does your mention of Rydberg constant have anything to do with my post? Because the constant I linked to is not the Rydberg constant and it is fundamental for any theory based on the standard model + gravity. The mass scale is provided by the heaviest stable, free, fundamental particle (the electron). The gravitational coupling constant is exactly as fundamental as the fine structure constant, and has no units.

If you are going to use GR for matter made of standard model particles, the dimensionless gravitational coupling constant needs to be put in for GR to make any predictions. It would even be needed to describe an electrovac universe of nothing be EM fields, because something must specify the strength of coupling to curvature. In practice, G (with units) is used, but to use only a dimensionless value for EM alone, you would still need something analogous to the gravitational coupling constant.

Last edited: May 2, 2015
5. May 2, 2015

### PAllen

Standard model mass based parameters can be made dimensionless by use of the Planck mass. I have seen complete lists of the SM parameters where all of them are dimensionless.

6. May 2, 2015

### bcrowell

Staff Emeritus
OK, but (a) that means you still have a dimensionful input parameter, which is the Planck mass; and (b) this seems to me to be a very poorly motivated thing to do, since the standard model doesn't include gravity. If it was going to be well motivated, the motivation would have to come from outside the standard model.

7. May 2, 2015

### Tweej

Okay, so that's interesting. G has no definition with regards to other fundamental constants because General Rel is a unitless theory. To assign it a set of units would be to arbitrarily pick the mass of one particle over another, as opposed to the charge of an electron in QM which is the smallest integer charge (not regarding quark 1/3 charges).

So what is it about General Rel which meant it was able to be derived entirely unitless? Is it because it's just very applied geometry coming into play?

Thank you very much for responding you two.

8. May 2, 2015

### PAllen

Except that the Planck mass is not one of the fundamental parameters of the standard model. The idea of all fundamental parameters being dimensionless is simply that the fundamental theory is what is independent of all units.

A (preferably) dimensionless gravitational coupling constant is needed to use any theory of gravity to make predictions about matter described by any theory of matter. Whether the gravity theory is Newton's or GR, after you specify matter you need a coupling constant of some. What might come out some unified theory of everything is fewer fundamental constants altogether. But needing gravity constant for GR has the same status as for Newtonian gravity.

9. May 2, 2015

### PAllen

To my mind, only the pure vacuum GR field equation is free of any parameters. Once you want to apply it to matter, you need one coupling constant of some type. Most sourced describe GR has having one fundamental parameter - which makes it very good compared to many theories.

10. May 6, 2015

### yogi

Einstein, in formulating General Relativity, did not attempt to derive G. G is put in from its measured value. G has units of volumetric
acceleration per unit mass - Einstein was convinced at the time he developed GR that the universe was Static - there being no known dynamic fields floating around to adapt to his idea of a static positively curved universe , Einstein hypothesized that inert mater bent static space. It was a masterful move - it provided a means to an end, that is Einstein was able to relate the motion of masses to Riemannian curvature. However, he had some doubts - he is quoted to have called the left half of the equation as made of fine marble (curved space) and the right half (the link to inert matter) as a house of straw. It is actually surprising that Einstein didn't make any modifications to GR after other discoveries were made. I point worth noting, Friedman equations can be used to express G in terms of Hubble parameters. Also, you will see in the two gravitational equations that are extracted from the General Theory, the term Lamda(R) also has units of volumetric acceleration, so it could apply to a curvature on the left side or acceleration on the right side. As far as results go, there is really no difference - if you have an analytical solution for G or an empirical one, GR theory works just the same and makes the same predictions. Nonetheless, the origin of G is of historical interest, at least to me, and there have been posts on these forums claiming to derive G, one of which is published in Electronics' World book of constants.

11. May 6, 2015

### Staff: Mentor

Are you referring here to Einstein adding the cosmological constant? If so, "inert matter" is not a good description of it.

12. May 7, 2015

### yogi

Actually, I was simply referring to the original formulation. Curved space solved the problem of not having a known field or an ether to speculate about . As it turns out, the cosmological constant as it appears in one of the two gravitational equations extracted from the General Theory, becomes all important in the situation where negative cosmic pressure equals rho(c^2/r)/3. There is then only one term left in the equation [Lambda(R)] which is then equal to (R double dot), and its solution is exponentially expanding de Sitter space. So while Einstein's initial concern in introducing Lambda was to balance against gravitational collapse, it seems to be generally viewed now as an essential vitality, but that is another subject.