Gravitational Constant: Why Is It Fundamental?

[Vishwasks001]

Hi,

I was wondering if there is any need in the theory of relativity for Newton's Gravitational Constant 'G' to remain a fundamental constant.

Constant in Coulomb's Law can be expressed in 'c' and 'pi', then why Constant in Newton's Law needs to be a fundamental constant??

 

[Dale]

Hi Vishwasks001, weldome to PF!

I would say that no dimensionful constant is fundamental. The only fundamental ones are the dimensionless ones like the fine structure constant:

http://math.ucr.edu/home/baez/constants.html

 

[PeterDonis]

In classical GR, the gravitational constant $G$ can be viewed as just a unit conversion factor, just like $c$ is. (More precisely, $G / c^2$ converts units of mass to units of length.)

In quantum gravity, it's not quite so clear. In quantum field theory, it doesn't make sense to give mass and length the same units; instead, they should have units which are the inverse of each other. So a constant that relates a mass to a length, which is what $G$ basically does, should have units of length squared, or inverse mass squared--i.e., it should not be dimensionless. This is closely related to the fact that the simplest quantum theory of gravity, a spin-2 field, is not renormalizable, because its coupling constant is not dimensionless (whereas field theories with a dimensionless coupling constant, such as QED, are renormalizable).

 

[Vishwasks001]

In classical GR, the gravitational constant $G$ can be viewed as just a unit conversion factor, just like $c$ is. (More precisely, $G / c^2$ converts units of mass to units of length.)

In quantum gravity, it's not quite so clear. In quantum field theory, it doesn't make sense to give mass and length the same units; instead, they should have units which are the inverse of each other. So a constant that relates a mass to a length, which is what $G$ basically does, should have units of length squared, or inverse mass squared--i.e., it should not be dimensionless. This is closely related to the fact that the simplest quantum theory of gravity, a spin-2 field, is not renormalizable, because its coupling constant is not dimensionless (whereas field theories with a dimensionless coupling constant, such as QED, are renormalizable).

I am not sure if I understand the term renormalizable. But I understand what you have said.Would it violate GR or QFT rules if G is expressed as a product or multiple of c?

 

[PeterDonis]

I am not sure if I understand the term renormalizable.

Here is an overview:

https://en.wikipedia.org/wiki/Renormalization#Renormalizability

Be warned, this is a complicated subject (and further questions on it should go in the Quantum Physics forum, not here).

Would it violate GR or QFT rules if G is expressed as a product or multiple of c?

Yes, because they are two different conversion factors. Different systems of units will have different ratios between these conversion factors, so there is no way to reduce them to just one (with the other being a fixed function of the one).

Or, if you view them as coupling constants, they are different, because they are associated with different interactions (G with gravity, c with electromagnetism), so you can't combine them into one.

 

[Vishwasks001]

Or, if you view them as coupling constants, they are different, because they are associated with different interactions (G with gravity, c with electromagnetism), so you can't combine them into one.

If somebody does, then what laws would it violate??

 

[PeterDonis]

If somebody does, then what laws would it violate??

The laws that say gravity and electromagnetism are different interactions with different coupling constants.

 

[Vishwasks001]

Thank you.