# Is coupling G actually a constant of nature or it can vary?

is coupling G (graviational coupling) actually a constant of nature or it can vary?

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That's actually a very good question. In general, physical constants such as G (but also c, $\hbar$, $\epsilon_0$) are believed to be constant. However, there are theories that these "constants" may have changed very slightly. Even if there is such a change, though, it is generally so slow that it would be barely noticable over the lifetime of the universe.

So far, as far as I'm aware, there is no conclusive evidence to support the variation of physical constants. See also for example Wikipedia.

It depends actually on what you mean... the Newton Gravitational constant is clearly a constant of nature...
BUT: there are Modified Theories of Gravity in which you consider G not as a constant anymore but as a field, thus depending on space and time... this is the so called Brans-Dicke theory, and is the first example of a scalar-tensor theory of gravity.

In practice what you do is to take the gravitational action (usual Einstein-Hilbert)
$$\mathcal{S}_{grav}=\frac{1}{16\pi G_N}\int\mathrm{d}^4x\sqrt{-g}R$$
where ##G_N## is the Newton gravitational constant, and consider ##G## as a field, therefore
$$\mathcal{S}_{grav}=\int\mathrm{d}^4x\sqrt{-g}\frac{1}{16\pi G(x)}R$$
Then you can rename the field as
$$\frac{1}{16 \pi G(x)}=\phi$$
and you have immediately
$$\mathcal{S}_{grav}=\int\mathrm{d}^4x\phi(x)\sqrt{-g}R$$
Then you can also add a kinetic term for ##\phi## to the action and you have the first scalar-tensor theory proposed, giving some modifications to the Einstein equation

It depends actually on what you mean... the Newton Gravitational constant is clearly a constant of nature...
BUT: there are Modified Theories of Gravity in which you consider G not as a constant anymore but as a field, thus depending on space and time... this is the so called Brans-Dicke theory, and is the first example of a scalar-tensor theory of gravity.

In practice what you do is to take the gravitational action (usual Einstein-Hilbert)
$$\mathcal{S}_{grav}=\frac{1}{16\pi G_N}\int\mathrm{d}^4x\sqrt{-g}R$$
where ##G_N## is the Newton gravitational constant, and consider ##G## as a field, therefore
$$\mathcal{S}_{grav}=\int\mathrm{d}^4x\sqrt{-g}\frac{1}{16\pi G(x)}R$$
Then you can rename the field as
$$\frac{1}{16 \pi G(x)}=\phi$$
and you have immediately
$$\mathcal{S}_{grav}=\int\mathrm{d}^4x\phi(x)\sqrt{-g}R$$
Then you can also add a kinetic term for ##\phi## to the action and you have the first scalar-tensor theory proposed, giving some modifications to the Einstein equation
As far as I know, this leads to no testable predictions that contradict GR. GR is the best theory of gravity we have, and in GR the gravitational constant is... constant.

As far as I know, this leads to no testable predictions that contradict GR. GR is the best theory of gravity we have, and in GR the gravitational constant is... constant.
Yes of course... Brans-Dicke theory was introduced to find a theory which respected also the Mach principle (which GR does not fully respect). This was only to point out an example of theories where ##G## is not constant.

Anyway Modified Gravity theories (in general, not necessarily Brans-Dicke) are used in an attempt to modify gravitation to account for Dark Matter and Dark Energy (standard model of cosmology, aka ##\Lambda CDM## is not completely satisfactory as it accounts for the acceleration but needs fine tuning and also is not so good at galactic scale in predicting the rotation curves of galaxies).

Nabeshin