Can the gravitational constant $G$ be calculated theoretically?

Question

We know all that the gravitational constant is $$G=6.67428±0.00067\times 10^{-11}\mathrm{m^{3} \:kg^{-1} s^{-2}}.$$ But can we calculate it theoretically?

Answer 1

You can't calculate the numerical value of Newton's constant from the first principle because it is a dimensionful constant – it has units – so the numerical value depends on the magnitude of the units. And because e.g. the kilogram is defined as the mass of a platinum prototype hosted by a French chateau (the kilogram has the "least objective" definition so far), it's clear that a "pure calculation" can't know how large the kilogram is, which also means that it can't determine the numerical value of Newton's constant which depends on the definition of a kilogram.

In other units, e.g. Planck units, people often set $G=1$ or $G=1/8\pi$. In that case, the constant may be calculated – I just did it. If one uses such units, there are other – dimensionless, and therefore potentially calculable from the first principles – constants of Nature such as the electron mass (in the unit of the Planck mass). String theory is the only framework in physics that allows one to calculate all these continuous dimensionless universal constants of physics. One may prove that for a given (stabilized) compactification of string theory, all of these constants are fully determined. In practice, physicists can't do that yet because they don't know how to choose the right compactification (which is just a discrete amount of currently uncertain information that must be inserted to the calculation).

Answer 2

No, as commented on above.

Worse, we don't know its value very well.

There are efforts underway to measure $G$ more accurately, as reported in Nature earlier this month:

It is one of nature’s most fundamental numbers, but humanity still doesn’t have an accurate value for the gravitational constant. And, bafflingly, scientists’ ability to pinpoint G seems to be getting worse. This week, the world’s leading gravity metrologists are meeting to devise a set of experiments that will try to set the record straight. This will call for precision measurements that are notoriously difficult to make — but it will also require former rivals to work together...

Answer 3

The gravitational constant can only be determined experimentally. It's empirically derived. There is no physical law we know of that dictates the strength of $G$. Rather, we have the observation that gravitational force is:

$F_g = G \frac{m_1 m_2}{r^2}$

Since we can know the other four numbers, we must solve for $G$ experimentally. For a brief background: http://en.wikipedia.org/wiki/Gravitational_constant#History_of_measurement

Answer 4

Like many, I read that G can't be calculated by a formula. I also read in relation to the Mach principle, that G could be related to the large universe values. Some also suggested it could be related to the ratio of mass of the proton and that of the universe. None produced a formula so far. I tried to find a way to do it, and once found that G=1/50c= 6.6713e-11, which is very near (c is the speed of light). But despite the agreement, it is just numerology- as the units are not the same in the equation. Few days back I seemed to have found one equation for G that is related to the universe values and it goes like this; Equate gravity Fg and centrifugal force Fc(double the radius) for a mass m and get; Fg=Gm^2/r^2; Fc=mv^2/(2r)2; put v=c to get; Gm/r=c^2/4, or; G=(r/m)*c^2/4= 6.56e-11, using the radius and mass of the known universe. This then relates G with the mass and radius of the observed universe and the constant speed of light.
I just came to know that Schrodinger in a 1925 (German paper) stated that G ≈ c^2 Ru/Mu wit no formula;Ru and Mu are the radius and mass of the universe.

Answer 5

But can we calculate it theoretically?

The question is more subtle than it sounds, because we always need to measure everything and, most importantly, compare within the same dimension.

Take the speed of light $c$ for example: it is defined on paper to be exactly equal to $299 792 458\ \,m\,s^{-1}$, and yet you will need an experiment, a mirror and the comparison between two values within the same dimension – i.e. the second and the time light needs to travel one metre – to determine what that practically means (in this case you will measure the metre rather than the speed of light).

Can we do something similar with $G$?

Sure. We can compare for example the elementary charge and the amount of electric charge needed to balance the gravitational force of $1\, kg$.

Let's imagine two negatively charged masses of 1 kg and let's call $n$ the number of equally-distributed electrons in excess in each mass. The electric repulsion is so that it balances exactly the gravitational attraction between the two masses (this will apply at any distance). Now you can simply define $G$ as

$$G = n^2 k_e e^2 \times\, kg^{-1}$$

Where $k_e$ is the Coulomb constant, $e$ is the elementary charge and $n$ is the number of electrons in excess in each mass.

You will still need an (almost impossible) experiment to count these electrons. But if you were able to determine $n$ you could use that as a constant and refer to the equation above for the official definition of $G$. It will not be very different from how the second and the kilogram are currently defined.

Interestingly enough, the number of electrons in excess ($n$) that you will find will be exactly equal to $\frac{1}{\sqrt{\alpha}} \approx 11.706$ times the amount of Planck masses contained in one kilogram.

Answer 6

Let's consider what can in principle could be calculated about G, given the fact that G is a dimensionfull constant whose value can be chosen completely arbitrarily, as made clear in the other answers. So, let's reformulate the problem. For an intelligent lifeform living on a planet, we can consider a unit system such that G is of the order of unity when expressed in terms of biologicaly relevant scales. E.g., we can express G as:

$$G \approx \frac{0.86} {\text{density of water}\times\text{ hour}^2}$$

The density of water is approximately the density of living organisms and an hour is the typical time scale at which microorganisms can replicate.

Since G is now of order unity, one can ask if this expression can be interpreted in a universal way. So, the question is if in a universe with different laws of physics where life could still arise, intelligent lifeforms would typically find G of the order of unity when expressed in terms of these biological units. I'm not sure such an argument can be made, but if this turns out to be true, then that would be an Anthropic explanation for the value of G.

One may speculate as follows. Life on some planet will be fighting against thermal equilibrium, it needs a steady flow of low entropy energy, which in our case are photons from the Sun. The intensity and the entropy per unit energy of photons then determines the typical metabolic activity that is possible to achieve. But this is also related to the scale of the solar system at which gravity is an important factor. Note e.g. that a cooler Sun means higher entropy photons and the planet having to be closer to the Sun. But I have to admit that I don't have a good argument at the moment to tie these things together in a convincing way.