# Gravitational constant units

February 21, 2016

G is the coupling of gravitational interaction. It has units and is not dimensionless because gravity is not renormalizable. Fine structure constant is given by \alpha = \frac{1}{4\pi\epsilon_{0}} \frac{e^{2}}{\hbar c} and is dimensionless since the quantum theory of EM interactions i.e QED is renormalizable.

In QFT calculations, if one tries to write down amplitude to higher orders in coupling constant (tree level Feynman diagrams -> higher order diagrams), you are plagued with infinities. You have to set a cut off to make sense for your theory. Now, the gravitational constant G has mass dimensions of -2. To make much more sense we see that \frac{1}{G^\frac{1}{2}} = [M] = M_{Planck} which is around 10^{19} m_{proton} . This says that at that scale, we might end up at some new coupling constant which will be dimensionless and hence we can hope for a theory of gravity which will be renormalizable. That is basically where we hit the domain of quantum gravity ! Also, this follow up explains why Fermi's theory of weak interactions was not renormalizable and eventually gave rise to Yukawa's theory where expectedly, coupling 'g' is dimensionless.

In summary, any interaction which has coupling constant mass dimension equal to 0 is renormalizable, mass dimension > 0 is super renormalizable (even better than renormalization) and if < 0, then it is notoriously not renormalizable.

Source: www.quora.com
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##### What are the units of gravitational constant.

There are two gravitational constants.
The 'real one', the Universal Gravitational Constant, is always known as G. As its name implies it works everywhere. *
The other one, acceleration due to gravity, is g. It is good only on our own planet.
Units of g are m s-2 . Units of G are m3 kg-1 s-2 .
*Some physicists are not quite sure about this. If one starts talking to you about "Planck-length gravity" you might want to remember an urgent appointment elsewhere.