When Do Gravitational Effects Become Relevant?

Question

Relativistic effects become relevant when $v\sim c$.

Quantum effects become relevant when $|\vec{p}|\sim\hbar c$.

But when do gravitational effects become relevant? It cannot be when the typical size of the system is about the size of the Schwarzschild radius $\left(d\sim\frac{GM}{c^2}\right)$—here, we'd be already in the strong field regime—, because gravity is obviously relevant in our solar system, where the Schwarzschild radius of the Sun is about 2 km, but the typical size is of millions of kilometers.

Any ideas for a useful threshold?

Answer 1

This is not a rigorous answer but I think when the energy of interaction $\sum_{\langle1,2\rangle}\frac{Gm_1m_2}{r_{12}}$ is comparable to the other scales in the problem, I would consider it relevant. For example, in atoms this is negligible compared the electric potential/kinetic energies, but for the sun and earth, it is not. Then again, energies are only physical upto a constant so we would have to modify this a little bit.

For example, for a human on earth, the characteristic energy scale for gravity would be $$ \Delta r\frac{\partial}{\partial r}\left(\frac{GMm}{r}\right) \approx mg\Delta x \approx 10kJ$$ which is comparable to stuff like our usual kinetic energies in the vertical direction. So it is relevant.

Answer 2

From a particle physics standpoint, the question is usually answered in the following way:

Energy curves spacetime the same as mass, because they are the same thing, expressed in different units. This means that as we force more and more energy into a bunch of fundamental particles, there comes a point where their gravitational interactions become as strong as, for example, the electrostatic interactions they are experiencing because of their charge. For all energy levels greater than this, gravity rules the world of particle interactions.

Luckily for us, the only way to jam enough energy into a cloud of, say, protons so that their gravitational attraction exceeds their electrostatic repulsion is by going way, way back in time to the earliest moments of the Big Bang. Only at the superhigh temperatures present then does gravity take over from all the other forces.