# When Do Gravitational Effects Become Relevant?

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?

• Quantum effects become relevant when $|\vec{p}|\sim\hbar c$. No, because the two sides don’t have the same dimensions. Jul 7, 2020 at 21:17
• When there is a lot of mass? Jul 7, 2020 at 22:19
• Sure, but how would you quantify "a lot"? @my2cts Jul 8, 2020 at 10:03

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.