Tensor density and the coefficient $\sqrt{-g}$

Usually it is claimed that we use the coefficient $$\sqrt{-g}$$ for the action in the curved spacetime, to make the integrand treats as a scalar but not as a scalar density under general coordinate transformation. Can we use any other covariant quantity with the weight -1 for this purpose? Or there is other reasons for this selection?

• In principle yes. Do you have another choice in mind? – Qmechanic Mar 23 at 8:07
• @Qmechanic No. I do not have any specific one in my mind. But I think there is possibility to find another scalar density function with weight -1 which is scalar under local Lorentz transformation, too. Since we can not find any other choice yet, is the reason for this particular choice? – Astrolabe Mar 23 at 8:13
• @Qmechanic And if we find alternatives, does our gravity theory (Einstein Gavity) change? Do its equations and principles save? – Astrolabe Mar 23 at 8:19