How is the Einstein field equation converted to its tensor form?

[TimeRip496]

∇²g₀₀ = 8πGρ

According to drphysics, as this is not a tensor equation, we need to change it such that it fits general relativity.

G_μv = 8πGT_μv

T₀₀ is the energy density. The conversion of the density(ρ) to T₀₀ is it done through E=mc²?

ρ=E/c²

And since is T₀₀, it will be over c⁴ instead of c². Is this how we add c⁴ to this equation(∇²g₀₀ = 8πGρ) as we convert it to its tensor form?

Source:

 

[PeterDonis]

The factor of ##c^4## is for unit conversion; it assumes that you are using conventional units of energy density for the RHS of the field equation and units of curvature (inverse length squared) for the LHS. A quantity with units of energy density is converted to units of curvature by multiplying by ##G / c^4##. (If the RHS is in units of mass density instead, the factor will be ##G / c^2##.)