# massless Newtonian gravitons

31 May 2023The most general Lagrange-density \(\mathcal{L}\) for classical Newtonian gravity obeying isotropy and linearity would be \(\mathcal{L} = \frac{(\nabla\Phi)^2}{2}+\lambda\Phi+\frac{m^2}{2}\Phi^2+4\pi G\rho\Phi,\) giving rise to the field equation \((\Delta - m^2)\Phi = 4\pi G\rho + \lambda,\) which only reduces to the conventional Poisson-equation if one requires scale-free potentials (setting \(m\) to zero) and no gravitational effect of empty space (setting the cosmological constant \(\lambda\) to zero). Why is it that one needs to require \(m\) to be zero in classical gravity, while it naturally follows to be zero in general relativity?