Show that Einstein's equation $$G^{\mu\nu}=R^{\mu\nu}-\frac12\mathcal{R}g^{\mu\nu}=\frac{8\pi G}{c^4}T^{\mu\nu}\tag{1}$$ can be written in the form $$R^{\mu\nu}=\frac{8\pi G}{c^4}\left(T^{\mu\nu}-\frac12\mathcal{T}g^{\mu\nu}\right)\tag{2}$$ where $\mathcal{T}=g_{\alpha\beta}T^{\alpha\beta}=T_{\a...

Consider a quantum particle in an infinite square well of width $2\pi$. In other words, the Hamiltonian is $H=p^2$, the position variable is restricted to $x\in[-\pi,\pi]$ and the wave function satisfies the boundary conditions $\psi(-\pi)=\psi(\pi)=0$. Is the parabolic function $\psi : [-\pi,\pi...