From the Bianchi identity and the symmetry properties of the Riemann curvature tensor, show that [math]\displaystyle{ \Box R_{\alpha\beta\gamma\delta}=0 }[/math] in a vacuum spacetime with [math]\displaystyle{ g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} }[/math].
Assume that one has a plane wave solution [math]\displaystyle{ h_{\mu\nu} }[/math] in the Lorenz gauge, but not in the TT gauge. Find a projector [math]\displaystyle{ P_{\mu\nu}^{\alpha\beta} }[/math] such that [math]\displaystyle{ h_{\mu\nu}^{TT}=P_{\mu\nu}^{\alpha\beta}h_{\alpha\beta} }[/math] and [math]\displaystyle{ P_{\mu\nu}^{\alpha\beta}h_{\alpha\beta}^{TT}=h_{\mu\nu}^{TT} }[/math].
Calculate the transformation of [math]\displaystyle{ h_+ }[/math], [math]\displaystyle{ h_\times }[/math] and [math]\displaystyle{ h_+\pm i h_\times }[/math] under rotations around z and under boosts along z.
From the expression of [math]\displaystyle{ T_{\mu\nu} }[/math] given in lecture, and assuming [math]\displaystyle{ ds^2=-dt^2+a^2(t)(d\vec x)^2 }[/math], obtain the Friedmann equations from the Einstein equations.
Show that a plane wave in conformal time [math]\displaystyle{ h_{\mu\nu}(c\eta-z)=h_{\mu\nu}^0 \exp(i k(z-c\eta)) }[/math] corresponds to a redshited wave when expressed in terms of t.
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