General Relativity: An Introduction for Physicists · Problem 7.20

The equivalence principle and spacetime curvature — Hobson, Efstathiou & Lasenby

Physics · conformal-transformation-curvature

Problem

A conformal transformation, such as that in Exercise 2.7, is not a change of coordinates but an actual change in the geometry of a manifold such that the metric tensor transforms as g~ab(x)=Ω2(x)gab(x),\tilde{g}_{ab}(x) = \Omega^2(x)g_{ab}(x), where Ω(x)\Omega(x) is some non-vanishing scalar function of position. Show that, under such a transformation, the metric connection transforms as Γ~abc=Γabc+1Ω(δca∂bΩ+δba∂cΩ−gbcgad∂dΩ).\widetilde{\Gamma}^a{}_{bc} = \Gamma^a{}_{bc} + \frac{1}{\Omega}\left(\delta^a_c\partial_b\Omega + \delta^a_b\partial_c\Omega - g_{bc}g^{ad}\partial_d\Omega\right). Hence show that the curvature tensor, the Ricci tensor and the Ricci scalar transform respectively as R~abcd=Rabcd−2(δ[caδd]eδbf−gb[cδd]egaf)∇e∇fΩΩ+2(2δ[caδd]eδbf−2gb[cδd]egaf+gb[cδd]agef)(∇eΩ)(∇fΩ)Ω2,R~bc=Rbc+[(N−2)δbfδce+gbcgef]∇e∇fΩΩ−[2(N−2)δbfδce−(N−3)gbcgef](∇eΩ)(∇fΩ)Ω2,R~=RΩ2+2(N−1)gef∇e∇fΩΩ3+(N−1)(N−4)gef(∇eΩ)(∇fΩ)Ω4,\begin{aligned} \widetilde{R}^a{}_{bcd} &= R^a{}_{bcd} - 2\left(\delta^a_{[c}\delta^e_{d]}\delta^f_b - g_{b[c}\delta^e_{d]}g^{af}\right)\frac{\nabla_e\nabla_f\Omega}{\Omega} \\ &\quad + 2\left(2\delta^a_{[c}\delta^e_{d]}\delta^f_b - 2g_{b[c}\delta^e_{d]}g^{af} + g_{b[c}\delta^a_{d]}g^{ef}\right)\frac{(\nabla_e\Omega)(\nabla_f\Omega)}{\Omega^2},\\ \widetilde{R}_{bc} &= R_{bc} + \left[(N-2)\delta^f_b\delta^e_c + g_{bc}g^{ef}\right]\frac{\nabla_e\nabla_f\Omega}{\Omega} \\ &\quad - \left[2(N-2)\delta^f_b\delta^e_c - (N-3)g_{bc}g^{ef}\right]\frac{(\nabla_e\Omega)(\nabla_f\Omega)}{\Omega^2},\\ \widetilde{R} &= \frac{R}{\Omega^2} + 2(N-1)g^{ef}\frac{\nabla_e\nabla_f\Omega}{\Omega^3} + (N-1)(N-4)g^{ef}\frac{(\nabla_e\Omega)(\nabla_f\Omega)}{\Omega^4}, \end{aligned} where NN is the dimension of the manifold.

Browse the problem library