Ricci scalar

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R = g^{ab} (\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + \Gamma^c_{ab}\Gamma^d_{cd} - \Gamma^d_{ac} \Gamma^c_{bd}).

Contents

Variation

Starting with the variation of the Ricci tensor,

\delta R_{\mu\nu} = \delta R^\rho{}_{\mu\rho\nu} = \nabla_\rho (\delta \Gamma^\rho_{\nu\mu}) - \nabla_\nu (\delta \Gamma^\rho_{\rho\mu})\,,

the Ricci scalar is

R = g^{\mu\nu} R_{\mu\nu}\,,

so that

\delta R\, = R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\,,
= R_{\mu\nu} \delta g^{\mu\nu} + \nabla_\sigma \left( g^{\mu\nu} \delta\Gamma^\sigma_{\nu\mu} - g^{\mu\sigma}\delta\Gamma^\rho_{\rho\mu} \right)\,,
= -R^{\mu\nu} \delta g_{\mu\nu} + \nabla^\mu \nabla^\nu \delta g_{\mu\nu} - g^{\mu\nu} \nabla^2 {\delta g}_{\mu\nu}\,, using \delta \Gamma^\sigma_{\nu\mu} = \frac{1}{2}g^{\sigma\delta}\left(\nabla_\mu {\delta g}_{\nu\delta} + \nabla_\nu {\delta g}_{\mu\delta} - \nabla_\delta {\delta g}_{\nu\mu}\right)\,,
= -R^{\mu\nu} \delta g_{\mu\nu} + \nabla^\mu \nabla^\nu \delta g_{\mu\nu} - \nabla^2 \delta \ln |g|\,, (see derivative of a logarithm of a determinant).

Weyl transformation

Under a Weyl transformation,

g_{ab} \to e^{2\omega} g_{ab}\,,

the Ricci scalar transforms as[1]

R \to e^{-2\omega} \left[ R - 2(n-1)\nabla^2 \omega - (n-1)(n-2) (\nabla \omega)^2 \right]\,,

so that

\sqrt{|g|}R\, \to e^{(n-2)\omega} \sqrt{|g|} \left[ R - (n-1)(n-2) (\nabla \omega)^2 - 2(n-1)\nabla^2 \omega\right]\,,

while

-2(n-1)\nabla^\mu \left[e^{(n-2)\omega} \nabla_\mu \omega\right] = -2(n-1)\left[ e^{(n-2)\omega} \nabla^2 \omega + (n-2) (\nabla \omega)^2 \right]\,

so

\sqrt{|g|}R \to e^{(n-2)\omega}\sqrt{|g|}\left[ R +(n-1)(n-2)(\nabla \omega)^2\right] -2 \sqrt{|g|}(n-1)\nabla^\mu \left[e^{(n-2)\omega} \nabla_\mu \omega\right]\,.


In 2 dimensions, this gives

R \to e^{-2\omega} [ R - 2\nabla^2 \omega]\,.


See also


References

[1]

  1. ↑ 1.0 1.1 C. Hull (1996). "String dynamics at strong coupling". Nucl.Phys.B 468: 113-154. arXiv:hep-th/9512181. DOI:10.1016/0550-3213(96)00096-X. 
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