- #1

Rabindranath

- 10

- 1

$$\Gamma'^{\lambda}_{\mu\nu} = \Gamma^{\lambda}_{\mu\nu} + \frac{1}{2}\{\delta^\lambda_\nu \partial_\mu\Omega + \delta^\lambda_\mu \partial_\nu\Omega - g^{\sigma\lambda}g_{\nu\mu}\partial_\sigma\Omega \}$$

The Riemann tensor is then defined in terms of the connection as

$$R'^{\lambda}_{\mu\nu\kappa} = \partial_\kappa\Gamma'^{\lambda}_{\mu\nu} - \partial_\nu\Gamma'^{\lambda}_{\mu\kappa} + \Gamma'^{\rho}_{\mu\nu} \Gamma'^{\lambda}_{\kappa\rho} - \Gamma'^{\sigma}_{\mu\kappa} \Gamma'^{\lambda}_{\nu\sigma}$$

which gives us the Ricci tensor as

$$R'_{\mu\kappa} = R'^{\lambda}_{\mu\lambda\kappa} = \partial_\kappa\Gamma'^{\lambda}_{\mu\lambda} - \partial_\lambda\Gamma'^{\lambda}_{\mu\kappa} + \Gamma'^{\rho}_{\mu\lambda} \Gamma'^{\lambda}_{\kappa\rho} - \Gamma'^{\sigma}_{\mu\kappa} \Gamma'^{\lambda}_{\lambda\sigma}$$

which is reasonable to study term by term. For the first term, we get

$$\partial_\kappa\Gamma'^{\lambda}_{\mu\lambda} = \partial_\kappa\Gamma^{\lambda}_{\mu\lambda} + \frac{1}{2}\{\delta^\lambda_\lambda \partial_\kappa\partial_\mu\Omega + \delta^\lambda_\mu \partial_\kappa\partial_\lambda\Omega - g^{\sigma\lambda}g_{\lambda\mu}\partial_\kappa\partial_\sigma\Omega \} = \partial_\kappa\Gamma^{\lambda}_{\mu\lambda} + \frac{1}{2} D \partial_\kappa\partial_\mu\Omega$$

where ##D = \delta^\lambda_\lambda## is the dimensionality. For the second term of the Ricci tensor, we get

$$\partial_\lambda\Gamma'^{\lambda}_{\mu\kappa} = \partial_\lambda\Gamma^{\lambda}_{\mu\kappa} + \frac{1}{2}\{\delta^\lambda_\kappa \partial_\lambda\partial_\mu\Omega + \delta^\lambda_\mu \partial_\lambda\partial_\kappa\Omega - \partial_\lambda (g^{\sigma\lambda}g_{\kappa\mu}\partial_\sigma\Omega) \} = \\

= \partial_\lambda\Gamma^{\lambda}_{\mu\kappa} + \frac{1}{2}\{2\partial_\kappa\partial_\mu\Omega - \partial_\lambda (g^{\sigma\lambda}g_{\kappa\mu})\partial_\sigma\Omega - g^{\sigma\lambda}g_{\kappa\mu} \partial_\lambda\partial_\sigma\Omega \}$$

For the third term of the Ricci tensor, we get

$$\Gamma'^{\rho}_{\mu\lambda} \Gamma'^{\lambda}_{\kappa\rho} = \Gamma^{\rho}_{\mu\lambda} \Gamma^{\lambda}_{\kappa\rho} + \frac{1}{4} \{ (D + 2) - 2 g^{\sigma\rho} g_{\kappa\mu}\partial_\sigma\Omega \partial_\rho\Omega \} + \frac{1}{4}g^{\sigma\lambda} (\partial_\mu g_{\lambda\sigma} \partial_\kappa\Omega + \partial_\kappa g_{\lambda\sigma} \partial_\mu\Omega) + \frac{1}{2}g^{\tau\rho}(\partial_\mu g_{\kappa\tau} + \partial_\kappa g_{\mu\tau})\partial_\rho\Omega - g^{\tau\rho}\partial_\tau g_{\kappa\mu} \partial_\rho\Omega $$

And for the fourth and last term, we get

$$\Gamma'^{\sigma}_{\mu\kappa} \Gamma'^{\lambda}_{\lambda\sigma} = \Gamma^{\sigma}_{\mu\kappa} \Gamma^{\lambda}_{\lambda\sigma} + \frac{1}{4} \{ 2D \partial_\mu\Omega \partial_\kappa\Omega + g^{\rho\sigma} g_{\kappa\mu} \partial_\rho\Omega \partial_\sigma\Omega \} + \frac{D}{4} \{ g^{\rho\sigma} \partial_\mu g_{\kappa\rho} \partial_\sigma\Omega + g^{\rho\sigma} \partial_\kappa g_{\mu\rho} \partial_\sigma\Omega - g^{\rho\sigma} \partial_\rho g_{\kappa\mu} \partial_\sigma\Omega \} + \frac{1}{4} \{ \partial_\mu\Omega g^{\tau\lambda} \partial_\kappa g_{\lambda\tau} + \partial_\kappa\Omega g^{\tau\lambda} \partial_\mu g_{\lambda\tau} - g^{\rho\sigma}g_{\kappa\mu}\partial_\rho\Omega g^{\tau\lambda}\partial_\sigma g_{\lambda\tau} \}$$

Putting it all together, I get the total Ricci tensor as

$$R'_{\mu\kappa} = R_{\mu\kappa} + \frac{1}{2}(D+2)\nabla_\mu \partial_\kappa \Omega - \frac{3}{2}\partial^\sigma g_{\kappa\mu} \partial_\sigma\Omega - \frac{1}{2}g_{\kappa\mu}\square\Omega + \frac{1}{4}(D+2) + \frac{1}{4}g_{\kappa\mu}\partial_\sigma\Omega\partial^\sigma\Omega + \frac{1}{2}g^{\sigma\lambda}\partial_\mu g_{\lambda\sigma}\partial_\kappa\Omega + \frac{1}{2}g^{\sigma\lambda}\partial_\kappa g_{\lambda\sigma}\partial_\mu\Omega - \frac{1}{4}g_{\kappa\mu}\partial^\sigma\Omega g^{\tau\lambda}\partial_\sigma g_{\lambda\tau} + \frac{1}{2}D\partial_\mu\Omega\partial_\kappa\Omega $$

where ##\nabla_\mu## is the covariant derivative, and ##\square = \partial^\sigma\partial_\sigma## is the d'Alembert operator. This seems to be wrong; as far as I can tell, from the formula for the Ricci tensor on https://en.wikipedia.org/wiki/Weyl_transformation, it should rather be something like

$$R'_{\mu\kappa} = R_{\mu\kappa} + \frac{1}{2}(2-D)\nabla_\mu \partial_\kappa \Omega - \frac{1}{2}g_{\kappa\mu}\square\Omega + \frac{1}{4} (D-2) \partial_\mu\Omega\partial_\kappa\Omega - \frac{1}{4} (D-2) g_{\mu\kappa} \partial_\sigma\Omega\partial^\sigma\Omega $$

with the metric in question.

Any suggestions on where I might have gone wrong in the above ##-## I feel a bit stuck for the moment. Thanks a lot in advance.