## Thursday, March 1, 2012

### lensing on an expanding metric

in relation to the last question: why is it possible to treat gravitational lensing on an expanding background metric as if it was Minkowskian, weakly perturbed by the gravitational potential? if this is related to a property of photons, to which other particle trajectories would this apply?

bonus question: what's $\sqrt[i]{i}$ with $i=\sqrt{-1}$?

1. I must admit at the beginning that this answer is more or less the result of a discussion with spirou and I hope I got it the right way:

The trajectory $\gamma$ of a photon has to be described by the geodesic equation $\nabla_{\dot{\gamma}} \dot{\gamma}$ in which the dot denotes the derivative with respect to a properly chosen path parameter.

Consider the metric $$ds^2 = a(\eta) \left(-(1-2\phi)d\eta^2 + (1-2\psi) dx^2 \right)$$ with the comoving coordinates $\vec{x}$ and the conformal time $\eta$. This metric is a conformal transformation of the comoving one:

$$ds^2 = -(1-2\phi)d\eta^2 + (1-2\psi) dx^2$$

The conformal transformation properties of the Christoffel symbols in the Levi-Civita connection and the fact that ultra-relativistic particles move on null geodesics $g_{\mu\nu}dx^{\mu} dx^{\nu}}=0$ allow to show that the transformed trajectory fulfills the transformed geodesic equation. Thus, null geodesics are conformally invariant such that the comoving metric can be used to describe the problem. Time-like geodesics are not conformally invariant such that for non-relativistic particles the problem is affected by the Hubble drag. Possibly neutrinos (if they are massless) might show the same properties like photons.

$$\sqrt[i]{i}= \left(\exp{(i \frac{\pi}{2})}\right)^{\frac{1}{i}}=\left(\exp{(i \frac{\pi}{2})}\right)^{-i} = \exp{(-i \cdot i \frac{\pi}{2})} = \exp{(\frac{\pi}{2})} \approx 4.81$$

2. it seems the bonus questions are too easy! wait until thursday...

3. there's not much to add to the answer... in fact it's the conformal invariance of null-geodesics which allows us to treat lensing on an expanding background as if space was Minkowskian with weak perturbations due to the Newtonian gravitational potentials. non-relativistic particles experience a frictional force in the comoving frame due to the Hubble-expansion, and it is exactly this mechanism by which e.g. dark energy models influence structure formation.