# Which metric is the main one in a metric-affine theory? $g^{\mu\nu}$ or its conformal counterpart?

For a metric-affine theory, I have found a conformal metric as $$h_{\mu\nu}=f(R)~g_{\mu\nu}.$$ My question is which metric is more important? For example in finding the weak field limit of such a theory, one can find two potentials by using these two metrics. I mean $\Phi_1=-(1/2) ~ g_{00}$ and $\Phi_2=-(1/2)~h_{00}$. Is the conformal metric only an instrument to solve the problem or contains a meaningful physics too?

In other words, which metric can be considered as the main metric that determines the physical properties of the problem.

• Dear @Qmechanic, Did U ban me?! How can I ask the question? – Perfect Fluid Jun 24 '18 at 15:10
• No. Q&A bans are caused by an automated SE script. Moderators can't hand out Q&A bans. In the future, after the ban lifts, be very careful to only ask questions that will be positively received. – Qmechanic Jun 24 '18 at 15:35
• @Qmechanic: Actually it is the second time:/ What should I do? My questions were not bad! – Perfect Fluid Jun 24 '18 at 17:08

The Jordan frame $g_{\mu\nu}$ is minimally coupled to matter while the Einstein frame $h_{\mu\nu}$ is minimally coupled to the Ricci curvature tensor. Wikipedia (June, 2018) writes that "there is currently heated debate about whether either, both, or neither frame is a 'physical' frame which can be compared to observations and experiment".