Let $ g_{\mu\nu} $ be the metric tensor of a spacetime manifold. Instead of expressing $ g_{\mu\nu} $ directly, I wish to express it as a product of two other tensors, say $ p_{\mu\alpha} $ and $ q^{\alpha\nu} $, such that $$ g_{\mu\nu} = p_{\mu\alpha} q^{\alpha\nu}. $$
Given the Ricci scalar $ R $ which is a scalar function of the metric $ g_{\mu\nu} $ and its first and second derivatives, my aim is to determine the variation of $ R $ with respect to $ p_{\mu\alpha} $, while treating $ q^{\alpha\nu} $ as independent.
In the usual approach of varying the Einstein-Hilbert action with respect to $ g_{\mu\nu} $, we get the Einstein field equations where the variation yields the Einstein tensor $ G_{\mu\nu} $. I'm interested in how this variation changes when varying with respect to $ p_{\mu\alpha} $ instead of $ g_{\mu\nu} $.
More specifically:
- How does one compute the variation $ \delta R $ when $ R $ is varied with respect to $ p_{\mu\alpha} $ instead of $ g_{\mu\nu} $?
- Would the resulting tensor still have the properties of the Ricci tensor, or would it be a different geometric object entirely?
Any insights would be greatly appreciated.