# What exactly is the difference between a Ricci-flat manifold and Minkowski space?

Minkowski space describes a flat spacetime and a Ricci-flat manifold is a solution of GRT without curvature. Shouldn't they be the same? So, what's the difference?

• A Ricci-flat manifold is a solution of GRT without curvature. No, it isn’t. – G. Smith Jul 11 at 17:17
• If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Jul 11 at 17:26

Ricci flat is not necessarily "truly" flat. Minkowski space has the entire Riemann curvature tensor vanishing, ie, $$R_{\mu\nu\rho\sigma}=0$$. However, being just Ricci flat is a weaker condition that only requires that the Ricci tensor $$R_{\mu\nu}$$ vanish.