The coupled Einstein equations (II)

Question

Let $M$ be a manifold with two metrics $(g,g')$, the two Levi-Civita connections are $(\nabla , \nabla')$, the mixed curvature tensor is:

$$R_{g,g'} (X,Y)=\nabla_X \nabla'_Y +\nabla'_X \nabla_Y-\nabla_Y \nabla'_X - \nabla'_Y \nabla_X-$$

$$-\nabla_{[X,Y]} -\nabla'_{[X,Y]}$$

The mixed Ricci curvature tensor is $Ric_{g,g'} (X,Y)=tr(Z \rightarrow R_{g,g'} (X,Z)Y)$. Then the coupled Einstein equations are:

$$Ric_{g,g'}=\mu (g+g')$$

$$Ric_g=\lambda g$$

$$Ric_{g'}=\lambda' g'$$

Comments

and then? why only a pair?