Microscopic origin of Einstein's field equations and the raison d'être for a positive cosmological constant
In the paradigm of effective field theory, one hierarchically obtains the effective action $\mathcal{A}_{\rm eff}[q, \cdots]$ for some low(er) energy degrees of freedom $q$, by integrating out the high(er) energy degrees of freedom $ξ$, in a path integral, based on an action $\mathcal{A}[q,ξ, \cdots]$. We show how one can integrate out a vector field $v^a$ in an action $\mathcal{A}[Γ,v,\cdots ]$ and obtain an effective action $\mathcal{A}_{\rm eff}[Γ, \cdots]$ which, on variation with respect to the connection $Γ$, leads to the Einstein's field equations and a metric compatible with the connection. The derivation \textit{predicts} a non-zero, positive, \cc, which arises as an integration constant. The Euclidean action $\mathcal{A}[Γ,v, \cdots]$, has an interpretation as the heat density of null surfaces, when translated into the Lorentzian spacetime. The vector field $v^a$ can be interpreted as the Euclidean analogue of the microscopic degrees of freedom hosted by any null surface. Several implications of this approach are discussed.