  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 $\xi$, in a path integral, based on an action $\mathcal{A}[q,\xi,
\cdots]$. We show how one can integrate out a vector field $v^a$ in an action
$\mathcal{A}[\Gamma,v,\cdots ]$ and obtain an effective action
$\mathcal{A}_{\rm eff}[\Gamma, \cdots]$ which, on variation with respect to the
connection $\Gamma$, 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
$\mathcal{A}[\Gamma,v, \cdots]$, 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.
