Microscopic origin of Einstein's field equations and the raison d'\^{e}tre for a positive cosmological constant

TL;DR

This paper derives Einstein's field equations from an effective field theory perspective by integrating out a vector field, revealing a natural emergence of a positive cosmological constant as an integration constant, and linking it to null surface microstates.

Contribution

It introduces a novel derivation of Einstein's equations from effective field theory by integrating out a vector field, providing a microscopic interpretation of the cosmological constant.

Findings

Einstein's equations emerge from an effective action after integrating out a vector field.

The cosmological constant appears as a positive integration constant.

The Euclidean action relates to the heat density of null surfaces.

Abstract

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.