  We study the incorporation of gravity into the trace dynamics framework for
classical matrix-valued fields, from which we have proposed that quantum field
theory is the emergent thermodynamics, with state vector reduction arising from
fluctuation corrections to this thermodynamics. We show that the metric must be
incorporated as a classical, not a matrix-valued, field, with the source for
gravity the exactly covariantly conserved trace stress-energy tensor of the
matter fields. We then study corrections to the classical gravitational action
induced by the dynamics of the matrix-valued matter fields, by examining the
average over the trace dynamics canonical ensemble of the matter field action,
in the presence of a general background metric. Using constraints from global
Weyl scaling and three-space general coordinate transformations, we show that
to zeroth order in derivatives of the metric, the induced gravitational action
in the preferred rest frame of the trace dynamics canonical ensemble must have
the form $$ \Delta S=\int d^4x (^{(4)}g)^{1/2}(g_{00})^{-2} A\big(g_{0i} g_{0j}
g^{ij}/g_{00}, D^ig_{ij}D^j/g_{00}, g_{0i}D^i/g_{00}\big), $$ with $D^i$
defined through the co-factor expansion of $^{(4)}g$ by
$^{(4)}g/{^{(3)}g}=g_{00}+g_{0i}D^i$, and with $A(x,y,z)$ a general function of
its three arguments. This action has "chameleon-like" properties: For the
Robertson-Walker cosmological metric, it {\it exactly} reduces to a
cosmological constant, but for the Schwarzschild metric it diverges as
$(1-2M/r)^{-2}$ near the Schwarzschild radius, indicating that it will
substantially affect the horizon structure.
