  The field equations associated with the Born-Infeld-Einstein action are
derived using the Palatini variational technique. In this approach the metric
and connection are varied independently and the Ricci tensor is generally not
symmetric. For sufficiently small curvatures the resulting field equations can
be divided into two sets. One set, involving the antisymmetric part of the
Ricci tensor $R_{\stackrel{\mu\nu}{\vee}}$, consists of the field equation for
a massive vector field. The other set consists of the Einstein field equations
with an energy momentum tensor for the vector field plus additional
corrections. In a vacuum with $R_{\stackrel{\mu\nu}{\vee}}=0$ the field
equations are shown to be the usual Einstein vacuum equations. This extends the
universality of the vacuum Einstein equations, discussed by Ferraris et al.
\cite{Fe1,Fe2}, to the Born-Infeld-Einstein action. In the simplest version of
the theory there is a single coupling constant and by requiring that the
Einstein field equations hold to a good approximation in neutron stars it is
shown that mass of the vector field exceeds the lower bound on the mass of the
photon. Thus, in this case the vector field cannot represent the
electromagnetic field and would describe a new geometrical field. In a more
general version in which the symmetric and antisymmetric parts of the Ricci
tensor have different coupling constants it is possible to satisfy all of the
observational constraints if the antisymmetric coupling is much larger than the
symmetric coupling. In this case the antisymmetric part of the Ricci tensor can
describe the electromagnetic field, although gauge invariance will be broken.
