  The field equations associated with the Born-Infeld-Einstein action including
matter are derived using a Palatini variational principle. Scalar,
electromagnetic, and Dirac fields are considered. It is shown that an action
can be chosen for the scalar field that produces field equations identical to
the usual Einstein field equations minimally coupled to a scalar field. In the
electromagnetic and Dirac cases the field equations reproduce the standard
equations only to lowest order. The spherically symmetric electrovac equations
are studied in detail. It is shown that the resulting Einstein equations
correspond to gravity coupled to a modified Born-Infeld theory. It is also
shown that point charges are not allowed. All particles must have a finite
size. Mass terms for the fields are also considered.
