Born-Infeld-Einstein theory with matter

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 the solution becomes singular on a spherical shell which implies that all charged particles must have a finite size. The space-time is therefore more singular than the Reissner-Nordstr\"om space-time, which is only singular at a point. Mass terms for the fields are also considered.