Corrections to Einstein's equations that become important at small curvatures are considered. The field equations are derived using a Palatini variation in which the connection and metric are varied independently. In contrast with the Einstein-Hilbert variation, which yields fourth order equations, the Palatini approach produces second order equations in the metric. The Lagrangian L(R)ϭRϪ␣ 2 /R is examined and it is shown that it leads to equations whose solutions approach a de Sitter universe at late times. Thus, the inclusion of 1/R curvature terms in the gravitational action offers an alternative explanation for the cosmological acceleration.

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 Rµν ∨ , 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 Rµν ∨

Following renewed interest, the problem of whether the cosmological expansion a †ects the dynamics of local systems is reconsidered. The cosmological correction to the equations of motion in the locally inertial Fermi normal frame (the relevant frame for astronomical observations) is computed. The evolution equations for the cosmological perturbation of the two-body problem are solved in this frame. The e †ect on the orbit is insigniÐcant, as are the e †ects on the galactic and galactic-cluster scales.

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.

In this paper we calculate the entropy of a thin spherical shell that contracts reversibly from infinity down to its event horizon. We find that, for a broad class of equations of state, the entropy of a non-extremal shell is one-quarter of its area in the black hole limit. The considerations in this paper suggest the following operational definition for the entropy of a black hole: S BH is the equilibrium thermodynamic entropy that would be stored in the material which gathers to form the black hole, if all of this material were compressed into a thin layer near its gravitational radius. Since the entropy for a given mass and area is maximized for thermal equilibrium we expect that this is the maximum entropy that could be stored in the material before it crosses the horizon. In the case of an extremal black hole the shell model does not assign an unambiguous value to the entropy. ͓S0556-2821͑98͒01710-X͔

Recently Flanagan [1] has argued that the Palatini form of 1/R gravity is ruled out by experiments such as electron-electron scattering. His argument involves adding minimally coupled fermions in the Jordan frame and transforming to the Einstein frame. This produces additional terms that are ruled out experimentally.Here I argue that this conclusion is false. It is well known that conformally related theories are mathematically equivalent but not physically equivalent. As discussed by Magnano and Sokolowski [2] one must decide, in the vacuum theory, which fame is the physical frame and add the minimally coupled Lagrangian in this frame. If this procedure is followed the resulting theory is not ruled out experimentally. The discussions in this paper also show that the equivalence between the generalized gravitational theories and scalar tensor theories discussed by Flanagan [3] is only mathematical, not physical.

The entropy of black holes in modified theories of gravity is examined in the Palatini formalism using the Noether Charge approach. It is shown that, if the gravitational coupling constant is properly identified, the entropy of a black hole is one-quarter of the horizon area in f(R) theories coupled to conformally invariant matter. If matter is present that is not conformally invariant the entropy is still proportional to the area of the black hole, but the coefficient is generally not one-quarter. The entropy of black holes in generalized dilaton theories and in theories with Lagrangians that depend on an arbitrary function of the Ricci tensor are also examined.

It is well known that it takes matter that violates the averaged weak energy condition to hold the throat of a wormhole open. The production of such ''exotic'' matter is usually discussed within the context of quantum field theory. In this paper I show that it is possible to produce the exotic matter required to hold a wormhole open classically. This is accomplished by coupling a scalar field to matter that satisfies the weak energy condition. The energy-momentum tensor of the scalar field and the matter separately satisfy the weak energy condition, but there exists an interaction energy-momentum tensor that does not. It is this interaction energymomentum tensor that allows the wormhole to be maintained.