It is argued that, under the assumption that the strong principle of equivalence holds, the theoretical realization of the Mach principle (in the version of the Mach-Einstein doctrine) and of the principle of general relativity are alternative programs. That means only the former or the latter can be realized—at least as long as only field equations of second order are considered. To demonstrate this we discuss two sufficiently wide classes of theories (Einstein-Grossmann and Einstein-Mayer theories, respectively) both embracing Einstein's theory (...) of general relativity (GRT). GRT is shown to be just that “degenerate case” of the two classes which satisfies the principle of general relativity but not the Mach-Einstein doctrine; in all the other cases one finds an opposite situation.These considerations lead to an interesting “complementarity” between general relativity and Mach-Einstein doctine. In GRT, via Einstein's equations, the covariant and Lorentz-invariant Riemann-Einstein structure of the space-time defines the dynamics of matter: The symmetric matter tensor Ttk is given by variation of the Lorentz-invariant scalar densityL mat, and the dynamical equations satisfied by Tik result as a consequence of the Bianchi identities valid for the left-hand side of Einstein's equations. Otherwise, in all other cases, i.e., for the “Mach-Einstein theories” here under consideration, the matter determines the coordinate or reference systems via gravity. In Einstein-Grossmann theories using a holonomic representation of the space-time structure, the coordinates are determined up to affine (i.e. linear) transformations, and in Einstein-Mayer theories based on an anholonomic representation the reference systems (the tetrads) are specified up to global Lorentz transformations. The corresponding conditions on the coordinate and reference systems result from the postulate that the gravitational field is compatible with the strong equivalence of inertial and gravitational masses. (shrink)

An analysis of the extension of the Hawking-Penrose singularity theorem to Riemann-Cartan U4 space-times with torsion and spin density is undertaken. The minimal coupling principle in U4 is used to formulate a new expression for the convergence condition autoparallels in Einstein-Cartan theory. The Gödel model with torsion is given as an example.

Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. This geometry can be characterized by an orthonormal coframe ϑ α and a (metric compatible) Lorentz connection Γ α β . These two potentials yield the field strengths torsion T α and curvature R α β . Evans tried to infuse electromagnetic properties into this geometrical framework by putting the coframe ϑ α to be proportional to four extended electromagnetic (...) potentials $\mathcal{A}^{\alpha }$ ; these are assumed to encompass the conventional Maxwellian potential A in a suitable limit. The viable Einstein-Cartan (-Sciama-Kibble) theory of gravity was adopted by Evans to describe the gravitational sector of his theory. Including also the results of an accompanying paper by Obukhov and the author, we show that Evans’ ansatz for electromagnetism is untenable beyond repair both from a geometrical as well as from a physical point of view. As a consequence, his unified theory is obsolete. (shrink)

It is well-known that Newton’s theory of gravity, commonly held to describe a gravitational force, can be recast in a geometrical form: Newton- Cartan theory. It is less well-known that general relativity, an apparently geometrical theory, can be reformulated in such a way that it resembles a force theory; teleparallel gravity does just this. This raises questions. One of these concerns theoretical underdetermination. I argue that these theories do not, in fact, represent cases of worrying underdetermination. On close examination, the (...) alternative formulations are best interpreted as postulating the same spacetime ontology. In accepting this, we see that the ontological commitments of these theories cannot be directly deduced from their mathematical form. The geometrical nature of a gravitational theory is not a straightforward consequence of anything internal to that theory as a theory of gravity. Rather, it essentially relies on the rest of nature (the nongravitational interactions) conspiring to choose the appropriate set of inertial frames. (shrink)

Evans attempted to develop a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for a theory that implements some of the ideas that have been (imprecisely) indicated by Evans and show that it yields two field equations. The second field equation is algebraic in the torsion and we can resolve (...) it with respect to the torsion. It turns out that for all physical cases the torsion vanishes and the first field equation, together with Evans’ unified field theory, collapses to an ordinary Einstein equation. (shrink)

Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The (...) Poincaré gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article. (shrink)