the stress–energy tensor. But the field energy of an electromagnetic field contributes a term to the stress–energy tensor which is traceless, so in Nordström's theory, electromagnetic field energy does not gravitate! Indeed, since every solution to the field equations of this theory is a spacetime which is among other things conformally equivalent to flat spacetime, null geodesics must agree with the null geodesics of the flat background, so this theory can exhibit no light bending. Incidentally, the fact that the trace of the stress–energy tensor for an electrovacuum solution (a solution in which there is no matter present, nor any non-gravitational fields except for an electromagnetic field) vanishes shows that in the general electrovacuum solution in Nordström's theory, the metric tensor has the same form as in a vacuum solution, so we need only write down and solve the curved spacetime Maxwell field equations. But these are conformally invariant, so we can also write down the general electrovacuum solution, say in terms of a power series. In any Lorentzian manifold (with appropriate tensor fields describing any matter and physical fields) which stands as a solution to Nordström's field equations, the conformal part of the Riemann tensor (i.e. the Weyl tensor) always vanishes. The Ricci scalar also vanishes identically in any vacuum region (or even, any region free of matter but containing an electromagnetic field). Are there any further restrictions on the Riemann tensor in Nordström's theory? To find out, note that an important identity from the theory of manifolds, the Ricci decomposition, splits the Riemann tensor into three pieces, which are each fourth-rank tensors, built out of, respectively, the Ricci scalar, the trace-free Ricci tensor and the Weyl tensor. It immediately follows that Nordström's theory leaves the trace-free Ricci tensor entirely unconstrained by algebraic relations (other than the symmetric property, which this second rank tensor always enjoys). But taking account of the twice-contracted and detracted Bianchi identity, a differential identity which holds for the Riemann tensor in any (semi)-Riemannian manifold, we see that in Nordström's theory, as a consequence of the field equations, we have the first-order covariant differential equation which constrains the semi-traceless part of the Riemann tensor (the one built out of the trace-free Ricci tensor). Thus, according to Nordström's theory, in a vacuum region only the semi-traceless part of the Riemann tensor can be nonvanishing. Then our covariant differential constraint on shows how variations in the trace