a kind of trace of the curvature tensor; is the Ricci scalar, the trace of the Ricci tensor; is the stress–energy tensor; is the trace of the stress–energy tensor; is the scalar field; is the Laplace–Beltrami operator or covariant wave operator, . The first equation describes how the stress–energy tensor and scalar field together affect spacetime curvature. The left-hand side, the Einstein tensor, can be thought of as a kind of average curvature. It is a matter of pure mathematics that, in any metric theory, the Riemann tensor can always be written as the sum of the Weyl curvature (or conformal curvature tensor) and a piece constructed from the Einstein tensor. The second equation says that the trace of the stress–energy tensor acts as the source for the scalar field . Since electromagnetic fields contribute only a traceless term to the stress–energy tensor, this implies that in a region of spacetime containing only an electromagnetic field (plus the gravitational field), the right-hand side vanishes, and obeys the (curved spacetime) wave equation. Therefore, changes in propagate through electrovacuum regions; in this sense, we say that is a long-range field. For comparison, the field equation of general relativity is simply This means that in general relativity, the Einstein curvature at some event is entirely determined by the stress–energy tensor at that event; the other piece, the Weyl curvature, is the part of the gravitational field which can propagate as a gravitational wave across a vacuum region. But in the Brans–Dicke theory, the Einstein tensor is determined partly by the immediate presence of mass–energy and momentum, and partly by the long-range scalar field . The vacuum field equations of both theories are obtained when the stress–energy tensor vanishes. This models situations in which no non-gravitational fields are present. The action principle The following Lagrangian contains the complete description of the Brans–Dicke theory: where is the determinant of the metric, is the four-dimensional volume form, and is the matter term, or matter Lagrangian density. The matter term includes the contribution of ordinary matter (e.g. gaseous matter) and also electromagnetic fields. In a vacuum region, the matter term vanishes identically; the remaining term is the gravitational term. To obtain the vacuum field equations, we must vary the gravitational term in the Lagrangian with respect to the metric ; this gives the first field equation above. When we vary with respect to the scalar field ,