  Only a severely restricted class of tensor fields can provide classical
spacetime geometries, namely those that can carry matter field equations that
are predictive, interpretable and quantizable. These three conditions on matter
translate into three corresponding algebraic conditions on the underlying
tensorial geometry, namely to be hyperbolic, time-orientable and
energy-distinguishing. Lorentzian metrics, on which general relativity and the
standard model of particle physics are built, present just the simplest
tensorial spacetime geometry satisfying these conditions. The problem of
finding gravitational dynamics---for the general tensorial spacetime geometries
satisfying the above minimum requirements---is reformulated in this paper as a
system of linear partial differential equations, in the sense that their
solutions yield the actions governing the corresponding spacetime geometry.
Thus the search for modified gravitational dynamics is reduced to a clear
mathematical task.
