The precise technical meaning of predictivity, interpretability, and quantizability is discussed in Ref. [7] and reviewed, to the extent necessary for a full understanding of the present paper, in Sec. II A.

Having to invoke specific matter field equations in order to decide on what constitutes a spacetime geometry only seems odd at first sight. For, in fact, this is precisely the reasoning that led Einstein from Maxwell electrodynamics—having properties which, at the time, contradicted the conceived model of space and time as much as superluminal neutrinos do now—ultimately to the Lorentzian geometry of spacetime.

This is true as long as one understands the Legendre duality map between covectors which are normal to admissible initial data surfaces and the vectors they correspond to, which is derived in Ref. [7] and concisely reviewed in Sec. II A.

The history of modified gravity theories is a long and winding one, from the recently heavily studied $f(R)$ modified gravity actions for Lorentzian manifolds, over various actions for nonsymmetric or scalar-tensor extensions of the gravitational field, dynamics for the affine connection rather than a metric, and finally to string-geometry-inspired proposals of various sorts. All these proposals share the pleasant feature of being inspired in some way; compelling, however, let alone without alternative in the sense of the present paper, appear few of these classes of theories.

This point has been made very lucidly, and very close to the spirit of the present paper, by Laemmerzahl in Ref. [17], who explores what can be learned about the spacetime geometry from the predictivity of linear matter field equations alone.

If the linear matter field equations are obtained from a linearization of nonlinear matter field equations around an exact solution ${\varphi }_0$ of the latter, the coefficients $Q$ of the linearized equation will generically depend also on ${\varphi }_0$.

Transformation of the entire equation as a tensor is guaranteed, for instance, if the equation is obtained by variation of a scalar action $S[G,\varphi ]$ with respect to the tensor field ${\varphi }^A$—note the corresponding position of the index $A$ in Eq. (1).

Since not even all degrees of freedom in the area metric $G$ enter the field $P$; see, e.g., Ref. [22].

This dependence from specific matter equations is not a weakness of the approach, but rather an insight: there are no viable or nonviable spacetime structures as such, but they prove their validity by enabling specific field equations to be predictive. The causality of Lorentzian manifolds is in fact the causality of Maxwell theory, and thus of all the other matter dynamics modeled in the image of Maxwell theory.

This choice is the general tensorial analogue to a choice of “mainly minus” signature $(+-\cdots -)$ in the special case of metric geometry.

If there are such superluminal particles, they can radiate off massless particles until they are infraluminal; see Ref. [13].

Employing the techniques for general tensorial spacetime geometries instead of those valid only for the special case of Lorentzian manifolds, this section follows closely the philosophy and calculation of Ref. [10].

The minus sign in the algebra equation involving two normal deformation operators is due to our normalization condition $P(n)=1$. In most standard texts on the hypersurface deformation algebra of Lorentzian manifolds, the normalization is chosen to be $P(n)=-1$, which results in a plus sign in the first algebra equation.

The change of the overall sign on the right-hand side of the Poisson algebra equations in comparison to the hypersurface deformation algebra is due to the action from the left of the deformation operators on hypersurface functionals as opposed to the insertion of the supermomentum and superhamiltonian in the right slot of the Poisson bracket when acting on a phase-space functional.

In the standard treatments of canonical general relativity, one chooses the metric $g_{\alpha \beta }$ and its conjugate momentum ${\pi }_{\mathrm{standard}}^{\alpha \beta }$ as the phase-space variables. It can be checked that changing the configuration variables from the metric to its inverse results in the canonically conjugate momentum ${\widehat{\pi }}_{\alpha \beta }=-g_{\alpha \gamma }{g}_{\beta \delta }{\pi }_{\mathrm{standard}}^{\gamma \delta }$, which explains the different sign in the supermomentum.

This certainly presents an entirely new angle on—and, more importantly, a physically well-founded approach to—the construction of gravitational dynamics for a nonsymmetric metric or for string geometries determined, for instance, by a metric $g$, a twoform potential $B$, and dilaton $\varphi$ from a purely canonical point of view.

For classical theories based on a connection formulation, this may require a reformulation in terms of purely tensorial objects whenever possible. Possibly some pure connection theories therefore escape the formalism presented here, but in this case, we feel a careful evaluation of their kinematical apparatus, in particular, the definition of observer frames, should yield the relevant information. This then certainly requires a case-by-case analysis.