Gravitational closure of matter field equations

The requirement that both the matter and the geometry of a spacetime canonically evolve together, starting and ending on shared Cauchy surfaces and independently of the intermediate foliation, leaves one with little choice for diffeomorphism-invariant gravitational dynamics that can equip the coefficients of a given system of matter field equations with causally compatible canonical dynamics. Concretely, we show how starting from any linear local matter field equations whose principal polynomial satisfies three physicality conditions, one may calculate coefficient functions which then enter an otherwise immutable set of countably many linear homogeneous partial differential equations. Any solution of these so-called gravitational closure equations then provides a Lagrangian density for any type of tensorial geometry that features ultralocally in the initially specified matter Lagrangian density. Thus the given system of matter field equations is indeed closed by the so obtained gravitational equations. In contrast to previous work, we build the theory on a suitable associated bundle encoding the canonical configuration degrees of freedom, which allows one to include necessary constraints on the geometry in practically tractable fashion. By virtue of the presented mechanism, one thus can practically calculate, rather than having to postulate, the gravitational theory that is required by specific matter field dynamics. For the special case of standard model matter one obtains general relativity.

Figure 1

Hyperbolicity cones of various polynomials. (a) The two hyperbolicity cones of a hyperbolic second degree principal polynomial, (b) hyperbolicity cones of a hyperbolic reducible fourth degree principal polynomial, and (c) a non-hyperbolic fourth degree principal polynomial, obviously featuring no hyperbolicity cones.

Figure 2

Gauss map sending $P$-null covectors to $P^{\#}$-null vectors. (a) Null surface of a hyperbolic reducible principal polynomial $P$ in cotangent space; with typical gradient (co-co-)vectors, and (b) null surface of the dual polynomial $P^{\#}$ in tangent space; containing, by definition, the gradient vectors to the $P$-null surface.

Figure 3

Positive energy cone ${\mathcal{O}}_x^{+}$ as the dual of the observer cone ${\mathcal{O}}_x$. (a) Cone covering all momenta of positive energy as unanimously judged by all observers, and (b) cone containing all tangent vectors to observer worldlines through one point.

Figure 4

Examples of positive energy mass shells. (a) Quadric mass shell of a second degree principal polynomial $P_x$ satisfying the three matter conditions, and (b) quartic mass shell of a fourth degree principal polynomial $P_x$ satisfying the three matter conditions.

Figure 5

Embedding of the immutable three-dimensional manifold $\mathrm{\Sigma }$ by a family of embedding maps $X_t$ into a smooth spacetime manifold $M$ of appropriate topology, yielding the leaves of a foliation of that spacetime into initial data hypersurfaces $X_t(\mathrm{\Sigma })$.