Gravitational closure of matter field equations

Abstract

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 provide consistent evolution equations to the coefficients of a given system of matter field equations. 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 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. a Corresponding author. Electronic address fps@aei.mpg.de 1 arXiv:1611.08878v3 [gr-qc]

Covariant constructive gravity: A step-by-step guide towards alternative theories of gravity

“…The causal structure of a given second-order EOM E A = 0 is closely related to the behavior of wave-like solutions in the infinite frequency limit (cf. [40]). We consider the WKB ansatz for the coordinate expression of a section G A ∈ Γ(F )…”

“…It can be shown (cf. [40], [41]) that Q (A 1 ...As)(B 1 ...Bs) (k a ) is subject to the general form…”

Symmetric gravitational closure

“…Without symmetry assumptions, the dynamics of a physically constrained class of matter actions restrict the possible dynamics of the underlying geometry so severely, that the dynamics for this geometry can be determined constructively [1]. More precisely, any matter action S matter [Φ, G), which is local in some matter field Φ and ultralocal in a tensor field G and satisfies three algebraic physicality conditions [2] constructively determines the causally compatible gravitational actions S gravity [G] for the tensor field G. Adding the latter action to the former closes the matter dynamics gravitationally, since variation of the total action with respect to Φ now recovers the stipulated matter field equations while its variation with respect to G yields the gravitational field equations for the pertinent tensorial geometry G sourced by the very matter field dynamics at play,…”

“…where φ is a scalar field and g a metric tensor on a four-dimensional manifold M, which satisfies the aforementioned three weak physicality conditions for the matter action if and only if the metric has Lorentzian signature. Gravitational closure of these Klein-Gordon dynamics was shown [1,3] to yield the well-known two-parameter family S κ,Λ gravity…”

“…for a one-form gauge potential A, which employs a fourth rank tensor field G abcd with the symmetries G abcd = G cdab and G abcd = −G bacd and a scalar density ω G constructed from it as its background geometry. While the mentioned countable set of gravitational closure equations is set up straightforwardly, see [1], the bottleneck is their actual solution.…”