Relativity without relativity

Abstract

We give a derivation of general relativity (GR) and the gauge principle that is novel in presupposing neither spacetime nor the relativity principle. We consider a class of actions defined on superspace (the space of Riemannian 3-geometries on a given bare manifold). It has two key properties. The first is symmetry under 3-diffeomorphisms. This is the only postulated symmetry, and it leads to a constraint linear in the canonical momenta. The second property is that the Lagrangian is constructed from a 'local' square root of an expression quadratic in the velocities. The square root is 'local' because it is taken before integration over 3-space. It gives rise to quadratic constraints that do not correspond to any symmetry and are not, in general, propagated by the Euler-Lagrange equations. Therefore these actions are internally inconsistent. However, one action of this form is well behaved: the Baierlein-Sharp-Wheeler (BSW [1]) reparametrisation-invariant action for GR. From this viewpoint, spacetime symmetry is emergent. It appears as a 'hidden' symmetry in the (underdetermined) solutions of the Euler-Lagrange equations, without being manifestly coded into the action itself. In addition, propagation of the linear diffeomorphism constraint together with the quadratic square-root constraint acts as a striking selection mechanism beyond pure gravity. If a scalar field is included in the configuration space, it must have the same characteristic speed as gravity. Thus Einstein causality emerges. Finally, self-consistency requires that any 3-vector field must satisfy Einstein causality, the equivalence principle and, in addition, the Gauss constraint. Therefore we recover the standard (massless) Maxwell equations.

The Superspace of geometrodynamics

“…Since, as already stated, all elements of Diff F (Σ) are isometries of G λ , it is natural to try to define a bundle connection on Riem(Σ) by taking the horizontal subspace Hor λ h at each T h Riem(Σ) to be the G λ -orthogonal complement to Vert h , as suggested in [35]. From (8) …”

“…Such manifolds are called spinorial. For each prime it is known whether it is spinorial or not, and the easy-to-state but hard-to-prove result is, that the only non-spinorial manifolds 8 are the lens spaces L( p, q), the handle S 1 × S 2 , and connected sums amongst them. That these manifolds are not spinorial is, in fact, very easy to visualise.…”

Scale-invariant gravity: geometrodynamics

“…This relational geodesic principle is implemented by best matching, which is explained in detail in [11] and PD, using an action that is homogeneous of degree one in the velocities.…”

“…The new theory, which we call conformal gravity, is a consistent best-matching generalization of the BSW action invariant under (8). It was proposed in the brief communication [17], which, however, treated only pure gravity and was written before the development of the special variational technique with free end points described in PD. The present paper uses the new method and develops three aspects: the Hamiltonian formulation, the coupling to the known bosonic matter fields and the physical and mathematical implications of conformal gravity.…”

“…York's method is to solve (3) in a conformally-invariant way and then to take (18) to determine φ forg ij given. An embedded hypersurface has constant mean curvature (CMC) 5 if…”