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 (Baierlein R F, Sharp D and Wheeler J A 1962 Phys. Rev. 126 1864) reparametrization-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.