New Approach to General Relativity

The Superspace of geometrodynamics

“…More precisely, the equivalence principle suggests to only take the metric as dynamical variable [76] representing the gravitational field (to which matter then couples universally), whereas diffeomorphism invariance, derivability from an invariant Lagrangian (alternatively: local energy-momentum conservation in the sense of covariant divergencelessness), dependence of the equations on the metric up to at most second derivatives, and, finally, four-dimensionality lead uniquely to the left-hand side of Einstein's equation, including a possibly non-vanishing cosmological constant [54]. Here we will review how this 'deduction' works in the Hamiltonian setting on phase space T * Riem(Σ), which goes back to [40,41,51,74].…”

“…This is necessary in order to interpret {−, H (α, β)} as a generator (on phase-space functions) of a spacetime evolution corresponding to a normal lapse α and tangential shift β. In other words, the evolution of observables from an initial hypersurface Σ i to a final hypersurface Σ f must be independent of the intermediate foliation ('integrability' or 'path independence' [40,41,74]). Therefore we placed the parameters (α , β ) outside the Poisson bracket on the right-hand side of (27), to indicate that no differentiation with respect to h, π should act on them.…”

From Magnetic Collapse to Gravitational Collapse: Levels of Understanding of Magnetism*†

“…"25 Simple as this derivation is, we today have one still simpler, which comes from Hojman et aZ. 26 It does not ask us to adopt one part of a Lagrangian, because it is quadratic in the electromagnetic field, and another because it is the simplest curvature-dependent scalar. Instead, it invites us to look at the "group" of deformations of a spacelike hypersurface (FIGURE 7).…”

New formulation of Horava–Lifshitz quantum gravity as a master constraint theory

“…spatial diffeomorphism symmetry is physically relevant. Such theories cannot obey the Dirac algebra which is the hallmark of space-time covariance and the embeddability of hypersurface deformations, and from which Einstein's geometrodynamics can be uniquely recovered [17,18].…”