Conformal geometrodynamics regained: Gravity from duality

Abstract

There exist several ways of constructing general relativity from ‘first principles’: Einstein’s original derivation, Lovelock’s results concerning the exceptional nature of the Einstein tensor from a mathematical perspective, and Hojman–Kuchař-Teitelboim’s derivation of the Hamiltonian form of the theory from the symmetries of space–time, to name a few. Here I propose a different set of first principles to obtain general relativity in the canonical Hamiltonian framework without presupposing space–time in any way. I first require consistent propagation of scalar spatially covariant constraints (in the Dirac picture of constrained systems). I find that up to a certain order in derivatives (four spatial and two temporal), there are large families of such consistently propagated constraints. Then I look for pairs of such constraints that can gauge-fix each other and form a theory with two dynamical degrees of freedom per space point. This demand singles out the ADM Hamiltonian either in (i) CMC gauge, with arbitrary (finite, non-zero) speed of light, and an extra term linear in York time, or (ii) a gauge where the Hubble parameter is conformally harmonic.

Introduction

In the golden years of the canonical approach to general relativity, one of the most profound thinkers on gravity, John Wheeler, posed the famous question  [1]: “If one did not know the Einstein–Hamilton–Jacobi equation, how might one hope to derive it straight off from plausible first principles, without ever going through the formulation of the Einstein field equations themselves?”. In response, Hojman, Kuchař and Teitelboim (HKT), in the aptly titled paper “Geometrodynamics regained”  [2], tackled the problem of deriving geometrodynamics directly from first principles rather than by a formal rearrangement of Einstein’s law. They succeeded in obtaining the canonical form of general relativity by imposing requirements onto a Hamiltonian formulation ensuring that it represents a foliated space–time.

Here I propose a different approach to Wheeler’s question. In short, I will look for what I refer to as dual symmetries in the gravitational phase space. Dual symmetries consist of two distinct sets of constraints, which I refer to as the dual partners. Each dual partner should be first of all a (spatially covariant) first class constraint–which by Dirac’s analysis means that each generates a (spatially covariant) symmetry–and secondly, to fix the partnership and establish duality, one partner must gauge-fix the other. In other words, dual symmetries should be ones for which one can always find a compatible space of observables. In Fig. 1, we see two first class constraint manifolds intersecting transversally, which illustrates the rather simple principle. Alternatively, this criterion amounts to searching for spatially covariant theories with two propagating degrees of freedom, which possess a gauge-fixing that is also consistently propagated. This dual role arises because in the Hamiltonian formalism, a gauge-fixing condition–represented as the vanishing of a given phase space function–also generates a transformation in phase space (the symplectic flow of said function).

The deeper reason for taking these first principles as the basis of my construction cannot, however, be fully appreciated by only considering the classical theory. As realized in the mid 60’s by Feynman, and later resolved simultaneously by Becchi, Rouet, Stora and Tyutin  [3], [4], theories with non-abelian symmetries require extra care upon quantization, so that pure gauge degrees of freedom do not propagate. The extended theoretical framework in which these redundancies are adequately taken into account is today known as BRST. In the Hamiltonian setting, the conditions required here for dual constraints imbue the extended BRST system with interesting properties. Namely, they ensure that with just the right gauge-fixing, the gauge-fixed, BRST-extended Hamiltonian possesses not only the symmetries related to the original system, but also those related to its gauge-fixing. Thus the results obtained here can be said to emerge out of broad symmetry requirements, but it is surprising that we do not have to demand in advance what symmetries the emerging theory should embody, they are, in a weak sense, self-selected.

In their seminal paper, Hojman–Kuchař–Teitelboim  [2] used the fact that the set of vector fields that generate the tangential and normal deformations of an embedded hypersurface in a Riemannian manifold produce a specific vector commutation algebra, i.e. a specific symmetry. They then sought constraints in the space of functionals of the spatial metric $g_{ab}$ and momenta ${\pi }^{ab}$, whose own commutation algebra (Poisson bracket) would mirror the hypersurface deformation algebra. Clearly, this derivation must assume the prior existence of space–time. With a few other requirements, they were eventually led to the (super)momentum constraint $H_a\left(x\right)={\nabla }_b{{\pi }^b}_a\left(x\right)=0$ and the (super) Hamiltonian constraint: $S\left(x\right)=R\sqrt{g}-\frac{{\pi }^{ab}{\pi }_{ab}-\frac{1}{2}{\pi }^2}{\sqrt{g}}=0$ (where $\pi =g_{ab}{\pi }^{ab}$), which are responsible for the entire dynamics of space–time in the Hamiltonian formulation of general relativity. That is, these two functionals have associated symplectic vector fields, which (due to their first class property) generate symmetry transformations in phase space. The Hamiltonian (or scalar) constraint $S\left(x\right)=0$ generates (on-shell) refoliations of space–times (i.e. a description of space–time by different sets of surfaces of simultaneity), while the momentum constraint (or diffeomorphism constraint) generates foliation-preserving spatial diffeomorphisms.

I cannot say if the HKT answer satisfied John Wheeler, or if any of the other construction principles that lead more or less uniquely to GR ever did. In the case of gravity there is an almost abundant variety of such principles. Very likely the most well-known was introduced by Lovelock  [5], building on earlier work by Weyl. Lovelock showed that the Einstein tensor is the unique second order, generally covariant, divergence-free tensor with 2 derivatives of the metric in 4-dimensions. Other construction principles have also been introduced, as early on as 1962 by Feynman  [6] based on the massless spin-2 character of the graviton, and as late as the thermodynamic-based derivation in 1995 by Jacobson  [7]. For some of these routes, I point the interested reader to box 17.2 of  [8], entitled “6 Routes to Einstein’s geometrodynamic law”.

There is some intrinsic value in possessing so many alternative ontologies for a theory, shifting the understanding of a theory between different principles. Quoting HKT: “The importance of alternate foundations of a basic physical theory cannot be overexaggerated. The conceptual reformulation of a theory may open a new path to its development or even lead to its modification. Thus, Feynman’s path-integral approach to quantum field theory led to the implementation of powerful approximation techniques, and Faraday’s reformulation of action-at-a-distance stationary electrodynamics in terms of the field concept developed into Maxwell’s electrodynamics.[…] In this spirit, believing in the potential fruitfulness of the canonical variational-differential approach to the general theory of relativity, we have undertaken the study of a Seventh Route to Einstein’s”. I believe the attempt to enlarge this catalog of alternative ontologies would alone justify the present work, although my truer motivations are “meta-theoretic”, as expressed earlier.

In the spirit of HKT, I will implement certain conditions on phase space functionals that will lead more or less directly to my results. Unlike HKT, I will not try to match algebras of constraints. In fact, I will not even assume the existence of space–time (and hypersurface embeddings therein), but only of the gravitational canonical phase space. Granted, there is already some semblance of time existing in phase space, which is fully fleshed out when one posits a Hamiltonian. It is however still surprising that the space–time structure that we know and love, with refoliation invariance, emerges from assumptions in which it was not present. In the interest of full disclosure, starting from my first principles I will not be able to recover the full range of solutions to Einstein gravity, only those that either have constant mean curvature foliations (i.e. those space–times that can be sliced with hypersurfaces with constant trace of the extrinsic curvature), $g_{ab}{\pi }^{ab}=c\sqrt{g}$, or which can be foliated by space-like surfaces for which the Hubble parameter is a conformal harmonic function, i.e. satisfies $({\nabla }^2-8R)H=0$.

The particular gauge fixing $g_{ab}{\pi }^{ab}=c\sqrt{g}$ has a distinguished status in GR, because it is the only one in which the solvability of the initial-value problem has been proven generically by York  [9]. But the truly special character of the trace of the momenta, $\pi ≔g_{ab}{\pi }^{ab}$, is that it serves a double role in the formalism. It can be seen as the maximal slicing gauge fixing for ADM, $\pi =0$, but also as a generator of spatial Weyl transformations,¹

$$\{\pi \left(\rho \right),g_{ab}\left(x\right)\}=\rho \left(x\right)g_{ab}\left(x\right)≕{\delta }_{\rho }{g}_{ab}\left(x\right)$$$$\{\pi \left(\rho \right),{\pi }^{ab}\left(x\right)\}=-\rho \left(x\right){\pi }^{ab}\left(x\right)≕{\delta }_{\rho }{\pi }^{ab}\left(x\right)$$

where I denoted the smearing $\int d^{3}x\rho \pi =\pi \left(\rho \right)$ and the canonical Poisson brackets $\{g_{ab}\left(x\right),{\pi }^{cd}\left(y\right)\}=\delta (x,y){\delta }_{\left(cd\right)}^{\left(ab\right)}$. This property, that $\pi$ generates a symmetry compatible with spatial diffeomorphisms, is at the heart of the success of the York method, and is part of what makes the entire construction of conformal superspace possible. It is the inspiration for one of the central criteria I use in my own construction principle, which can be encapsulated in the question: ‘Are there other sets of symmetries that gauge-fix each other, besides Weyl and refoliations?’

In favor of the $3+1$ representation, which I take as a starting point, Dirac wrote  [10]: “A few decades ago it seemed quite certain that one had to express the whole of physics in four-dimensional form. But now it seems that four-dimensional symmetry is not of such overriding importance, since the description of nature sometimes gets simplified when one departs from it”. It is my duty now to show the reader why such requirements make physical and mathematical sense, and why indeed they lead to the results claimed here. I aim to show not only that one can obtain a refoliation symmetry without any assumption of an underlying space–time, but that the requirements are natural in a physical sense.

It is my hope that providing this new first principles derivation of gravity (in these two very special gauges), I extend HKT’s philosophy of considering the 3+1 Hamiltonian formalism not only as a technical tool, but as a self-consistent means of arriving at consistent theories. The present results set Hamiltonian 3+1 gravity as an equivalent ontology to, and independent from, space–time.