Hypersurface foliation approach to renormalization of ADM formulation of gravity

Eur. Phys. J. C Hypersurface foliation approach to renormalization of ADM formulation of gravity I. Y. Park 0 1 0 Department of Applied Mathematics, Philander Smith College , Little Rock, AR 72223 , USA 1 Department of Physics, Hanyang University , Seoul 133-791 , Korea We carry out ADM splitting in the Lagrangian formulation and establish a procedure in which (almost) all of the unphysical components of the metric are removed by using the 4D diffeomorphism and the measure-zero 3D symmetry. The procedure introduces a constraint that corresponds to the Hamiltonian constraint of the Hamiltonian formulation, and its solution implies that the 4D dynamics admits an effective description through 3D hypersurface physics. As far as we can see, our procedure implies potential renormalizability of the ADM formulation of 4D Einstein gravity for which a complete gauge-fixing in the ADM formulation and hypersurface foliation of geometry are the key elements. If true, this implies that the alleged unrenormalizability of 4D Einstein gravity may be due to the presence of the unphysical fields. The procedure can straightforwardly be applied to quantization around a flat background; the Schwarzschild case seems more subtle. We discuss a potential limitation of the procedure when applying it to explicit time-dependent backgrounds. - The quantization of 4D Einstein–Hilbert action has been a long-standing problem. (See [1] for a review.) One-loop renormalizability of pure 4D gravity (i.e., gravity without any matter field coupled) was established in [2]. However, the presence of matter fields does not preserve the one-loop renormalizability [3,4]. Also, it was subsequently shown [5] that two-loop and higher order diagrams of the pure gravity require proliferation of counter terms, thereby leading to unrenormalizability. Needless to say, the lack of renormalizability has been a serious obstacle that has delayed (or even blocked) progress in many fundamental issues such as the black hole information paradox. There have been several approaches to the quantization of gravity. The first was through the Hamiltonian formulation of gravity [6,7]. (Earlier discussions can be found, e.g., in [8–11].) The obstacle in this approach was the complexity of the Hamiltonian constraint; a different approach based on a different set of variables was proposed and is now known as loop quantum gravity [12–14]. Still another approach is based on adding higher derivative terms in the action [15,16]. In this work, we show by conducting the 3 + 1 splitting in the Lagrangian formulation that the dynamics of the pure 4D gravity is effectively reduced.1 to the dynamics of pure “3D gravity.” (As is well known, the genuine 3D gravity does not have a graviton. Our “3D” theory – originating from 4D gravity – is not the genuine 3D gravity; as we will see, it has two propagating degrees of freedom that are inherited from 4D. It holographically represents the original 4D system as shown in Fig. 1.) Since pure 3D gravity is known to be renormalizable [19,20], our result implies that the 4D Einstein gravity is renormalizable: renormalizability of the 4D gravity is achieved essentially by removing all of the unphysical degrees of freedom in the particular manner which we will describe in detail in the main body. Some of the analyses in this work have been repeated from different angles in [21–23]. It may be good to clearly state at this point the task undertaken in the present work. The equivalence between the usual formulation and ADM formulation of general relativity was questioned in [24]. In our view, it is a legitimate concern given, e.g., that the ADM formulation may not generally be applicable to an arbitrary spacetime but is most useful in dealing with a globally hyperbolic spacetime. The ADM formulation should be applicable to a “locally hyperbolic” spacetime 1 To our pleasant surprise, we recently have found the work of [17] where a Hamilatonain reduction led to lower dimensional volume. There also is an earlier related work [18] We will comment on these further in the conclusion. Fig. 1 a 4D scattering; b projection onto 3D hypersurface when the issue under consideration is local in nature such as renormalizability. Furthermore, once one takes the gauge in which the shift vector is set to zero, the analysis becomes essentially that of the usual formulation in the synchronous gauge. It is certainly possible to repeat all of the analysis in this work without referring to the ADM formulation but rather within the usual Lagrangian and Hamiltonian formulations with the synchronous gauge. Since gauge-fixing is a subtle issue in general, we clearly state the task undertaken in this work: quantization associated with the physical states dictated by the ADM formulation with the synchronous type gauge. The focus of the present work is the possibility that the alleged unrenormalizability may be due to the presence of unphysical degrees of freedom in the 4 (...truncated)