Session T15: Classical Gravity: Mathematical Formalism

The Dirac--Bergmann algorithm: Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance

The Dirac--Bergmann algorithm is a recipe for converting a singular Lagrangian system into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories---general relativity, electromagnetism, Yang Mills, string theory, etc. The Dirac--Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac--Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this work I analyze a finite--dimensional system that contains all of the major steps in the algorithm. The system includes primary and secondary constraints, first and second class constraints, restrictions on Lagrange multipliers, and both physical and gauge degrees of freedom. This relatively simple system provides a platform for discussing the Dirac conjecture, constructing Dirac brackets, and applying gauge conditions.    

An Invariant Characterization of the Levi-Civita Spacetimes

In the years 1917–1919 Tullio Levi-Civita published a number of papers presenting new exact solutions to Einstein’s equations. These works, while partially translated, remains largely inaccessible to English speaking researchers. Furthermore, the age and structures of these papers have often resulted in more contemporary works citing these papers in confusing or incorrect ways. It is not uncommon to find the series of Levi-Civita's papers cited by the general heading of the entire series or by referencing to only the first article in the series. Here, we clarify the structure of Levi-Civita's works on exact solutions that are of current interest. We review and characterize these solutions and their properties. We present them in a modern readable manner as well as show a new additional alternative form to one solution.   

Analysis of Levi-Civita Spacetimes using the Cartan-Karlhede Algorithm

Here, a brief introduction to the Cartan-Karlhede algorithm is given. This algorithm allows for the (local) characterization of spacetimes and is applied to several of the Levi-Civita spacetimes. This method allows for the identification of the limits of several distinct spacetimes as a single solution, as well as uniquely characterize these solutions as a reference for future work.   

Local and covariant flow relations for OPE coefficients in Lorentzian spacetimes

As their spacetime points approach coincidence, the n-point functions of local quantum field theories can be approximated to arbitrary precision by their operator product expansions (OPEs). The coefficients of OPEs are c-number distributions which contain important information about both the quantum fields and the physical states. Under variations of the interaction parameters, Hollands et al. have shown the OPE coefficients of renormalizable Euclidean QFTs satisfy "flow equations". These Euclidean flow equations have been proven to hold order-by-order in perturbation theory, but remain mathematically well defined under very general assumptions for any value of the interaction parameters.  The flow equations, therefore, potentially provide a non-perturbative approach to obtaining OPE coefficients. However, there exist serious obstacles to deriving flow relations for OPE coefficients on Lorentzian spacetimes in a manner compatible with locality and covariance. In this talk, I describe these issues and our resolutions to them for a solvable toy model: Klein-Gordon theory with the mass viewed as an "interaction parameter". Our approach to obtaining local and covariant flow relations for the Klein-Gordon OPE coefficients on Lorentzian spacetimes generalizes to interacting QFTs.   

Boundary conditions and proof of the scalar, vector, tensor decomposition theorem in cosmological perturbation theory

Cosmological perturbations play a vital role in the study of the anisotropy of the cosmic microwave background radiation and in large-scale structure formation. The standard procedure for treating these perturbations is to decompose them into scalar (S), vector (V) and tensor (T) components, and assume that the three sectors independently satisfy the fluctuation equations. Ordinarily, this procedure is carried out in a convenient gauge  in a background geometry with vanishing spatial 3-curvature. We have carried out a general SVT decomposition in a cosmology with an arbitrary spatial 3-curvature and expansion radius in a completely gauge invariant way that involves no choice of gauge at all. We have been able to prove the decomposition theorem in this general case, by first manipulating the fluctuation equations so that the various sectors separate out at a higher derivative level; and then finding appropriate boundary conditions under which the solving of these equations leads to the form that is required of  the decomposition theorem. With these boundary conditions we thus justify the use of the decomposition theorem in the standard Einstein gravity based cosmology. In addition we establish the decomposition theorem in the alternate conformal gravity theory.   

The geometry of field sheet foliations of spacetime in force-free electrodynamics

In this talk, I will present the geometry of foliations of a spacetime whose distributions are kernels of a force-free electromagnetic field. Force-free electrodynamics (FFE)  in curved spacetime is the natural setting for black hole magnetospheres. In particular, the Blandford-Znajek mechanism is an example of FFE. I will conclude the talk by presenting strategies for obtaining exact solutions for FFE.   

Quantum Corrections to Classical Time Dilation

At the intersection of quantum mechanics and relativity lies the possibility for a clock to move along a superposition of two distinct classical trajectories — perhaps these trajectories correspond to different speeds or locations in a gravitational field of the clock. It is then natural to ask: what time dilation would such a quantum clock observe? Using covariant time observables, I will introduce a formulation of relational quantum dynamics that allows for a probabilistic formulation of relativistic time dilation. This framework will then be used to describe quantum time dilation effects that occur when a clock moves in a superposition of different relativistic momenta and is at rest in a spatial superposition of an external gravitational field. I will argue that these time dilation effects may be observable with present-day technology and offer a new test of fundamental physics in the regime where quantum coherence and relativistic effects play an important role.