Roh-Suan Tung

Hamiltonian boundary term and quasilocal energy flux

The Hamiltonian for a gravitating region includes a boundary term which determines not only the quasilocal values but also, via the boundary variation principle, the boundary conditions. Using our covariant Hamiltonian formalism, we found four particular quasilocal energy-momentum boundary term expressions; each corresponds to a physically distinct and geometrically clear boundary condition. Here, from a consideration of the asymptotics, we show how a fundamental Hamiltonian identity naturally leads to the associated quasilocal energy flux expressions. For electromagnetism one of the four is distinguished: the only one which is gauge invariant; it gives the familiar energy density and Poynting flux. For Einsteins general relativity two different boundary condition choices correspond to quasilocal expressions which asymptotically give the ADM energy, the Trautman-Bondi energy and, moreover, an associated energy flux (both outgoing and incoming). Again there is a distinguished expression: the one which is covariant.

Scalar field cosmology: I. Asymptotic freedom and the initial-value problem

The purpose of this work is to use a renormalized quantum scalar field to investigate very early cosmology, in the Planck era immediately following the big bang. Renormalization effects make the field potential dependent on length scale, and are important during the big bang era. We use the asymptotically free Halpern–Huang scalar field, which is derived from renormalization-group analysis, and solve Einstein’s equation with Robertson–Walker metric as an initial-value problem. The main prediction is that the Hubble parameter follows a power law: , and the universe expands at an accelerated rate: a ∼ exp t1 − p. This gives ‘dark energy’, with an equivalent cosmological constant that decays in time like t−2p, which avoids the ‘fine-tuning’ problem. The power law predicts a simple relation for the galactic redshift. Comparison with data leads to the speculation that the universe experienced a crossover transition, which was completed about seven billion years ago.

SCALAR FIELD COSMOLOGY II: SUPERFLUIDITY, QUANTUM TURBULENCE, AND INFLATION

We generalize the big bang model in a previous paper by extending the real vacuum scalar field to a complex vacuum scalar field, within the FLRW framework. The phase dynamics of the scalar field, which makes the universe a superfluid, is described in terms of a density of quantized vortex lines, and a tangle of vortex lines gives rise to quantum turbulence. We propose that all the matter in the universe was created in the turbulence, through reconnection of vortex lines, a process necessary for the maintenance of the vortex tangle. The vortex tangle grows and decays, and its lifetime is the era of inflation. These ideas are implemented in a set of closed cosmological equations that describe the cosmic expansion driven by the scalar field on the one hand, and the vortex–matter dynamics on the other. We show how these two aspects decouple from each other, due to a vast difference in energy scales. The model is not valid beyond the inflation era, but the universe remains a superfluid afterwards. This gives rise to observable effects in the present universe, including dark matter, galactic voids, nonthermal filaments, and cosmic jets.

Stationary untrapped boundary conditions in general relativity

A class of boundary conditions for canonical general relativity is proposed and studied at the quasi-local level. It is shown that for untrapped or marginal surfaces, fixing the area element on the 2-surface (rather than the induced 2-metric) and the angular momentum surface density is enough to have a functionally differentiable Hamiltonian, thus providing definition of conserved quantities for the quasi-local regions. If on the boundary the evolution vector normal to the 2-surface is chosen to be proportional to the dual expansion vector, we obtain a generalization of the Hawking energy associated with a generalized Kodama vector. This vector plays the same role for the stationary untrapped boundary conditions that the stationary Killing vector plays for stationary black holes. When the dual expansion vector is null, the boundary conditions reduce to those given by the non-expanding horizons and null trapping horizons.

Quasilocal energy flux of spacetime perturbation

A general expression for quasilocal energy flux for spacetime perturbation is derived from covariant Hamiltonian formulation using functional differentiability and symplectic structure invariance, which is independent of the choice of the canonical variables and the possible boundary terms one initially puts into the Lagrangian in the diffeomorphism invariant theories. The energy flux expression depends on a displacement vector field and the 2-surface under consideration. We apply and test the expression in Vaidya spacetime. At null infinity the expression leads to the Bondi type energy flux obtained by Lindquist, Schwartz, and Misner. On dynamical horizons with a particular choice of the displacement vector, it gives the area balance law obtained by Ashtekar and Krishnan.

Energy and angular momentum in strong gravitating systems

A quasilocal framework of stationary and dynamical untrapped hypersurfaces is introduced to generalize the notions of energy and angular momentum of isolated and dynamical trapping horizons to general strong gravitating systems.

The Hamiltonian boundary term

iopas120323 2 / 36 ● The Hamiltonian for interacting classical fields with quite general theories of dynamic geometry generates the evolution of a spatial region along a time-like vector field. ● It includes a boundary term which determines the value of the Hamiltonian. From this value one obtains the quasi-local quantities: energy-momentum, angular-momentum/center-of-mass. ● The Hamiltonian boundary term also directly controls the boundary term in the variation of the Hamiltonian. From the latter one obtains the associated built in boundary conditions and an expression for energy flux. ● Here we extend our preferred boundary term choice for Einstein’s GR (which we had identified in 2005) to select a unique boundary term expression for any dynamic geometry gravity theory along with interacting classical fields.