14 research outputs found
Path integration in relativistic quantum mechanics
The simple physics of a free particle reveals important features of the
path-integral formulation of relativistic quantum theories. The exact
quantum-mechanical propagator is calculated here for a particle described by
the simple relativistic action proportional to its proper time. This propagator
is nonvanishing outside the light cone, implying that spacelike trajectories
must be included in the path integral. The propagator matches the WKB
approximation to the corresponding configuration-space path integral far from
the light cone; outside the light cone that approximation consists of the
contribution from a single spacelike geodesic. This propagator also has the
unusual property that its short-time limit does not coincide with the WKB
approximation, making the construction of a concrete skeletonized version of
the path integral more complicated than in nonrelativistic theory.Comment: 14 page
Dynamics of Tachyonic Dark Matter
Usually considered highly speculative, tachyons can be treated via
straightforward Einsteinian dynamics. Kinetic theory and thermodynamics for a
gas of ``dark'' tachyons are readily constructed. Such a gas exhibits density
and pressure which, for the dominant constituent of a suitable
Friedmann-Robertson-Walker spacetime, can drive cosmic evolution with features
both similar to and distinct from those of a standard dark-energy/dark-matter
model. Hence, tachyons might bear further consideration as a cosmic dark-matter
candidate.Comment: 16 pages, 1 figur
Topics in Black-Hole Physics: Geometric Constraints on Noncollapsing, Gravitating Systems and Tidal Distortions of a Schwarzschild Black Hole
This dissertation consists of two studies on the general-relativistic theory of black holes. The first work concerns the fundamental issue of black-hole formation: in it I seek geometric constraints on gravitating matter systems, in the special case of axial symmetry, which determine whether or not those systems undergo gravitational collapse to form black holes. The second project deals with mechanical behavior of a black hole: specifically, I study the tidal deformation of a static black hole by the gravitational fields of external bodies.
In the first paper I approach the problem of geometric constraints determining gravitational collapse or non-collapse through the initial-value formalism of general relativity. I construct initial-value data representing noncollapsing, nonsingular, axisymmetric matter systems and examine the constraints imposed on this construction by the initial-value equation derived from the Einstein field equations. The construction consists of a nonsingular, momentarily static interior geometry with nonnegative mass-energy density, matched smoothly to a static, vacuum exterior geometry (described by a Weyl solution of the Einstein field equations) at a boundary surface. The initial-value equation is found to impose restrictions on the choice of the boundary surface for such a system. Two such constraints are obtained here, appropriate to spherical and toroidal interior-region topologies. These constraints are studied by applying them to simple examples of Weyl exterior geometries. The "hoop conjecture" for the general geometric-constraints problem states that a system must collapse to a black hole unless its circumference in some direction exceeds a lower bound of the order of the system's mass. The examples examined here show, however, that the constraints derived in this study are not generally correlated with any simple measure of system size, and thus that they do not embody the hoop conjecture.
The second paper examines the tidal distortion of a Schwarzschild black hole by bodies ("moons") suspended above the horizon on "ropes." A solution of the Einstein field equations is constructed describing this configuration, using the Weyl formalism for axisymmetric, static, vacuum geometries. The intrinsic geometry of the tidally deformed black-hole horizon is obtained from this solution; I construct embedding diagrams to represent the shape of the horizon and the tidal bulges raised on it for both weak and strong perturbations. The relations among the masses of the hole and moons, the binding energy of the system, and the rope density and tension are calculated from the solution and shown to be mutually consistent. Also, the Riemann curvature tensor representing the tidal fields near the horizon is calculated. This solution is found to agree with a previous calculation by Hartle of black-hole tides, in the limit of perturbing moons far from the horizon. In the opposite case of moons very near the horizon, this solution approaches the static limit of the distorted horizon in Rindler space calculated by Suen and Price. The results of this study thus support the use of the Rindler approximation to Schwarzschild spacetime in calculating static black-hole tides, and its extension to dynamical situations.</p
Localized Particle States and Dynamics Gravitational Effects
Scalar particles--i.e., scalar-field excitations--in de Sitter space exhibit
behavior unlike either classical particles in expanding space or quantum
particles in flat spacetime. Their energies oscillate forever, and their
interactions are spread out in energy. Here it is shown that these features
characterize not only normal-mode excitations spread out over all space, but
localized particles or wave packets as well. Both one-particle and coherent
states of a massive, minimally coupled scalar field in de Sitter space,
associated with classical wave packets, are constructed explicitly. Their
energy expectation values and corresponding Unruh-DeWitt detector response
functions are calculated. Numerical evaluation of these quantities for a simple
set of classical wave packets clearly displays these novel features. Hence,
given the observed accelerating expansion of the Universe, it is possible that
observation of an ultralow-mass scalar particle could yield direct confirmation
of distinct predictions of quantum field theory in curved spacetime.Comment: 12 pages, 5 figure
Black hole formation from colliding bubbles
Some indication of conditions that are necessary for the formation of black
holes from the collision of bubbles during a supercooled phase transition in
the the early universe are explored. Two colliding bubbles can never form a
black hole. Three colliding bubbles can refocus the energy in their walls to
the extent that it becomes infinite.Comment: 12 pages, NCL93-TP13 (RevTeX
Quantum Dynamics of Lorentzian Spacetime Foam
A simple spacetime wormhole, which evolves classically from zero throat
radius to a maximum value and recontracts, can be regarded as one possible mode
of fluctuation in the microscopic ``spacetime foam'' first suggested by
Wheeler. The dynamics of a particularly simple version of such a wormhole can
be reduced to that of a single quantity, its throat radius; this wormhole thus
provides a ``minisuperspace model'' for a structure in Lorentzian-signature
foam. The classical equation of motion for the wormhole throat is obtained from
the Einstein field equations and a suitable equation of state for the matter at
the throat. Analysis of the quantum behavior of the hole then proceeds from an
action corresponding to that equation of motion. The action obtained simply by
calculating the scalar curvature of the hole spacetime yields a model with
features like those of the relativistic free particle. In particular the
Hamiltonian is nonlocal, and for the wormhole cannot even be given as a
differential operator in closed form. Nonetheless the general solution of the
Schr\"odinger equation for wormhole wave functions, i.e., the wave-function
propagator, can be expressed as a path integral. Too complicated to perform
exactly, this can yet be evaluated via a WKB approximation. The result
indicates that the wormhole, classically stable, is quantum-mechanically
unstable: A Feynman-Kac decomposition of the WKB propagator yields no spectrum
of bound states. Though an initially localized wormhole wave function may
oscillate for many classical expansion/recontraction periods, it must
eventually leak to large radius values. The possibility of such a mode unstable
against growth, combined withComment: 37 pages, 93-
Is Quantum Spacetime Foam Unstable?
A very simple wormhole geometry is considered as a model of a mode of
topological fluctutation in Planck-scale spacetime foam. Quantum dynamics of
the hole reduces to quantum mechanics of one variable, throat radius, and
admits a WKB analysis. The hole is quantum-mechanically unstable: It has no
bound states. Wormhole wave functions must eventually leak to large radii. This
suggests that stability considerations along these lines may place strong
constraints on the nature and even the existence of spacetime foam.Comment: 15 page