arXiv subjects
Publications and source records attributed to Peter Collas.
We derive exact solutions to the one-dimensional Schr\"odinger equation for compact support parabolic and hyperbolic secant potential barriers, along with combinations of these types of potential barriers. We give the expressions for transmission and reflection coefficients and calculate some dwell times of interest.
We discuss three ways of obtaining the Born approximations for Coulomb scattering: The standard way, making use of a convergence factor ("screening"), Oppenheimer's way using cylindrical (instead of spherical) coordinates, and finally Landau and Lifshitz' way. The last one although it does require some background from the theory of generalized functions is nevertheless a very instructive and important technique deserving more exposure to physicists.
In these informal lecture notes we outline different approaches used in doing calculations involving the Dirac equation in curved spacetime. We have tried to clarify the subject by carefully pointing out the various conventions used and by including several examples from textbooks and the existing literature. In addition some basic material has been included in the appendices. It is our hope that graduate students and other re- searchers will find these notes useful.
Using the Newman-Penrose formalism we calculate the positive energy momentum eigenstates of the Dirac equation for a plane polarized grav- itational wave pulse. We then consider Dirac particles whose spins are polarized in each orthonormal coordinate direction prior to the arrival of the gravitational wave pulse, and determine how the spins are changed by the pulse after its departure.
We show how to embed spacelike slices of the Schwarzschild wormhole (or Einstein-Rosen bridge) in R3. Graphical images of embeddings are given, including depictions of the dynamics of this nontraversable wormhole at constant Kruskal times up to, and beyond, the "pinching off" at Kruskal times \pm1.
We find exact Fermi coordinates for timelike geodesic observers for a class of spacetimes that includes anti-de Sitter spacetime, de Sitter spacetime, the constant density interior Schwarzschild spacetime with positive, zero, and negative cosmological constant, and the Einstein static universe. Maximal charts for Fermi coordinates are discussed.
An improved version has been submitted with the title: Recessional velocities and Hubble's law in Schwarzschild-de Sitter space. arXiv:1001.1875
This paper has been withdrawn by the authors, an improved version with the title "Exact Fermi coordinates for a class of spacetimes", arXiv:0912.2779, replaces it.
For spacetimes with timelike Killing fields, we introduce a "Fermi-Walker-Killing" coordinate system and use it to prove a Liouville Theorem for an appropriate volume element of phase space for a statistical mechanical system of particles. We derive an exact relativistic formula for the Helmholtz free energy of an ideal gas and compare it, for a class of spacetimes, to its Newtonian analog, derived both independently and as the Newtonian limit of our formula. We also find the relativistic thermodynamic equation of state. Specific examples are given in Kerr spacetime.
Examples of axially symmetric solutions to Einstein's field equations are given that exhibit anomalous "negative frame dragging" in the sense that zero angular momentum test particles acquire angular velocities in the opposite direction of rotation from the source of the metric.
We prove a theorem that gives an easily verifiable necessary and sufficient condition for a reference frame with fixed spacelike coordinates to be nonrotating in the sense of Walker. Applications are discussed.
We exhibit timelike geodesic paths for a metric, introduced by Bonnor [11] and considered also by Steadman [13], and show that coordinate time runs backward along a portion of these geodesics.
We calculate the transformation and inverse transformation, in the form of Taylor expansions, from arbitrary coordinates to Fermi-Walker coordinates in tubular neighborhoods of arbitrary timelike paths for general spacetimes. Explicit formulas for coefficients and the Jacobian matrix are given.
We use Fermi coordinates to calculate the canonical partition function for an ideal gas in a circular geodesic orbit in Schwarzschild spacetime. To test the validity of the results we prove theorems for limiting cases. We recover the Newtonian gas law subject only to tidal forces in the Newtonian limit. Additionally we recover the special relativistic gas law as the radius of the orbit increases to infinity. We also discuss how the method can be extended to the non ideal gas case.