44 research outputs found
Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients
This thesis describes the development of some basic mathematical tools of
wide relevance to mathematical physics. Transmission and reflection
coefficients are associated with quantum tunneling phenomena, while Bogoliubov
coefficients are associated with the mathematically related problem of
excitations of a parametric oscillator. While many approximation techniques for
these quantities are known, very little is known about rigorous upper and lower
bounds. In this thesis four separate problems relating to rigorous bounds on
transmission, reflection and Bogoliubov coefficients are considered, divided
into four separate themes:
1) Bounding the Bogoliubov coefficients;
2) Bounding the greybody factors for Schwarzschild black holes;
3) Transformation probabilities and the Miller--Good transformation;
4) Analytic bounds on transmission probabilities.Comment: 263 pages; 5 figures; PhD Thesis (Victoria University of Wellington,
June 2009
Bounding the greybody factors for scalar excitations of the Kerr-Newman spacetime
Finding exact solutions for black-hole greybody factors is generically
impractical; typically one resorts either to making semi-analytic or numerical
estimates, or alternatively to deriving rigorous analytic bounds. Indeed,
rigorous bounds have already been established for the greybody factors of
Schwarzschild and Riessner-Nordstrom black holes, and more generally for those
of arbitrary static spherically symmetric asymptotically flat black holes.
Adding rotation to the problem greatly increases the level of difficulty, both
for purely technical reasons (the Kerr or Kerr-Newman black holes are generally
much more difficult to work with than the Schwarzschild or Reissner-Nordstrom
black holes), but also at a conceptual level (due to the generic presence of
super-radiant modes). In the current article we analyze bounds on the greybody
factors for scalar excitations of the Kerr-Newman geometry in some detail,
first for zero-angular-momentum modes, then for the non-super-radiant modes,
and finally for the super-radiant modes.Comment: 22 page
Near-horizon geodesics for astrophysical and idealised black holes: Coordinate velocity and coordinate acceleration
Geodesics (by definition) have an intrinsic 4-acceleration zero. However,
when expressed in terms of coordinates, the coordinate acceleration can very easily be non-zero, and the coordinate velocity can
behave unexpectedly. The situation becomes extremely delicate in the
near-horizon limit---for both astrophysical and idealised black holes---where
an inappropriate choice of coordinates can quite easily lead to significant
confusion. We shall carefully explore the relative merits of
horizon-penetrating versus horizon-non-penetrating coordinates, arguing that in
the near-horizon limit the coordinate acceleration is best
interpreted in terms of horizon-penetrating coordinates.Comment: V1: 25 pages, 10 figures. V2: Now 28 pages; slight change in title
and abstract; some discussion and extra references added. Closely resembles
published versio
Buchdahl-like transformations for perfect fluid spheres
In two previous articles [Phys. Rev. D71 (2005) 124307 (gr-qc/0503007), and
gr-qc/0607001] we have discussed several "algorithmic" techniques that permit
one (in a purely mechanical way) to generate large classes of general
relativistic static perfect fluid spheres. Working in Schwarzschild curvature
coordinates, we used these algorithmic ideas to prove several
"solution-generating theorems" of varying levels of complexity. In the present
article we consider the situation in other coordinate systems: In particular,
in general diagonal coordinates we shall generalize our previous theorems, in
isotropic coordinates we shall encounter a variant of the so-called "Buchdahl
transformation", while in other coordinate systems (such as Gaussian polar
coordinates, Synge isothermal coordinates, and Buchdahl coordinates) we shall
find a number of more complex "Buchdahl-like transformations" and
"solution-generating theorems" that may be used to investigate and classify the
general relativistic static perfect fluid sphere. Finally by returning to
general diagonal coordinates and making a suitable ansatz for the functional
form of the metric components we place the Buchdahl transformation in its most
general possible setting.Comment: 23 page