Cormac Breen

Mode-Sum Prescription for Vacuum Polarization in Black Hole Spacetimes in Even Dimensions

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function f(r). As an explicit example of our method, we show plots for vacuum polarization of a massless scalar field in the Schwarzschild-Tangherlini spacetime for even d = 4, ..., 10. However, the method presented applies straightforwardly to massive fields or to nonvacuum spacetimes.

Hadamard Renormalization of the Stress Energy Tensor on the Horizons of a Spherically Symmetric Black Hole Space-Time

We consider a quantum field which is in a Hartle-Hawking state propagating in a general spherically symmetric black hole space-time. We make use of uniform approximations to the radial equation to calculate the components of the stress tensor, renormalized using the Hadamard form of the Green’s function, on the horizons of this space-time. We then specialize these results to the case of the ‘lukewarm’ Reissner-Nordstrom-de Sitter black hole and derive some conditions on the stress tensor for the regularity of the Hartle-Hawking state.

Mode-Sum Prescription for the Vacuum Polarization in odd Dimensions

We present a new mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in odd dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher dimensions. Remarkably, the regularization parameters can be computed in closed form in arbitrary dimensions and for arbitrary metric function f(r). In fact, we show that inspite the increasing severity and number of the divergences to be regularized, the method presented is mostly agnostic to the number of dimensions. Finally, as an explicit example of our method, we show plots for vacuum polarization in Schwarzschild-Tangherlini spacetime for odd d = 5, ..., 11.

Vacuum Polarization on the Brane

We compute the renormalized expectation value of the square of a massless, conformally coupled, quantum scalar field on the brane of a higher-dimensional black hole. Working in the AADD brane-world scenario, the extra dimensions are flat and we assume that the compactification radius is large compared with the size of the black hole. The four-dimensional on-brane metric corresponds to a slice through a higher-dimensional Schwarzschild-Tangherlini black hole geometry and depends on the number of bulk space-time dimensions. The quantum scalar field is in a thermal state at the Hawking temperature. An exact, closed-form expression is derived for the renormalized expectation value of the square of the quantum scalar field on the event horizon of the black hole. Outside the event horizon, this renormalized expectation value is computed numerically. The answer depends on the number of bulk space-time dimensions, with a magnitude which increases rapidly as the number of bulk space-time dimensions increases.

Extended Green-Liouville asymptotics and vacuum polarization for lukewarm black holes

We consider a quantum field on a lukewarm black hole space-time. We introduce a new uniform approximation to the radial equation, constructed using Green Liouville asymptotics. We then use this new approximation to construct the renormalized vacuum polarization in the Hartle-Hawking vacuum. Previous calculations of the vacuum polarization rely on the WKB approximation to the solutions of the radial equation, however the non-uniformity of the WKB approximations obscures the results of these calculations near both horizons. The use of our new approximation eliminates these obscurities enabling us to obtain explicitly finite and easily calculable values of the vacuum polarization on the two horizons.

Investigating secondary students beliefs about mathematical problem-solving

ABSTRACT Many studies over the past 30 years have highlighted the important role of students’ beliefs for successful problem-solving in mathematics. Given the recent emphasis afforded to problem-solving on the reformed Irish secondary school mathematics curriculum, the main aim of this study was to identify Irish students’ (n = 975) beliefs about the field. A quantitative measure of these beliefs was attained through the use of the Indiana Mathematical Belief Scale, an existing 30-item (five-scale) self-report questionnaire. A statistical analysis of the data revealed that students who were further through their secondary education had a stronger belief that not all problems could be solved by applying routine procedures. In contrast, the same students held less positive beliefs than their younger counterparts that they could solve time-consuming problems and that conceptual understanding was important. The analysis also indicated that gender had a significant impact on three of the five belief scales.