Stress-energy tensor in colliding plane wave space-times: An approximation procedure

In a recent work on the quantization of a massless scalar field in a particular colliding plane wave space-time, we computed the vacuum expectation value of the stress-energy tensor on the physical state which corresponds to the Minkowski vacuum before the collision of the waves. We did such a calculation in a region close to both the Killing-Cauchy horizon and the folding singularities that such a space-time contains. In the present paper, we give a suitable approximation procedure to compute this expectation value, in the conformal coupling case, throughout the causal past of the center of the collision. This will allow us to approximately study the evolution of such an expectation value from the beginning of the collision until the formation of the Killing-Cauchy horizon. We start with a null expectation value before the arrival of the waves, which then acquires nonzero values at the beginning of the collision and grows unbounded towards the Killing-Cauchy horizon. The value near the horizon is compatible with our previous result, which means that such an approximation may be applied to other colliding plane wave space-times. Even with this approximation, the initial modes propagated into the interaction region contain a function which cannot be calculated exactly and to ensure the correct regularization of the stress-energy tensor with the point-splitting technique, this function must be given up to adiabatic order four of approximation.Comment: 27 pages, Latex file plus three figures in PostScrip

Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

Particle creation in a colliding plane wave spacetime: wave packet quantization

We use wave packet mode quantization to compute the creation of massless scalar quantum particles in a colliding plane wave spacetime. The background spacetime represents the collision of two gravitational shock waves followed by trailing gravitational radiation which focus into a Killing-Cauchy horizon. The use of wave packet modes simplifies the problem of mode propagation through the different spacetime regions which was previously studied with the use of monocromatic modes. It is found that the number of particles created in a given wave packet mode has a thermal spectrum with a temperature which is inversely proportional to the focusing time of the plane waves and which depends on the mode trajectory.Comment: 23, latex, figures available by fa

Stress-energy tensor in the Bel-Szekeres space-time

In a recent work an approximation procedure was introduced to calculate the vacuum expectation value of the stress-energy tensor for a conformal massless scalar field in the classical background determined by a particular colliding plane wave space-time. This approximation procedure consists in appropriately modifying the space-time geometry throughout the causal past of the collision center. This modification in the geometry allows to simplify the boundary conditions involved in the calculation of the Hadamard function for the quantum state which represents the vacuum in the flat region before the arrival of the waves. In the present work this approximation procedure is applied to the non-singular Bel-Szekeres solution, which describes the head on collision of two electromagnetic plane waves. It is shown that the stress-energy tensor is unbounded as the killing-Cauchy horizon of the interaction is approached and its behavior coincides with a previous calculation in another example of non-singular colliding plane wave space-time.Comment: 17 pages, LaTex file, 2 PostScript figure

On the determination of the leptonic CP phase

The combination of data from long-baseline and reactor oscillation experiments leads to a preference of the leptonic CP phase $\delta_{\rm CP}$ in the range between $\pi$ and $2\pi$. We study the statistical significance of this hint by performing a Monte Carlo simulation of the relevant data. We find that the distribution of the standard test statistic used to derive confidence intervals for $\delta_{\rm CP}$ is highly non-Gaussian and depends on the unknown true values of $\theta_{23}$ and the neutrino mass ordering. Values of $\delta_{\rm CP}$ around $\pi/2$ are disfavored at between $2\sigma$ and $3\sigma$, depending on the unknown true values of $\theta_{23}$ and the mass ordering. Typically the standard $\chi^2$ approximation leads to over-coverage of the confidence intervals for $\delta_{\rm CP}$. For the 2-dimensional confidence region in the ($\delta_{\rm CP},\theta_{23}$) plane the usual $\chi^2$ approximation is better justified. The 2-dimensional region does not include the value $\delta_{\rm CP} = \pi/2$ up to the 86.3\% (89.2\%)~CL assuming a true normal (inverted) mass ordering. Furthermore, we study the sensitivity to $\delta_{\rm CP}$ and $\theta_{23}$ of an increased exposure of the T2K experiment, roughly a factor 12 larger than the current exposure and including also anti-neutrino data. Also in this case deviations from Gaussianity may be significant, especially if the mass ordering is unknown.Comment: 25 pages, 12 figures. Matches version which is to appear in JHEP. New appendix with the first anti-neutrino results from T2K is adde