Tevian Dray

The gravitational shock wave of a massless particle

The (spherical) gravitational shock wave due to a massless particle moving at the speed of light along the horizon of the Schwarzchild black hole is obtained. Special cases of our procedure yield previous results by Aichelburg and Sexl[1] for a photon in Minkowski vpace and by Penrose [2] for sourceless shock waves in Minkowski space. A new derivation of the (plane) shock wave of a photon in Minkowski space [1] involving explicit calculation of geodesics crossing the shock wave is also given in order to clarify the underlying physics. Applications to quantum gravity, specifically the possible effect on the Hawking temperature, are briefly discussed.

Junction conditions for null hypersurfaces

The authors derive an expression for the singular part of the stress-energy tensor on a hypersurface in spacetime in terms of the discontinuities in fundamental forms associated with the surface for both the non-null case, where the results are standard, and the null case. They then derive the minimum conditions which must be satisfied in order to glue two spacetimes together along such a hypersurface. In both cases, the essential requirement is only that the naturally induced (possibly degenerate) 3-metrics on the hypersurface must agree.

The octonionic eigenvalue problem

We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.

Particle production from signature change

We consider the (massless) scalar field on 2-dimensional manifolds whose metric changes signature and which admit a spacelike isometry. Choosing the wave equation so that there will be a conserved Klein-Gordon product implicitly determines the junction conditions one needs to impose in order to obtain global solutions. The resulting mix of positive and negative frequencies produced by the presence of Euclidean regions depends only on the total width of the regions, and not on the detailed form of the metric.

Detecting the rotating quantum vacuum

We derive conditions for rotating particle detectors to respond in a variety of bounded spacetimes and compare the results with the folklore that particle detectors do not respond in the vacuum state appropriate to their motion. Applications involving possible violations of the second law of thermodynamics are briefly addressed.

Scalar field equation in the presence of signature change

We consider the (massless) scalar field on a two-dimensional manifold with metric that changes signature from Lorentzian to Euclidean. Requiring a conserved momentum in the spatially homogeneous case leads to a particular choice of propagation rule. The resulting mix of positive and negative frequencies depends only on the total (conformal) size of the spacelike regions and not on the detailed form of the metric. Reformulating the problem using junction conditions, we then show that the solutions obtained above are the unique ones which satisfy the natural distributional wave equation everywhere. We also give a variational approach, obtaining the same results from a natural Lagrangian.

The effect of spherical shells of matter on the Schwarzschild black hole

Based on previous work we show how to join two Schwarzschild solutions, possibly with different masses, along null cylinders each representing a spherical shell of infalling or outgoing massless matter. One of the Schwarzschild masses can be zero, i.e. one region can be flat. The above procedure can be repeated to produce space-times with aC0 metric describing several different (possibly flat) Schwarzschild regions separated by shells of matter. An exhaustive treatment of the ways of combining four such regions is given; the extension to many regions is then straightforward. Cases of special interest are: (1) the scattering of two spherical gravitational “shock waves” at the horizon of a Schwarzschild black hole, and (2) a configuration involving onlyone external universe, which may be relevant to quantization problems in general relativity. In the latter example, only an infinitesimal amount of matter is sufficient to remove the “Wheeler wormhole” to another universe.

The Exceptional Jordan Eigenvalue Problem

We discuss the eigenvalue problem for 3 ×3 octonionic Hermitian matrices which is relevant to theJordan formulation of quantum mechanics. In contrast tothe eigenvalue problems considered in our previous work, all eigenvalues are real and solve theusual characteristic equation. We give an elementaryconstruction of the corresponding eigenmatrices, and wefurther speculate on a possible application to particle physics.

Angular momentum at null infinity

The definition of angular momentum recently given by Penrose (1982) is analysed on I. It is shown that this definition is essentially a supertranslation of previous definitions. The origin dependence of Penroses angular momentum is shown to have the correct form. Based on Penroses expression, the authors then define a momentum for all elements of the BMS Lie algebra, which is the first such expression with the property that its flux vanishes identically in Minkowski space.

Why is Ampère’s law so hard? A look at middle-division physics

Because mathematicians and physicists think differently about mathematics, they have different goals for their courses and teach different ways of thinking about the material. As a consequence, there are a number of capabilities that physics majors need in order to be successful that might not be addressed by any traditional course. The result is that the total cognitive load is too high for many students at the transition from the calculus and introductory physics sequences to upper-division courses for physics majors. We illustrate typical student difficulties in the context of an Ampere’s law problem.