# Belinsky, Khalatnikov, Lifshitz ## Oscillatory approach to a singular point in the relativistic cosmology \[Links: [Inspire](https://inspirehep.net/literature/61185)\] # Gutzwiller ## Energy Spectrum According to Classical Mechanics \[Links: [DOI](https://doi.org/10.1063/1.1665328)\] \[Abstract: The phase integral approximation for the Green's function is investigated so as to yield an approximate expression for the density of states per unit interval of energy. This quantity is shown for negative energies (bound states) to depend only on the periodic orbits, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories. A particle in a periodic box of one, two, and three dimensions is discussed first to demonstrate how the approximate density of states contains a continuous background besides the δ‐function spikes of the discrete spectrum. Then we examine the situation in a spherically symmetric potential where special problems arise because the quasiclassical propagator has to be evaluated at a focal point of the classical trajectory. With the help of the Helmholtz‐Kirchhoff formula of diffraction theory, the amplitude is shown to remain finite at the focus. The orbits which remain entirely in a region of Coulombic potential yield a spectrum of Balmer terms with appropriately reduced degeneracy. However, the orbits which penetrate the screening charge give discrete levels obeying the Bohr‐Sommerfeld conditions with the correct degeneracy. The continuous background in the approximate density of states can be discussed on the basis of the formulas derived in this paper. This is necessary as an introduction to the problem of a particle in a potential where the motion is not multiply periodic.\] # Hawking, Penrose ## The Singularities of gravitational collapse and cosmology \[Links: [inspire](https://inspirehep.net/literature/55131)\] \[Abstract: A new [[0225 Singularity theorems|theorem on space-time singularities]] is presented which largely incorporates and generalizes the previously known results. The theorem implies that space-time singularities are to be expected if _either_ the universe is spatially closed _or_ there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface) _or_ there is a point _p_ whose past null cone encounters sufficient matter that the divergence of the null rays through _p_ changes sign somewhere to the past of _p_ (i. e. there is a minimum apparent solid angle, as viewed from _p_ for small objects of given size). The theorem applies if the following four physical assumptions are made: (i) Einstein’s equations hold (with zero or negative cosmological constant), (ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’), (iii) there are no closed timelike curves, (iv) every timelike or null geodesic enters a region where the curvature is not specially alined with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.) In common with earlier results, timelike or null geodesic incompleteness is used here as the indication of the presence of space-time singularities. No assumption concerning existence of a global Cauchy hypersurface is required for the present theorem.\] # Kinnersley, Walker ## Uniformly Accelerating Charged Mass in General Relativity \[Links: [DOI](https://doi.org/10.1103/PhysRevD.2.1359)\] \[Abstract: We discuss a type-{22} solution of the Einstein-Maxwell equations which represents the field of a uniformly accelerating charged point mass. It contains three arbitrary parameters $m$, $e$, and $A$, representing mass, charge, and acceleration, respectively. The solution is a direct generalization of the Reissner-Nordstrom solution of general relativity and the Born solution of classical electrodynamics. The external "mechanical" force necessary to produce the acceleration appears in the form of a timelike nodal two-surface extending from the particle's world line to infinity. This does not prevent us from regarding the solution as asymptotically flat and calculating the radiation pattern of its electromagnetic and gravitational waves. We find as well a maximal analytic extension of the solution and discuss its properties. Except for an extra "outer" Killing horizon due to the accelerated motion, the horizon structure closely resembles the Reissner-Nordstrom case.\] ## Refs - [[0336 C-metric]] # Kulish, Faddeev ## Asymptotic conditions and infrared divergences in quantum electrodynamics \[Links: [Inspire](https://inspirehep.net/literature/60986)\] \[Abstract: A definition which is free of [[0295 Infrared divergences in scattering amplitude|infrared divergences]] is proposed for the S matrix of a relativistic theory of interacting charged particles. This is achieved by a modification of the asymptotic condition and the introduction of a new space of asymptotic states. This state differs from the Fok space, but is separable and relativistically and gauge invariant. The mass operator has no nonvanishing discrete eigenvalues.\]