C. K. Raju

Products and compositions with the Dirac delta function

The need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation, junction conditions and curved shock waves. Earlier definitions are briefly reviewed, and new definitions are proposed using non-standard analysis. Basic properties are established, and some products and compositions with the delta distribution are explicitly evaluated. With these definitions, the domain of validity of the nonlinear differential equations of classical field theory can be extended to include Rankine-Hugoniot equations are derived from the Euler equations. An immediate application to quantum field theory is pointed out.

Junction conditions in general relativity

An analytical formalism is developed to deal with the occurrence of jump discontinuities in the gmu nu or their derivatives across a hypersurface Sigma . It is shown that the equations of relativity remain meaningful at Sigma , even when Sigma does not inherit a unique intrinsic geometry, so that the gmu nu are discontinuous across Sigma in natural coordinates. The spherically symmetric surface layer at the Schwarzschild-Minkowski junction is used to illustrate these techniques, and to establish rigorously the existence of C0 solutions of the Einstein equations and the conservation equations. The possible validity of relativity at the microscopic level is examined, and it is concluded that, if relativity is valid at the microscopic level, then it is likely that the gmu nu are not globally continuously differentiable.

Classical time-symmetric electrodynamics

A brief review of the classical aspects of the absorber theory of radiation is presented. The divergences which arise from the use of time-symmetric electrodynamics are pointed out. It is shown that the earlier difficulties can be removed by attributing differing signal velocities to be advanced and retarded interactions. This difference in signal velocities is interpreted as arising from the extended, shell-like structure of charged particles. This leads to a new calculation of the absorber response. Absorption due to the time-symmetry normalisation factor is described. It is concluded that retarded radiation is approximately consistent in the Einstein-de Sitter model, whereas in the closed Friedmann model it is likely that retarded radiation is dominant during expansion, and advanced radiation during contraction. The theory predicts that advanced radiation exists in small amounts and can be detected experimentally.

Quantum-Mechanical Time

We present a brief exposition of the orthodox axiomatic approach to q.m., indicating the relation to the text-book approach. We explain why the usual axioms force a change of logic. We then explain the attempts to derive the Hilbert space and the probability interpretation from a new type of ‘and’ and ‘or’ or a new type of ‘if’ and ‘not’. Included are the Birkhoff-von Neumann, Jauch-Piron, and quantum logic approaches, together with an account of their physical and mathematical obscurities.

Probability in Ancient India

Publisher Summary A first course on probability (at the high-school level) typically begins with an account of the theory of permutations and combinations needed for calculating probabilities in games of chance, such as dice or cards. This theory of permutations and combinations first developed in India, although an account of its history is usually missing in stock presentations of combinatorics. The theory of permutations and combinations was basic to the Indian understanding of meter and music. From the earliest Vedic tradition, there is a continuous tradition linking the first accounts of permutations and combinations with those of Bhaskara II. Apart from the ability to work with large numbers, and to calculate permutations and combinations, and weighted averages, there is also needed the ability to work with fractions having large numerators and denominators. The traditional Indian understanding of mathematics, using zeroism, dispenses with the need for convergence, limits, or supertasks, and rehabilitates the frequentist interpretation of probability, in the sense that it provides a fresh answer to a long-standing philosophical difficulty in the Western tradition.

Retarded gravitation theory

We propose a Lorentz-covariant theory of gravity, and explain its theoretical origins in the problem of time in Newtonian physics. In this retarded gravitation theory (RGT), the gravitational force depends upon both retarded position and velocity, and the equations of motion are time-asymmetric retarded functional differential equations. We explicitly solve these equations, under simplifying assumptions, for various NASA spacecraft. This shows that the differences from Newtonian gravity, though tiny within the solar system, are just appropriate to explain the flyby anomaly as a νc effect due to earth’s rotation. The differences can, however, be large in the case of a spiral galaxy, and we show that the combined velocity drag from a large number of co-rotating stars enormously speeds up a test particle. Thus, the non-Newtonian behaviour of rotation curves in a spiral galaxy may be explained as being due to velocity drag rather than dark matter. RGT can also be tested in the laboratory. It necessitates a reappraisal of current laboratory methods of determining the Newtonian gravitational constant G. Since RGT makes no speculative assumptions, its refutation would have serious implications across physics.We propose a Lorentz-covariant theory of gravity, and explain its theoretical origins in the problem of time in Newtonian physics. In this retarded gravitation theory (RGT), the gravitational force depends upon both retarded position and velocity, and the equations of motion are time-asymmetric retarded functional differential equations. We explicitly solve these equations, under simplifying assumptions, for various NASA spacecraft. This shows that the differences from Newtonian gravity, though tiny within the solar system, are just appropriate to explain the flyby anomaly as a νc effect due to earth’s rotation. The differences can, however, be large in the case of a spiral galaxy, and we show that the combined velocity drag from a large number of co-rotating stars enormously speeds up a test particle. Thus, the non-Newtonian behaviour of rotation curves in a spiral galaxy may be explained as being due to velocity drag rather than dark matter. RGT can also be tested in the laboratory. It necessitates a re...