Alan Macdonald

An elementary construction of the geometric algebra

We give a simple, elementary, direct, and motivated construction of the geometric algebra overRn.

Universal One-Way Light Speed from a Universal Light Speed Over Closed Paths

This paper gives two complete and elementary proofs that if the speed of light over closed paths has a universal value c, then it is possible to synchronize clocks in such a way that the one-way speed of light is c. The first proof is an elementary version of a recent proof. The second provides high precision experimental evidence that it is possible to synchronize clocks in such a way that the one-way speed of light has a universal value. We also discuss an old incomplete proof by Weyl which is important from an historical perspective.

Clock synchronization, a universal light speed, and the terrestrial redshift experiment

This paper (i) gives necessary and sufficient conditions that clocks in an inertial lattice can be synchronized, (ii) shows that these conditions do not imply a universal light speed, and (iii) shows that the terrestrial red‐shift experiment provides evidence that clocks in a small inertial lattice in a gravitational field can be synchronized.

Einstein’s hole argument

In general relativity, space and time are inseparable from a gravitational field: no field, no spacetime. This is a lesson of Einstein’s hole argument. We use a simple transformation in a Schwartzschild spacetime to illustrate this. On the basis of the general theory of relativity … space as opposed to “what fills space” … has no separate existence. … There is no such thing as an empty space, i.e., a space without [a gravitational] field. … Spacetime does not claim existence on its own, but only as a structural quality of the field. Albert Einstein, 1952.

Comment on “The Cosmic Time in Terms of the Redshift,” by Carmeli et al.

The time-redshift relation of Carmeli et al. differs from that of the standard flat ΛCDM model by more than 500 million years for 1 ≤ z ≤ 4.5.

A new statement of the second law of thermodynamics

A new statement of the second law of thermodynamics is given. The law leads almost effortlessly, for very general closed systems, to a definition of absolute entropy S, a demonstration that ΔS≥0 in adiabatic processes, a definition of temperature, and a demonstration that dS≥δQ/T in quasistatic processes. Entropy is given a clear physical meaning.

Entanglement, Joint Measurement, and State Reduction

Entanglement is perhaps the most important new feature of the quantum world. It is expressed in quantum theory by the joint measurement formula. We prove the formula for projection valued observables from a plausible assumption, which for spacelike separated measurements is a consequence of causality. State reduction is simply a way to express the joint measurement formula after one measurement has been made, and its result known.

The fundamental theorem of geometric calculus via a generalized riemann integral

Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative ∇VF exists and is Lebesgue integrable. We also give sufficient conditions that ∇VF exists.

On the Marzke-Wheeler and Desloge constructions

There is no indication of time dilation of clocks or of length contraction of rods in Marzke and Wheelers clock or in Desloges metrosphere.

The form of magnetic work in a fundamental thermodynamic equation for a paramagnet

Magnetic work takes two forms in the thermodynamics of a paramagnet as developed in many textbooks. We observe that in the case when the lattice energy is excluded, the form δW=BdM cannot be used in a fundamental thermodynamic equation. This shows that there are thermodynamic systems with no fundamental thermodynamic equation.