RELATIVITY AND ACCELERATION.
First, an explanation. Why include a more or less standard treatment of a theory developed over 90 years ago? There are two reasons. Firstly, there still seems to be confusion in published articles and letters on the subject, particularly concerning the 'twins paradox' and other problems involving acceleration. It is still widely believed that acceleration can only be treated by the General Theory, and it may be a surprise to some readers that Special Relativity is perfectly adequate for this purpose, and can even be used via the principle of equivalence to predict the properties of a given gravitational field. The General Theory alone, however, gives the relationship between gravity and matter in the equation G = 8 pi T.
A simple treatment of acceleration may already be available somewhere, but I have yet to find it. For example, in the book 'Gravitation' by Misner, Thorne and Wheeler there is something similar to the extended reference frame for an accelerating observer developed here, but it goes under the name of 'a Fermi-Walker transported tetrad' and occurs in chapter 6 of a 1279 page book written at a fairly advanced level and likely to be understood only by specialists in this area.
The second reason is to try to present an unusual treatment of the subject, not necessarily sticking to the original formulation or the more common examples. Nowhere in the present series are there observers on trains watching flashes of light! Light is avoided almost completely to highlight the fact that the theory concerns the properties of space and time, and the properties of light are not of central importance.
I must make a confession, that I have read almost nothing written by Einstein. The early development of a theory may be interesting to many people, but is liable to contain errors. According to one account Einstein believed that two clocks at sea level would run at different rates if one is at the north pole and the other at the equator, but this can easily be shown to be false, as appears to be confirmed by experiment.
For now, here is a diagram of an extended reference frame which could in principle be constructed by an accelerating observer, drawn as seen by a non-accelerating observer. A full derivation of this can be found in standard text books such as 'Gravitation' but instead of this mathematical approach an explanation is attempted which gives some idea of the reason for this form of reference frame.
In general the effect of acceleration is zero when the accelerated object is observed from an inertial reference frame. It is only when trying to explain a situation from the point of view of an observer undergoing acceleration that it must be taken into account. An observer in an inertial reference frame can describe everything in terms of velocity using Special Relativity. This is done routinely in high energy physics where Special Relativity is used to analyse particles in collisions involving accelerations of more than 1030 earth gravities. It is nevertheless interesting and instructive to consider what the universe looks like to an observer undergoing acceleration, so this will be covered.