The Semi-Symmetric Metric Connection series

Part I – The Background
The intention is to describe the work I did for many years (starting about 1989/1990 or so) on the semi-symmetric metric connection (SSMC) as a possible extension to the (Riemannian) geometry of GR, which was done in order to see whether it might be able to model the addition of the electromagnetic field to GR in a geometrically-unified way, given that the SSMC adds only a single new object to Riemann geometry – to wit, an object of precisely a most very-highly suggestive form, namely, a 4-vector. I spent a fair bit of time trying to nut this idea out over the years, but I have never quite got there… It always seems as though there is some clever trick or an important insight that is dangling just beyond reach… I hope that by making these explorations public and open – and having to clarify and explain to others what I had tried to do all those years ago (and since) – it might lead to someone else examining these ideas and might possibly nudge them to have a try, and yield a coherent mathematical theory which, one hopes, could be tested – both for internal consistency and for empirical validity.
Part II: Mathematical Preliminaries
This post establishes some basic definitions for some mathematical operations that will be required in the next post (Part III) where we will examine the geometry of the SSMC as a precursor to seeking field equations that may be implicitly contained in the geometry. I will work using the component notation (as opposed to the more elegant component-free form) for tensors, as well as working within a co-ordinate frame (as opposed to non-coordinate frames) which simplifies the number of symbols used.
Part III: The Geometry
OK, with the mathematical preliminaries suitably dealt with, let’s go…
As noted previously in Part II, we’ll be using the tensor component notation and assuming a co-ordinate basis (together with the misuses of geometrically precise language that comes with that choice; my bad).
Part IV: General Relativity
With the underlying geometry of the space defined by the semi-symmetric metric connection (SSMC) having been explored, we’re now in a position to examine how Einstein derived his field equations for GR. We will be seeking to follow similar physically-motivated reasoning, such as he used for GR, in our search for candidate field equations which might add electromagnetism to GR based on the geometrical properties of the SSMC.
Part V: Testing General Relativity (pending…)
In this fairly brief Part, we examine the many tests which GR has passed over the last century since it was proposed – all with flying colours, so far. This is done to remind ourselves not only of the magnitude of Einstein’s achievement, but also the limits of what can be claimed by the theory. In physics, we always need to confront theory with experiment. And the degree and the extent to which theory succeeds is the degree and extent to which we tentatively accept the theory as an approximately-correct description of ‘reality’ (whatever that is). This is especially important when seeking to extend a theory (as is being attempted here) because the proposed extension should pay very careful attention to the degree to which the initial theory has succeeded in confronting experiment. And therefore, we might add as a logical corollary, no more than that, until such time as new experiments extend the scope of what can be claimed.
Part VI: Towards Field Equations (pending…)
Following the preparatory work done in previous posts, we’re now in a position to begin the search for field equations for a geometrically-unified theory of gravitation and electromagnetism based on the semi-symmetric metric connection (SSMC). We are motivated by a primary appeal to physics as our guiding principle, as opposed to more usual formal methods. Those formal methods, while they proved immensely useful post hoc for General Relativity, might not be so helpful in the search for field equations, as was the case with Einstein’s own attempt at a unified field theory.
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