Many people struggle with, and even rail against, Einstein's Special Theory of Relativity. The way it is usually taught or presented often seems to make it appear to be ever so complex, far too abstract and opaque, and even downright "hokey".* My experience certainly allows me full empathy for such struggles.

However, through my journey through these struggles, I did find the kernel, the "missing piece", even the "gem", so to speak, behind the theory. I suspect that this kernel may be what is missing in the understanding of those that struggle with, or, maybe, even rail against, Einstein's Special Theory of Relativity. I know I could have easily fallen into this category.

Through this (hopefully rather short) series I shall endeavor to illuminate this kernel, and, hopefully, develop a better approach to introducing Einstein's Special Theory of Relativity.

First, though, I wish to share with you my journey with this theory, since I suspect many who struggle, and maybe even many who rail against, this theory may be able to identify with, at least, how my journey began.

# My Journey: A Confession

### First Exposure, and Impressions

I first learned about Einstein's Special Theory of Relativity from a book I read on Einstein and his theories in Junior High School (about 7^{th} or 8^{th} grade, what is sometimes termed "Middle School"). The explanation involved such things as "time dilation", "space contraction", "Lorentz transformations", and a strange way to "add" velocities—all stuff that is, I'm sure, quite familiar to any that have learned of the subject.

My immediate feeling was "What?!? This can't possibly be the way the universe *really* works!" While I didn't doubt that all this "stuff" actually matched the experimental observations, I was *sure* there had to be a better explanation: There had to be something happening "under the hood", so to speak, that was the *real* reason behind these appearances.

Now, unlike most of the "crackpots" and "cranks" out there (no offense intended to any), I knew that if I was going to "get to the bottom of this" I was going to have to learn all I could about all aspects of Physics, and about as much of the experimental evidence, and the nature of the experiments as I possibly could. That's when I decided to get a Ph.D. in Theoretical Physics. (Back in the fifth or sixth grade I had already decided that since I was good at both mathematics and science I was going to be a physicist. I'm sure that my dad being a Nuclear Physicist had some influence as well. :) He never tried to push any of us [my brothers and I] into any particular field, but he did expect excellence.)

### High School and Undergraduate Physics: No Change

Later, in High School, taking a Physics class, where we could progress at our own pace, I and one other student went all the way through the lesson packets, all the way to Special Relativity.

Once again, it was "time dilation", "space contraction", "Lorentz transformations", and a strange way to "add" velocities. The only substantive difference was that I learned how all this could be derived from the Lorentz Transformations. However, none of this changed my original assessment of the "hokey" theory. Oh, sure, everything "worked", some (most?) of it seemingly by "magic" or pure chance, but what was the *reality*?

Still later, I'm off in college, learning Physics, Mathematics, etc. On my way toward a Ph.D. in Theoretical Physics. I devoured my Physics classes, took more Mathematics and Computer Science classes than required (getting Minors in both Mathematics and Computer Science). All the while with a "bifurcated mind", as I call it: While I couldn't take any of the Physics at "face value", since I was questioning it all, I still had to learn it well, since there would be no other way I could see to solve this "mystery"!

Again, another class on "Modern Physics": Special Relativity and Quantum Mechanics. Once again, Special Relativity was taught in essentially the very same way, with all the same "players". Oh, sure, we spent a lot more time on the subject, and did a lot more exercises, but no further insights.

I think Minkowski spacetime was introduced, and spacetime invariants, and such, but not one single thing that showed any greater "light" upon the subject. I was still left with my dilemma, and my "bifurcated mind".

### Graduate School: No Change, 'till General Relativity

OK. Skipping to graduate school, actually taking my courses toward my Ph.D. I was still of a "bifurcated mind"—that's a long time to have to keep that up, basically having to do "double duty" in all my Physics classes!

Then, I get into the class on General Relativity: Everything changed! The "shackles" fell away, and I was free, at last!

Now, what made the difference? True, there are a lot of very new concepts in General Relativity: Curved spaces, Manifolds, general coordinates and general coordinate transformations, Tangent spaces and their dual spaces, Differential Geometry, etc. etc. etc. So what changed? What did it?

The difference was I finally learned a very general and fundamental concept pertaining to any space where one can "measure" lengths and angles: The "metric", AKA the inner/dot product.

Hadn't I learned about the dot and/or inner product before, way back in vector calculous, and abstract algebra classes? Sure, but they always were "positive definite"! They were always only applicable to Euclidean Geometry!

I knew that such things (leading to the Pythagorean Theorem, and such) were fundamental to Euclidean Geometry. What opened my mind was the introduction of a different kind of "metric", inner/dot product: One that is not positive definite, but indefinite (a so called "pseudo-metric").

Everything else proceeded, absolutely naturally, from this extraordinarily simple concept, this most tiny little "change" to what I already knew about (Euclidean) Geometry. This was what was "happening 'under the hood', so to speak, that was the *real* reason behind these appearances." Since all other choices of this fundamental piece of Geometry lead to different "appearances", the only choice left was to accept whatever one matches Nature—the character of the Universe around us: Science must accept however the Universe shows us it works, regardless of how "odd" we may think it to be—it is what it is.

### The Alternate Teaching Idea, for Einstein's Special Theory of Relativity

Some time later, while still working on my Ph.D. (though I was all done with course work, and just working on my Dissertation) I attended a Colloquium given by a visiting professor. He presented an alternative approach to teaching Einstein's Special Theory of Relativity. A method he found to be the easiest, with the highest success rate—in terms of students actually "getting it".

The method involved introducing the metric (AKA inner/dot product) in an explicit way, since it is usually rather invisible in much of Euclidean Geometry, because it is the simple identity (when using the usual othonormal coordinate representations). Once the students were comfortable with the metric, he would show it in its most general form, without imposing the usual positive definiteness restriction of Euclidean Geometry.

The next step is to enumerate all the possible non-equivalent forms such a metric can take (really a very small set, indeed), and show how each form has distinguishable characteristics.

All that then remains (before getting into the more usual aspects of Special Relativity) is to determine which of the small number of possible forms match the Universe in which we dwell—since the Universe "is what it is", regardless of our "feelings" on the matter. After all, the purpose of science is to find out what this Universe is *actually* like, regardless of any desires, on our part, for what we may think the Universe *should* be like.

I recognized this as what I had needed from the beginning (though I had found it on my own, and probably benefited all the more for having gone through my own personal journey).

# The Road Forward

### What I'm Going to Attempt With This Series

I have tried, in times past, to find teaching materials or web sites that present Einstein's Special Theory of Relativity in this "metric centric" manner. I wish I knew who this professor was (it may be in my notes, somewhere, in some box, in the shed, being eaten by silverfish). So, if any readers know of some material on this subject, or anyone that teaches like this, I would be most grateful for any pointers.

So, lacking any such materials, I have actually attempted to teach someone (a Special Relativity contrarian) using something akin to this "metric centric" approach. Unfortunately, either due to my own imperfect methods, or something else, I haven't succeeded—at least not with that individual, yet.

Partly as an attempt to refine my approach, and partly to see if anyone, here, can point me to better methods and/or materials, and, more specifically and immediately, due to the "prodding" and encouragement of friends here on Science 2.0, I'm going to attempt to present a "metric centric" approach to Einstein's Special Theory of Relativity.

How far we go, and how quickly we proceed will depend upon available time, and reader interest and feedback.

I will do my best, and I hope I receive a lot of good, constructive feedback, so we can make this the best attempt possible.

So, next time, we shall take a closer look at vector spaces and the inner/dot product in such spaces (so called inner product spaces).

* More recently, I have actually seen a far more opaque, overly formal, and downright "hokey" approach to Einstein's Special Theory of Relativity. :-{ So I no longer consider the more usual approach to the subject to be the worst. :-/

Articles in this series:

What is the Geometry of Spacetime? — Introduction (this article)

What is the Geometry of Spacetime? — What is Space? — Inner-Product Spaces (next article)

What Is The Geometry Of Spacetime? — What Kinds Of Inner-Products/Metrics? (third article)

Yeah!

This sounds like it will be a fun series.

I think most people find the geometric view much easier once they learn it. I'd be curious if it is a good place to start though. Interesting idea.

I always had trouble looking back at SR after seeing some GR. When SR is taught we are often given 'vectors' like (ct,x,y,z) and work out the proper time or whatnot between two events. But then in GR we learn that the geometric vectors are really in a tangent space at each point. This is easy for me to visualize for a vector field, but while I can (usually) do the math fine I don't quite get why the stuff we did before with "vectors" made with coordinate labels in SR actually worked. How is anything involving just coordinates of derivatives with respect to coordinates actually a vector? Is the velocity four-vector even a real vector then? If in GR we can always choose our coordinates to have the metric at a point be just diagonal (-1,1,1,1), how and why exactly does this affect the coordinate form of the metric at neighboring points if each tangent space is separate?

The more I think about it, it seems more "natural" to have the metric be diagonal (-1,1,1,1) for all the tangent spaces, and instead describing geometry with how vectors warp while parallel transporting around. I don't know how to make this mathematical, but I've heard this is essentially how the vierbein (or tetrad or frame field) way of writing GR works. I've always wanted to ask you about this, but was worried Doug might see it as a way to pursue generalizing his way of abusing quaternions to get the spacetime dot product to work in curved spacetime.