[Note: Since the creation of footnotes and cross references can be so tedious on this site, I shall go ahead and publish this version of this article without such, at first. However, I shall endeavor to add such in as I can find the time. I apologize for the lack of "polish" or completeness this will produce, at this time, but feel it will help me get this out to you in a more timely manner. (Especially, since otherwise I fear I may not be able to complete it in the time I have available.)]
Well, we have seen how, if we have a vector space (at least very locally) that has the additional structure of an inner or dot products (giving us an [pseudo-]inner-product space), that we obtain a (pseudo-)metric, and that there are invariant characteristics of such (pseudo-)metrics that can be distinguished by experimental means—thus leaving one to either accept or reject observational evidence for the nature of our particular space-time.
While there may well be some portions of this (admittedly mathematical) argument that could use some "shoring up", I suspect that the greatest barrier for this audience is one of "motivation": Why should we care about all this (mathematical) "mumbo-jumbo"?
I must apologize. I appear to have focused too much on getting the "rigorous" parts of the argument "in line", and failed to provide sufficient motivation for why any of this "matters".
If I ever attempt this again, I shall reverse the order of things: I shall endeavor to provide the motivation first, and only provide the rigor as the audience may demand.
So, in an attempt to, at least, partially make up for my blunders with this audience, let's, now, provide at least some of the answers to the question:
Why does any of this Matter?
In a very fundamental sense, Physics is about trying to explain the physical world (and, by extension, the Universe) around us to such an extent as we can predict the results of experiments/observations to as exacting a degree as possible. We call such "explanations" (as have successfully predicted myriad experiments/observations, so far) "Theories".
So, Physics is not just about experiments/observations, nor is it just about "neat" theories (explanations—often rather highly mathematical, admittedly): It is about the interplay between the two. (This is also why most theories [explanations] are highly mathematical: It tends to facilitate the generation of predictions that can be compared to the results of experiments/observations to as exacting a degree as possible.)
Therefore, when confronted with a question, such as the subject of these articles, of "What is the Geometry of Spacetime?" we must approach such from both perspectives as well.
I have, essentially, already approached this from the theory/explanation direction, ending with the point that one can determine "which explanation fits" our world/universe by experimental/observational means.
Now I shall try to proceed from the observational/experimental evidence (for all but the final determination of "which explanation fits") in my attempt to provide at least some of the motivation for the (mathematical) pieces that were used in the previous articles.
Why Vectors or Vector Spaces?
That is a very good question, though one that can be easily overlooked by those of us that have been "brought up" on such things.
Certainly the use of vectors and vector spaces (spaces where objects, called vectors, exist with the specific characteristics of what we call "vectors") in physics and mathematics is a somewhat modern concept, along with some of the (mathematical) notation we tend to use these days. However, the concept that certain physical "things" (like "force", "motion", "velocity", etc.) have both a "magnitude" (a measurable "amount" of something) and a "direction" (in some concept of "space") goes back at least as far back as Galileo, and was certainly used my Newton, even though the ability to express such concepts in a succinct notational form wasn't available until some time later.
Why do you suppose that is? Does such a notion come naturally to human beings in this world of ours? Does such seem to have utility in at least some aspects of trying to understand and explain phenomena in this world?
Yes. It had indeed been shown by observation that some phenomena had both "magnitudes" and "directions" that are important at least in their description (such as the motion of the wind).
What of other aspects of "vectors"? What of the ability to multiply a vector by an "amount" to get another vector that points in the same direction, but has a different "magnitude"? What about the ability to "add" two vectors, even if they don't have the same "direction", to get a new (net) vector quantity?
Consider actions like blowing on a toy boat while a wind is blowing, or having two oxen pulling a cart instead of one, or two people pushing on the same object while not having both pushing in the same direction?
I think, given enough thought about experiences in this world, around us, the reader will see that the (mathematical) characteristics of these mathematical entities called vectors do, actually, fit with careful observation of characteristics of "directional" phenomena we can experience. They aren't just mathematical abstractions, made up by "ivory tower" mathematicians for their amusement (and the bemusement of others). They are careful abstractions intended to provide a rigorous framework for working with such directional entities we experience.
Now, when we talk (rather glibly) about "direction" and "magnitude" of vectors, we should remember that such characteristics do not exist "in a vacuum", so to speak, when referring to directional phenomena all around us. I'm certain the reader is well aware that we can measure, almost independently, the direction of the wind vs. its magnitude (speed), or the direction of the sun vs. how bright it is shining.
We can even determine "angles" between the directions of very different directional phenomena (like where a clock tower is, vs. where the sun is in the sky, vs. the direction the wind is blowing, vs. the direction you are traveling on a road).
This brings up the next question:
Why Inner or Dot Products?
The Pythagoreans (followers of the philosopher Pythagoras) already had a "formula" for calculating a diagonal for two perpendicular lengths, long ago. It worked very well for any such case in their world (and ours). Of course, part of the "problem" is you have to be able to determine what "perpendicular" is.
Most of us think we know what "perpendicular" ("right angle") is (at least when we see it). Of course, the Pythagoreans had figured out how to construct such using tools they had readily available to them (straight edge and "compass" [drafting] or "dividers")
Years later, Euclid defined his geometry. Again, this wasn't just some mathematically abstract construction, by Euclid, for his amusement, but a careful abstraction designed (as far as Euclid knew) to bring mathematical rigor to the geometry he, and others, observed in the world around them, and us.
In fact, Euclid's geometry was considered to be so "intuitively true" to the physical geometry around us that once people constructed vectors (and spaces of such) it was typical to simply assume that one was referring to his geometry. Hence, the magnitude of a vector was "certainly" "always" to be given by the Pythagorean construction.
Even the idea that one could have vectors without being able to get the magnitude was not considered until more recent times (as mathematicians tore apart axiomatic systems, and rebuilt them so as to form hierarchies of such systems). This is why allowing an "inner product" to be anything other than "positive definite" (as required by Euclid's geometry) is not "strictly allowed", and is referred to using the (somewhat pejorative) moniker of pseudo-inner product.
However, in our case, we don't wish to artificially restrict what the answer to the question of "What is the Geometry of Spacetime?", so we do allow for pseudo-inner products, in addition to the positive definite inner products mandated by Euclid's geometry.
But Why did You Talk About "Curved" Spaces, etc.?
Well, again, this was me trying to be rigorous in generality. I suppose I must apologize once again.
All the above about vectors, Euclid's geometry, (pseudo-)inner products all assumed that the total geometry is that of a vector "space"—applicable over all distances, without limit.
However, again, observational/experimental evidence shows that this is only a good approximation at rather human scales (though as our instruments continue to become increasingly precise, we can measure deviations at ever shorter length scales). So, by only requiring the applicability of vectors, and (pseudo-)inner products of such vectors, to highly localized regions (even down to single mathematical points in spacetime), we have a construct that can even withstand these most modern of observations.
So, What are We Left With?
Therefore, the only experimental/observational evidence we have yet to match, in order to determine "What is the Geometry of Spacetime?", are those that determine the invariant characteristics of a metric constructed using the (pseudo-)inner product of the (potentially highly localized) vector space of local spacetime.
The fact that Euclid's geometry works so incredibly well, locally, over the spacial dimensions of spacetime is a very strong constraint. For four (4) space-time dimensions, this means that the three (3) dimensions of space must all act the same, and be a positive-definite (Euclidean) subspace of the whole. This means that even though we have four (4) dimensions, we really are only left with the freedom of a two dimensional space (one of time, and the other of space—all spacial dimensions together).
This ends up leaving us with three (3), and only three (3) possibilities for the (local) Geometry of Spacetime:
- Euclid's geometry over all (four ) dimensions of spacetime (Euclidean Spacetime, or simply Euclidean Space, since there is no distinction for time);
- The three (3) dimensions of Space is Euclidean, but there is no measure of "length" of time like there is for space—Space and Time are quite separate (Galilean/Newtonian Space and Time); or
- Space and Time is combined in a non-positive definite, though invertible way as "derived" by Einstein (Lorentzian/Einsteinian/Minkowskian Spacetime).
Now, all that remains is to compare these three possibilities to experiments/observations to determine which matches our world/universe best.