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    What Is The Geometry Of Spacetime? — Wrap-Up: Why Does Any Of This Matter?
    By David Halliday | December 30th 2013 02:36 PM | 20 comments | Print | E-mail | Track Comments
    I'm sorry I've been away so long:  The "press of life" got the "better of me".  Sometimes, life just "gets in the way".  (No excuses.  I simply had other "things" that needed to take priority.  Unfortunately, I don't see that changing any time soon.  So, unfortunately, I cannot promise to have an especially active presence here for the foreseeable future.)

    [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.)]

    Review

    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?

    Fair enough.

    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:

    1. Euclid's geometry over all (four [4]) dimensions of spacetime (Euclidean Spacetime, or simply Euclidean Space, since there is no distinction for time);
    2. 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
    3. 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.

    Comments

    Bonny Bonobo alias Brat
    Now, all that remains is to compare these three possibilities to experiments/observations to determine which matches our world/universe best.
    What a great article David, you've explained everything brilliantly! Now all those past articles about vectors and dot products make a lot more sense to me. I hope the press of life eases for you soon and we can now go on to determine which of these three possibilities best suits our universe. Happy New Year!
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Halliday
    Thank you, Helen.

    "Determin[ing] which of these three possibilities best suits our universe" is actually an "exercise left to the reader".  However, some hints:

    1. The "Galilean/Newtonian Space and Time" and the "Lorentzian/Einsteinian/Minkowskian Spacetime" should both be eminently familiar to most readers, here.  The first is the "Space and Time" of good old classical Newtonian physics (with its universal time, and unlimited velocities, including the instantaneous propagation of Newton's Gravity).  The second is the Spacetime of Einstein's Special and General Theories of Relativity (complete with the universal speed of only one type of "thing" [which we call "light", or "luxons", or anything that has no "rest" frame of reference]).
    2. The first one, "Euclidean Space", is simply a four (4) dimensional version of what our three (3) dimensions of Space appear to be like (at least locally).  It has no velocity that is unchanged by transformations, in that space, since all transformations (that preserve the Metric) are simply rotations.  In fact, since all rotations are allowed, there is no "causality" of any kind that observers could agree upon.  The truth is that we could move as freely in whatever direction you may wish to label "Time" as we could move in all other directions.  No problem with time travel in this kind of "Spacetime", since all directions/dimensions are simply "Space".

    So, do you know of any evidence that can be used to choose between the three possibilities?

    David

    Bonny Bonobo alias Brat
    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).

    David, I think you have done a very good job of showing us some of the basic geometry of spacetime and why vectors or vector spaces and inner dot products can be so useful for this spacetime geometry and predictive calculations and why it must include curved, very tiny or very large spaces and times. However I can't help feeling that there are potentially still infinitely more forces that may interact in potentially infinitely more dimensions and manners that we still don't know or understand and therefore our spacetime geometry in an infinite universe will need to constantly add more formulae and/or tools to the predictive equations as we discover them. Also if spacetime constants turn out not to be constants throughout the universe that will add another confusing variable or dimension to the equation making it even more messy and potentially totally chaotic. Wouldn't it be like trying to measure the butterfly effect in chaos theory with the butterfly's wing?

    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:
    1. Euclid's geometry over all (four [4]) dimensions of spacetime (Euclidean Spacetime, or simply Euclidean Space, since there is no distinction for time);
    1. 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
    1. 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.

    You say that you 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 however none of these possibilities are perfect for every situation. Negative or 'pseudo' inner dot products and negative or reverse time for example are nice in theory as you have shown but not in practice and how can you geometrically explain and predict such things as instantaneous catalystic chemical interactions and instantaneous spooky action and entanglement at a distance in quantum mechanics using spacetime geometry? 

    If we simply imagine a drop of radioactive, highly coloured and chemically concentrated rainwater falling into a massive, moving, salty ocean then in theory spacetime geometry should be able to accommodate and calculate this instantaneous effect or force in every direction upon every wave and shore and ocean bed and even upon every other atom and/or chemical in that ocean and somehow I doubt if that will ever be possible simply because of the myriads of interactive factors and/or forces involved including gravity, sunlight, wind, moonlight, heat, cold, land and water run-offs, evaporation etc and this water drop in an ocean analogy is not even vaguely catering for the infinite forces, chemicals, factors, variables and volumes which we would expect in an infinite universe with no boundaries, shores or ocean beds where even the constants we have identified are probably not really constants!

    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    What are you talking about?

    First off, even a very tiny, finite, continuous Spacetime (the Spacetime Continuum [heard in a deep, rich voice, with reverberation]), with only a finite number of dimensions (evan as few as just two [or even just one, though such is a rather boring case]), can already handle an uncountably infine number of possibilities.  No need to go off into some "mystical beyond" sort of thing.

    Quantum Mechanics (even with all of its "spooky action at a distance", and other "mysticisms") needs nothing more, and works in the same kind of Spacetime as we have found works for everything else.

    Even if the "constants" are not constant, but vary in both space and time, that can also work just fine within the same kind of Spacetime.

    So why do you think there is no one of the three possible spacetimes that can fit all the experimental and observational data that humanity has amassed so far?

    David

    P.S.  I never said that Spacetime Geometry, alone, explains everything, all by itself.  While the dynamic Spacetime Geometry of General Relativity, alone, seems to do a remarkable job of explaining all gravitational interactions, even it doesn't pretend to explain Everything.

    Bonny Bonobo alias Brat
    So why do you think there is no one of the three possible spacetimes that can fit all the experimental and observational data that humanity has amassed so far?
    I never said that David! I'm just wondering about the likelihood of these 3 possibilities for the geometry of spacetime being able to accommodate ALL future experimental and observational data that we either discover or more likely that we don't discover because we can't observe or measure them for whatever reason. 

    We humans may be like goldfish in a glass bowl of water observing and measuring the universe we live in according to our limited observations and experience made by our limited intellects using our limited sensory measuring systems of a limited observable horizon. Naturally we would then find it difficult to imagine other space or spacetime forms in different dimensions that conform to different constants and rules. For example if Webb et al are correct and the fine-structured constant alpha is not constant then other constants may also not be constant and if so what implications does that have for spacetime geometry? 

    In physics, a dimensionless physical constant is a physical constant that is dimensionless – 'having no units attached, so its numerical value is the same under all possible systems of units. An example is the fine structure constant α, with the approximate value 1/137.036. Dimensionless physical constants are a subset of fundamental physical constants, which are mostly dimensional, for example, the speed of light c, vacuum permittivity ε0, Planck's constant h, or the gravitational constant G'. 

    If any or all of these constants vary throughout the universe then how can all three of your possibilities for measuring the geometry of spacetime still handle all of that?

    You are claiming that there are only three possibilities for the geometry of spacetime :-
    1. Euclid's geometry over all (four [4]) dimensions of spacetime (Euclidean Spacetime, or simply Euclidean Space, since there is no distinction for time);
    1. 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
    1. Space and Time is combined in a non-positive definite, though invertible way as "derived" by Einstein (Lorentzian/Einsteinian/Minkowskian Spacetime).
    Regardless of the time dimension being present or not aren't you assuming that all of these spacetime geometry measurements must always contain three dimensional dot points of space somewhere that must always relate or be derived in some way to or from some form of direction and/or magnitude or force but maybe they don't? Maybe there are areas within the multiverse where three dimensional space simply does not or cannot exist, like within a black hole for example? Maybe there is pseudo negative or inverse or even a one dimensional space somewhere? Can all three of these geometry of spacetime possibilities still cater for all other possible space dimensions that we still currently can't observe or even imagine but that we realize probably must exist somewhere in an infinite multiverse?
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Halliday
    OK, Helen:

    Somehow I knew that you were going to be bringing up the "multiverse".  It just seemed like your previous comment was being strongly influence by such.

    Let's first dispense of "ALL future experimental and observational data that we ... don't discover because we can't observe or measure them for whatever reason".

    First off, science is about attempting to explain that which we can observe, whether by simply looking out at the world/universe around us, or through experimental means.  So, if we cannot observe or measure it—for whatever reason—then we cannot do science on such.  That is certainly not to say that such "does not exist"—not by a long shot.  It is simply that such are outside the reach or realm of science (until, at some point, when we are able to observe or measure such, if ever).

    One of the problems with some versions of the "multiverse" is that all the universes are completely isolated from one-another—there can be no interaction between universes.  This form of "multiverse" is quite outside the reach or realm of science, and will always remain so.  (At least so long as we are confined within our particular universe.  But if the universes are actually isolated, how would it ever be possible to not be confined within our particular universe?)

    On the other hand, there are other versions of the "multiverse" that do allow interactions between the universes:  Gravity "leaking" from one universe into another, universes "bumping into" each-other, universes (quantum) interfering with each-other, etc.  This form may be within the reach or realm of science.  However, in almost all such cases, what we really have are multiple universes, like unto what we know as such, embedded within a higher dimensional Universe (the embedding space).

    However, "theorists" (I use that term loosely, hence the quotes) often don't "like" to have the universes embedded in some higher Universe, because, then, the embedding Universe "imposes" certain "structure" upon the embedded universes—so the embedded universes are not quite as "free" to be so very "different".

    In either case, the individual universes of the "multiverse" are certainly "free" to have different numbers of dimensions; to be smooth enough as to have vector spaces (tangent to their "manifold"), or simply not smooth enough, so they have no vector spaces; to have inner products, if they have vector spaces, or not, even if they do have vector spaces; and, if they have inner products, to have any of the possible kinds, given their number of dimensions.  Of course, this is all in addition to having different "constants" (whether actually constant, or not), and even different "forces" of interaction.  (While our universe appears to only need four forces to explain "everything we know", Quantum Mechanics [QM] has no problem with an infinite number of different forces.  We already have the form worked out, just "plug in" the symmetries.)

    But remember, I was asking about what best fits our universe, not some other, hypothetical, universe (especially not one we have no ability to observe or measure)!

    On the question of "ALL future experimental and observational data that we [can] discover":  As I already thought I explained, having "constants" that are not constant has no bearing on the question of which of the three cases best fit our universe.  The Geometry of our Spacetime does not completely determine all constants or all interaction.  All the Geometry of our Spacetime does is provide the framework for everything else.  (Such a framework does impose constraints upon the interactions that are allowed, however.  So it is a rather important "key".)

    Additionally, at least if we actually live in a Spacetime Continuum, then the local Geometry of Spacetime must be consistent over all of Space and Time.  (This is due to the non-continuous nature of numbers of dimensions, and the kinds or types of inner product one may have.  You cannot actually go smoothly from one to another.)

    Now, if we do away with the Continuum, then we can have other fun, like having the number of dimensions vary with position (with some restriction that is rather "entropy" like), as well as having some variation in the inner products (at least for those positions with more dimensions).  But that's getting on to quite another subject.

    Does this help put such "musings" into perspective?

    David

    Bonny Bonobo alias Brat
    But remember, I was asking about what best fits our universe, not some other, hypothetical, universe (especially not one we have no ability to observe or measure)!

    David, you were asking what best fits our universe and I must say I like the idea you mentioned above that even multiverses theoretically form part of an all encompassing universe but I apologise for digressing with my musings about what best fits future measurements of a hypothetical universe. I also did this because in my opinion our universe is still only hypothetical because our past, present and future measurements and our spacetime geometry are all based upon assumptions that we still can't really prove can we? We just think that we can until we find evidence to the contrary one day.

    I have just reread all of your spacetime geometry articles again and realized that I didn't understand some of the maths in the previous article or the relevance of all of those different types of positive, negative or pseudo inner dot products for helping to explain and improve our understanding of Einstein's special relativity by providing the 'missing piece' or 'gem' behind the theory. BTW a lot of the links within the article seem to have stopped working. I am also confused because in your first spacetime geometry article you said :-

    '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.'

    That is why I keep mentioning the universal constants that possibly aren't constants and you keep saying that this has no bearing on the question of which three cases best fit our universe but surely they have relevance to explaining special relativity? Aren't the assumptions that the speed of light in a vacuum  is a constant which has implications for the observer and that the universal physical laws are unchanged both vital components of special relativity and therefore the spacetime geometry must also rely upon them being constants when calculating corresponding space, light or time vector inner dot products? This Britannica article says :-

    By use of a four-dimensional space-time continuum, another well-defined flat geometry, the Minkowski universe (after Hermann Minkowski), can be constructed. In that universe, the time coordinate of one coordinate system depends on both the time and space coordinates of another relatively moving system, forming the essential alteration required for Einstein’s special theory of relativity. The Minkowski universe, like its predecessor, contains a distinct class of inertial reference frames and is likewise not affected by the presence of matter (masses) within it. Every set of coordinates, or particular space-time event, in such a universe is described as a “here-now” or a world point. Apparent space and time intervals between events depend upon the velocity of the observer, which cannot, in any case, exceed the velocity of light. In every inertial reference frame, all physical laws remain unchanged.
    It then specifically describes special relativity as :-
    Special relativity is limited to objects that are moving at constant speed in a straight line, which is called inertial motion. Beginning with the behaviour of light (and all other electromagnetic radiation), the theory of special relativity draws conclusions that are contrary to everyday experience but fully confirmed by experiments that examine subatomic particles at high speeds or measure small changes between clocks traveling at different speeds. Special relativity revealed that the speed of light is a limit that can be approached but not reached by any material object; it is the origin of the most famous equation in science, E = mc2, which expresses the fact that mass and energy are the same physical entity and can be changed into each other.....
    Einstein described how at age 16 he watched himself in his mind’s eye as he rode on a light wave and gazed at another light wave moving parallel to his. According to classical physics, Einstein should have seen the second light wave moving at a relative speed of zero. However, Einstein knew that Maxwell’s electromagnetic equations absolutely require that light always move at 3 × 108 metres per second in a vacuum. Nothing in the theory allows a light wave to have a speed of zero. Another problem arose as well: if a fixed observer sees light as having a speed of 3 × 108 metres per second, whereas an observer moving at the speed of light sees light as having a speed of zero, it would mean that the laws of electromagnetism depend on the observer. But in classical mechanics the same laws apply for all observers, and Einstein saw no reason why the electromagnetic laws should not be equally universal. The constancy of the speed of light and the universality of the laws of physics for all observers are cornerstones of special relativity.
    Anyway, that's why I was musing about the best spacetime gemetry fit for measuring a future hypothetical universe with universal constants that were not constants :)

    So David how do negative or pseudo inner dot products improve our understanding of Special Relativity and provide the best fit for our universe and why? Is it because they provide us with a spacetime geometrical representation of vectors combined in a non positive definite though invertible way that can also then be extended mathematically to include and represent the inverse energy of the speed of light and mass in spacetime as potential pseudo negative inner dot products or tensors?
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    I'm glad to see that we have, apparently, "whittled" the issues down to this "non-constant constants" issue.  I also see that you haven't taken my "word" for this "issue" being a non- issue.

    Actually, that's just fine, since, after all, science is, in a fundamental sense, about not having to take anyone's word for anything (in science).*

    Now, the only "constant of the universe" (whether actually constant, or otherwise) that truly relates to Spacetime Geometry (at least of the portion we are dealing with right now) is the speed of light.  (Since Spacetime Geometry actually forms the "scaffolding", so to speak, of all interactions, and the "forces" thereof, they actually depend far more upon the Spacetime Geometry than the reverse.)

    In fact, the only reason the speed of light is involved is because, historically, we humans have tended to use different units of measure for what we call "space" or "distance" vs. what we call "time", so we "need" to be able to convert the units of measure from one into the other.  (Converting between momentum and energy, and such, is highly related.)**

    Now, the "constancy of the speed of light" that you quote from the Britannica article doesn't actually have much of anything to do with whether the speed of light is constant, or the same over all space and time, but, almost exclusively, is strictly about whether the speed of light is "constant" (the same) for all inertial frames of reference.  (It is unfortunate that the term "constant" is so often used for these two, rather different situations.)

    The only way that the "constancy of the speed of light" has anything at all to do with the question of whether the speed of light could vary from one location to another (in Spacetime) goes back to the definition of Einstein's Special Relativity, where all of (his) Spacetime is one, single vector space.  (It's not explicit in the definitions, but it is clearly implicit in the nature of the transformations and other manipulations.)  However, once we allow for Spacetime to be "curved" (actually, a manifold), as I did early on in my presentation, this is certainly no longer anything close to a requirement.

    In fact, very early on, in working with General Relativity, Einstein, and others, came face to face with the realization that General Relativity allowed (some even thought it "required" or "forced") the speed of light to vary with location within space and/or time!  However, since then, it has been shown (rather easily, actually) that one may transform in and out of coordinate systems for which the speed of light is quite constant, or quite variable (since such are actually rather arbitrary, and one is quite free to choose coordinate systems almost arbitrarily [about the only requirement is that the coordinate transformations must be sufficiently "smooth" or "differentiable"]).

    The fact is that so long as one is not working with a Spacetime Geometry that is all one, single vector space (so one is working with a Spacetime manifold, where the only vector spaces are defined at each [and every] individual location in the Spacetime manifold), one has this same freedom with (essentially) all "unit-full" "constants of the universe" ("constants" that have units, as opposed to unit-less "constants").

    This is why all of the best treatments of questions of whether "constants of the universe" vary always emphasize the need to look at unit-less (or dimensionless) quantities.  However, once there, one is not dealing with anything that has any affect on Spacetime Geometry, but only on the interactions, and their descriptions, within the scaffolding provided by the Spacetime Geometry.

    David

    *  If someone truly decides to take noone's word for anything, within science, they are taking upon themselves quite a burden:  They must redo all theory, and all the predictions of such, as well as having to reproduce all observational and experimental evidence (both for and against any theoretical, or even hypothetical proposition).  So, perhaps needless to say, most of us are, usually, rather judicious about what we will or will not take on someone's word, within the realm(s) of science.

    **  There are some differences in the "need" for transforming between the units of "space" or "distance" vs. those of "time" for different choices in the Geometry of Spacetime or Space and Time (or purely Space), in terms of the three choices we are left to consider (for four dimensions):

    1. The Galilean/Newtonian Space and Time has very little interaction between what one calls "space" or "distance" vs. what one calls "time".  So one can actually get by with never defining a "unit conversion" between whatever we use to call our units of measure between these two.  The truth is that the fact that the pseudo-metric (AKA pseudo-inner product) is singular (not invertible), in this case, tends to keep Space and Time, and the entities that "reside" within each, as quite distinct.  (One also has to keep track of two different "metrics", so to speak, so as to be able to handle things that would otherwise involve the inverse of the Spacetime metric.  However, humanity was able to deal with such things rather well for many centuries, before learning that such is only a low velocity approximation.)
    2. The Lorentzian/Einsteinian/Minkowskian Spacetime certainly has a "need" to transform between the units of what we call "time" and the units we call "space" or "distance".  However, it also has a very natural, "ready made", conversion "factor":  The one and only "velocity" or "speed" that is unchanged by all transformations that preserve the Spacetime metric (inner-product).  (It is also, not coincidentally, the "velocity", or Spacetime direction, that is a "null" direction of the metric [inner-product]:  The direction of non-zero vectors that have an inner-product with themselves that is zero.  [Something quite impossible in a, more usual/normal, positive definite metric or inner-product.])
    3. The Euclidean Space also "needs" to be able to transform between the units that may be used for any one direction vs. any other.  However, it seems rather unlikely that beings (if any could exist) would evolve different units for different directions in such a space where all directions are so identical.  It would be like us using different units for North/South distances vs. East/West distances.  However, considering that under certain circumstances even humans have used different units for up/down directions vs. horizontal.  So, I suppose, it's not all that inconceivable.
    Bonny Bonobo alias Brat
    OK, I get the message David. It is only the constant speed of light for observers in inertial frames of reference that matters for special relativity. As Einstein's Special Relativity for Dummies (ie me) explains :-
    Special relativity includes only the special case (hence the name) where the motion is uniform. The motion it explains is only if you’re traveling in a straight line at a constant speed. As soon as you accelerate or curve — or do anything that changes the nature of the motion in any way — special relativity ceases to apply. 
    Einstein’s theory was based on two key principles:
    The principle of relativity: The laws of physics don’t change, even for objects moving in inertial (constant speed) frames of reference.
    The principle of the speed of light: The speed of light is the same for all observers, regardless of their motion relative to the light source. (Physicists write this speed using the symbol c.)
    The genius of Einstein’s discoveries is that he looked at the experiments and assumed the findings were true. This was the exact opposite of what other physicists seemed to be doing. Instead of assuming the theory was correct and that the experiments failed, he assumed that the experiments were correct and the theory had failed.

    Oh well, back to swimming in circles in my goldfish bowl :)

    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.'
    Sorry if I'm being very thick but in the introduction to this series you said you would illuminate the 'missing piece', the 'gem' or 'kernel' that is missing in the understanding of those that struggle with Einstein's Special Theory of Relativity. Well I don't feel that I struggle to understand Einstein's Special Theory of Relativity, he made the SR theory fit the data and even he probably didn't have the ether foggiest idea as to why these axioms that the SR theory is based upon were happening but I am still struggling to understand exactly what your gem or kernel for us understanding Einstein's SR theory in this geometry of spacetime series is?
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    Halliday
    Helen:

    I haven't reread my first article to see whether I gave a name or label to this "'missing piece', the 'gem' or 'kernel'".  However, it is simply the metric (or pseudo-metric, for those that wish to be pedantic), also known as (AKA) the inner or dot product (or pseudo-inner product, for the same pedants).

    Unfortunately, when Minkowski showed how all of Special Relativity derived from this simple "kernel" (a non-positive definite, pseudo-metric), Einstein most definitely didn't like his development.  I suppose he felt that it was too far removed from his two postulates, or too far removed from the experimental evidence he wished (I suppose) to have others anchor to.  (In this article I've tried to show how this metric idea is anchored in even older observational and experimental evidence, where the only need for more modern experimental evidence is in distinguishing between the final three candidates.  Besides, Einstein didn't actually base his postulates so much on experimental evidence anyway.  His "principle of the speed of light" ["The speed of light is the same for all observers, regardless of their motion relative to the light source"] was actually based more upon his greater "belief" or "trust" in Maxwell than in Newton.)

    In any case, he must have "warmed up" to the metric (or pseudo-metric) idea, since it takes a central role in General Relativity (which is why I didn't actually "know" about this "'missing piece', the 'gem' or 'kernel'" until I learned General Relativity.  (Even the advanced [Classical {AKA Newtonian}] Mechanics class tended to skirt the issue of a "metric", even though it was all about Differential Geometry [the same mathematical language of General Relativity], in handling non-inertial reference frames in Newtonian Mechanics.  The problems are two fold, in Newtonian Mechanics: #1 The overall [spacetime] "metric" is singular [AKA "degenerate"], so it cannot be inverted; and #2 The "sub-metrics" for the [rather separate] space and time "sub-spaces" are both "isomorphic" to the identity [because they are positive definite], so they tend to "disappear" easily.)

    So, is this the "missing piece" for you?

    David

    Bonny Bonobo alias Brat
    So, is this the "missing piece" for you?
    Yes and no David. I now understand that you are showing us is that the metric or pseudo metric or inner or dot product  or pseudo inner or dot product is the missing piece or gem you were talking about in the introduction and hopefully why but in the process of you demonstrating that you have broadened the overall jigsaw picture for me and made me aware of many more missing pieces that I would need to find and understand in order to understand the bigger picture you are painting 

    I understand that Einstein's theories of special and general relativity predict the distance between events in spacetime and that although they are fundamentally different in their mathematical structure, inertia is a defining link between the two. Funnily enough I just found  this paper called 'Einstein's vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions' possibly written by our own Science20 blogger Doug Sweetster (I can't see an author but the URL contains Sweetster in it) that describes how :-

    'Special relativity dictates the transformation rules for observers who change their inertia, assuming the system observed does not change. The field equations of general relativity detail the changes in distance due to a system changing its inertia from the vacuum to a non-zero energy density. A quaternion product necessarily contains information about the metric, but also has information in the 3-vector. This additional information about quaternion products will suggest a provocative link between metrics and inertia consistent with both special and general relativity.' 

    Wiki describes how 'in mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.'?

    Is that also correct? Where do quarternions fit into this overall big picture or jigsaw and the gem or pseudo dot inner product or metric that you have shown us? This quarternion cheat sheet says that 'a quaternion can be represented by a vector of length four: ˙q = (q0, qx, qy, qz)' and that 'the dot-product (inner product) of two quaternions is their usual vector dot-product: ˙p· q˙ = p0q0 + pxqx + pyqy + pzqz' 

    Are quarternions maybe the ring that can potentially hold and even display the pseudo metric or inner dot product gem by linking special and general relativity in new ways? The Wiki quotations about quarternions are quite dramatic (for mathematicians) and even call them an unmixed and positive evil :-
    • "Time is said to have only one dimension, and space to have three dimensions. […] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Rowan Hamilton (Quoted in R.P. Graves, "Life of Sir William Rowan Hamilton").
    • "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." — Lord Kelvin, 1892.
    • "I came later to see that, as far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work." Oliver HeavisideElectromagnetic Theory, Volume I, pp. 134–135 (The Electrician Printing and Publishing Company, London, 1893).
    • "Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in every-day life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols." — Ludwik Silberstein, preparing the second edition of his Theory of Relativity in 1924.
    • "… quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist." — Simon L. Altmann, 1986.

    So David, do you agree that quaternion products suggest a provocative link between metrics and inertia consistent with both special and general relativity? Could they be the ring that holds your gem and also an unmixed or positive evil and if so where are the hobbits when we need them to protect mathematicians and scientists from this potential evil?
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    The "paper" you link to (in the beginning of your comment) does appear to be our very own Doug Sweetser.  If you go to the top level of the site (by following the Quaternion s Question and Answer link at the bottom of the page) you will find the following at the bottom:

    This forum was set up and moderated by Douglas Sweetser, probably back in the 1990s.  It is currently a read-only archive.  None of the content on this server has been reviewed by the webmaster.

    The quote you have from William Rowan Hamilton (Quoted in R.P. Graves, "Life of Sir William Rowan Hamilton") sure looks like the kind of thing that probably inspired Doug.  :-)

    I also like the quote from Ludwik Silberstein (preparing the second edition of his Theory of Relativity in 1924).  While I do not quite agree that "those older mathematical languages continue ... to offer conspicuous advantages in the restricted field of special relativity", I do agree that "in science as well as in every-day life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols."

    So, unfortunately, I do not "agree that quaternion products suggest a provocative link between metrics and inertia consistent with both special and general relativity".  There are better languages, that have greater utility (such as being able to handle both more and fewer dimensions).  Some of these other languages even have the quaternions as a special case.  (I tried to get Doug to investigate at least one of these other languages, but, at least at the time, I was unsuccessful.)

    David

    P.S. While you are (at least somewhat) correct that "Einstein's theories of special and general relativity predict the distance between events in spacetime and that although they are fundamentally different in their mathematical structure, inertia [as in inertial mass?] is a defining link between the two", I'm not so sure I would emphasize "inertia" as "a defining link between the two".

    While both theories do treat inertial mass in, essentially, the same way, I would say that the most important link between the two is that the (pseudo-)inner product space of Special Relativity is the same as the (pseudo-)inner product space of the Tangent Space (at each point of the manifold) within General Relativity, and, hence, Special Relativity is a special case of General Relativity:  That of a rigidly flat spacetime.

    Bonny Bonobo alias Brat
    While both theories do treat inertial mass in, essentially, the same way, I would say that the most important link between the two is that the (pseudo-)inner product space of Special Relativity is the same as the (pseudo-)inner product space of the Tangent Space (at each point of the manifold) within General Relativity, and, hence, Special Relativity is a special case of General Relativity:  That of a rigidly flat spacetime.
    OK David then that looks like a gem to me because it now seems to make perfect sense, so thank you for your kind patience :) 

    I hope you will find time to write more blogs this year as your series has been very stimulating, enjoyable and educational. Of course some of it went way over my head, especially in the comments sections but I still feel that I learnt a lot and I am keen to keep reading, learning and thinking about local spacetime geometry and where it can best be applied to make sense of our perceived universe.
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    I'm glad that this has all been helpful.  That makes it all worthwhile.  :)

    David

    The Stand-Up Physicist
    Hello David:

    I wish you could quit the day job.  Heck, I'd like to quit mine :-)  Well, back to physics in these tiny time windows.

    You can guess what my geometry hang-up is all about: spinning tops.  Earth spins, the Earth revolves around the Sun, our atoms do things with angular momentum we don't fully understand.  What concerns me about discussions of inner products and metrics is they often neglect to account for how big a spinning thing is.  This is one reason I don't buy into all the work in higher dimensions: the inner product stuff travels to higher dimensions very well, but the cross product does not.  While I admit it may be natural to have a bias toward stuff that moves in a line, one's choices about geometry to use must embrace spinning tops in step number one, and not just consider it an add-on, at least in my opinion.  

    Spinning tops appear to enjoy abusing our idea of what should be intuitively true, so I am a big fan.
    Halliday
    Hello Doug:

    If you're a big fan of "spinning tops" and other things that "abuse our idea of what should be intuitively true", then you should be a bigger fan of Differential Geometry!

    Metrics are actually secondary in Differential Geometry.  Even if one doesn't have a metric (so one has trouble comparing lengths in different directions, and trouble comparing how different different directions are) one still has the ability to handle "spinning" things, and more, via the true generalization of the true meaning of the "cross product".  (The "cross product" is actually a "bastardized" version that is only available in three dimensions, having been mapped from bi-vectors back into a "vector" that doesn't transform like a "true" vector, so it is referred to as a "pseudo-vector".  How "dumb" is that?)

    Anyway, again, perhaps you would be wise to learn more about Differential Geometry.  Especially since the language of Quantum Mechanics is, in a sense, at least, closer to the language of Differential Geometry than it is to the language of Newtonian mechanics or other three dimension only "languages".

    David

    i vote for 3. einsteinian/lorentzian/minkowskian space time. (add in kaluza-klein and newtonian geometry ----nathan rose of epr) though someone wrote it first in 20's)
    hestenes approach is also interesting----quaternions and such. or you can do spinning tops---harmonic oscilattors. y s kim on arxiv. or try t n palmer
    www.arxiv.org/abs/1309.2396.

    Halliday
    ishi:

    You are quite correct that "3. einsteinian/lorentzian/minkowskian space time" is the best match to our (apparently) four (4) dimensional spacetime.  However, "newtonian geometry", at least as I understand what is meant by that term, is 2. Galilean/Newtonian Space and Time.  (I do note that you reference "nathan rose of epr", so I suppose he may have had a different meaning for the term "newtonian geometry".)

    Of course, if one adds "kaluza-klein", then one is talking about more than four (4) spacetime dimensions.  (Additionally, "kaluza-klein" only encompasses the addition of Electromagnetism [EM], so one would need something more if one were to try and use something like this to handle all known forces.  On the other-hand, one can simply augment the Tangent Space of General Relativity with "fiber spaces" to handle all of the non-gravitational forces.  The Differential Geometry thus handles all of the "first quantization" "wave equation" level portions of Quantum Mechanics, though it may be desirable to not have complex "fiber spaces" be simply "tacked" onto a real spacetime manifold.)

    I'm quite familiar with "hestenes['] approach", since I worked for him, for a few years.  (That's a whole other story, altogether.)

    Palmer's paper, that you provided a link to, does look intriguing.  Is the paper at http://arxiv.org/abs/1301.6091 one of the works of "y s kim" that you were alluding to?

    David

    hi david---i am 'rusty' and also not a specialist in this stuff at all (my background is more in stat mech/complexity/math bio.

    when i said newtonian gravity i was actually thinking of the reformulation due to cartan which makes it look in some ways like general relativity. Nathan Rosen (which i mispelled above) of the EPR paradox also reviewed this stuff if i recall in a paper in Phys Rev or Rev Modern physics around 1945.
    I am interested int this stuff partly because of ways one can view a statistical mechanical system as a field theory (and also via the Wick trnsformation as related to quantum fields---where temperature becomes 'imaginary time' ), and also as kind of fluid mechanical system. Hence on can have newtonian mechanics reformulated into a curved space analogy, and conversely general relativity can be reformulated in flat space time. (I have also been interested in stuff like Edward Nelson's attempts to reformulate quantum theory as a kind of classical diffusion theory with i think little or no sucess; I think de Broglie in his last book also tried this.)

    I mentioned Hestenes and Y S Kim (and you found him on arxiv) because I really dont understand Hestene's view he appears to be trying to generate quantum theory from some sort of discrete statistical theory using quaternions and such rather than real or complex numbers. (There's Joy Christian ---also on arxiv---of Perimieter institute who also tries to make quantum theory classical using hypercomplex numbers. Another discrete theory of course G T'Hooft 'cellular automota interpretation of quantum mechanics', or the Vaxjo interpretation by Khrennikov. There's tons of this stuff and I really can't digest it or judge it.)
    Kim has some discussions of harmonic oscillators as being in a sense the fundamenntal particles from which you can get theories of everything (though i don't think he puts it that way). (of course in a sense this is like string theory, since strings describe harmonic oscillators) .

    I will have to spend more time on your paper since the math you use of differential geormetry, while it occurs in mathematical biology, complexity theory, etc. is much less commonly used there (which is msotly differential equations and statistics----it does appear in reformulations called 'information dynamics' by Ariel Caticha for example.) ---ishi

    correction------i meant newton-cartan theory (not einstein-cartan)