I don't think people in these forums object to discussing the idea of a "block universe" per se. I don't. The only think I would object to in your post above is your use of the term "space-space" to describe the block universe; I would use the term "spacetime". Using that term does not require adopting any concept of observers "moving" along worldlines; it simply recognizes the fact, which you mention, that the metric of spacetime is not positive definite. The term "space" unqualified implies a positive definite metric. Or at least, using the term "space" instead of "spacetime" for a manifold that does not have a positive definite metric is a great way to invite confusion, IMO. But that's more a question of terminology than physics.
There is one technical correction I would make: it is not correct to say that the X4 vector for a moving observer "slants" while the three spatial vectors remain perpendicular. The space vector in the direction of motion "slants" as well. Your diagrams show this, or at least a consequence of it (the tilting of the planes of simultaneity).
With the above corrections/clarifications, I personally have no problem with the concept of a block universe, because I don't view it as a final statement about "reality". I view it as a model. As a model, it simplifies a lot of calculations and makes a lot of relativistic physics easier to visualize.
What exactly is the fourth dimension?
Are you saying the fourth dimension is just another spatial dimension? But how can these be viewed in reality?
Yes. To us it is different for a few reasons, among which are 1) Our consciousness "sees" just a continuous sequence of 3-D cross-sections of the 4-D space as our consciousness moves at the speed of light along our 4th dimension world line and 2) Four dimensional objects are vastly longer along the 4th dimension, X4, (perhaps billions or trillions of miles long) as compared to the lengths along X1, X2, and X3.
First, the 4th spatial dimension is implied by considerations depicted in the side sketches of my post. The fact that special relativity describes a situation in which different observers "live in different planes of simultaneity", each one including different cross-sections of 4-D objects, implies a spatial 4th dimension. Beyond that, we actually have views of the 4th dimension in this sense: We experience a sequence of cross-sections of the 4th dimension. If you merge those cross-sections together you have the 4-dimensional picture. So, the 4th dimension is as real as our normal X1, X2, and X3. (I'm avoiding any discussion of philosopher's viewpoints that deny the existence of an external reality.)
As you travel a 1000 mile interstate cross country you experience a piece of it at a time. You have no problem understanding that the whole stretch of interstate actually exists "out there", even though you cannot see it all at once. After traveling it, you can merge the whole path together to form the mental picture of the highway as it stretched for 1000 miles.
Remembering the continuous historical past sequence of experienced 3-D images, you can merge them together to get a kind of 4-dimensional image in your mind. However, the brain does not seem to be equiped with the tools for actually visualizing a 4-dimensional object. This makes your question difficult to deal with.
You say that I am moving through the fourth dimension at the speed of light. I certainly don't feel like I'm moving at the speed of light through any dimension.
Thanks for jumping in. I didn't do your comments justice at the end of the other thread where we worked on this subject. I felt like I was unfairly hi-jacking the topic from the original poster. Also, you had me boxed into a corner, and I couldn't quite figure out the appropriate response. Not to say anything of my pushing the boundaries of our forum rules.
I have to agree with you here. I was uncomfortable using a term that is not standard in the literature. I have no business offering a new terminology here. I was of course wanting to emphasize the character of the 4th dimension, X4, as not that of time, but rather having the same quality of X1, X2, and X3.
The only reason I would hesitate to agree here is that again I wanted to make it clear that an indefinite metric can be associated with four spatial dimensions. I agree that folks hearing the term "space" unqualified would be assuming that one is referring to a manifold having a space defined by a positive definite metric. But, I agree that in discussions among physicists about special relativity it is best to use the "spacetime" terminology while adding the qualifier that with L4 space we can have four spatial dimensions.
My introductory sentence about the X4' slant may have left that impression, but I intended the following statement referring to both X4' and X1' slanting to make clear that it is the combined slanting of X4' and X1' with Lorentz boosts that accounts for the notion that the 4th dimension is "different." And of course my sketches make it explicitly clear.
I agree physicists most likely can never prove any model for external reality (many philosphers would not even permit any external reality--much less the exact nature of such a reality). We can attempt to identify models that are consistent with established physical theory.
The block universe seems at least to be a model that is consistent with special and general relativity and can be comprehended. A model that describes the 4th dimension as time or a mixture of time and space is quite vague and difficult to describe in an explicit way, although it easily fits a Max Tegmark approach if one wishes to regard reality as some sort of mathematical reality.
Same comment here as with the term "space" instead of "spacetime". Under what definition of the term "spatial dimension" do you claim that X4 is a spatial dimension? The only generally accepted definition of the term "spatial" that I'm aware of requires the dimension in question to be part of a positive definite metric. So you're going to need to clarify what you mean by "spatial".
This is another area of apparent confusion. The "block universe" model *is* a model that describes the 4th dimension as "time"; although a better way of putting it, IMO, would be that it is a model that describes what we call "time" as a fourth dimension in a manifold with a non-positive-definite metric, which we call "spacetime". There's no need to repudiate the term "time"; you just have to clarify that "time" does not require a "flow" of time; that's a separate idea that is not required to understand spacetime (though it may help one to visualize some problems).
It's kind of like watching a movie. You are caught up in the continuous evolving of the succesion of movie scenes without any thought of a long strip of movie film moving through the projector. We would have a more direct analogy of course if the movie film were strung out in space along a straight line and you were doing the moving along the length of the strip watching that same movie, experiencing the same movie-going experience without any thought of your motion through space.
So, a bundle of 4-dimensional neurons is analogous to the long strip of movie film. But, the neurons extend for billions or trillions of miles along ones 4-dimensional world line with consciousness flying along at the speed of light watching the movie.
Hmm, I didn't spot this part before. The idea of "your consciousness moving along your worldline at the speed of light" is, as I said in a previous post, an *addition* to the basic idea of the block universe; it is not necessary to the basic model. In the basic model, the speed of light is just a conversion factor that comes in because, for historical reasons, we measure the X4 dimension in different units than the X1, X2, X3 dimensions. Interpreting this conversion factor as a "speed" implicitly depends on interpreting X4 as "time" in the common sense of that term, instead of just as another dimension of the manifold. The same goes for the idea of "moving" along your worldline; in the basic block universe model, there is no motion.
Also, adding the idea of "motion" is not necessary to understand our conscious experience, as Julian Barbour has pointed out in a number of papers:
You can capture what we experience simply by saying that there is a collection of "snapshots", each one capturing an "instant" of our experience, and that the collection of these snapshots has a structure: some snapshots are "earlier" or "later" than others, and some snapshots that are "later" contain data that is correlated with others that are "earlier" (this is what a "memory of a past event" is). What we normally think of as "time" emerges as a derived concept from all this; there is no need for it as a fundamental concept.
Yes. You are right. It is not necessary to the basic 4-dimensional spatial universe populated by 4-dimensional objects to bring in time. However, since the obvious question arises about how time relates to such a model, it seemed fruitful to pose a possible explanation for the perception of time flowing.
I have Barbour's book along with others on the concept of "block time" and am well aware of what you bring in here. I only avoided further discussion along those lines to avoid a more extended tortuous discussion. There have been two approaches to bringing time into the picture of the 4-D spatial universe. The first is the traveling of consciousness along the 4th dimension at light speed (this leads to zombies and solipsism, which Einstein cautioned against).
The other concept puts consciousness simultaneously along the entire world line of an observer. This was the concept of which Einstein seemed to refer. I have not seen the book, but evidently Fred Hoyle wrote a novel in which observers existing with their 4-dimensional consciousnesses accompanying their entire 4-D material structure were at the mercy of a devious super hyperspatial being who was at a console of buttons allowing him to stimulate the consciousness arbitrarily at one point along a world line, then another. He could fiendishly cause the observer's focus of attention to jump from one point to another, randomly, up and down the world lines. The observers had no awareness at all about what was going on. At any given station along a world line the observer is only aware of what information is presented at that point, i.e., the normal memories, hopes and desires, etc., at that point.
That is contrary to the concept I have described in the original post. I thought I had made it clear with the example of calibrating distance along the interstate with time markers. So, it is exactly the reverse of what you've said here. The 4th dimension is spatial in the same sense as X1, X2, and X3. We are able to calibrate physical distance along a world line using the conversion factor, t = X4/c. It is actually just for historical reasons that time has come into usage as a 4th dimension.
And of course there is no physical motion of 4-D material objects in a spatial block universe. However, there is the psychological impression of motion arising from the interaction of consciousness with the 4-dimensional object. The impression of time flowing results. This impression of the flow of time can be represented mathematically as the focus of consciousness moving along the world line at the speed of light.
I don't argue with that concept. I was quite familiar with it and acknowledge that is consistent with the 4-D spatial block universe. Again, I hesitated to expand my post with extended discussions that might account for many of the implications of the interaction of consciousness with the 4-spatial dimension block universe. If the use of the term "block universe" causes confusion, given the different versions of this concept, I suppose we could emphasize the 4th spatial dimension version by using something like "spatial block universe." But, I don't want to be the author of new terminology.
Hmm, interesting, I hadn't heard about this book. If you can find a link to a review, summary, etc., please post.
Again, you are using the term "spatial" here without an adequate definition. You need to explain how X4 can be a "spatial" dimension when the metric is not positive definite. (The question does not arise for X1, X2, X3 because on any spacelike slice there is an induced metric involving just X1, X2, X3 on that slice which *is* positive definite, so there is a clear definition of how those dimensions are "spatial". You can't do this with a 3-surface that has X4 as one of its dimensions.)
Or it can be represented, as Barbour does, as an ordering relation on the "snapshots" that does not require "flow" at all. This has the advantage of not requiring the concept of the "speed" at which consciousness is "moving" along a worldline. Again, interpreting the conversion factor c as a "speed" requires you to already have a concept of "time", so you can't use speed to define the concept of time as we experience it.
I finally found the reference in Paul Davies's book "About Time" (paper back pg. 41). Hoyle's book is "October First Is Too Late." It has been quite some while since I read the Davies book and my memory of it mixed the initial musings of Davies with Hoyle's theme. Davies had similar thoughts of jumping in time, even in his youth. He would muse over pushing magic buttons that would transport him randomly to different times: "...I would have no subjective impression of randomness, because at each stage the state of my brain would encode a consistent sequence of events." He continued, "It is but a small step from this wild fantasy to the suspicion that maybe someone else--a demon or fundamentalist-style deity perhaps--is pressing those buttons on my behalf, and I, poor fool, am totally oblivious to the trickery..."
Davies then describes Hoyle's book: "Hoyle also imagined some sort of cosmic button-pusher, but one who fouled things up and got different bits of the world out of temporal kilter." ..."Hoyle's fictional scientist caught up in this nightmare has no truck with the notion of time as an 'ever-rolling stream', dismissing it as a grotesque and absurd illusion."
I googled "October First Is Too Late" and found reviews on Amazon. Also, you can find Chapter 14 in pdf format. The sub script to the Chapter 14 title is a quote by Hoyle:
The 'science' in this book is mostly scaffolding for the story, story-telling in the
traditional sense. However, the discussions of the significance of time and of the
meaning of consciousness are intended to be quite serious, as also are the contents of
chapter fourteen. Fred Hoyle, 14 July 1965
Again, I maintain that the indefinite metric is related to the slanting of the X4' and X1' axes. It is not at all related uniquely to time. We can have slanted spatial axes without any need for time.
How do you define or characterize the quality of X4? Well, how do you characterize the quality of X1, X2, and X3? The existential quality of the physical space along different directions is not dependent on the geometric orientation of the axes. You don't turn a coordinate axis into time by rotating it. The thing that associates X4 with time is that 4-D objects are vastly longer in X4 as compared to X1, and consciousness interacts along the X4 world line in a manner that results in psychological illusions of a flow of time. And that flow is associated with a continuous sequence of 3-D images having a quality of change and movement. Certain 4-D objects, like mechanical clocks, have a periodic repetitive geometry that provides a measure of distance along X4 (which is particulary useful when calibrating the 4-D positions as t = X4/c).
Further, when considering the arbitrary different cross-sections of 4-D space, there can be no comprehensible meaning given to various vague kinds of mixtures of space and time as the cross-sections for different observers cut through the same 4-dimensional space at different angles.
That is certainly a concept that is consistent with the spatial 4-dimension universe. Nevertheless, we as thinking observers do have a psychological sense of time passing, notwithstanding the illusion aspect you refer to. The idea of consciousness moving along the world line at speed c aids in the description of that psychological phenomena.
The notion of time and time passing was anchored in the psyche of man centuries before special relativity. Likewise the concept of motion and speed. Special relativity and the geometry of Lorentzian space provided, perhaps for the first time, a logical and easily comprehended world view that encompases space and time. It leads logically to the understanding that one must focus attention on the understanding of consciousness in order to understand time, rather than looking to the 4th spatial dimension.
You can have non-orthogonal spatial coordinates, yes, but that's not what the "slanting" of X1' and X4' relative to X1 and X4 is doing. X1' is still orthogonal to X4', just as X1 is orthogonal to X4. The only reason X1' and X4' look slanted is that they are drawn in the X1-X4 frame; if you drew everything in the X1'-X4' frame then X1 and X4 would look slanted.
In any case, you still haven't justified the term "spatial" when used in reference to a non-positive-definite metric. That's the key issue with that terminology, and nothing in the rest of your post appears to me to address it.
Agreed. I didn't mean to imply that I thought the idea of consciousness moving along the worldline was not useful, just that it was not fundamental to the "block universe" idea.
That is correct, of course. I'm not suggesting anything at all contrary to that. Nevertheless those Lorentz boosts require a metric resulting in the invariances of special relativity theory. And some new quality, such as "time", is not at all necessary to arrive at the successful metric, one that results in, X4^2 - X1^2 = X4'^2 - X1'^2. Why the need to insert the quality of time to achieve relationships that are all about space?
I think the situation is just the opposite. I think the burden is heavily on the mathematician to demonstrate that when you have a space with indefinite metric the space then fails to maintain the same existential quality in all directions. Just having a different sign in the signature does not automatically signal a change in the quality of space. There is especially no rationale for assigning time as a new quality for the spatial 4th dimension.
However, you've brought a very useful discussion to the table. You've crystallized this issue of how the 4th dimension should be chacterized. I know my comments do not resolve the issue. We may need to start again, looking at this issue from the ground up. Perhaps begin carefully building up the mathematical machinery step-by-step, considering the quality of space in each direction as we go. Look carefully at what (if any) quality can be ascribed to the manifold, the topology, what spaces are available, and what linear vector spaces are available and what can be said about the quality of the spaces in each of the directions.
Sure. And again, the consciousness moving along the world lines at c can lead to bizarre implications that hardly anyone would like.
You're correct that one does not need to add a new quality like "time" to X4 to justify the indefinite metric. And there is a use of the term "space" in mathematics that is general enough to encompass the way you are using it here. But AFAIK that usage is not common in relativity physics; the more precise technical term, "manifold", is used instead to convey what I think you are trying to convey (that spacetime has a certain structure, dimensionality, topology, etc., without reference to the metric).
It is true that in quantum physics, the term "space" is used much more generally, as it is in mathematics; it basically means any larger structure that objects of interest "live" in. For example, quantum state vectors are elements of a Hilbert space. But this usage of "space" is highly abstract and certainly does not imply any easily comprehended relationship with what we commonly think of as space; most quantum Hilbert spaces, for example, have an infinite number of dimensions.
In some ways it does. For example, in a manifold with positive definite metric, the only way for a vector to have zero norm is to be the zero vector. In a manifold with non-positive-definite metric, there is an infinite number of vectors with zero norm (the null vectors). This leads to some properties that may seem counterintuitive: for example, every null vector is orthogonal to itself.
This has already been done; it's called "differential geometry".
I've drawn on Paul Davies's cosmic button-pusher to present a fanciful example that may get across the point I've been trying to make. I'm not offering this as a proof for the spatial 4th dimension, but rather, hopefully, a clarification of the concept.
I see two questions regarding this:
(1) The cosmic button-pusher starts out by setting up a Euclidean space with a positive definite metric. Then, somehow, the mere act of granting consciousness to the blue observer changes the metric. How does that happen? Of course we know what the actual metric of spacetime is, so we know in advance what the answer is supposed to be, but how does that answer arise logically, within the scenario, as a result of granting the blue observer consciousness? I don't see any logical connection there. Put another way, since we already know that the metric of spacetime is not positive definite, your introduction of a hypothetical Euclidean space whose metric magically gets changed when an observer is granted consciousness is superfluous; it can be eliminated without changing anything else.
(2) You've used something like the second picture before in another thread, and I pointed out an issue there which, IIRC, you never responded to. Yes, for the particular set of values you chose, you can write down a "Pythagorean theorem" that appears to apply, but it only applies to particularly chosen sets of values; it does not apply generally. For example, draw a similar right triangle with sides blue X1, blue X4, and red X1 (instead of red X4); in other words, drop a perpendicular from the blue X1 axis to the red X1 axis (instead of from the blue X4 axis to the red X4 axis). You can still write down a "Pythagorean-style" equation: (red X1)^2 = (blue X1)^2 - (blue X4)^2. The problem is, now the LHS is a *side* of the right triangle, *not* the hypotenuse! So the cosmic button-pusher has *not* restored a Euclidean metric; he has merely picked out a particular set of values that can be rearranged in a certain way.
i just jumped in to say wow that someone sharing my thoughts. i was always a bit unhappy that 4th dimension had different units. sometimes i think light as 2d structures in 3d space (may be thats why wave-particle duality... just kidding) but what i thought was all the 3d objects that we know to move at C in 4-d space not just consciousness.
Good to have you jumping in here, to have one who can relate to the issue at hand. Don't be a stranger.
I knew I was not giving an accounting of how the slanted X1' pops out. Just didn't want to make the post so long and involved. The other part of the story would involve the cosmic guy's apprentice. Before adding in the consciousness the cosmic guy tells the apprentice to add in some other 4-dimensional objects to the positive definite manifold, then cosmic guy walks away leaving the apprentice to play with the toy universe. But, the apprentice is actually quite ingenious and puts objects in with very special orientations. He has developed an elaborate set of rules about how all of the objects should be arranged on the positive definite manifold.
To picture the kinds of geometric patterns the apprentice used in placing the objects, just think things like Feynman diagrams, processes involving conservation laws, etc.
Thus, when the consciousness is turned on and sent on a trip along the blue guy's world line, the only way the consciousness could acquire any comprehension of the continuous sequence of 3-D cross-section views of the 4-dimensional manifold with embedded objects would be to psychologically adjust his X1 cross-section view so as to be in sync with a Lorentz boosted view of the universe. That's just because the apprentice formed the patterns in just the right way to produce the unique invariances that are normally associated with Lorentz boosts. If the blue guy did not view the universe across a Lorentz boost view there would not be the kind of correlation in the sequence of events unfolding around him that could produce a comprehensible experience. It's kind of analogous to the difference between listening to random noise and music. If you are in an environment of a loud audible random noise, yet there is a lone violin playing a melody somewhere in the background, your brain has a way of filtering the violin melody so that you comprehend the sound in spite of the noise. The symmetry of geometric patterns present in the 4-D spatial universe makes possible some kind of correlation within the brain that plays some kind of role in the ability of the consciousness to recognize and comrehend. For the blue guy it was a matter of psychologically adjusting his cross-section view to the proper Lorentz boost that makes for an intelligible continuous sequence.
The cosmic button-pusher initially did not realize what his apprentice had done, but he quickly discovered the benefit of switching to a new metric so he could recognize the invariances and appreciate the local physics that resulted. He was quick to realize why the blue guy's consciousness automatically began scanning 4-D space in the slanted X1' direction. But, it is important to note that it was not necessary for the cosmic button-pusher to change metrics for his bird's eye view of the universe. The cosmic button-pusher could happily muse over the varied patterns of objects placed on the positive definite metric space. He just wouldn't see the physics that the apprentice had built into the patterns--which are only apparent if you change the metric so as to recognize the invariances associated with the Lorentz boost. Arranging objects does not change the intrinsic mathematical properties of the manifold--it's topology, etc.
The whole point of the story is to try to explain in what sense you can start with a positive definite manifold, yet then orient objects in a way that leads to the selection of an indefinite metric to make the orientation of objects intelligible. The quality of the four spatial dimensions did not change at all in that process. And there was certainly no rationale for regarding the 4th dimension as "time."
I don't agree with that. You give me any pair of observers moving relative to each other at any speed you wish. There is always a rest frame for which both observers are moving in opposite directions at the same speed. So, a symmetric space-time diagram can always be found that works in general for any pair of observers. And for example the Lorentz time dilation equation may be derived directly from the Pythagorean Theorem.
And yes, you're right; it's the differential geometry and associated mathematical machinery. However, all through the physics Master's and PhD curriculum, in all of the functional analysis, tensor analysis, group theory, set theory, QM courses, classical field theory, special relativity, general relativity, and cosmology courses, none of my professors ever discussed manifolds in this context. I tried only two or three times to discuss this with my doctoral relativity advisor, but he was quite annoyed that I would allow myself to get so distracted from doing real physics. And he was right in terms of how I should have spent my time in that phase of education.
In which case the manifold is no longer positive definite. Put another way, your claim that the manifold started out positive definite is not justified: "positive definite" is supposed to describe the actual physical metric that describes actual physical intervals, not an unobservable starting point that you then throw away and that plays no part in predicting any actual measurements.
As dimensions in a topological manifold, no. As dimensions in a metrical space, yes, X4 *did* change; you started out saying the metric was positive definite but as soon as any actual physical measurements were made it changed to non positive definite. You can't just handwave away that change in metric structure.
For *that particular pair of observers*, in *that particular frame*. A real metric is not like that; it gives the right answer for *all* pairs of observers, in any frame and any state of motion, without any special setup required each time.
I'm sorry you went through that kind of experience. I avoided it because I didn't study differential geometry at all in school, which may have been a good strategy for actually being able to learn something about it.
But that doesn't mean the subject can't be learned. I learned it mainly from Misner, Thorne, & Wheeler, which I also had the advantage of not having to learn in school. But that may not be the best up to date source. Others here at PF could give better advice than I on where to look.
Correct. But the whole point is that the quality of the X4 did not change. The apprentice simply added objects to the space without changing the fundamental quality of the space--it remains characterized as four spatial dimensions.
Wait. The cosmic button-pusher established a displacement vector, V, and established its magnitude as invariant with respect to both orthogonal coordinates using the positive definite metric.
Of course the measure of X4 changed with the selection of a new metric that could account for the kinds of symmetries present in the new geometry associated with the objects added into the space by the apprentice. But the whole point is that the character of the space itself did not change. We started with four spatial dimensions and did nothing to change the spatial character of the 4th dimension. Now, if we could not rely on a one-to-one mapping, then things might be different.
I didn't say that the derived equation was a derivation of the metric (you boxed me in on that one once before--good job, too). I just said that the cosmic button-pusher knew that he had the right form for the metric after checking that result. After all, he knew that he had reciprocal coordinate systems, i.e., the red was the dual of the blue. He understood the implications of the contravariant-covariant relationship that was manifest. And that observation was the inspiration for the apprentice's choice of rules for positioning the new objects--he wanted to utilize the metric,
You are to be highly commended for your accomplishments. There's a lot to be said for learning in your own way at your own pace and having the ability to dig deep when something really interests you and picking up the pace as you wish. You seem to be more knowlegeable than I (and keep boxing me into corners with your insight), especially for all the course work I've had. But, I have all the necessary resources at hand with just about all of my text books, notes and other literature. I enjoy reading new books from time to time as well, such as Naber's "The Geometry of Minkowski Spacetime", Penrose's "The Road To Reality" and B. Crowell's "General Relativity" (excellent formal approach by Ben--that guy knows what he is doing--you should check it out on the internet).
And as soon as the metric is changed to respect the Lorentz symmetry instead of the Euclidean symmetry, it is impossible to maintain those supposedly established magnitudes for all displacements. You can, by carefully choosing only certain displacements, make it seem as though the magnitude is invariant for those particular displacements. But there's no way to do it for *all* displacements. It's not possible; it would amount to equating a positive definite metric with a non positive definite metric. It can't be done. So your claim that X4 remains a "spatial" dimension when the metric changes simply can't be sustained.
So you don't think that the metric is part of "the character of the space itself". That viewpoint is not inconsistent, but it's also not very common, and as I said earlier, trying to describe things this way will increase confusion, not reduce it. The standard viewpoint in relativity views the metric as part of "the character of the space itself", because you can't describe all of the physics without it. You can describe *some* properties without it, as you note: for example, the topology of the manifold. But you can't describe *all* properties that are needed for physics.
One property in particular that you can't describe without the metric is causality: without the metric there is no way to tell whether a given pair of events is timelike, null, or spacelike separated, so you don't know what causal relationships are possible or forbidden between them. This is one big reason why the standard viewpoint considers the metric to be "part of the character of the space itself".
I have, I agree it's a great site and pedagogical resource. It gets linked to fairly frequently around here.
No. That has never been my point. I've not implied that you can have the same measurement of distance going from one metric to another. All you have to do is to start with the Lorentz transformations, compute the products, and there's no way the physical distances from one point to another come out the same.
The mathematicians have given us the mathematical machinery that describes mathematical objects and relationships. They don't a priori give us the physical reality. The physicist takes the manifold, topology, set theory, group theory, linear vector spaces, etc., and uses them to describe reality as he envisions it, with the requirement that any models developed will be consistent with established theories of physics (unless new analysis can prove otherwise after experimental confirmation). So, the metrics of the mathematician do not automatically give us the physical character and quality of different directions in physical space. We have the abstract mathematical space of the mathematician, and we have the physical space envisioned by the physicist. Some physicists envision the 4th dimension as some kind of physical time. Other physicists say that is not comprehensible; there is no basis to assume any different physical character and quality to the 4th dimension that would make it any different than the normal X1, X2, and X3.
Of course we need the L4 space with its metric to make intelligible the physics hiding in the manifold. That's why the cosmic button-pusher was initially confused with the arrangement of 4-dimensional objects placed by the apprentice. He initially viewed the assortment of 4-D objects in the context of his original Euclidean metric induced on the manifold. But once he switched over to the relevant L4 metric all of the invariances came into play manifesting the illusion of physical laws (resulting from the apprentice's ingenious placement of the 4-dimensional objects).
In spite of this situation, it is not correct (in the view of the initial post here) to say that the metric accounts for the X4 as being either physical time or physical space. But yes, the metric is intimately associated with the revelation of the physics manifest on the manifold. But: It is the physics first (the very special arrangement of the 4-D objects) that prompts for a successful selection of a metric. The L4 metric has been revealed to us, but only because the 4-D objects have been arranged in that very special way.
A couple more comments: The metric does not place a preference on the quality of the dimensions. The traditional view among physicists is the one PeterDonis has been advocating. Although the mathematical system applied in desribing special relativity theory does not force this view, Minkowski himself embraced it. Most physicists embraced the idea of time as the 4th dimension. However, that did not mean that they did not embrace the idea of the block universe. Weyl wrote: "The world does not happen, it simply is." Einstein apparently subscribed to this view as well (everyone always references Einstein's letter to the wife of his close friend, Besso, at the time of Besso's passing).
Typical of sentiments in the early years of special relativity is the commentary from the writings of Sir James Jeans on Space-Time unity (book "Physics and Philosophy): "The physical theory of relativity suggests, although without absolutely conclusive proof, that physical space and physical time have no separate and independent existences; they seem more likely to be abstractions or selections from something more complex, namely a blend of space and time which comprises both.
This is exactly the view that I've tried to refute with the beginning post of this thread.
So, the two versions of block time: 1) A four-dimensional universe all there at once with physical time as the 4th dimension and 2) A four-dimensional universe all there at once with the 4th dimension as just another "physically spatial" dimension (we use "physically spatial" to avoid the confusion of the meaning of the mathematical abstract space that implies particular metrics--metrics that really do not force X4 to be either physical time or physical space). This physically spatial 4th dimension could be accompanied either by a 3-D consciousness moving along a world line at speed c, or it could be accompanied by a consciousness that exists simultaneously all along the world line.
By the way, a point made by Einstein ("Albert Einstein - Philosopher-Scientist", Library of Living Philosophers, Edited by Paul Schilpp": "First a remark concerning the relation of the theory to 'four-dimensional space.' It is a wide-spread error that the special theory of relativity is supposed to have, to a certain extent, first discovered, or at any rate, newly introduced, the four-dimensionality of the physical continuum. This, of course, is not the case. Classical mechanics, too, is based on the four-dimensional continuum of space and time. But in the four-dimensional continuum of classical physics the subspaces with constant time value have an absolute reality, independent of the choice of the reference system. Because of this [fact], the four-dimensional continuum falls naturally into a three-dimensional and a one-dimensional (time), so that the four-dimensional point of view does not force itself upon one as necessary. The special theory of relativity, on the other hand, creates a formal dependence between the way in which the spatial co-ordinates, on the one hand, and the temporal coordinates, on the other, have to enter into the natural laws."
And again, he is expressing a concept we've tried to refute in this thread. He could just as easily regarded the 4th dimension of Newton as a physically spatial dimension with time as a parameter. Nevertheless, his view gives me an opening to make a point of rebuttle to the notion that the metric detemines the physical nature of a dimension.
In Einstein's example here we have a positive definite metrice (the X4' and X1' axes are not slanted symmetrically in accordance with the Lorentz boost at all. If a positive metric is associated with physically spatial coordinates only, then Einstein could not have his "time" as a 4th dimension in the Newtonian Euclidean 4-D space.
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