A Case for the 4-D Space-Space Block Universe

Snip3r

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.

bobc2

Good to have you jumping in here, to have one who can relate to the issue at hand. Don't be a stranger.

bobc2

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.

PeterDonis

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.

bobc2

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).

PeterDonis

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.

bobc2

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.

bobc2

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.

PeterDonis

The actual metric we use in physics is not "of the mathematician". Mathematicians can come up with zillions of different metrics to put on a 4-D manifold. It's physicists who tell us that the metric that actually applies to our actual universe is the Lorentz metric.

Except that it has opposite sign in the metric. Do you know of any physicists who deny that, other than those who multiply it by i, which just moves the difference from one place (the sign of the term in the metric) to another (the kind of number used for the coordinate)? This is a physical difference, not a mathematical difference. Without it you don't have causality as we observe it, as I said before.

bobc2

Minkowski was Einstein's professor of mathematics.

That's exactly what I said in a previous post. And some physicists choose to let X4 represent physical time and others choose to let X4 represent a physically spatial dimension. This is more natural because it is easily comprehensible. The notion of a "mixture of time and space" is not a concept that can be physically comprehensible. We have our common sense notion of what physical space is from everyday experience. We also have a fairly definite psychological feeling about time. But picturing time as a physical dimension and then space-time as a mixture of space and time is not nearly as natural to handle conceptually as 4 physically spatial dimensions. It is not conceptually difficult to envision time as closely associated with consciousness.

Again, the metric associated with the Lorentz boosts we see diagrammed in the space-time diagrams is directly related to the orientation of the four coordinates of the 4-D physical space. There is nothing that implies the metric demands a physical time for X4.

I did not invent the block universe, nor the consciousness moving along world lines. Of course the ideas were present in the Davies and Hoyle references mentioned earlier. The earliest notion of a block universe without time of which I'm familiar was offered by Einstein's Princeton colleague, Kurt Godel (one of the foremost among mathematicians and logicians who gave us the Incompleteness Theorem). He solved Einstein's with world lines curving back on themselves and used the example to demonstrate a block universe without time.

Yes. Dramatic physical phenomena are manifest which are intimately related to the use of the L4 metric. But, again, the physical quality (other than the obvious geometric characteristics) of the 4th dimension are not among these. A concept of a block universe with four physical spatial dimensions is consistent with the L4 metric and special relativity.

PeterDonis

Yes. So what?

References, please? I'm particularly curious to see in what, if any, contexts the word "spatial" is used to refer to X4 or its equivalent. This discussion is mostly about terminology, so the actual usage of actual physicists is important.

Really? I seem to have no trouble comprehending it. Nor, I suspect, do lots of other relativity physicists who use the Lorentz metric all the time.

I could ask here, what does "physical time" mean? But that's really beside the point. I am not objecting to the usage of the word "space" as in "4-D manifold" per se; I am only objecting to the specific usage of the word "spatial" to imply that there is no physical difference between X4 and the other three dimensions. There is. It's a metrical difference, not a topological difference, but it's still a genuine physical difference.

If all you are saying is that spacetime diagrams help to visualize relativity problems, I have no objection to that. But spacetime diagrams are tools; you don't have to adopt any particular position on whether or not X4 is "spatial" in order to use spacetime diagrams.

No, he found a solution of Einstein's field equation that had closed timelike curves through every point. That's not the same thing as a "block universe without time". In a purely "spatial" universe (i.e., one with a positive definite metric), there are no such things as timelike curves, and there are no potential causal paradoxes associated with closed curves in any dimension (because there's no causality to begin with). Do you think Godel's solution would have been such a big deal if all it showed was that we can have spatial circles through every point? It was the fact that there were closed *timelike* curves through every point that was the big deal. But you can't even form the concept of timelike curves as distinct from spacelike curves at all, let alone closed ones, without admitting that there is a physical difference associated with one of the dimensions.

PeterDonis

So you think that causality is not a "physical quality" or "physical phenomenon" of any consequence, so there's nothing wrong with calling a timelike dimension "spatial". As I've said before, this terminology of yours is highly nonstandard and is likely to cause a lot of confusion if you try to use it. There's a reason physicists pay close attention to the difference between timelike and spacelike (and null) curves; your suggested terminology basically ignores it.

bobc2

Of course these are physical qualities. In the block universe model these characteristics arise from the geometry and orientations of the 4-dimensional objects that populate the 4-D space. The problem here is that a complete geometry-based physical theory is not yet available. Einstein was unable to complete his program. Kaluza tried. The string theorists are working on it. What I'm saying is that the causality is inherent in the geometry. And yes, we need the L4 metric to represent this mathematically. Still, just because we have some physical concepts implied by the L4 metric, that does not require we ascribe a physical time quality as representing the 4th dimension.

There's far worse confusion about the 4th dimension with the continuing representation that it is time. And the idea that you are reading time directly when you look at a clock is one of the confusion factors commonly sustained.

No. I just recommend qualifiers when using the standard terminology. Time-like, Space-like and null cone have specific mathematical and physical meaning that does not need to be compromised. And there is no damgage done when clarifying the distinctions among the competing concepts related to the 4th dimension in special relativity theory.

bobc2

I'll respond to your other comments later. In the meantime please discuss what you mean by

a) Time as the 4th dimension.

b) What is meant by "a mixture of time and space?" We know what that means mathematically, and we can easily see it mathematically when it is represented on a space-time diagram--its just a mathematical cross-section of a mathematical space-time.

But please offer some kind of description that would give us a concept of the mixture of space and time for which we would have no trouble comprehending. Discuss the quality or character of that mixture. I can comprehend in some sense the quality of space based on experience with X1, X2, X3. I can comprehend the notion of time from my direct psychological experience with time passing. How do you mix those two concepts and come up with a comprehensible concept? They are so different. The mathematical description is useful and necessary, but that alone does not provide a comprehensible concept.

If I had been deaf, blind and without the sense of touch all my life, I may not be capable of having a comprehensible concept of space. My care giver could tell me when he is moving me from place to place, but the concept of space should be much more difficult to comprehend for someone in that state.

I could comprehend a notion of time in that state. But, the notion of a mixture of space and time would be hopeless, even if my care giver could teach me mathematics, i.e., differential geometry and special relativity.

PeterDonis

In standard Minkowski coordinates on spacetime, one dimension has a different sign in the metric. That's the one we call "time".

Yes, that's what it means.

I may not be able to. There is no guarantee that every concept that can be described mathematically can be described in a way your intuition will comprehend. That's why we have mathematics.

Why do you need to? The mathematics provides enough of a description to predict the results of experiments. Why must there be a way of comprehending it that fits with your intuition?

For "comprehend" in the sense of "visualize", yes. But that's not the only way you could approach the concept.

But if you knew the mathematics, you would have a way of dealing with the concept that did not require you to "comprehend" it in the sense you mean.

bobc2

For now, here is one reference: "A World Without Time - The Forgotten Legacy Of Godel and Einstein" (a Discover magazine best seller).

Here is the writeup from the back cover of the book: In 1942, the logician Kurt Godel and Albert Einstein became close friends; they walked to and from their offices every day, exchanging ideas about science, philosophy, politics, and the lost world of German science in which both men had grown up. By 1949, Godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist. Einstein endorsed this result reluctantly, but he could find no way refute it, and in the half-century since then, neither has anyone else. Even more remarkable was what happened afterward: nothing. Cosmologists and philosophers alike have proceeded as if this discovery was never made. In "A World Without Time," Palle Yourgrau sets out to restore Godel to his rightful place in history, telling the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue from undeserved obscurity the brilliant work they did together.

I have another reference, a paper written by Godel for inclusion in a book dedicated to Einstein, the one I mentioned earlier edited by Arthur Schilpp. Schilpp mangaged to collect writings from a number of physicists, mathematicians and philosophers, all dedicated to commenting on Einstein's life and work, which formed one volume in Schilpp's series on Living Philosophers. I'll have to dig it out again and locate some pertinent excerpts from which Godel is describing his solutions to Einstein's equations of general relativity in the context of "destroying time."

bobc2

Here is an excerpt from a review of the Godel-Einstein book. I give it just because it explicitly refers to the "spatial dimension."

Yourgrau manages to convey fairly clearly what exactly Gödel demonstrated in his short paper, taking Einstein's theory of relativity and focussing on the knotty time-issue, presenting a world model in which he could show "that t, the temporal component of space-time, was in fact another spatial dimension". The implications and consequences are profound and far-reaching...