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

PeterDonis

Was there supposed to be a link here? I'm not seeing one.

Edit: Ok, I see that the last paragraph of the post is supposed to be the excerpt. It would still be nice to have a link to the entire review.

PeterDonis

You do realize that that last parenthetical phrase by itself is a minus for credibility...

The wording of the description of Godel's "remarkable proof" is already a red flag: the author is being way too dogmatic about what Godel's proof "means". The author appears to be a philosopher, not a physicist, which IMO is another reason to take what he says about the application of such proofs to physics with a grain of salt. John Stachel, a physicist, reviewed the book, and his review is online here:

http://www.ams.org/notices/200707/tx070700861p.pdf

He appears to have similar misgivings about the author's flat assertions about what Godel actually proved. This review, btw, also has an interesting quote from Einstein:

"Time and space are fused in one and the same continuum, but the continuum is not isotropic. The element of spatial distance and the element of duration remain distinct in
nature…"

bobc2

Yes. I was just getting ready to post Stachel's reference when I saw your post. Stachel (Boston University physicist?) is definitely in the strongest disagreement with Godel's logic on the role of time in relativity, as are many physicists--particularly those of the logical positivist bent. And Stachel has no regard at all for Yourgrau (yes, he is definitely a philospher). I think that Stachel's view is representative of the majority of physicists today.

However, I have a hunch that more and more physicists--at least among those interested in the subject at all--are beginning to have second thoughts about the concept of time and the 4th dimension in special relativity. Probably most physicists are too involved in their own specialty to give much thought to these questions. My advisor certainly thought it a waste of my time to worry about it. He had no time for those kinds of discussions, and he devoted all of his time to relativity and published often in Phys Rev.

You would need more detail from Einstein to be sure exactly what he meant here. He could be referring to 4-dimensional physical world lines that are accompanied by conscious over the entire world line in a way that results in a local psychological flow of time experience. By the way, Stachel was in agreement with Godel that there is no global time in Godel's universe model. Stachel thought that was irrelevant, because the flow of time is local--it is associated with local world lines.

One should not lose track of the requirement that material objects must extend into the 4th dimension if different observers simultaneously exist in different cross-sections of 4-D, each including different instantaneous 3-D volumes of the same 4-D object. And it is not comprehensible that those different 3-D volumes are made of some kind of mixture of space and time--they must indeed be cross-sections cutting across at different angles. (see my sketch illustrating this in the very first post of this thread)

There is no argument about whether the cross-sections are spatial--they are of course--X1', X2', and X3'. How you could get those different 3-D volumes to all come out as spatial in all three dimensions when they were cut at totally different angles from a 4-D object without a spatial quality in the 4th dimension defies logic. I don't think you can cite a reference where Einstein dismissed the notion of 4-D objects. What would it mean to have a 4-dimensional object whose 4th dimension is not spatial?

The reason the continuum is not isotropic is because of the way 4-D objects extend into the spatial 4th dimension (X4)--extending billions and trillions of miles--as compared to the relatively miniscule extent along X1, X2, and X3. And then the consideration of the different relationships of X4 and X1 with respect to the light cone.

bobc2

PeterDonis, You never did offer an explanation as to how you could have simultaneous different cross-section views (by two different observers) of the same object unless that object was actually a 4-dimensional object.

PeterDonis

I agree we can't be sure without more context (there is a bit more context in Stachel's review), but I don't see anything to indicate that Einstein was talking about the psychological flow of time at all. It looks to me like he was making a direct reference to the fact that "time" as a dimension of the 4-D manifold is different, physically, from space.

Yes, in the sense that Godel's spacetime can't be sliced up into a "stack" of spacelike hypersurfaces of "constant time" that each worldline crosses only once.

Basically, yes. I think he was also referring to causality and the limitation of signal speed to the speed of light. See below.

This is how things are usually described, but strictly speaking, it's not true. These "instantaneous 3-D volumes" are not direct physical observables; nobody actually observes a full 3-D volume at an "instant of time", because of the finite speed of light. The instantaneous 3-D volumes are mathematical constructs, and as useful as they are as part of a model, one needs to be careful about what they might or might not imply about the nature of "reality".

What we actually observe is what is in our past light cone, and in so far as objects we interact with must "extend into the 4th dimension", all we can directly observe is that they extend into our past light cone. As we look further into the past, we can consider intersections of the interior of our past light cone with surfaces of "constant time" as being snapshots of a portion of "space" at that time in the past, but it's still only a portion, and the "space" is still a mathematical construct. The same goes for the different "spaces" that are formed by slicing through our past light cone at different "angles", as would be done by someone just passing by us now at some appreciable relative velocity.

This sort of thing is what I think Stachel was getting at when he talked about "time" being local. Once you have the mental model of spacetime as a "stack" of 3-D spaces formed into a single 4-D manifold, it's natural to ask questions like "what is happening in the Andromeda galaxy *right now*"? But these questions have no well-defined physical meaning. Once you realize that different states of motion correspond, in the model, to "slicing" the 4-D manifold into a stack of 3-D spaces at different angles, it's natural to ask further questions like "since what time 'now' is in the Andromeda galaxy can change depending on how I'm moving, doesn't that mean that the whole, entire worldline of the Andromeda galaxy must already 'exist', so the Andromeda galaxy really extends through the 4th dimension as well as the other three"? But those questions don't have any well-defined physical meaning either. As I said in an early post in this thread, I view the "block universe" as a *model*, not as a claim about "reality". As a model, it clearly distinguishes between X4 and the other three dimensions.

bobc2

You are probably right about that.

And that was the fundamental basis for Godel’s thesis. His string of logic supposedly deduces that if you don’t have global time, then you don’t have time at all, that is as a global 4th dimension. Any notion of a 4th dimension (either time or spatial) must exist in that form over the entire 4-dimensional space. Yes, you can have a local time concept having to do with consciousness associated with the local world line. But, that is not a description of time as a 4th dimension. That is a description of consciousness coupled to a 4-dimensional object that extends into a spatial 4th dimension.

I don’t think Stachel thought carefully through Godel’s position.



I wouldn’t say it is not true. It could be true, because it is consistent with the special theory of relativity. We already have our 3-D prototype for space, and we can easily comprehend an extension of that concept to include the 4th dimension along with the general idea of a close link between time and consciousness as the consciousness couples to the 4th dimension (either moving along world lines at light speed or else coupling simultaneously all along the entire world line).

Of course we must acknowledge that it is possible that it is not true, i.e., the assertion that objects exist as 4-dimensional. I have no problem with that.

It’s just that a spatial 4th dimension seems more natural than a time 4th dimension. For the realists, the thrust of physics is comprehending what is “out there” as something physically real, existing as a space populated by “real” objects. It is not necessary to observe those objects to know they are there if measurements along with supporting analysis sufficiently imply an existence. And that extence concept must at the very least be compatible with special and general relativity.

This is in contrast to the idealistic notion of “things” existing in the mind (a concept dangerously close to solipsism). My graduate philosophy of physics professor told our class that the class room next door did not exist until he entered that room and observed it (nothing exists unless it is being observed). He had PhDs in physics and philosophy and seemed to be a disgruntled ex-physicist. He was always in attendance at our physics department Friday colloquium.

This has never been an absolute criterion for implying the existence of real objects. We have a scientific method capable of inferring objects that are not directly observable. If I began enumerating those objects it would make for a very long list. And physicists routinely take into account the speed of light in constructing configurations of objects throughout space. A physicist friend at The University of Texas who works on the RHIC Star project at Brookhaven told me that they had data that showed that the gold nucleons in their collision were flattened into a near-disk just before the instant of collision with another gold nucleus. (did the disk of gold nuclei really exist in the reference frame of the lab?) It reminded me of George Gamow's limmerick: There once was a man named Fisk; Whose fencing was exceedingly brisk; So fast was his action; The Fitzgerald Contraction; Reduced his rapier to a disk.

Of course we must always be careful when describing a key component of external objective reality. It would appear that much more care would be warranted when considering notions of time as a 4th dimension.

I don’t think physicists depend on that as a criterion for the existence of real objects in the world “out there.” That would mean that as I sit here, at a given instant, only objects that existed in my past really existed (but right now they don’t exist), because I get the evidence belatedly. This implies that objects do not exist in my present volume of simultaneity, because a finite amount of time will elapse before the light arrives. The sun does not exist right now because it will be another 8.3 minutes before the light arrives.

So it seems that you reject the concept of the simultaneous volumes (“planes”) of simultaneity described by special relativity. That would seem to imply that you would reject the concept of time dilation and length contraction. Particularly length contraction, since both ends of the beam in the simultaneous space of special relativity could not be observed simultaneously.

When starting this thread, I probably should have stipulated that we are only interested in discussing the relative merits of a 4th dimension having the essence of space as compared to the 4th dimension having the essence of time--and discussing this with the stipulation that, either way, we base our discussion on the assumption that we have accepted the concept of a real objective external world. That means we would agree that every observer experiences existence in an extended real external world of 3 dimensions.

I don’t consider that a valid criterion for the existence of objects, whether those objects exist in angled slices of space-time or not.

Within the context of special relativity, and with the criterion that all observers occupy a 3-dimensional space as described by the theory of special relativity, those questions have very well defined meaning. I think the ideas you’ve drawn on come out of the legacy of the old Vienna Circle of logical positivists, the philosophy of which seems to have been largely discredited—even by current philosophers. The logical positivists denied reality for anything not directly observed. They denied metaphysical concepts, generally.

The point I’ve been trying to make is that, given that every observer occupies an external 3-dimensional world, and given the theory of relativity, then a spatial 4th dimension is implied.

I agree with you that if you cannot assume a real external world for a given observer, then the kind of analysis we've been doing here is not meaningful.

Yes. But, it is a model that is not prominent at all as compared to the model with time as the 4th dimension. My point has been that the 4th spatial dimension concept is far more natural and consistent with special and general relativity.

PeterDonis

I did respond indirectly to this when I talked about the finite speed of light. Neither blue nor brown directly observe red at R1 or R2; they *reconstruct* that those were the respective positions of red in their frames simultaneous with the event of their crossing, based on light signals they receive later on. Normally this reconstruction is glossed over in SR because there's no need to go into it when solving practical problems; but if we're going to talk about fundamentals then the fact that the reconstruction is necessary is crucial, because it reveals an extra assumption that must hold for the reconstruction to happen as you illustrate: nothing must happen to red between R1 and R2 that alters his trajectory.

For example, suppose red fires a rocket between R1 and R2. Blue has been keeping track of red's motion and, when blue and brown pass, blue sends brown a message saying that red is at R1 simultaneous with their crossing, in blue's frame. Brown knows his velocity relative to blue, so he can calculate that red is at R2 simultaneous with the crossing in brown's frame. But when brown receives light signals that red emits after R1, he will see that his calculation was wrong: red's worldline never actually goes through the event R2, and if brown reconstructs his surface of simultaneity at the event where he passed blue, he will find, on the basis of later light signals, that red actually crossed that surface at some other event, R3.

The point of all this is that the view that red's worldline between R1 and R2 must "already exist" when blue and brown cross is a *model*, not reality; and the model may need to be updated as more data comes in. In theoretical problems, when we're just trying to study the logical implications of a model, we can stipulate that things like red firing his rocket don't happen; but in the real world, we can't do that. We have to recognize that there is always more data that hasn't come in yet about what is happening "now" outside our past light cone; and until we receive that data, we can't treat those events as having "happened".

PeterDonis

Not true. The concept of timelike vs. spacelike (vs. null) worldlines is local. Godel's universe still has local timelike worldlines and local light cones and local causality; it just so happens that in this universe the topology of the "time" dimension is closed, like a circle, instead of open, like the real line.

The local concept of timelike vs. spacelike vs. null worldlines does not require a concept of consciousness.

I think he understood it quite well; he just also understood its limitations.

Again, you keep on using the word "spatial" but you have never justified it, at least not as you are using it here. Nobody is disagreeing that we can treat the universe as a 4-dimensional manifold, and nobody is disagreeing that, in every respect except the non-positive-definite metric, this manifold has the properties of a "space". But you are trying to go further and claim that the X4 dimension is *no* different from the others, even with the metric taken into account; and that is simply not justified. Unless you can justify it, you should not be using "spatial" in this sense.

If you are just talking about a theoretical model, sure, this is fine. But if you are trying to talk about the actual universe, then no, you cannot make this claim as you are making it. See my previous post about red and blue and brown; until blue and brown receive red's light signals from R2, they can't know whether red's worldline actually passes through R2. They can't even know for sure that red still *exists* at R2--his rocket may have exploded, or he may have fallen into a black hole. You can't just ignore these possibilities if you are talking about fundamentals.

I am not making any such claims. I am merely pointing out the limitations of what any given observer can know at a given point on his worldline. I am not contesting the fact that other objects exist and that we can know about them by observing them, and that we can *assume*, for practical purposes, that they exist even when we don't observe them; but such assumptions are always subject to falsification if new data comes in (for example, if blue and brown see light signals from the explosion of red's rocket before he reached R2).

Once again, I am not making any such claims. I am merely claming that, until we get light rays from the sun 8.3 minutes from "now", we do not *know* for certain what the sun's state was "now". The sun could have exploded just 10 seconds ago as I sit here "now" writing this; we won't find out about it for another 8 minutes, so we are going on the best evidence we have right now if we say the sun exists "now", but that doesn't change the fact that we would be wrong if the sun had exploded 10 seconds ago. I am not trying to say anything mysterious; I am merely pointing out that, when talking about fundamentals, we must be careful to recognize the limitations of what we can know.

When have I said any such thing? I am only saying that those planes of simultaneity are *constructs* in a model, and the model only gets validated (or changed) progressively as we receive light signals from new regions of spacetime that were previously outside our past light cone.

Observing time dilation and length contraction can be done purely by means of light signals; there is no requirement to observe things instantaneously. Read Einstein's original thought experiment about this; he carefully states everything in terms of light signals.

I am not questioning that a real external world exists. See above.

It is a valid criterion for asserting *knowledge* of whether something exists; otherwise your model is asserting things which might later be falsified.

Good, then define it. As soon as you do, I will be able to propose a different definition, and there will be no experiment we can possibly run that says either one of us is right or wrong. Of course we can arbitrarily adopt any simultaneity convention we want to; but the convention will be just that, arbitrary. It will have no physical meaning, in the sense that no experimental result can possibly be affected by choosing one convention over another.

I may sound like a Vienna Circle positivist to you, but that's because you're not reading carefully enough. I am not saying that it is never justified to talk about things that can't be directly observed. Blue and brown can certainly talk about what red is doing between R1 and R2, even though they haven't seen any light signals from that portion of red's worldline yet. Who's to stop them? But until they *do* receive those light signals, they don't *know* for certain whether what they were saying was right or wrong.

4th dimension, yes. "Spatial", no; see above.

bobc2

It seems that there is this concept that one cannot know that he is existing in a volume (plane) of simultaneity, because the only things experienced (observed) directly in the mind are thoughts based on information arriving from the past. There is no external existence within ones instantaneous Lorentz space that can be directly observed until sometime later—then it is too late—it is already in your past by then. This idea is certainly compatible with the Vienna logical positivist school and gave strength to the idealists like Berkeley who maintained that reality existed within the mind (ultimately, the mind of God). The idea is favored by solipsists as well (Einstein warned about falling into the trap from which there would be “…no escape from solipsism”).

Here, we will appeal to Hawking’s more flexible approach he refers to as “model-dependent realism: the idea that a physical theory or world picture is a model (generally of a mathematical nature) and a set of rules that connect the elements of the model to observations. This provides a framework with which to interpret modern science.” This is from “The Grand Design” by Stephen Hawking and Leonard Mlodinow. Actually, PeterDonis, I believe the ideas you are advancing here are generally much more in line with Hawking. So, I should not be accusing you of advocating logical positivism or solipsism. (Perhaps I owe you and apology on that count.)

So, here is a space-time diagram with a sequence of events. The diagram represents a sort of sequence of thought experiments in which observers in a Lorentz space send messages back and forth as they move through 4-dimensional space. At the end of the sequence of experiments, they get back together and compare notes to see if there is a basis for knowing that each other existed in any of the Lorentz space simultaneous external world volumes.

In the sketch below the brown, light brown, and blue observers are together. They are initially together in the brown rest frame, synchronizing space-time markers, displays of actual distance traveled in 4-dimensional space, referenced from a point assigned as zero distance. The distances displayed (and photographed at selected points) will correspond to distances traveled along the respective world lines.

All three are together at event 1 (events in the space-time diagram are brown, light brown, or blue circles) where they synchronize their distance markers). It is planned that the light brown guy and the blue guy will move to a new position that puts them in brown’s instantaneous plane (3-D volume) of simultaneity at event 8. Light brown and blue have used Lorentz transformations to assure that their distance markers display the same values at event 8 as the brown guy’s markers display at event 2. Light brown and blue both transmit pictures of their displayed values so that brown can validate their numbers when he (brown) arrives at event 3 in the space-time diagram (brown calculates how far he has traveled along the 4th dimension since leaving event 2.

At event 9 the light brown guy transmits a photo of his distance display, which is received by blue at event 14 and received by brown at event 4. The blue and brown guys do calculations that demonstrate that events are still occurring in agreement with theoretical physics. At this point the brown guy is able to confirm that the light brown guy existed in his plane of simultaneity back when he (the brown guy) was at event 3). From the data received from the blue guy at event 4 he is also able to determine that the blue guy was also in his (brown’s) plane of simultaneity back at event 3.

Just one experiment doesn’t seem enough, so they continue acquiring data.
The blue guy arrives at the brown guy’s position at event 5. So, here the blue guy and the brown guy simultaneously occupy the same position at the intersection of their X4 axes. Special relativity tells them that if the light brown guy really still exists, then the light brown guy must exist at event 11 in blue’s instantaneous 3-D space volume, while simultaneously existing at event 12 in brown’s simultaneous space. However, as PeterDonis points out, they can’t really be sure, because they have no way of getting information from those events instantly while they are at event 5. They must wait until later for confirmation from the light brown guy.

Brown gets his confirmation when he arrives at event 6, receiving the picture of light brown’s event 12 photo of his distance traveled along the 4th dimension, which is exactly the same distance that the brown guy recorded for his own trip when at event 5. Thus, brown concludes that the light brown guy must have been in his simultaneous space which included both event 5 and event 12 simultaneously.

The blue guy has to wait until his event 16 for confirmation that light brown was at event 12 simultaneously with event 5 (when both blue and brown guys were simultaneously at event 5). Of course the blue guy saved a copy of brown’s distance position along brown’s X4 dimension when they were together at event 5. So, now he had confirmation that light brown was in brown’s simultaneous space, i.e., both brown and light brown were in the simultaneous space of events 5 and 12.

But, now, blue asks whether the light brown guy was in his (blue’s) simultaneous space when blue was at event 5. Fortunately, light brown included a photo of his X4 position corresponding to event 11. Light brown and blue both used Lorentz transformations to figure out what each other’s positions should be along their respective X4 axes when blue arrived at brown’s position, event 5. Light brown transmitted his computations that he had made about what blue’s X4 position should be when light brown was at event 11. And blue computed the X4 reading that light brown should have when he (blue) was at event 5.

Light brown and blue wanted to be sure science was working right, so they took photos of their respective X4 distances corresponding to blue’s simultaneous space at blue’s event 16. Light brown did calculations (by prearranged agreements) at event 13. At event 17 blue found that light brown was in his (blue’s) simultaneous space when blue was at event 16 and light brown was at event 13.

Without including it in the space-time diagram we have all three observers get together at the end of the experiments and review all of their data. They conclude that sure enough, when the brown guy and blue guy were at event 5, the light brown guy simultaneously existed at event 12 (in brown’s simultaneous space) and event 11 (in blue’s simultaneous space). They then conclude that the light brown guy is actually a 4-dimensional object, and-- by extension--they all are. Thus, we have a model in which objects are 4-dimensional extending into a 4th spatial dimension.

Further, once it is recognized that we have a 4-dimensional spatial universe populated by 4-dimensional objects, it is obvious that the objects do not move. There is a perception of motion related to 3-dimensional cross-sections of the 4-dimensional objects, and this is compatible with the notion of consciousness moving along the 4-dimensional world line of an observer, the consciousness having a deep connection to the perception of the "flow of time." Finally, it implies two possible models of consciousness: 1) A 3-D consciousness that moves at light speed along the observer's world line, or 2) A 4-D consciousness that is coupled to the 4-dimensional material structure of the observer over the full extent of the observer world line.

These experiments (albeit thought experiments) would satisfy the criteria for a good model fit, at least in agreement with Hawking’s criteria:

1. Is elegant
2. Contains few arbitrary or adjustable elements
3. Agrees with and explains all existing observations
4. Makes detailed predictions about future observations that can disprove or
falsify the model if they are not borne out.

To satisfy number 4 in a more convincing manner, The three observers could have sat down together and planned the entire set of experiments in advance, predicting all of the readings that would be observed at each of the events in the scenario.

PeterDonis

In general, yes, what I've been saying is consistent with Hawking's model-dependent realism. As I've said before, I'm certainly not trying to claim that the only things that "exist" are those things we directly observe.

I have no problem in general with your scenario; as you've specified it, all three observers could, as you say, plan the whole sequence of experiments in advance, and (provided nothing external interfered) collect the data to confirm that their model was correct.

The only thing I would add is to take note of that parenthetical statement I put in about something external interfering. In the real world, there is always the possibility of something external interfering, and when that happens, we can't know about it until light signals from the external event reach us. For example, suppose that, when brown, light brown, and blue actually run the experiments as you specify, something unexpected happens: a powerful laser pulse comes in from the negative x direction and hits light brown at event 12. The impulse of the laser pushes light brown into a new trajectory that is moving to the right (i.e., in the positive x direction) with the same velocity as blue, relative to brown.

Light brown knows immediately at event 12 that his trajectory from event 12 on will not match what they all predicted before the experiments were begun. But brown will not know this, and will not be able to base any predictions on it, until event 6, when he sees the light from event 12 and realizes he was hit by the laser pulse. And blue will not know it until event 16, when the light from event 12 reaches him (and of course brown can't send him a signal any faster).

Now, how will brown and blue reconstruct the effect of the laser pulse? Brown, reconstructing events after he sees the light pulse from event 12, will say that light brown was hit by the laser simultaneously with event 5, i.e., at the same time as brown and blue were passing each other. But to blue, event 5 happens *before* event 12, so even though blue and brown are co-located at event 5, they will not agree on whether or not light brown had been hit by the laser pulse "at the same time" as they were passing each other. This is why statements like "event A and event B happened at the same time" can't have any physical meaning unless they are qualified (by referring them to a specific observer's frame); two observers can disagree on them without having any way of settling their disagreement by making any further observations.

bobc2

Exactly. And that is the whole point behind the understanding the 4-dimensional universe. After the laser intervenes, Blue and Brown guys will both recognize (later -- after the fact) that the laser impulse at event 12 could not possibly occur in each of their planes of simultaneity corresponding to those planes in which their X4 axes intersect. They would do the special relativity calculations (later, after the fact) and, in retrospect, know that blue did not yet have a laser event in his 3-dimensional world.

One has to be careful with the language. I agree that when we say that the two observers, blue and brown, are together at "the same time," different people could come away with different interpretations about the meaning of that phrase. In this instance it certainly does not mean that the blue guy and red guy would each be reading the same proper time on their world line clocks--we can't use the phrase "same time" in that sense (just as we have different "age times" in the reunion of the twin paradox).

But, there is no confusion in understanding that there are two different events, 11 and 12, and that as blue and brown exist together (next to each other), event 11 is in brown's world but it is not in blue's world; and event 12 is in blue's world but it is not in brown's world. And it would appear that we can only have such a circumstance if the light brown object is 4-dimensional (allowing different 3-D cross-section views) and extends into the 4th dimension.

PeterDonis

I do not agree with this way of stating it. I would state it as follows: until blue or brown receive light signals from events 11 or 12, they don't know what happened at those events, so they can't say anything about them as *actual events*. They can *predict* what they think will happen at those events, based on what they know prior to receiving light signals from them; but those predictions are only tentative, and do not require any belief that the events are "real" or "actually happened" prior to receiving light signals from them.

Blue and brown can build *models* of their "worlds", and in those *models*, yes, event 11 is "in blue's world at event 5" but event 12 is not, while event 12 is "in brown's world at event 5" but event 11 is not. But those models are just that, models; the "worlds" in them do not have to be "real worlds" in order to serve their function as models. There are certainly "real events" 11 and 12, but blue and brown, at event 5, *don't know* what those actual, real events are; they only know what their models say about them.

It may well seem that I'm belaboring this, but I'm only doing so to make what I think is a fundamental point about what the "block universe" model does or does not claim. As I understand it, the "block universe" model does *not* claim that, for example, events 11 or 12 have to be "real" or have "actually happened" from blue or brown's point of view at event 5. All that the "block universe" model requires is that blue and brown have a *model* of events other than those along their own worldlines, such as events 11 and 12; in other words, blue and brown have a *model* of light brown traveling along his worldline, and that model will recognize that different events on light brown's worldline (11 and 12) are seen as "simultaneous" to blue and brown at event 5. But the model must also recognize that no physical experience blue or brown can have at event 5 can possibly be affected by what happens at events 11 or 12, so there is no need or reason for blue or brown to claim that events 11 or 12 have "already happened" or "are happening *now*" at event 5.

Similar remarks apply to what blue or brown might say after receiving light signals from event 12 showing the laser pulse hitting light brown and changing his motion. Blue or brown can certainly reconstruct events and show that event 12 was simultaneous with event 5 in brown's frame, but happened *after* event 5 in blue's frame. But what, if anything, should they conclude from this? My answer is, nothing of any consequence. The fact still remains that no physical experience blue or brown could have had at event 5 could possibly have been affected by what happened at event 12.

In other words, these 3-D "worlds" you are talking about, as tempting as they are conceptually to our intuitions, are actually not the right things to concentrate on, even in the "block universe" model. The crucial "boundaries" in spacetime are not surfaces of simultaneity, but light cones; and the crucial relationships between events are not simultaneity relationships but causal relationships--timelike, spacelike, or null separation. Events 11 and 12 are both spacelike separated from event 5, and *that* is the crucial fact about them. The "block universe" model is not supposed to change that.

bobc2

But, the whole point of the three observers planning out the experiments, with predictions about the outcomes, was to show that the light brown guy actually existed, in a real physical sense, at events 11 and 12, in the simultaneous planes of blue and brown while blue and brown were next to each other. The implication of the light brown guy existing as a 4-dimensional object is profound and cannot be dismissed as a mere mathematical example--dismissed as a model without relevance to a real world.

When blue and brown are next to each other, either the light brown guy exists as a real world object or he doesn't. Then, if he does exist in the real external world of blue and brown when they are next to each other, one must reconcile the data that shows light brown existing in the real world of blue as well as the real world of brown. All of the data they analyzed after the fact leads to the conclusion that the light brown guy existed at events 11 and 12, even though it was impossible for blue and brown to confirm that at the time when blue and brown were next to each other and in their respective planes of simultaneity.

Given the difficulty of the concepts that we have been wrestling with, I don't think you are belaboring this at all. We have clearly been representing two different assessments of the significance of the scenario outlined with my sketch above. I believe the issue of the existence of physical objects in a real external world is crucial to the foundations of physics. And if those objects exist as 4-dimensional objects occupying 4-dimensional space, this is indeed profound and is certainly worth discussion.

DaleSpam

bobc2, I have noticed references by you to the idea that it is possible to nave a non-positive definite metric in four spatial dimensions. Do you have a mainstream scientific reference for that.

I always thought that a non-positive definite metric implied at least one spatial dimension (positive signature) and one temporal dimension (negative signature). I don't see how you can possibly get a non-positive definite metric if they are all spatial and therefore all have a positive signature.

PeterDonis

See, here is where you appear to me to be going beyond what the model justifies. You use the word "when". This implies a simulaneity convention. Which one, blue's or brown's? In other words, in the sentence just quoted above, are you talking about light brown at event 11, or light brown at event 12? Or both? Or neither? Or the entire length of light brown's worldline from the past light cone of event 5 (where blue and brown cross) onward? As you appear to be viewing things, the statement quoted above has no well-defined meaning until you have specified exactly one of the above possibilities.

Whereas my answer is, it doesn't matter. The latest event on light brown's worldline that blue and brown can have direct evidence of at event 5 is the event where light brown passes through event 5's past light cone. *All* of light brown's worldline beyond that is unknown to blue and brown at event 5. That's the physical fact. That does not mean that portion of light brown's worldline does not "exist"; it simply means that blue and brown, at event 5, don't *know* what that portion of light brown's worldline consists of. That means that there are things that our normal, conventional ways of speaking would lead blue and brown to say, at event 5, that from the standpoint of physics, simply can't be said from their actual knowledge at that event.

For example, suppose the laser pulse I postulated hits light brown at event 12 and destroys him, instead of just changing his motion. Brown and blue, when they later find out about this, would have to disagree, if they follow your prescription, about whether light brown was killed "at the same time" as they passed each other. Brown would say yes, he was killed at that time; blue would say no, he wasn't killed until "later", so at that event, light brown was "still alive". But if you were blue or brown, would you care? The primary fact about event 12 as it relates to event 5, as I said before, is that the two are spacelike separated; nothing blue or brown could do at event 5 would have kept light brown from getting killed at event 12. Given that inescapable fact, if you were blue or brown, would you bother arguing about whether light brown was killed "at the same time" as you passed each other, or not?

The point of all this is that when you assert what you asserted in the quote above, you are asserting something whose truth might depend on events outside blue or brown's control at event 5. Brown, on your view, would have to say that no, light brown did *not* exist as a real-world object when brown and blue passed; he had just been killed that instant. Whereas blue, on your view, would have to disagree, and say that no, light brown *did* exist as a real-world object when brown and blue passed; he wasn't killed until "later". But on my view, if they are arguing about this at all, they are missing the whole point; they are arguing about something for which there simply is no invariant "fact of the matter"; whether light brown still "existed" at event 5 is frame-dependent.

Yes, when they receive light signals from the appropriate events, blue and brown can now know for sure what happened to light brown at events 11 and 12, including light brown being killed by the laser pulse at event 12. So their model of that portion of spacetime is now in full correspondence with what actually happened in that portion of spacetime. And even then, whether or not light brown "still existed" at event 5 will be frame-dependent; brown and blue will reconstruct different answers to that question, if they insist on treating their frame-dependent "viewpoints" and surfaces of simultaneity as being "real" instead of just features of the model. And they will therefore be arguing about something which, in my view, is pointless to argue about.

And, once again, I am not saying and have never said that the real external world, and the objects in it, do not exist. I have merely been trying to draw a critical distinction between those real, actual objects, and what really, actually happens to them, and our *knowledge* of those objects and happenings when we are spacelike separated from them.

bobc2

I'm afraid I cannot cite a reference that specifically discusses spatial coordinates with an indefinite metric--other than those in the context of a purely abstract mathematical application. I've used an indefinite metric quite often in mechanics applications, but, again, that is in the context of an N-dimensional abstract space--although the coordinates do relate to physical translational displacements and rotations on a real structure such as a space shuttle subsystem (a space with around 20,000 to 100,000 or more coordinates is not unusual). In fact slanted coordinates are used, having some similarity to our L4 space, but my coordinates were not resticted to the symmetric angular positions about a 45-degree line ("photon worldline"). However, a dual space is utilized (leading to a covariant-contravariant-like situation similar to coordinates associate with Lorentz boosts and the indefinite metric).

It is interesting to me that of the large number of references I've encountered on the subject of block universe, four-dimensionalism (a philosopher's favorite), presentism, eternalism, etc., it is quite rare to see an author explicitly refer to the 4th dimension as "spatial." You often find comments about 3-dimensional objects physically extending into the 4th dimension, but that 4th dimension is usually identified only as time (if identified explicitly at all).

It has always seemed elemental to me that if you have a material object extending into a 4th dimension, maintaining its material character with the extension, that 4th dimension would necessarily be spatial. That does not mean that you could not have time associated with that dimension in some manner (consciousness coupling, etc.), but how do you have a material 4-D objects existing without the 4-dimensional space?

But back to your question. I don't think the indefinite metric specifies the essence of the 4th dimension. That signature is there because it is required to give you the orientation of the X1 coordinate and X4 coordinate with the constraint that they must always be rotated symmetrically about the worldline of the photon. That orientation has nothing to do with whether X4 is time or spatial.

The other geometric aspect of that is that nature has populated 4-dimensional space with 4-dimensional objects that are billions and trillions of miles long, always oriented in a 4-dimensional direction confined to the inside of a light cone. So, it's these geometric aspects of the 4-dimensional world that account for the special nature of the 4th dimension.

So, the signature of the metric is required to account for the geometry, and our measure of 4-dimensional distance (interlinked with our physics, the conservation laws) in this unusual 4-D world.

DaleSpam

I'm sorry, but this reply is not very informative. I still don't see how it is mathematically possible to have a (1,3) signature with four spatial dimensions. Your inability to produce a mainstream physics reference makes me suspect this is speculative; I would strongly urge you to critically examine your position.