Starship Asterisk*

Seriously, around 1990 I remember the balloon was a bubble. The bubble expanded, and small bubbles bulged out here and there from the big bubble; when things went way over the event horizon, the small bubble was effectively pinched off into its own little expanding universe. These pinchings were effectively little bangs. What ever happened to that line of conjecture, theory, or analogy?

The whole point of using the balloon analogy was to explain to a layman how to visualize the universe with respect to the big bang and the expansion of space. The layman has difficulties visualizing a 4-D expanding universe that does not have a center. So, we tell the layman to imagine himself as a 2-D creature on the surface of a balloon. This creature can only see sideways, i.e., in 2-D, just as we can see only in 3-D. If you can come up with a better explanation or analogy that doesn't confuse the layman, please do so.

Gary

To question these dimensional analogies further:

I remember way back when, we had problems of the type "John has x apples. If Mary takes . . . , etc., etc. How many apples does John have left?" In many cases involving quadratic solutions, there were two answers, one positive, one negative. We were always told that the "real" answer was the positive one and the negative answer was just a meaningless artifact of the way that mathematics works.

In later years, other math problems also had "meaningless" answers to be ignored because "that's just the way the math works out but it has no significance in the real world".

Given many of the cosmological theories floating around which indicate 5, 7, 13, 26 or whatever-happens-to-be-fashionable-nowadays dimensions, how do we know that these are not artifacts of the mathematics that don't really mean anything but exist just because "that's just the way the math works out"?

Is there any "real" (whatever the definition of real is) evidence of the existence of dimensions greater than the number that we as humans actually experience, or is all of this just the result of playing with mathematics?

I'm not trying to denigrate anyone; I'd just like to get a feel for what is actually known from actual evidence, versus what is conjecture that just happens to be borne out by purely mathematical results that may or may not be germane in the real world.

iampete wrote:Is there any "real" (whatever the definition of real is) evidence of the existence of dimensions greater than the number that we as humans actually experience, or is all of this just the result of playing with mathematics?

The idea that the Universe is four-dimensional in structure, called spacetime, is fundamental to General Relativity. GR is strongly supported by multiple, independent observations and experiments. Google GR, read the Wikipedia article, and you'll get many examples.

I imagine more of an inflating ocean-type expanse that's getting 'bigger' as the material of matter and energy is fondled and grasped by gravity and other forces as you say 'interacts' if they are in proximity to eachother.

so then a big bang may not even be a true title if the matter/energy of stars and galaxies first started by the slow process of the CMB baking (in a sense) matter (and that matter clumping) and the chemical reactions of the earliest universal molecules becoming more and more intricate... in that sense I can understand the dimensional planes without making a certain point the center of the all. If space existed within a state of being just that, open space, without the energy churning furnaces of the stars and such, then its possible that matter made a slow start, cooked by the surrounding CMB... God has one of those new microwave ovens. Our universe is but a cookie from that oven. The galaxies many many chocolate chips... mmmmm



Is space just this inflating fabric of spacetime material that is made even larger by the interactions of matter, energy, gravity and whatever other forces at work we know and don't know of...? Will this fabric be taken outside and shook in the wind to get all the dirt out...? O.o!

I like the bubble theory... little universes popping up and splitting off. Its like the entire cosmos is cooking and the material being cooked off is creating everything around us thats forming in space.

Chris Peterson wrote:But really, it is the expansion of space itself, not the material that space contains.

This remark keeps on resounding in my head, for a few days. We observe that there is a relation between distance and average speed of the galaxies/clusters flying away from us, or the other way around: who is moving and who is not moving is impossible to say. There is a definite preference for increasing the distance. The BBT supplements this idea nicely and Steven Weinburg explains the 3 K radiation and the general ration between He and H. Most people happy and they continue with their every day live. Then Chris points out: space itself is expanding, and my every day is disrupted. "What does it mean?" "Which conequences?" Now i have some ideas i would like to share.

On a short distance scale the expansion speed is low and thus galaxies, which have a speed of their own, may approach us. In statistical terms: the variance in speed of galaxies within a cluster is large compared to the mean expansion velocity on the short distance. Nevertheless the underlying expansion of space is still there.

Your statement "space itself expands" fits within this image/idea. I may mis-interpret your statement, when i conclude that the fabric of space itself is expanding: Take two points in space, wait for a while, and the distance between those two points will have increased. At least at a very long time scale, of Mega of Giga years. But if the expansion of space is an generic property of space itself, not just on intergalactic scales, the consequence is that the distance between e.g. jupiter and the sun must have increased. Not much, since the distance between jupiter and the sun is small, but it will. Even the distance between the earth and the moon might increase due to the expansion of space, not just by the dissipation of inertia due to the tides.

"Ah", you (or somebody else) might say, "but the suns gravity field will prevent that jupiter will move away from the sun. " My argument is: "If the gravity between clusters of galaxies can not prevent that these will move away from each other, than the suns gravity certainly will not be able to prevent that jupiter will move away. "

The expansion of space might be expressed as a strain rate: the change in time of the strain of space. When this strain rate is constant, it is easy to integrate it in time and determine the actual strain over e.g. 1E6 years. Multilply it with the distance between jupiter and the sun, and one can calculate how many meters or micrometers jupiter will have moved away from the sun. Moreover, apply this method on the earth moon system and measure the elongation of the distance between the earth and the moon. The laser reflectors which were placed on the moon during the Apollo flights, are still there. An opportunity to test the expansion of space on the short distance scale.

If space expands, we must be able to measure it, even on a short distance, with sufficiently accurate equipment.

A challenge is set!

henk21cm wrote:If space expands, we must be able to measure it, even on a short distance, with sufficiently accurate equipment.

The trouble with this idea, I think, is that you can only measure distances between two pieces of matter. There is no physical ruler that directly measures the "fabric" of space. You can measure a distance between two pieces of matter at two particular times and try to account for the difference or lack thereof, but none of the measurements are distinguished and labeled "due to physical motions" or "due to the expanding universe". On one hand you have differences that can be explained by classical physics. On the other hand you have differences that can be explained by general relativity. With good enough measurements, you can judge for yourself whether classical physics and general relativity together account for everything you observe. So far I'm satisfied that Einstein's general relativity on top of Newton's law (with a nod to Kepler's laws which are derived from Newton) explain gravitation and motion. As far as I know, nothing yet fully explains expansion, but it looks like it happens within Einstein's spacetime framework.

Looking for the expansion of space in the earth-moon distance tasks you with looking for a miniscule amount of space expansion among a whole menagerie of classical effects of much greater magnitude. That's why cosmologists go fishing in a bigger pond where they might actually observe fish.

Last I heard, the only general relativistic effects observable in our own solar system (so far) are the bending of light by gravity (we can view a star directly behind the sun during an eclipse) and the warping of time by gravity (we can measure the precession of Mercury's orbit).

iampete wrote:Is there any "real" (whatever the definition of real is) evidence of the existence of dimensions greater than the number that we as humans actually experience, or is all of this just the result of playing with mathematics?

We humans actually experience two spatial dimensions. Everything we see can be expressed in two-dimensional coordianates on a surface surrounding us. Yet we accept "real" evidence of a third dimension, as our experience takes us beyond vision into travel. When we see a doorway and walk through it, we have experienced the third dimension. Even though we have not experienced it visually like the first two dimensions, we believe it is "real". A skeptical philosopher might question the validity of the third dimension, arguing that travel is a separate class of thing (perhaps only the result of mathematics) that is unlike the first two dimensions in its characteristics; after all, once we have walked through the doorway, all we see is still two-dimensional. Indeed, the third dimension requires the concept of distance, whereas the first two dimensions required only the concept of direction. Yet we accept the three dimensions as equivalent, and the mathematics agrees.

So what's the problem going from three dimensions to four? I know you can't visualize four, but you can't really visualize three either and that doesn't stop anyone from thinking they can.

For the good stuff, forget about astronomy, general relativity, and anything larger than an atom. Mess your head up trying to visualize quantum states. Electrons now now superstrings, multi-dimensional entities that loop through even more dimensions, and none of these are the three or four dimensions that we "accept". This is a result of the mathematics, but they didn't just leave the equations alone in the lab one night to create a horrific result on their own; they used mathematics to try to explain observed phenomena.

apodman wrote: With good enough measurements, you can judge for yourself whether classical physics and general relativity together account for everything you observe.

The thought that space itself is stretched was interesting. The strain rate is 2.5E-18 [1/s]. Take 30 years, that is 1 G s. Over a period of 30 years the strain of the space fabric would have been 2.5E-9 [-]. Take the distance between earth and moon: 4E8 m. The elongation after 30 years would have been of the order of 1 m. Currently measurements point to 0.02 m/y, being the same order of magnitude, due to tides.

Other people had similar thoughts and they have found an answer, i.e. the effect is
annihilated by gravity!, see http://en.wikipedia.org/wiki/Hubble's_l ... e_velocity. (Since the url has a ' in it, phpbb does not show it as a link).

You are right about the Einsteinian view of the universe: space curved by mass. That is a more important aspect.

apodman wrote:Last I heard, the only general relativistic effects observable in our own solar system (so far) are the bending of light by gravity (we can view a star directly behind the sun during an eclipse) and the warping of time by gravity (we can measure the precession of Mercury's orbit).

Not just by the precession of the Mercury perihelion. In Chicago the Mössbauer effect has been used to demonstrate the influence of mass on time. At the top of the university building time is running faster than at the basement.

So I picture the earth-moon system in its little gravitational well, complete with grid lines. I picture the emptier space further out, also with grid lines. Okay, now the emptier space is expanding and its lines are stretching and getting further apart as time goes forward. The distances between various gravitational wells expand with the grid. Down in a well, we sort of hide from the expansion effect, and the grid lines around us do not stretch or move apart.

Now my Physics is weak when you speak of expansion in terms of strain; but if you have a zone that is not expanding embedded in a zone that is expanding, doesn't there have to be strain at the transistion? I'm pretty good at visualizing these things, but the shape and expansion characteristics of the transition zone as it stretches are beyond me. As an alternative to visualization, I could learn to perform the necessary transformations on those 4x4 Cartesian tensors that carry gravity by trading a dose of time for a dose of x,y,z and plot the results, but I got a D in that course a long time ago. So it's left for someone else to visualize and me to appreciate.

And I've been thinking again. If a gravity well produces a node of unexpanded space, wouldn't something the size of a galaxy leave a trail of pinched space as it moves? I guess I'm only asking about the motion due to its own momentum within its local environment, not its observed motion due to the expansion of space. Or am I forgetting the first principle of Relativity again, that there is no "motion" for any one object measured against a fixed grid of space? Like Captain Janeway said, "Temporal Mechanics give me a headache."

iampete wrote:Is there any "real" (whatever the definition of real is) evidence of the existence of dimensions greater than the number that we as humans actually experience, or is all of this just the result of playing with mathematics?

apodman wrote:... trading a dose of time for a dose of x,y,z ...

When time was introduced as a fourth dimension, it wasn't just sitting over here on the left with clock-like characteristics while x, y, and z sat there on the right with yardstick-like characteristics. Rather, it came with a precise mathematical conversion from time to distance. The factor, the universal exchange rate, is equal to the speed of light. Not just the intergalactic speed limit, the speed of light shows up throughout electromagnetic physics and here as the binding glue that makes three yardstick dimensions and one clock dimension part of a single coherent dimensional system. I can't testify myself, but the four dimensions of space, space, space, and time were all supposed to have been created together at the big bang (before which there was no space, no time, and no "before").

Per my limited recollection, if you want a distance in three dimensions, you take the square root of (x²+y²+z²). If you want a spacetime distance in four dimensions, you take the square root of (x²+y²+z²-c²t²). Note the minus on the time versus the plus on the other dimensions. Using "c" (the speed of light) in this expression is the only factor that works with all the other equations of spacetime physics, and it apparently represents the actual four-dimensional mechanics of the universe. The other notable feature of this expression is that "t" behaves mathematically as perpendicular to x, y, and z!

apodman wrote:

iampete wrote:Is there any "real" (whatever the definition of real is) evidence of the existence of dimensions greater than the number that we as humans actually experience, or is all of this just the result of playing with mathematics?

. . . the four dimensions of space, space, space, and time were all supposed to have been created together at the big bang

. . . (x²+y²+z²-c²t²)

. . . "t" behaves mathematically as perpendicular to x, y, and z!

My original question (as is obvious from the answers thus far) was not stated clearly enough.

From my point of view, I "experience" four dimensions, 3 spatial dimensions plus time. I cannot "picture" said 4 dimensions as an orthogonal co-ordinate system, but I like to think that I can handle that at a conceptual level. In addition, there seems to be clear evidence (enough for my purposes, anyway) that there are 4 unique dimensions.

My original question was intended to inquire about various stories one can read of science news in non-technical publications (e.g., newspapers, magazines like Science News, SciAm, etc.) that is dumbed down for laymen in which some new work in cosmological theories propose the universe to be a multi-dimensional space of n (where I seem to recall n=7, 13 or more, at various times). I also recall reading somewhere that some recent string theory came up with something like, I believe, 27 dimensions.

So, to rephrase the question: Is there any evidence for dimensions greater than the "basic" 4, or are these just artifacts that arise from people playing with math in various ways? Also, if there is no current evidence, can someone describe (in layman terms) what evidence of dimensions > 4 might look like or consist of?

While the following links may shed some light on the plethora of dimensions in the modern universe, I can't easily tell whether any of the theory relies on evidence or just mathematical construction. So I'm sorry I'm no help.

http://www.damtp.cam.ac.uk/user/gr/public/qg_ss.html

http://web.wt.net/~cbenton/kabbalah/string.htm

http://en.wikipedia.org/wiki/Superstring_theory

http://en.wikipedia.org/wiki/String_theory

http://www.sukidog.com/jpierre/strings/extradim.htm

apodman wrote: . . . So I'm sorry I'm no help. . .

Quite the contrary, the references are great.

While they may not explicitly answer my question, I had not seen either the Cambridge site or the sukidog site before and I expect to spend some time reading and following some of the links they provide.

Thank you very much.

G'day Apodman,

The picture you wrote down for a gravity well is the same as what i'm visualizing. In a 2D analogon: just a membrane that is stretched in two perpendicular directions. The gravity well is formed by the famous bowling ball Carl Sagan once used, lying on that membrane.

apodman wrote:Now my Physics is weak when you speak of expansion in terms of strain;

I will explain that. Please do not feel offended when the level of my explanation is rather basic. I might tell things you already knew.

Visualize a rubber band of 1 m long. Fix one end at a wall. Then apply sufficient force to the other end, that it will elongate to 1.1 m. Its extension is 0.1 m. The strain is the extension divided by the original length: 0.1m/1m = 0.1 [-]. Strain has no units, it is length divided by length.

Release the force and the rubber band will jump back to its original length of 1 m. Now fetch from your toolbox a motor with a gearbox. This motor will pull at the free end of the rubber band and stretch it with a velocity of 1 mm/s. When we let this motor run for 60 seconds, it will have extended the rubber band by 0.06 m, so its strain is 0.06. Since the motor has a constant speed, you can divide the strain by the time needed to create this strain: 0.06/60 s = 0.001 1/s. This is the strain rate: unit 1/s. You would have found the same answer if you would have divided the velocity of the motor by the length of the rubber band straight away: (1 mm/s) / 1m = 0.001 1/s.

Note the same type of division in the H0 Hubble constant:
70 km/s /Megaparsec = 2.5E-18 1/s.

you further wrote:but if you have a zone that is not expanding embedded in a zone that is expanding, doesn't there have to be strain at the transistion?

What you mean is a gradient in strain, a zone where there is a transition between the gravity dominated well and the "free" expansion of the space time fabric. What will follow is an oversimplification of the universe. Suppose the strain caused by the gravity well and the strain by the expansion of the space fabric both follow the rules of a linear elastic model. I know this is not true, follow my motto: "start simple, complicate later". From theorems of the theory of elasticity it is known that for a linear elastic material strains may be added up linearly.

As you might imagine, the strain caused by the gravity well is larger close to the center of the mass and it decreases if the distance to the center increases. This is a gradient in strain, and it is not easy to accomplish in a rubber band. The direction of the strain of the gravity well is opposite to the direction of the strain of the expanding space fabric. The latter is a constant.

Now suppose the strain of the gravity well can be mathematically approximated by: (this is far from the real shape, i do not know what particular shape it has)

gs(r) = -g0 /(r-rm)

where g0 is a constant depending on the mass, r is the distance between the center of the mass and a point in 'space' and rm is the radius of the mass. r > rm, otherwise it is nonsense. gs is the gravity strain. As you can see, if r gets bigger and bigger, gs(r) approaches 0. So at large distances from the mass, there is hardly any strain. Close to the mass there is a lot of influence.

The strain of the space fabric is (for simplicity sake) a constant, e.g. +fs (fabric strain).

Since we simplified space to a linear elastic material we can add up the strains.

gtotal = gs(r) + fs = -g0/ (r-rm) + fs.

Far away from the gravity well, the strain is fs, since gs approaches 0. There is a point r0, where the fabric strain is equal to minus the gravity strain:

r0 = rm + g0/fs.

At that point there is no strain. Closer than r0 to the mass the strain is negative, a gravity dominated region, further away than r0 the strain is positive, a fabric dominated region. If you look at the formula for gtotal, there is a smooth transition zone between both regions.

apodman wrote:I could learn to perform the necessary transformations on those 4x4 Cartesian tensors that carry gravity by trading a dose of time for a dose of x,y,z and plot the results, but I got a D in that course a long time ago. So it's left for someone else to visualize and me to appreciate.

That is definitively a "complicate later" excercise. My oversimplified description might help you visualize it in a much simpler way. Of coarse, the space fabric is not a linear elastic material. The gravity well is not linear, since Einstein needed non-linear tensor algebra to derive to its conclusions. So my numbers and shapes will not hold, it is more the general idea.

And you continued:

If a gravity well produces a node of unexpanded space, wouldn't something the size of a galaxy leave a trail of pinched space as it moves?

Visualize Carl Sagans bowling ball on bowling lane (made by a thin rubber membrane) speeding to the pins. As it moves, it indents the lane, but the lane jumps back when the ball has passed. And you were referring to the peculiar velocity of the galaxy, not just velocity due to the general expansion.

Hope this help you visualize the expanding space fabric.

henk21cm wrote:What you mean is a gradient in strain, a zone where there is a transition between the gravity dominated well and the "free" expansion of the space time fabric.

Indeed. Thank you for the analysis.

Too bad that the space recovers after the bowling ball has passed. I was allowing myself to speculate some fiction based on the effects of crossing your previous path of unexpanded space. Oh, "well".

And now I'm picturing Carl Sagan walking down the aisle at the field house after speaking at my graduation 33 (!) years ago. The only grid was the hockey rink. The floor was concrete. I sat on the aisle and Dr. Sagan's robe touched mine as he passed. The floor did not deform as he stepped.

Food for thought:

What if space isn't expanding at all?

What if the speed of light is just slowing down and the laws of physics are slowly changing? Maybe things are getting smaller instead of farther away. We're told everything is made of pretty much nothing anyway.

Stuff that was right next door to us just a moment after "the big bang" but is now "at the other side of the universe" is now still right there, but it takes light 13 billion years to get from there to here.

It's all relative.

Like I said, just food for thought.

-Noel

NoelC wrote:What if space isn't expanding at all?

What if the speed of light is just slowing down and the laws of physics are slowly changing? Maybe things are getting smaller instead of farther away.

As a Mathematician or Physicist, I would favor a theory that requires the simplest set of equations to explain. Whether "simple explanation" truly equates to "physical reality" is a greater question I can't answer. But I am happy with a theory that just expands distance over time while a lot of other things stay the same. I think a theory (which one could surely construct) that requires adjustment of all the equations of Physics is hard to deal with even if it happens to turn out to be "physical reality".

All physical properties (not counting quantum properties) I know about are made of various powers of Mass, Distance, Time, and Charge. In Newton's universe, these were all invariant. Special and General Relativity ruined the invariance for the first three (is Charge still untouched?), so I need to pick one to hang my hat on if I'm going to try to visualize the relationships among them. For me (until I get a vivid picture of something better), space is expanding and that is that.

This strikes me as similar to explaining retrograde motion in a heliocentric system rather than with epicycles. The epicycle explanation works fine mathematically (as far as it goes), but it is ungainly and we accept the simpler heliocentric model as physical reality.

More vague memories are returning. Galilean spacetime was an improvement over Newtonian spacetime (despite Galileo's death the year before Newton's birth) but still didn't work as well as Einsteinian spacetime. Lorenz's transformations were made for Galilean spacetime, but are now used for Einsteinian spacetime - same equations supporting a different physical explanation. Thus the mathematical and conceptual ends of a theory may progress toward the same explanation but not necessarily in perfect synchrony.

Then there's this link:

http://en.wikipedia.org/wiki/Philosophy ... e_and_time

NoelC wrote: What if the speed of light is just slowing down and the laws of physics are slowly changing?

25 years ago i read the book written by Heinz Pagels: The cosmic code. He pointed out that when a daemon was tinkering with the physical constants , the current universe could not exist. The margins between which tinkering was possible without huge consequences, were narrow. Specially on atomic and subatomic scale. Reduce the gravitational constant by a factor 10, and no galaxies could be formed. (Or was it 10%? i do not remember).

So if the laws of physics are changing in time, images from the past must provide us with a different window on the universe and its previous physical constants. As far as i know these constants are the same even in the most remote systems. Two years ago there was some discussion about the fine structure constant, α (being 1/137...), which was slightly different in a very remote quasar. Data were contradicted later. Maybe it is resolved, maybe not.

NoelC, i do agree that a concept "space itself is expanding", is outlandish. Remember, the same would have been valid for Isaac Newton, when he would hear that his laws are just approximations in the far field. "We are both people too old to embrace new concepts immediately."

Why were insects so large 100 million years ago? Why does the Earth go through ice ages? I see no reason to conclude subtle things aren't happening. Does the incredible red shift of ultra-distant quasars indicate they are moving away, or maybe just that they emitted light at a different frequency (relative to today) back then.

Any of this is probably moot, insofar as virtually all the things that we put labels on - those things we hold as constants in our brains: Distance, time, dimensions, etc. - are just the observed effects of something else in our tiny little corner of the cosmos at this moment in time, anyway.

Indeed, we're made of the stuff. And so everything we see and say and write is conceived in our own frames of reference. It is only because we have incredible similarities to one another that we can communicate these concepts. Who's to say that time didn't pass more quickly today than during a day last week? Certainly seems to go by faster to me.

It really comes down to this: We probably can't ever understand everything. Indeed we would likely need brains as complex as the universe itself to fully grok the universe. And so we must approximate and make models.

And that's okay.

-Noel

starnut wrote:The whole point of using the balloon analogy was to explain to a layman how to visualize the universe with respect to the big bang and the expansion of space. The layman has difficulties visualizing a 4-D expanding universe that does not have a center. So, we tell the layman to imagine himself as a 2-D creature on the surface of a balloon. This creature can only see sideways, i.e., in 2-D, just as we can see only in 3-D. If you can come up with a better explanation or analogy that doesn't confuse the layman, please do so.
Gary

Seed cake. Souffle. (Soo-flay). Cancer growing through a tissue.
The idea is that all parts of the cake, souffle or growth grow at a constant
rate all through the volume. Well, it does if you imagine a very *large*
cake or souffle.
The "4-d" aspect of the cosmos is merely Time.
The reason there is no "centre" to the cake is because it was at the centre
at the beginning of Time. The centre isn't *miles* away, it's thirteen
milliard *years* away, in that direction, the direction of "past".
From every point in the cosmos, the "centre" is equidistant.
So, think of a seed cake that never ends. One where all points look like
a central point, but where, a long, long time ago, it all began.
Does any of this help?
Oh, the analogy of the seed cake came from a lady. I doubt any man would
ever have thought of it.
SDM.

SittingDownMan wrote: . . . The centre isn't *miles* away, it's thirteen
milliard *years* away, in that direction, the direction of "past". . .

OK, more conceptualization/visualization problems!!

It seems to me that the "center" of the universe at the time of the big bang (EDIT: actually just after the big bang) was a point in 4-space, think of it as the (0,0,0,0) point. Given the co-ordinate system defined with that as the origin, right now, every point in space has co-ordinates of (i,j,k,14x10E12). Just at a conceptual level, it seems to me that it should be possible to find a point in space that corresponds to (0,0,0,14x10E12), i.e., where the center of the universe was at the time of the big bang (EDIT: actually just after the big bang). As a conceptual matter, this seems to be analogous to taking the projection of a point in 3-d space onto the x-y plane.

As a practical matter, of course, finding this point is likely to be impossible for a myriad of reasons. What I don't understand is why it is said that this point in 3-D space doesn't even exist.

Can anyone help me with this?

iampete wrote:What I don't understand is why it is said that this point in 3-D space doesn't even exist.

Can anyone help me with this?

A 3-D sphere has a 3-D center, but its 2-D surface has no preferred 2-D center point on that surface.

Likewise, a 4-D supersphere has a 4-D center, but its 3-D supersurface has no preferred 3-D center point on/in that supersurface.

===

I'm not sure, but I also think there may be a problem with your turning the 0,0,0 space coordinates into i,j,k values over time. We might be violating one of the ideas of Einsteinian spacetime again by thinking of i,j,k as absolute positions with respect to the origin point rather than positions that can only be measured relative to other positions.

NoelC wrote:Why were insects so large 100 million years ago?

The paleontological record suggests that the oxygen content of the atmosphere was as high as 30% during some periods (instead of the 21% we have now), which would support much larger insects than we have today.

Why does the Earth go through ice ages?

Changes in solar output, Earth's orbit and axial orientation, changes in the composition of the atmosphere, the configuration of the continental land masses... it's a confluence of many factors.

I see no reason to conclude subtle things aren't happening. Does the incredible red shift of ultra-distant quasars indicate they are moving away, or maybe just that they emitted light at a different frequency (relative to today) back then.

Changing the frequency of the emitted light won't change the positions of the absorption spectra, the way redshift does. Also, if physics was different enough back then to alter the way the objects we observe emit light, then it would probably be different enough that the objects couldn't be there to emit the light in the first place... or at least not in the form we observe them.

Any of this is probably moot, insofar as virtually all the things that we put labels on - those things we hold as constants in our brains: Distance, time, dimensions, etc. - are just the observed effects of something else in our tiny little corner of the cosmos at this moment in time, anyway.

It could be even worse, if the Holographic Principle holds.

apodman wrote: . . . A 3-D sphere has a 3-D center, but its 2-D surface has no preferred 2-D center point on that surface.
Likewise, a 4-D supersphere has a 4-D center, but its 3-D supersurface has no preferred 3-D center point on/in that supersurface. . .

Thanks, that makes sense to me; I can even comprehend a "picture" of that in my mind!

Yet, there's still a conceptual disconnect here for me. If one takes a projection of a 3-D sphere onto a 2-D surface, one can determine the center of the 2-D projection. Why would that analogy not work when trying to project a 4-D sphere into the 3-D space that we observe?

apodman wrote: . . . I'm not sure, but I also think there may be a problem with your turning the 0,0,0 space coordinates into i,j,k values over time. We might be violating one of the ideas of Einsteinian spacetime again by thinking of i,j,k as absolute positions with respect to the origin point rather than positions that can only be measured relative to other positions.

This I have a problem with, even at a conceptual level. I will continue to do some more reading to see if I can find something helpful to me. To date, all the stuff I've been able to find is either too elementary to satisfactorily explain anything or way beyond my comprehension level. If you have further suggestions for layman type reading material beyond what you provided earlier, I'd be interested to see them.

I appreciate your helpful and non-condescending responses in all your previous replies.