Raisin bread model of space-time

[jordankonisky]

The impact of a bowling ball on the shape of a stretched gridded sheet is often used as a tool to help some of us to better visualize how the presence of a mass influences the curvature of spacetime? However, while this kind of diagram is helpful, I think that it is misleading because it depicts the perturbation of only a single slice of the space-time continuum. In real fact, the massive object is totally enmeshed in a space-time continuum. Rather, I visualize a raisin enmeshed in a bread loaf in which the loaf represents the full continuum of space-time and no part of the raisin exists, indeed can exist, outside the space-time continuum. Am I thinking about this in the right way?

 

[Mister T]

The visualization of the rubber sheet is a tool used to explain the math. Likewise, the raisin bread analogy is often used to explain that each raisin will see all others moving away from it, so the observation of all raisins moving away from you doesn't demonstrate you're at the center.

These visualizations can never really be the correct way to think about gravity or the expanding universe. The only correct way to think about them is to think about the math. So in some sense when you leave out the math you can't be thinking about them in a way that's entirely correct. It will always be at best only partially correct.

I like what Dirac said when asked to explain his work in non-mathematical terms. He said that it would be like a blind man trying to appreciate the beauty of a snowflake. As soon as he touches it, it disappears.

 

[PeterDonis]

I visualize a raisin enmeshed in a bread loaf in which the loaf represents the full continuum of space-time

If we imagine the loaf to be spacetime, an individual object is not a single raisin; it's a whole sequence of raisins in a line extending through the loaf. And if the loaf represents the entire universe, the lines representing two different objects move apart as you move "up" the loaf (i.e., in the future time direction); that's the expansion of the universe.

 

[Rudi Van Nieuwenhove]

In the "raisin bread model", the raisins do not expand (as the universe expands). It is often mentioned that the space within gravitationally bound systems, such as galaxies, does not expand. However, I have never heard a good explanation for this. Saying that space in a galaxy does not expand because of some gravitational bonds between the stars is hardly an explanation. In some magical, unexplained way, the masses in a galaxy are able to stop the overall expansion of space locally.
Suppose you have a big single mass, floating more or less isolated through space. Then one could assume that the space around it follows the global expansion of the universe. Next, put a tiny mass in a large orbit around this mass. Now, by some magic, the space between the tiny mass and the large mass suddenly stops expanding.
My feeling (not scientific, I agree), is that the space within a galaxy or a cluster of galaxies, is different in a way from the space around it as if some kind of "bubble" exists in space with different properties. Maybe one could see this as some sort of dimple in space time.

 

[PeterDonis]

It is often mentioned that the space within gravitationally bound systems, such as galaxies, does not expand. However, I have never heard a good explanation for this.

The explanation is that space does not expand, period. The universe expands, but that is not the same as "space expanding". The spatial scale factor increases with time in a particular set of coordinates used in cosmology, but that is not the same as "space expanding" either. In short, "space expanding" does not refer to any actual physical process, so there is no need to "explain" why this nonexistent process takes place in some places but not others. In gravitationally bound systems, individual parts of the system are not flying apart; in the universe as a whole, galaxies are flying apart. That's all there is to it.

My feeling (not scientific, I agree), is that the space within a galaxy or a cluster of galaxies, is different in a way from the space around it as if some kind of "bubble" exists in space with different properties

Please review the PF rules on personal theories. This kind of speculation is out of bounds here at PF.

 

[Rudi Van Nieuwenhove]

I find your answer highly confusing; In every textbook and article, it is clearly stated that space expands (see also wikipedia ; https://en.wikipedia.org/wiki/Metric_expansion_of_space , "Over time, the space that makes up the universe is expanding "). If space would not have expanded since the big bang, we wouldn't be here. So my answer remains: Why is most of the universe expanding, while some parts are not?

 

[Ibix]

Also from your Wikipedia link:

Due to the non-intuitive nature of the subject and what has been described by some as "careless" choices of wording, certain descriptions of the metric expansion of space and the misconceptions to which such descriptions can lead are an ongoing subject of discussion in the realm of pedagogy and communication of scientific concepts.

Be very, very wary of intuitive explanations of anything in advanced physics. You might want to check out reference 5 on that page - the maths may go over your head, but terms like "expanding space fallacy" should be clear enough.

Our current observations suggest that the universe is spatially flat (given a certain definition of "space" that I won't get in to). Our models say that such a universe is infinite in extent. How can something infinite be getting bigger? Certainly distances between distant galaxies are increasing (the return leg of any round trip will always be longer), but that is not the same thing.

The reason why galaxies don't expand is similar to the reason that two coins sitting on a rubber sheet will move apart if the sheet is stretched but won't disintegrate. They're too tightly bound by their own internal forces.

 

[Rudi Van Nieuwenhove]

In fact, we don't really know if our university is infinite. Models can suggest that this is the case, but we will never get experimental proof of this. If the universe is finite, then I can quote J. Peacock from ref 5 (you indicated); "This is most clear-cut in the case of closed universes, where the total volume is a well-defined quantity that increases with time; so undoubtedly space is expanding in this case".
Regarding the cosmological red-shift, contradicting statements seem to exist. Also from wikipedia https://en.wikipedia.org/wiki/Redshift , one can read "As a result, the wavelength of photons propagating through the expanding space is stretched, creating the cosmological redshift.". It keeps me wondering who is right.

 

[Ibix]

The maths is right. Verbal descriptions are highly suspect. But your own link refutes your claim that "In every textbook and article, it is clearly stated that space expand".

 

[Rudi Van Nieuwenhove]

Let us suppose that the universe is not infinite. This is a valid hypothesis, in view of the fact that we have no experimental evidence that the universe is infinite. In that case, space itself must have expanded (ref Peacock). Then, I ask myself why tightly bound systems do not expand. The hydrogen atom has now probably still the same size as billions of years ago. If the expansion of space would be an intrinsic property (still assuming a finite universe), why was the space in between the electron and the proton not increasing? Just saying that they are a bound system is not convincing to me (it looks like a circular argument). If the size remains the same, space has not expanded within the system.

 

[PAllen]

An example of the non-universality of the notion of expanding space is the following exact quote from Steven Weinberg, author, among many other things, of Gravitation and Cosmology:

"...how is it possible for space, which is utterly empty, to expand? How can nothing expand? The answer is: space does not expand. Cosmologists sometimes talk about expanding space, but they should know better"

from:

New Scientist, Martin Rees & Steven Weinberg(1993)

 

[Rudi Van Nieuwenhove]

Thanks for the comment (though it does not really answer my previous question).
Of course, this statement does not take into account quantum physics: In quantum field theory space is not empty at all but filled with a vibrant sea of virtual particles. Maybe that in this context, space can expand after all.

 

[rootone]

Although the universe is expanding as a whole, there can locally be objects which don't expand.
On the scale of galaxies and clusters, gravity is strong enough to overcome the general expansion.
The stars and other constituent parts of galaxies are said to be gravitationaly bound.
At the scale of atoms gravity becomes irrelevant, here the two nuclear forces bind particles together even more powefully.

 

[PeterDonis]

In quantum field theory space is not empty at all but filled with a vibrant sea of virtual particles.

This is also an ordinary language description that does not really capture the actual physics. And no, the actual physics does not make "space expanding" anything physically real.

 

[weirdoguy]

In quantum field theory space is not empty at all but filled with a vibrant sea of virtual particles.

No, read this:
https://www.physicsforums.com/insights/misconceptions-virtual-particles/
https://www.physicsforums.com/insights/vacuum-fluctuation-myth/

 

[PAllen]

A clear, detailed, answer to questions like scale at which universe expansion becomes apparent, and how to understand it in GR terms, is provided in the following paper. This discussion is much more rigorous than many debating papers on this topic:

http://philsci-archive.pitt.edu/10033/1/Granada2011_paper.pdf

 

[pervect]

The impact of a bowling ball on the shape of a stretched gridded sheet is often used as a tool to help some of us to better visualize how the presence of a mass influences the curvature of spacetime? However, while this kind of diagram is helpful, I think that it is misleading because it depicts the perturbation of only a single slice of the space-time continuum. In real fact, the massive object is totally enmeshed in a space-time continuum. Rather, I visualize a raisin enmeshed in a bread loaf in which the loaf represents the full continuum of space-time and no part of the raisin exists, indeed can exist, outside the space-time continuum. Am I thinking about this in the right way?

Raisins (representing events) embedded in a loaf (representing space) is a common popularization of how to visualize space. But it's really oriented to expaining "curved space" and not "curved space-time". Which is common to most popularizations, due to the abstractness of curved space-time.

As far as "expanding space" goes, one interesting paper that I think makes some good points is "Expanding space, the root of all evil?" <<Link>>

My summary of the paper would be that the concept of expanding space has its detractors, and can be and frequently is mis-used and leads to notable misconceptions (many of which we try to straighten out here on PF, with varying degrees of success). But it has supporters, as well, amongst them I would include the authors of this paper, who attempt to present the idea of "expanding space" in a way that they claim is less likely to cause confusion.

As far as curvature goes, most discussions of curvature really only visualize the curvature of space, and not space-time. The "rubber sheet" would need to be not a spatial rubber sheet, but a space-time diagram, for the analogy to represent space-time curvature. Unfortunately, the concept of a space-time diagram seems to be hard to get people to utilize, for reasons that I don't really understand.

There's another analogy to curved space (and not space-time) due to Einstien that I think has some merit, the "heated slab" approach. See for instance , A Einstein, "Relativity, the special and general theory" <<link>>.

I believe that the expanding slab idea has some limitations as to what it can model, but it serves as a conceptual model that may help people who are extremely used to envisioning only Euclidean geometries. A generalization of the approach may be general enough to handle small sections of actual space-time, for instance in Straumann, "Reflections on Gravity" <<link>>. In the generalization, one includes the idea that gravity affects clocks as well as rulers.

While the idea is helpful and probably less prone to misconceptions than the rubber sheet analogy, it does have some limits, though I haven't seen these limts discussed in the literature. For instance, the relativity of simultaneity is a basic feature of special relativity that is frequently not understood is not well-represented by this kind of model, which tries to "tweak" familiar notions of geometry into a more general form. Another concern is handling global topology.

I was going to comment more on curved space-time geometry as opposed to just spatial geometry, but, I think the post has already gone on long enough , and possibly too long.

One final note. Popularizations are not a replacement for learning the full theory - which, ultimately, involves a lot of math. They do the best they can to present some features of the theory in a way that's understandable without the math, but they are not the full theory. Frequently people criticize the full theory by criticizing the popularizations (because the populariations are all they understand). Well, the popularizations ARE flawed, in many cases. The flaws can be minimized by applying them only to the situations for which they are intended and suitable for, but one should not expect a full understanding based only on popularizations :(.

 

[maroubrabeach]

All analogies can be useful, but their usefulness is always limited in some way or fashion.
And I have seen the word "space"used many times, when I'm pretty sure it should be spacetime.
Albert Einstein's at one time teacher said it like this......
"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".

— Hermann Minkowski
The main point with the spacetime expansion, is that we do see over large scales a recessional velocity of galaxies, that is evidenced in a cosmological redshift.

 

[Ibix]

The main point with the spacetime expansion

The problem with "expanding space" is the "expanding", not the "space". The scale factor applies to a particular set of spacelike slices, each one of which has a larger scale than its predecessor. We are definitely not talking about spacetime as a whole expanding.

 

[Rudi Van Nieuwenhove]

No, read this:
https://www.physicsforums.com/insights/misconceptions-virtual-particles/
https://www.physicsforums.com/insights/vacuum-fluctuation-myth/

Well, that is the opinion of one author. I'm sure there are many other opinions on this. In https://www.physicsforums.com/insights/misconceptions-virtual-particles/ the author writes that virtual particles "cannot cause anything, interact with anything or affect anything". The Casimir force, which has been experimentally demonstrated seems to disprove this statement.

 

[Nugatory]

The Casimir force, which has been experimentally demonstrated seems to disprove this statement.

There are perfectly satisfactory ways of calculating the Casimir force between two plates without using the concept of virtual particles - so observations of the Casimir effect do not demonstrate the existence of virtual particles.

 

[Rudi Van Nieuwenhove]

That is interesting. Could you give a reference to this (from a refereed journal)?

 

[weirdoguy]

Well, that is the opinion of one author.

No, that is how QFT is done. Did you take any course on that? Here is a very illuminating discussion about the article:
https://www.physicsforums.com/threads/misconceptions-about-virtual-particles-comments.865706/

 

[Rudi Van Nieuwenhove]

No, that is how QFT is done. Did you take any course on that? Here is a very illuminating discussion about the article:
https://www.physicsforums.com/threads/misconceptions-about-virtual-particles-comments.865706/

Hi, yes I took courses on QFT and have some books at home. So, all other effects such as vacuum polarization, lamb shift, ... can be explained without virtual particles?
Even when the concept of a virtual particles just comes from a representation of some terms in an integral, I prefer to think of space as a soup of virtual particles instead of a collection of mathematical objects (invented by mathematicians or physicists) floating (or doing their "job") somehow in space. To me reality is not identical to mathematics. Somehow, these mathematical terms must correspond to something physical.

 

[PeterDonis]

all other effects such as vacuum polarization, lamb shift, ... can be explained without virtual particles?

Yes.

I prefer to think of space as a soup of virtual particles

As a heuristic this can work (although it is limited). But you always have to be careful with heuristics, and you always have to be aware that they're just heuristics--they aren't telling you "this is how it is", they are only telling you "this is an analogy that might work in some cases".

Somehow, these mathematical terms must correspond to something physical.

As you state it this is obviously way too strong. Consider a simpler example from Newtonian mechanics: a center of gravity. We can calculate the center of gravity of any system in Newtonian mechanics; but that does not mean there must be something physical corresponding to it. It's just a useful abstraction in our analysis.

 

[Rudi Van Nieuwenhove]

Yes.
As a heuristic this can work (although it is limited). But you always have to be careful with heuristics, and you always have to be aware that they're just heuristics--they aren't telling you "this is how it is", they are only telling you "this is an analogy that might work in some cases".
As you state it this is obviously way too strong. Consider a simpler example from Newtonian mechanics: a center of gravity. We can calculate the center of gravity of any system in Newtonian mechanics; but that does not mean there must be something physical corresponding to it. It's just a useful abstraction in our analysis.

Hi, I agree, I stated it too strongly. We came to the subject of virtual particles in this thread because of the discussion on expanding space and the following arguments that there is nothing to expand since space is completely empty. Then I replied that space is filled with a sea of virtual particles, a statement which was refuted by the experts. However, I think that the experts can agree that quantum field theory is a valid and successful theory and in this theory the vacuum contains quantum fields or quantum fluctuations (even in the absence of any real matter). So, again, we come back to the fact that there is no such thing as "empty space".

Yes.
As a heuristic this can work (although it is limited). But you always have to be careful with heuristics, and you always have to be aware that they're just heuristics--they aren't telling you "this is how it is", they are only telling you "this is an analogy that might work in some cases".
As you state it this is obviously way too strong. Consider a simpler example from Newtonian mechanics: a center of gravity. We can calculate the center of gravity of any system in Newtonian mechanics; but that does not mean there must be something physical corresponding to it. It's just a useful abstraction in our analysis.

 

[PAllen]

Hi, I agree, I stated it too strongly. We came to the subject of virtual particles in this thread because of the discussion on expanding space and the following arguments that there is nothing to expand since space is completely empty. Then I replied that space is filled with a sea of virtual particles, a statement which was refuted by the experts. However, I think that the experts can agree that quantum field theory is a valid and successful theory and in this theory the vacuum contains quantum fields or quantum fluctuations (even in the absence of any real matter). So, again, we come back to the fact that there is no such thing as "empty space".

However, a more complete summary of the discussion you've been involved in is as follows:

1) You raised the question of understanding bound systems in the context of expanding space.
2) It was pointed out by several people here that the expanding space notion is an inessential feature of understanding GR cosmology, and has no associated physics.
3) You argued that expanding space was universally accepted. Several people here gave multiple references to establish that this is not so. Basically, some find the notion useful, some don't, but it is not an essential feature of the math of cosmology. One passing comment I posted (by Weinberg, who is presumably one of the world's top experts on both QFT, and a substantial contributor to cosmology) mentioned emptiness of space. However, this was just passing thought in a long article, and not relevant to all the other links debating reasons to like or dislike the notion of expanding space. You jumped on this, which IMO is complete red herring to the discussion of expanding space. The notion of vacuum is irrelevant to the interpretation of GR based cosmology models in terms expanding space or not. It is not even tangentially mentioned in the substantive links you were given.
4) On your question of understanding bound systems, I gave you a detailed yet fairly accessible link answering exactly this question, which it is not clear you even looked at. (see my post #16). Among other interesting derivations in that link is that there is no upper limit to the size of gravitatioally bound systems, nor any effect at all large scale local physics, unless expansion is accelerating. (To me, this strongly limits the utility of expanding space notion, because this notion would suggest that such effects should exist for linear expansion rate). For parameter ranges characteristic of our universe, it derives that large galaxy clusters are not bound (and any smaller structure is), consistent with observation.

On reviewing the above, can you state what questions remain unclear to you, that you want to discuss further.

 

[Rudi Van Nieuwenhove]

Hi, I'm grateful for all the information I got and I tried to read most of the referenced articles. I think I understood the bound systems (also due to the information in the References).

In post #8, I wrote "If the universe is finite, then I can quote J. Peacock from ref 5 (you indicated); "This is most clear-cut in the case of closed universes, where the total volume is a well-defined quantity that increases with time; so undoubtedly space is expanding in this case"."

I still do not see what is wrong with this reasoning. In my mind it is contradicting the statement that "expanding space notion is an inessential feature of understanding GR cosmology, and has no associated physics". Could you please explain what is wrong with Peacocks simple argument?

I agree now that the notion of vacuum is irrelevant within the framework of GR based cosmology. However, in the inflation theory (Alan Guth), it is an essential ingredient.

 

[Mister T]

In post #8, I wrote "If the universe is finite, then I can quote J. Peacock from ref 5 (you indicated); "This is most clear-cut in the case of closed universes, where the total volume is a well-defined quantity that increases with time; so undoubtedly space is expanding in this case"."

I still do not see what is wrong with this reasoning.

In Post #4 you gave an example.

In my mind it is contradicting the statement that "expanding space notion is an inessential feature of understanding GR cosmology, and has no associated physics". Could you please explain what is wrong with Peacocks simple argument?

It's descriptive, not essential. There's a contradiction between the claim that space is expanding and the claim that it's not, but there's no contradiction when something is descriptive but not essential. Being descriptive means it's intended as an aid to understanding. Being essential means it's a necessary part of the explanation.

You have a number of physicists here on this forum who are telling us that the notion that space expands is a purely descriptive notion that they don't agree with. They have given us references to statements by prominent physicists telling us that space doesn't expand. You have given us references to prominent physicists who tell us that space does expand. The only way to reconcile this apparent conflict is to understand that it's a purely descriptive notion. It has no bearing on the actual physics.

You can take their word for it, learn the actual physics for yourself as opposed to learning the description of the physics, or wait in the hopes that someone can give you an explanation that you find satisfactory. None of that will change the physics.

Have you thought about focusing on the phenomenology instead? What is it that you see happening that makes you think space is or is not expanding? We have evidence that galaxies are separating at speeds that are increasing, and that they have been separating for the last 13.7 billion years. Anything else?

 

[maroubrabeach]

The explanation is that space does not expand, period. The universe expands, but that is not the same as "space expanding". The spatial scale factor increases with time in a particular set of coordinates used in cosmology, but that is not the same as "space expanding" either. In short, "space expanding" does not refer to any actual physical process, so there is no need to "explain" why this nonexistent process takes place in some places but not others. In gravitationally bound systems, individual parts of the system are not flying apart; in the universe as a whole, galaxies are flying apart. That's all there is to it.
Please review the PF rules on personal theories. This kind of speculation is out of bounds here at PF.

As a poor old retired lay person, I must say that at first glance, much about this thread confused me.
Firstly, I have seen the use of "space" as in space expanding used many times. Isn't spacetime a more accurate and accepted term?
Secondly, I have generally learned that we observe a recessional velocities of galaxies as interpreted by cosmological redshift, as a part of GR and leading to the BB model.
In saying that, I was also at one time informed that we could just as legitimately view the spacetime expansion, as a "shrinking rules" concept...I thought this was totally ugly though and out right rejected it.

Within the expanding spacetime/Universe picture, it is logically explained that this expansion is observed over large scales, and that over smaller scales like galaxies, and even clusters and groups of galaxies, are gravitationally bound.ie, gravity is actually overcoming whatever it is making spacetime expand over the larger scales.
On smaller scales still, the forces of EMF, and the strong and weak nuclear forces also act in opposition to the large scale expansion.

On the observed accelerating expansion, I see that as simply a result of the subsequent lesser density of the universe as it expands, while the reason behind the acceleration or what we call DE, is pretty well constant over all spacetime, acting against a universe/spacetime that is getting less dense. Does that make sense?

 

[PeterDonis]

we come back to the fact that there is no such thing as "empty space".

In the context of cosmology, "empty space" includes quantum fields in their vacuum states. So there is such a thing as "empty space" in cosmology.

 

[PAllen]

Hi, I'm grateful for all the information I got and I tried to read most of the referenced articles. I think I understood the bound systems (also due to the information in the References).

In post #8, I wrote "If the universe is finite, then I can quote J. Peacock from ref 5 (you indicated); "This is most clear-cut in the case of closed universes, where the total volume is a well-defined quantity that increases with time; so undoubtedly space is expanding in this case"."

I still do not see what is wrong with this reasoning. In my mind it is contradicting the statement that "expanding space notion is an inessential feature of understanding GR cosmology, and has no associated physics". Could you please explain what is wrong with Peacocks simple argument?

I agree now that the notion of vacuum is irrelevant within the framework of GR based cosmology. However, in the inflation theory (Alan Guth), it is an essential ingredient.

I will attempt to address this. However, the capsule summary is that 'expansion of space' means different things to different people and has no precise definition.

With regard to Peacock, note that his discussion of the closed universe is to justify 'expansion of the universe', which he then treats as equivalent to 'global expansion of space' as distinct from 'local expansion of space'; the latter he argues is fundamentally misleading in all cases. He further argues that galaxies recede in an expanding universe simply because they always were and are able to continue the motion from their initial conditions - NOT because of space expanding between them. Rather than join Peacock in using two flavors of expanding space (one of which he, of course, argues against), I prefer expanding universe, as he titles his first section, and not bother with expanding space at all.

I will make an analogy for 1 x 1 spacetime, that (like any analogy) is limited, but still gets across many points. Consider a spacetime shaped like a cone with the apex down, and cosmological time being distance up from the apex. We will compare this to the a similar model where the sides of the cone bend to asymptote an imagined enclosing cylinder, and also to the case where they bend outward faster than linear.

Considering first the simple cone, first consider the family of geodesics moving up from the apex. They recede from each other because they started receding from each other and in this geometry, such geodesics continue to have growing distance between them. Consider also other geodesics, e.g. one parallel to one of these. Such geodesic will remain parallel to one of the apex family, without growing distance between between them (this feature actually carries over to cosmologies with non-accelerating expansion). Also note that there is no difference in nature of the growing distance between the apex geodesics and the growing distance between any two arbitrary, non parallel geodesics you can imagine on the cone. There is no basis to isolate the apex geodesics as not really moving, with space growing between them, while other non-parallel geodesics are really moving relative to each other. For me, the best description of all of this is that 'expansion of the universe' is a statement about global geometry of spacetime allowing for the existence of an isotropically expanding congruence of world lines. The red shift and growth of distance between them is no different from the so called peculiar motion against the same geometric background. To me, adding the notion of expanding space is as if one insisted on imagining a rubber band placed on the cone that stretches as you move it up the cone. It doesn't add anything to understanding the geometry of the cone, or to why the apex geodesics have growing distance between them. The latter is simply due to the geometry of the cone. One final note is that one can slice the cone many different ways, getting complex patterns of 'growth' and 'shrinkage' depending on how you do it (you can even find a slicing where the elliptical cross sections oscillate between growth and shrinkage). Thus, even the 'expanding universe' notion is dependent on how you slice the manifold. However, invariants like the existence and behavior of an isotropically expanding congruence remain the same.

Now consider the case of the cone with sides bending to asymptote a cylinder. In this geometry, the apex geodesics have ever slower growth of distance between them, and a geodesic initially parallel to an apex geodesic starting near the apex will ultimately intersect the apex geodesic. To stay the same distance from the apex geodesic, such an initally parallel world line will have to deviate from being a geodesic (physically, it would have to maintain proper acceleration away from the apex geodesic to maintain its distance). Rather than talking about declining expansion of space, I simply see this case as a different overall geometry, with different geodesic behavior.

Finally, consider the case cone modified to bend outward ever faster than linear. The only point I want to make in this case is that a geodesic initially parallel to an apex geodesic will have its distance from that apex geodesic increase. A non-geodesic world line would be required to maintain fixed distance, in this case requiring proper acceleration toward the apex geodesic. This is analogous to the similar fact about cosmologies with accelerated expansion.

One final technical caveat - it is well known that distance in general relativity has no unique, preferred definition. In the above, I am using a specific geometric construction known as fermi-normal distance, which is different from what is commonly used in cosmology.

I have no idea how much all of this will help or confuse.

 

[Rudi Van Nieuwenhove]

Thanks for this nice explanation. This was really helpful.

 

[Mister T]

I have no idea how much all of this will help or confuse.

I think it would make a good INSIGHT article, especially if illustrations were added.

(I found the part about the cone slices hard to follow, but overall I loved it!)