## Saturday, May 29, 2021

### What does the universe expand into? Do we expand with it?

[This is a transcript of the video embedded below.]

If the universe expands, what does it expand into? That’s one of the most frequent questions I get, followed by “Do we expand with the universe?” And “Could it be that the universe doesn’t expand but we shrink?” At the end of this video, you’ll know the answers.

So here’s that little bit you need to know about General Relativity. First of all, Einstein used from special relativity that time is a dimension, so we really live in a four dimensional space-time with one dimension of time and three dimensions of space.

Without general relativity, space-time is flat, like a sheet of paper. With general relativity, it can curve. But what is curvature? That’s the key to understanding space-time. To see what it means for space-time to curve, let us start with the simplest example, a two-dimensional sphere, no time, just space.

That image of a sphere is familiar to you, but really what you see isn’t just the sphere. You see a sphere in a three dimensional space. That three dimensional space is called the “embedding space”. The embedding space itself is flat, it doesn’t have curvature. If you embed the sphere, you immediately see that it’s curved. But that’s NOT how it works in general relativity.

In general relativity we are asking how we can find out what the curvature of space-time is, while living inside it. There’s no outside. There’s no embedding space. So, for the sphere that’d mean, we’d have to ask how’d we find out it’s curved if we were living on the surface, maybe ants crawling around on it.

One way to do it is to remember that in flat space the inner angles of triangles always sum to 180 degrees. In a curved space, that’s no longer the case. An extreme example is to take a triangle that has a right angle at one of the poles of the sphere, goes down to the equator, and closes along the equator. This triangle has three right angles. They sum to 270 degrees. That just isn’t possible in flat space. So if the ant measures those angles, it can tell it’s crawling around on a sphere.

There is another way that ant can figure out it’s in a curved space. In flat space, the circumference of a circle is related to the radius by 2 Pi R, where R is the radius of the circle. But that relation too doesn’t hold in a curved space. If our ant crawls a distance R from the pole of the sphere and you then goes around in a circle, the radius of the circle will be less than 2πR. This means, measuring the circumference is another way to find out the surface is curved without knowing anything about the embedding space.

By the way, if you try these two methods for a cylinder instead of a sphere you’ll get the same result as in flat space. And that’s entirely correct. A cylinder has no intrinsic curvature. It’s periodic in one direction, but it’s internally flat.

General Relativity now uses a higher dimensional generalization of this intrinsic curvature. So, the curvature of space-time is defined entirely in terms which are internal to the space-time. You don’t need to know anything about the embedding pace. The space-time curvature shows up in Einstein’s field equations in these quantities called R.

Roughly speaking, to calculate those, you take all the angles of all possible triangles in all orientations at all points. From that you can construct an object called the curvature tensor that tells you exactly how space-time curves where, how strong, and into which direction. The things in Einstein’s field equations are sums over that curvature tensor.

That’s the one important thing you need to know about General Relativity, the curvature of space-time can be defined and measured entirely inside of space-time. The other important thing is the word “relativity” in General Relativity. That means you are free to choose a coordinate system, and the choice of a coordinate system doesn’t make any difference for the prediction of measurable quantities.

It’s one of these things that sounds rather obvious in hindsight. Certainly if you make a prediction for a measurement and that prediction depends on an arbitrary choice you made in the calculation, like choosing a coordinate system, then that’s no good. However, it took Albert Einstein to convert that “obvious” insight into a scientific theory, first special relativity and then, general relativity.

So with that background knowledge, let us then look at the first question. What does the universe expand into? It doesn’t expand into anything, it just expands. The statement that the universe expands is, as any other statement that we make in general relativity, about the internal properties of space-time. It says, loosely speaking, that the space between galaxies stretches. Think back of the sphere and imagine its radius increases. As we discussed, you can figure that out by making measurements on the surface of the sphere. You don’t need to say anything about the embedding space surrounding the sphere.

Now you may ask, but can we embed our 4 dimensional space-time in a higher dimensional flat space? The answer is yes. You can do that. It takes in general 10 dimensions. But you could indeed say the universe is expanding into that higher dimensional embedding space. However, the embedding space is by construction entirely unobservable, which is why we have no rationale to say it’s real. The scientifically sound statement is therefore that the universe doesn’t expand into anything.

Do we expand with the universe? No, we don’t. Indeed, it’s not only that we don’t expand, but galaxies don’t expand either. It’s because they are held together by their own gravitational pull. They are “gravitationally bound”, as physicists say. The pull that comes from the expansion is just too weak. The same goes for solar systems and planet. And atoms are held together by much stronger forces, so atoms in intergalactic space also don’t expand. It’s only the space between them that expands.

How do we know that the universe expands and it’s not that we shrink? Well, to some extent that’s a matter of convention. Remember that Einstein says you are free to choose whatever coordinate system you like. So you can use a coordinate system that has yardsticks which expand at exactly the same rate as the universe. If you use those, you’d conclude the universe doesn’t expand in those coordinates.

You can indeed do that. However, those coordinates have no good physical interpretation. That’s because they will mix space with time. So in those coordinates, you can’t stand still. Whenever you move forward in time, you also move sideward in space. That’s weird and it’s why we don’t use those coordinates.

The statement that the universe expands is really a statement about certain types of observations, notably the redshift of light from distant galaxies, but also a number of other measurements. And those statements are entirely independent on just what coordinates you chose to describe them. However, explaining them by saying the universe expands in this particular coordinate system is an intuitive interpretation.

So, the two most important things you need to know to make sense of General Relativity is first that the curvature of space-time can be defined and measured entirely within space-time. An embedding space is unnecessary. And second, you are free to choose whatever coordinate system you like. It doesn’t change the physics.

In summary: General Relativity tells us that the universe doesn’t expand into anything, we don’t expand with it, and while you could say that the universe doesn’t expand but we shrink that interpretation doesn’t make a lot of physical sense.

1. @Dr. Hossenfelder
This is the sort of answer one would like when asking 'But how?? Why???' to explain things in finer detail, if such answers are available.
Thanks for this video/post.
I've got a bit of work to do to understand this one, but it's still fascinating.

2. Brilliant explanation, doctor.

3. Thank you so much for explaining space-time curvature in a way that an amateur physicist can understand. I don't think I quite grasped what it meant until now.

4. Also, nice potato. :)

5. Sabine, you stated that "...General Relativity tells us that the universe doesn’t expand into anything..."

However, if according to the Big Bang theory, the universe allegedly began in a state of being that is smaller than the dot between these two brackets [ . ] and is now approximately 93 billion light years in diameter, then whatever it is that is giving-way to the presence and expansion of that dot...

(be it pure and utter "nothingness" or whatever you wish to call it)

...is what the universe is expanding into.

Clearly, "something" is making room for the presence of not only our universe, but a possible multiverse of other universes, and that "something" appears to be infinite in nature.

I get it that there are many counter-intuitive features of our modern understanding of reality, but the idea that the universe is not expanding into anything, seems to be an unnecessary assault on common sense.
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1. Keith, the same rationale that holds now also held when the universe was a "dot". It was not expanding into anything back then and it isn't now. Also, it's not an "assault on common sense". As the article quite nicely explains, any embeddings make things more complicated, are unverifiable and thus of no practical value. To the best of our current knowledge.
Don't get me wrong, if for psychological reasons, you need to imagine that the universe is a rubber balloon thats being inflated by God's breath in his living room, that is fine. It's just not our current understanding of physics.

2. Think of an infinitely large plane or sheet of material. This is such that points on this are always separating away from each other. Since this is an infinetly large plane, does it need to expand into anything?

3. If there is space for the Universe to expand into and it's a vacuum, then there'll be subatomic particles popping in and out of existence, so wouldn't that vacuum be part of the Universe also? Maybe I've missed something but this seems to be a paradox.
I'm tempted to settle for the answer (as per the video above) SPACE-TIME IS A POTATO, and the Universe is what is not the potato, which is, to say, arguably not nothing.

4. Lawrence Crowell wrote:
"Think of an infinitely large plane or sheet of material. This is such that points on this are always separating away from each other. Since this is an infinetly large plane, does it need to expand into anything?"

The very existence of an "...infinitely large...sheet of material..." implies some sort of "arena of nothingness" in which the infinitely large sheet of material is suspended. Indeed, an "arena" that could accommodate an infinite number of those "sheets of material" and never run out of room for more.

We can't import graphics here, but picture one of those images that represent the totality of our universe as being a "bubble-like" phenomenon surrounded in pure blackness.

Well, it's that unbounded blackness part of the image that I am referring to.
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5. This comment has been removed by the author.

6. I meant, outside the Universe is what is NOT the potato.
That typo makes my silly joke even worse.
That aside, I'm still stranded on that paradoxical shore. I'll look into the information in the comments and see if I can understand more.

7. A space can exist without being embedded in another space. At least this is the case in pure mathematics. A torus, think of the 2-dimensional donut shape, is a manifestation of its embedding. It has an extrinsic curvature induced by the embedding. This curvature is positive on the outside, where it is elliptical, and negative inside the hole where it is hyperbolic (saddle shaped) and the total curvature is zero. For this reason the torus can be cut open and a path that leaves a side appears in the opposite side. We can take this further and think of this as tiling up a 2-dimensional plane. A tessellated plane is a sort of covering geometry.

Complex variables is a two dimensional plane formed by the independent real and imaginary part of any complex number. For a function on that sheet with a singular point there is a cyclic condition 2Ï€×residues or equivalently a branch cut is set on the plane so that any path that crosses that it appears in another copy of the complex plane. These are Riemann sheets.

8. C Thompson wrote:
"If there is space for the Universe to expand into and it's a vacuum, then there'll be subatomic particles popping in and out of existence, so wouldn't that vacuum be part of the Universe also? Maybe I've missed something but this seems to be a paradox."

Hello C Thompson,

As a metaphorical tool to help visualize what I am speculatively talking about, picture that "bubble-like" image of the universe that I mentioned to Lawrence Crowell and then realize that the surrounding blackness in which that bubble is suspended would be "absolute nothingness" and would not contain any subatomic particles popping in and out of existence.

And the point is that the not-so-empty vacuum that you are thinking of resides only on the inside of the bubble and is part of the interconnected space-time fabric that binds the bubble together.
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9. Keith, there is nothing in the Big Bang theory that requires the universe to have itself been very small. Indeed, the universe could have been infinite then, and so would still be infinite now.

The observable universe (which is finite) was once a very dense plasma. But there's no reason it couldn't have been an infinitesimal part of a larger whole. In fact, we know the universe has to be much larger than the observable universe (at least 500 times bigger).

10. Scott wrote:
"Keith, there is nothing in the Big Bang theory that requires the universe to have itself been very small. Indeed, the universe could have been infinite then, and so would still be infinite now."

Hi Scott,

We seem to throw around the word "infinite" in ways that are simply inappropriate.

Do you truly believe that matter is infinite in the sense that there is absolutely no end to it?

Furthermore, you mentioned that we "know" that the universe has to be at least 500 times larger than what we observe.

Well, I say that even if it were 500 trillion times larger, it would still be nothing but a "speck" compared to actual infinity.

And lastly, when it comes to the standard description of the Big Bang, according the Phys.org (and pretty much any other source):

"...In short, the Big Bang hypothesis states that all of the current and past matter in the Universe came into existence at the same time, roughly 13.8 billion years ago. At this time, all matter was compacted into a very small ball with infinite density and intense heat called a Singularity. Suddenly, the Singularity began expanding, and the universe as we know it began..."

The point is that if you're trying to introduce a new way of visualizing how the universe came into existence, then you're up against a deeply ingrained model (with zillions of graphics and videos) that will not be easy to replace.
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11. "...surrounding blackness in which that bubble is suspended would be "absolute nothingness"..."--Keith Gill

This seems to be claiming that something which by definition does not exist (nothingness) actually exists.

The space-time we are in exists. It is all we know of/can observe. Religions postulate things outside our universe which cannot be observed by us, and some make the claim that the outside parts can communicate inside our universe. My claim is that anything which can communicate with or otherwise affect anything in this universe is part of this universe.

As an example, H.G. Well's Invisible Man should have been blind. To see things his retina would have to absorb photons, but they were supposed to be transparent. Similarly, anything used to effect things in this universe (photons, gravity waves, etc.) would have to interact with objects in this universe and thus would be part of it.

Nothingness cannot interact with anything and is not part of this universe, or of any universe. By definition it is not something we are embedded in. It literally has no existence.

I have a feeling that Philosophy is largely arguments over semantics, and largely a waste of time compared to engineering, physics, and math--but it can be hard to resist. I will fine myself a donation to some charity for not resisting this time.

12. Do you truly believe that matter is infinite in the sense that there is absolutely no end to it?

We currently have no evidence one way or another. Either is possible.

Furthermore, you mentioned that we "know" that the universe has to be at least 500 times larger than what we observe.

Well, I say that even if it were 500 trillion times larger, it would still be nothing but a "speck" compared to actual infinity.

That is true. It could be finite.

And lastly, when it comes to the standard description of the Big Bang, according the Phys.org (and pretty much any other source):

The point is that if you're trying to introduce a new way of visualizing how the universe came into existence, then you're up against a deeply ingrained model (with zillions of graphics and videos) that will not be easy to replace.

I am not 'introducing a new way', my description is exactly what current science says and has said.

As Sabine's video explains, thinking of the universe, past or present, as a 'ball' isn't a requirement of the physics and adds nothing to the explanation.

13. JimV wrote:
"This seems to be claiming that something which by definition does not exist (nothingness) actually exists."

Yes, and it is forever giving-way (making room for) that which we call "reality."

Indeed, I am speculatively suggesting a sort of "reification" of infinity itself.
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14. There is Trotman's argument that positive matter plus negative gravitational potential energy sum to zero. Even if the space is infinite it will still sum to zero.

15. @JimV
Greetings. I see no reason not to succumb to philosophy here, as long as it's somehow appended to the subject at hand. 'What is outside the Universe, is it nothing?' is a discussion of physics and ontology, as well as philosophy. If we're going to waste time, we may as well be having interesting discussions about 'nothing' or whatever else on this blog. Your comments are thought-provoking.

@Keith D. Gill
Thanks for your description of vacuum as part of the Universe, that makes sense.

16. Scott wrote:
"As Sabine's video explains, thinking of the universe, past or present, as a 'ball' isn't a requirement of the physics and adds nothing to the explanation."

It may not be a "requirement" for explaining the physics of the universe.

However, whatever the ultimate explanation truly is, it must also include the reason for the fact that at least from our present perspective in which we are standing on this flying orb of reality called a planet and gazing out into all directions away from it, it certainly does *appear* as if the universe is a spherical or "bubble-like" phenomenon.
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6. @Dr. Hossenfelder There's a fake 'Sabine Hossenfelder' account on your YouTube channel that's spam-posting. I already reported it.

1. Thanks for this, I didn't notice!

7. I like Ned Wright’s cosmology tutorial and I recommend it to anyone. To look at expansion I recommend the second page http://www.astro.ucla.edu/~wright/cosmo_02.htm with the tear drop light path shown in red. This illustrates the path of a photon emitted at the very earliest moment of the universe. The photon reaches a point in the future because when it was emitted by its source that source was not moving faster than light. Now, all these local light cones for objects expanding away are skewed in a funny way, and if you print this diagram out and draw pictures you can convince yourself a photon emitted later by a source along a path angled more than 45 degrees from the vertical can never reach you. This is why we can see galaxies with z = v/c > 1m where the CMB surface of last scatter has z = 1100. That is a factor by how much it is being frame dragged away by this expansion of space.

For those who do not like this idea of objects being frame dragged at a speed greater than c the diagram below this has this placed in a wedge. The flat spatial surfaces of simultaneity become hyperboloids in the light cone demarked by this wedge. Our tear drop shape is split int two, and this is because the path is from the very beginning and at the same spatial point as the observer. A photon from a source not exactly at the beginning will emerge from a place not on this light cone in a single path. This though does illustrate that we can in principle detect signals from the earliest moments of the universe. In fact if we take that to be a Planck unit of time and consider it redshifted by this expansion to the scale of the CMB this distance is around 2 trillion light years. Beyond this we run into limits with quantum gravity. A purely classical geometry though is completely projective, and this does has a relationship to projective geometry.

The observable universe is possibly a low energy vacuum pocket in an inflationary spacetime. The large vacuum energy of this inflationary de Sitter spacetime is unstable and collapses into a low energy configuration. This generates an endless number of these pocket worlds. The mass-energy gap is what generated radiation and matter in the observable universe. It is possible for these pockets to interact with each other. If they do then they will exchange vacuum energy. Think of two vessels of water with a connected between them Turn on a valve for a few seconds and their water level equilibrates some. This could result in some non-Gaussian signature in the CMB.

There is then a boundary between the observable expanding world and this exterior inflationary spacetime. Even though for an observer this boundary is moving at v ~ 10^{55}c, it leads to an uncomfortable proposition. This tends to run foul of cosmological relativity, where there would be a central position and a boundary --- even if in accessible by most observers. I wrote about how such pocket worlds might “pop off” the inflationary manifold https://arxiv.org/abs/1205.4710, using projective geometry.

The continued accelerated expansion is due to a vacuum energy, which by gravitation drives this repulsion. Whether embedded as a pocket in the inflationary spacetime or a bubble that pops off this continues.

1. I thought I would write a caveat about my paper I reference above. It is not quite the same as I indicated above. The physics though is related.

2. Thanks, I enjoyed the idea of pockets and comparison with vessels of water.

Lawrence (or anyone else who bothers), sorry to bother, but is there any unitary source you can recommend which coherently develops all the necessary relations and ideas in proper math (w/o dumbing them down) suitable for proper operational GTR, QM and standard model understanding (with all the necessary mathematical apparatus, like Hilbert spaces, necessary Riemannian geometry, and so on)? Something like Feynman's lectures but going to the very end? Granted undergrad math is not a problem.

Or there are no 'free meals' here and one has to go through proper textbooks and exercises which deal with every field separately (so kind of a university curriculum) to cover all holes in knowledge? I am asking as I learn stuff a bit funky and standard textbooks do not usually work (i.e. it's not a problem with self-study capacity as such but with personal reaction to structure in them, as there were fantastic manuals in one correspondence school that worked magic and did't require rote memorization and developed all the relations coherently making standard textbooks at the time obsolete, also spoiling me forever, I guess).

Feynman's lectures have that same kind of magic (each as a coherent mostly self-sufficient piece developing some deep principle through relations), but stop abruptly. I've currently started Roger Penrose "Road to Reality" but don't know whether it will work (or click with me). There are Landau books in Russian but for those parts that I could check I didn't enjoy the way he exposes relations (very different from Feynman's approach, who always starts from relations and understanding), he rather makes stress to definitions and proper formalisms and goes from there. So if you (or anyone else) have some thoughts or recommendations would greatly appreciate those.

Still naively looking for 'one book to rule them all, and in the darkness bind them'. :-)

3. I wrote on stackexchange this little derivation that uses only Newtonian mechanics. https://physics.stackexchange.com/questions/257476/how-did-the-universe-shift-from-dark-matter-dominated-to-dark-energy-dominate/257542#257542 This is curious, where the symmetry of the spacetime, its local symmetry in the sense of General relativity, with the global symmetry of special relativity results in a very Newtonian-like structure.

I referenced Ned Wright's website above. He is at UCLA and gives a pretty clear idea of basic cosmology. You can read some of this.

4. Thanks, I'll check it out.

I guess I didn't express myself clearly. The question was more general, as most topics developed here require a solid structure of knowledge and I constantly bump into holes, so was wondering what can be used to install such structure coherently to close those holes or is there any such source at all.

For now I'm sticking with Penrose's text to get the structure (it started to pick up, some initial exercises with conformal projections were a bit unusual, but I like that he often looks at things from different and multiple perspectives) and I actually intuitively don't like the idea of inflation (not based on anything, just initial bias) so predisposed to see if it can be approached differently and Roger seems to go this way.

5. Penrose loves conformal geometry. The group theory basis for twistor theory is SU(2,2) of conformal geometry. He has proposed the idea of Weyl complexity in the evolution of the universe, and now he has his CCC cosmology. Conformal geometry is nice, but the problem is that it is broken.

Conformal symmetry in a way is analogous to the principle of least action and the conservation of phase space volume for optical light fronts. We might think of conformal geometry as a pure condition on spacetime. Penrose also penned work extending Petrov's solution types, based on conformal symmetries that are eigenvalues of Killing vectors, and these are the Penrose, Petrov, Pirani solutions. However, once you put sources of gravitation, eg matter and fields, within the spacetime of consideration things get a bit mucked up.

6. "Conformal symmetry in a way is analogous to the principle of least action and the conservation of phase space volume for optical light fronts."
That's the part I intuitively enjoy the most (and prematurely, as I'm not sure if 1) it is what I think it is 2) I can pull it out). I think it attempts to naturally develop an intuition about TR. Particularly, because it prioritizes interaction. So it's kind of begins with interaction and consequent light lines, then builds up calibration curves according to invariant (e.g. like Minkowski metric in STR, stressing the relation) and develops from there. So I take that approach (with conformal geometry) as an attempt to generalize it even further and to think that way.

But thanks. I am not a big fan of cosmology so far (beyond some basic principles explicating GTR), and most of it is over my head (not sure I want to fix that), but I also hope to clarify the required math apparatus to think about SM, TR and QM. I also found some Lev Okun intro-level books building intuition about SM, so will see what works.

8. Nice and concise explanation, thanks.

I do wonder about the rationale for the requirement of a 10 dimensional space. I can embed a curved n-dimensional thing (a line, a surface) into a flat n+1-dimensional space (circle on a plane, sphere in 3-D space), so why does a curved 4-dimensional thing suddenly require ten dimensions?

NB: "You don’t need to know anything about the embedding pace" -- missing 's'.

9. Cool video! It once was an important realization to me that no 'space embedding' is needed at all to properly formulate TR, as Max Born's beautifully expressed replying to Newton's Principia in his "Einstein's Theory of Relativity":

[N] "Absolute, true and mathematical time flows in itself and in virtue of its nature uniformly and without reference to any external object whatever. It is also called duration."
[N] "Absolute space, in virtue of its nature and without reference to any external object whatsoever, always remains immutable and immovable."
[B] "The definitive statement, both in the definition of absolute time as in that of absolute space, that these two quantities exist 'without reference to any external object whatsoever' seems strange in an investigator of Newton's attitude of mind. ... But what exists 'without reference to any external object whatsoever' is not ascertainable, and is not a fact."

It can be seen above how ontologies of 'absolute time' and 'absolute space' appear in thinking, when they are stated outside of relation with anything, just postulated as obvious (also what many philosophers are doing, Greek and otherwise). And how Max Born dissects the essence out of it and shows that they are not even required!

Nowadays, it seems that many theorists are back to such postulates (sometimes out of convenience, it seems). Unfortunately cannot comment on most of them as they are way over my head, but treat such 'necessities' as very suspicious by containing implicit postulates and therefore inherently confusing.

Concerning the model to describe the curvature, I thoroughly enjoyed the simple and clear description by Feynman in FLP (feynmanlectures.caltech.edu/II_42.html), which gives a proper intuition about curvature w/o the need to go into intricate details of Riemannian geometry with scary names:

"Can’t we get around all of these components by using a sphere in three dimensions? We can specify a sphere by taking all the points that are the same distance from a given point in space. Then we can measure the surface area by laying out a fine scale rectangular grid on the surface of the sphere and adding up all the bits of area. According to Euclid the total area A is supposed to be 4Ï€ times the square of the radius; so we can define a “predicted radius” as √A/4Ï€. But we can also measure the radius directly by digging a hole to the center and measuring the distance. Again, we can take the measured radius minus the predicted radius and call the difference the radius excess,

r[excess] = r[measured] − √(measured area/4Ï€),

which would be a perfectly satisfactory measure of the curvature. It has the great advantage that it doesn’t depend upon how we orient a triangle or a circle."

PS Feynman's lectures are now available in audio! feynmanlectures.caltech.edu/flptapes.html

10. Just like neither we nor galaxies are expanding, neither is Brooklyn expanding.

11. "...those coordinates have no good physical interpretation. That’s because they will mix space with time. So in those coordinates, you can’t stand still. Whenever you move forward in time, you also move sideward in space. That’s weird and it’s why we don’t use those coordinates."

OTOH isn't it less general to single out time from curved spacetime to make time special? (I have seen 't Hooft slides somewhere online with time maybe closed and curved in a moibus strip.)

12. Dr. Hossenfelder, regarding expansion of the universe you state "you are free to choose whatever coordinate system you like. It doesn’t change the physics." So it depends on the chosen coordinates whether the space between the galaxies expands or whether the galaxies move away from each other. So being coordinate dependent its not true physics that space expands in the sense that new space is created.
But then - if I see it correctly - the quibbling question remains how is the increasing size of the universe (proportional to the increasing scalefactor) which is true physics compatible with the fact that expanding space isn't true physics. Or am I missing something?

13. Or we just live inside a gigantic black hole..

1. ... which then means that the cosmological principle doesn't hold. We have reason to think it holds though.

14. Once you start talking about embedding spaces its turtles all the way up.

1. I thought of theoretical extra dimensions in Kaluza-Klein theory, etc. as 'tiny turtles' that could go all the way down.

15. >It took Einstein to convert that obvious insight into a scoentific theory.

Whilst Einstein dreamt up General Covariance, the British physicist and mathematician, William Clifford had already declared after hearing Riemanns lectures on generalising the intrinsic curvature of surfaces to higher dimensional spaces, that both space and force should be understood in terms of curvature. This was fifty years before Einstein came up with General Relativity. I'm astonished that this is not more widely known. I'd be even more astonished if Einstein hadn't known of it. After all, if Clifford could learn from Germany's scientific establishment, then there is no reason why Einstein could not learn from Britain's. After all, the scientific world then was so much smaller than it is now. And I'd be surprised if such an audacious suggestion as Cliffords didn't alight in a few prepared minds. Even Riemann thought there were physical applications of his work. But he didn't take the cosmological view, rather the opposite. He thought that curvature would manifest itself in the small. It turns out that his insight was just as true as Cliffords and Einsteins. After all, whilst everyone knows that curvature is implicated in General Relativity, far fewer seem to know that it is also implicated in Quantum Mechanics.

1. I think he did it on his own. Once you relativize simultaneity (something many scientists of his day could not even stand thinking about, although they could easily formulate STR, Poincare is an example, his Science and Hypothesis, 1902, is basically converging to STR but constantly dancing around the relativization, Poincare was well aware of it, he could not just get 'absolute time' out of his head, i.e. cannot befriend the idea that simultaneity can be relativized) - you get into a deep rabbit hole.

And GTR is a natural generalization of STR (it's even in the name), an attempt to generalize the results in inertial frame of reference to non-inertial. It's more curious that most scientists who were technically brilliant and capable of formulating it, have also some blockages in thinking exactly coming from postulates of ontologies like 'absolute space' and 'absolute time' (e.g. "Yeah we measure it like this, *but* in reality it *should be*..."). And it took a courage and brilliance of Einstein to question that.

Of course, it's easy to talk in retrospect. But it doesn't seem so far fetched to see how GTR development naturally follows once one goes into relativization of simultaneity. As, basically, it leads to cancellation of absolutes of space and time.

16. I still have diffulties to understand why the expansion is not important on "galaxy scale". Hubble's constant is roughly 70 km/s/Mpc. The diameter of our own galaxy ist some 35 kpc. Ok, the Hubble flow is not very big but also not zero. Is the Hubble flow within galxies reduced, due the mass of the galaxy? Is thus the "Hubble constant" a function of the space-time curvature?

1. The "Hubble constant" simply isn't defined for galaxies. It doesn't exist. It's not a thing. The Hubble constant is itself a statement about the expansion of space on large scales. Galaxies just don't expand. Imagine you glue pennies to the surface of a balloon and inflate the balloon. That won't change the size of the pennies - they are way too strongly bound.

2. Or like soap bubbles on an ever increasing surface of water. The bubbles keep their respective sizes and stick together, like galaxies and galaxy clusters. Fascinating. Thank you for explaining.

3. Heiko, you can still ask, why behave galaxies like pennies on the balloon. The reason is that the force (due to the accelerated expansion of the universe) which tries to stretch things is too tiny to stretch solar systems, galaxies and even galaxy cluster. In other words the gravitational bondage of these systems resists stretching. Only supercluster consisting of many smaller galaxy clusters are sufficient loosely bonded so that they participate in the expansion of the universe.

17. If a galaxy sized cluster of noninteracting test masses existed the expansion of the universe would expand this cluster. The Hubble constant is H = 68Km/sec-Mpc from CMB data or 74km/sec-Mpc based on supernova SN1 data. This discrepancy is potentially a very interesting development. The square of this is H^2 = 8Ï€GÎ›/3c^3 where Î› ≃ 10^{-52}m^{-2} is the cosmological constant and is a curvature defined as Î›g_{Î¼Î½}. This means the expansion is a form of gravitation.

Now consider that cluster of test masses as a cluster of stars that attract by their mutual gravitation. This mutual gravitation is locally much stronger than this cosmological repelling gravity. Because of this galaxies and of course stellar and planetary systems remain intact. This cosmological gravitational repulsion is very weak on a small scale and disrupts structures only on the scale of galaxy superclusters.

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1. You are sort of on the right track.

2. I see the "Quantum Void" as a "void" filled with a space medium that is the basic field of our universe (and beyond). The Big Bang was just what created the stuff we know as matter and energy out of the space medium. So it is just the created matter that is expanding into the vastness of the space medium. The space medium is the zero point energy. In this scenario there would have to be something close to an infinity of space medium filling the "void".

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5. Johnathan, you did not seem to read my comment very well or did not understand it. I definitely did not say anything like "an infinite/eternal expanse of empty space". My comment is very similar to your starting comment, i.e.
You: "zero point field and the quantum void are the same things and that they are the Embedding Space that the universe expands into"
Me: "So it is just the created matter that is expanding into the vastness of the space medium. The space medium is the zero point energy."
The biggest difference may be you seem to believe in non-existent virtual particles that were invented to make the math work - vs - I believe in a real medium that particles, forces and 'universes' can be created out of.
Undoubtedly in this realm these statements are pretty much just our opinions on the subject and we like to believe we have the answers.

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19. " Do we expand with the universe? No, we don’t. Indeed, it’s not only that we don’t expand, but galaxies don’t expand either. It’s because they are held together by their own gravitational pull. They are “gravitationally bound”, as physicists say. The pull that comes from the expansion is just too weak. "

If 'the pull is to weak', then there is definately a pull. And if there is a pull from the expansion, then there must be a counter-pull from the local gravity of the galaxy to maintain the non-expansion of the galaxy.
(Pull being a slight simplification of terms of course).

That sounds like a case of extremely unlikely finetuning, no ?

1. "That sounds like a case of extremely unlikely finetuning, no ?" No its not. Its just that the attractive "pull" due to matter and the repelling "pull" from Dark Energy are in "competition". Galaxies and cluster win, supercluster lose.

2. ... at least for the time being, so it seems ...

Different galaxies have different strenghts of gravity. And yet if they were in equivalent locations, (same amount of pull from expansion), then they couldn't *both* maintain non-expansion.

21. Dear Sabine, in keeping with Paul Dirac's LNH idea, could we say that in relation to a scale invariant yardstick of around 10^6 meters, the Universe as expanded by a factor of 10^20 since the big bang while atoms have shrunk by the same factor ? If our yardstick (the meter) and everything else around us shrinks at the same rate (because everything is made of these shrinking atoms) then we would not notice it. The only thing we would measure is an acceleration in the Universe expansion rate, which is exactly what is happening. According to you, is this at all plausible ?

1. I meant: "the observable Universe has expanded .."

22. Many people have trouble with getting their heads around the question of "What does the universe expand into?" However, these same people have no trouble in understanding how time extends (expands) into the future. They understand that the past has already occurred, that we are in the present and that, although the future does not yet exist, it will be with us shortly. Time is one component of space-time, so just as time expands (extends) into the future so space similarly expands. I know this analogy is not perfect as we perceive the future as being time added on, rather than the existing time being expanded.

23. Maybe no one is interested now, but the way I’ve always made sense of the Big Bang was to remember that Spacetime was created then. That has a number of implications. But, foremost it means that our modern science had no physical counterpart prior to the BB. It wasn’t a black hole, or a singularity, or anything else we now sensibly describe. It means that whatever the universe was prior to the event; to any imagined observer, the BB occurred everywhere at once. To an interior occupant it would appear as if the entire universe was undergoing some change simultaneously.

24. Is the universe expanding only along its three spatial dimensions and not along the dimension of time?

1. Proper time - the time which the clock shows - is invariant. It flows just second per second. Time is also not an observable which could react somehow (expand) with the system.
To see the connection with the expanding universe space you can imagine a stack of time-slices. According to the expansion this stack will be growing. Thereby a slice represents space and the stack represents space-time.

25. Dear Dr. Hossenfelder,

I think that your explanation of this topic is very good.
I have one question. You stated

"Now you may ask, but can we embed our 4 dimensional space-time in a higher dimensional flat space? The answer is yes. You can do that. It takes in general 10 dimensions."

Is that true? Someone nows where to find the statements of these results? Is 10-dimenional manifold an R^n? are isometric embeddings?

Thank you very much.

26. I found this video very interesting and more informative than a similar one posted on YouTube by another theoretical physicist.

I would be grateful if you would reply to my email sent to your work email address.

27. I have always thought about space expansion as if all the existing space was initially within the singularity, and then the BB blew it out, so the actual size of the Universe increased, but local distances did not change. If you like the balloon surface analogy, it is as if you count the number of rubber molecules on the balloon surface and start pumping water inside: it expands, but the number of molecules stays the same, so the "space density" decreases. But after considering how this would constrain physical phenomena to local distances, making the expansion virtually unobservable from inside the expanding space, I am now inclined to think that the "space density" is (roughly) maintained, while the "amount of space" increases. Coming back to the balloon surface analogy, instead of counting rubber molecules on the balloon surface (that conserves) you have to count water molecules under the balloon surface. So, expansion appears as space is constantly being created everywhere, not as it stretches uniformly. This brings the question where the necessary (dark?) energy comes from, and considering higher dimensions becomes more justified.

28. Thank you.

Are there Riemannian or Lorentzian results?

29. There must be at least a few hydrogen atoms between galaxies.
Does the density of them decrease, locally (say on the scale of a few or few tens kiloparsecs), as the universe expands. Does this depend on the original density (say mean distance of one meter, one billion meters, one parsec, 100 parsec?

30. Hello Sabine, I wonder if there is a small problem with the site.
This morning you approved my comment on an old article, "How to Live Without Free Will". It has yet to appear in the comment thread and neither has the exchange between yourself and Alberto that drew my attention in the first place.
Perhaps my argument was so persuasive that you are hiding it. Just kidding, but wonder what's up.
Best, Don

1. Don,

If a comment thread has more than 200 comments you have to scroll down all the way and click on "Load More" to see the newest ones. (Or search the page for "Load More" and jump there). This is a longstanding problem with Blogger comments, and we have all complained about it for years and so on, but it still hasn't improved and the only thing I can do about it is turning comments entirely off.

31. Ah me, please disregard last comment re "How to Live Without Free Will" --- I forgot about the 'Load More' feature.
Thanks.

1. Yes, well, of course I only saw this after I replied to it... I'll leave it up because it's good to remind people of this every once in a while.

32. Nash theorem is a global theorem in Riemannian geometry. Also, the paper that you mention is on Riemannian geometry. Riemannian and Lorentzian geometry are totally different. Indeed, this has been stressed by many people, including Gromov himself. The main difference comes from the naturality of Riemannian metrics, in contrast with Lorentzian: any manifold admits a smooth enough Riemannian structure. Also, the isometry group are totally different: while for a Riemannian manifold is compact, for a Lorentzian manifold is non-compact, which mathematically precludes many analogous developments, at least it complicated them (harmonic analysis).

I am not aware of the corresponding theorem for Lorentzian signature. If there is a nice reference where one can see how a four dimensional C^5 manifold is isometrically embedded (globally) in a R^n with metric n-1 flat, or the corresponding result for four dimensional solutions of Einstein equations. The statement from Hossenfelder is very precise ( 4 in 10 dimensional isometric embedding of Einstein solutions). Thus I would like to know if this is true or not and then if someone has a nice reference where it is explain the methodology of embedding.

Remark. if 4 can be embedded in 10, then it can be embedded in 11, 12,.....arbitrary large dimension. Hence the concept of extrinsic space where universe expands is meaningless.

33. - Charon: "Scientists of century 21st believed the universe expands, ha ha ha!".
- Grisha: "Based on scientific evidence?"
- Charon: "Well, naive scientific evidence, of course. They even thought the universe was undergoing an accelerated expansion, ha ha!"
- Grisha; "But, isn't that what we observe nowadays?"
- Charon: "Yes, observation of light is always tricky, at the end of the day it always manages to fool us with false evidence. We already know that so called expansion of the universe isn't real, but a naive conclusion taken from false evidence. The observation of red shifts of light of remote galaxies and galaxy clusters is just an illusion. Today we know that red shifts of light are systematically augmented against any eventual blue shift, because of the application of wrong and obsolete Doppler formulas."
- Grisha: "Yes sir, I studied that lesson yesterday. Doppler effect is a self-similar phenomenon, and remote astronomical sources of light, which exhibit Gaussian distributions, show us augmented red-shifts against eventual shrunk blue shifts. We'd say a remote star, with Gaussian distribution of its photosphere plasma granules, exhibits an asymmetric Doppler profile. The more remote the star is, the more pronounced that Doppler profile asymmetry. That leads to a false conclusion of watching a receding star. "
- Charon: "Go travel to the past and tell them they are wrong, ha ha!"
- Grisha; "Sir, they even believed the speed of light in a vacuum is a universal constant!"
- Charon: "True, they were good believers."
- Grisha; "A good believer is a bad scientist?"
- Charon; "I didn't say that, kid."
- Grisha; "And what about the curvature of the universe, sir?"
- Charon; "Kid, you can't curve what doesn't exist!. Anyway, they used a wrong conception of spacetime. The correct formulation of spacetime is space times time (s x t). Take a Minkowski spacetime reference frame, let the time axis be the abscissa in order to avoid time paradoxes. Then, plot any spacetime curve into it, and perform a definite integral in any time interval. The region (area) bounded by its graph is the correct notion of spacetime that is fruitful for calculations."
- Grisha; "Is the universe infinite?"
- Charon; "I've told you that a thousand times, kid. Of course the universe is not infinite, but has undefined borders. An empty space (no matter in it) is nonsense. You can imagine a universe with just two particles in it. Then, space is only defined between those two particles, they can only attract each other (there is only one degree of freedom in that system), they can't repel each other because there is no further space than the space between them. So, matter produces space and time. Understood, kid?"
- Grisha; "Understood, master, sir."

1. "You can imagine a universe with just two particles in it. Then, space is only defined between those two particles, they can only attract each other (there is only one degree of freedom in that system), they can't repel each other because there is no further space than the space between them."
To hold a particle on a circular path, a force is required that is permanently directed to the center of rotation.
This does not mean that the force is necessarily a pull-force, a push-force will also do. That's what Euler's e^(i*Pi)=-1 means.

34. Sabine, you say:

“There’s no embedding space. So, for the sphere that’d mean, we’d have to ask how’d we find out it’s curved if we were living on the surface, maybe ants crawling around on it.

One way to do it is to remember that in flat space the inner angles of triangles always sum to 180 degrees. In a curved space, that’s no longer the case. “

But this result of inequality with 180 degrees is only a proof for curvature if we already start with the conviction that the space is curved. Because, how can be determine the angles of a triangle in practice by measurement? The only way we can do it in space is that we define three points and send light beams from one point to the other one and then measure the angle between them. What does this tell us? If we do not assume the curved space a priori, we may refer to the classical (non-Einsteinian) knowledge that light beams are curve if matter is around. That is the consequence of the reduction of c in a gravitational field by which a light beam undergoes refraction. And if we are faced by a triangle built by those refracted beams, the angles will of course not sum up to 180 degrees.

So, this way of proof is a kind of a self-fulfilling prediction.

35. @Dr. Hossenfelder:
I'm writing this here because I don't use Twitter. The article from the Wall Street Journal that you re-tweeted a few days ago about transgender women being transferred into women's prisons is misleading and transphobic.
Consider that anyone who goes to the effort of outing themselves as transgender, making an application to move into a women's prison and to live life as a woman isn't doing it just to access other women to violate. Also, a woman needn't need to alter her physique or hormonal state in any way to be a woman (or a non-binary person) as they already *are* not male, whatever their body is like. Transgender people don't always have the means and/or the desire to alter themselves to be more 'biologically female'.

The assumption that anyone with a penis is a potential threat (as a criminal or a civillian) is damaging to both transgender women and to cisgender men.
The article makes no mention of incarcerated transgender men, of the discomfort they may feel at being forced to live among cisgender women.
I'm writing you this as I hope you would not wish to cause hurt and discomfort to any transgender, non-binary and queer followers you have (I myself am queer and a trans ally) - given that your YouTube videos get tens of thousands of views it's exceedingly likely you have them. I admire you as a human being as well as an entertainer and physicist, I hope you reconsider sharing such content.
Thanks and regards,
Colleen

36. Sabine,

Thank you for an excellent explanation of how the universe expands and why most mathematicians and physicists view embedding spaces as redundant. Some work I did years ago kept nagging at my memory until I recalled it just today. Below I take a contrary position and explain why I would not only say that embedding spaces do matter, but that not using them is unforgivably sloppy mathematics.

The first point is the observation that if you dismiss embedding spaces, you must also dispense with the concept of thickness. A balloon, for example, has a thickness and so possesses an inside and outside surface. Most mathematicians long ago decided they did not like these identically shaped and thus informationally redundant parallel surfaces. For example, they replaced dual-surface balloons with the concept of 2-spheres that assume a single infinitesimally thin two-dimensional surface. All topologies that discard embedding spaces must merge inside and outside surfaces, since only by eliminating the separation between these two surfaces can they dispense with all mention of an embedding space.

As Sabine aptly described, the easiest way to discard thickness is to focus solely on connectivity between components and never mention thickness. Thus a cube becomes a connectivity relationship between six squares, with no mention of 3D volume. Similarly, a square becomes a connectivity relationship between four lines, with no mention of 2D area. Even a line can become a “linked” relationship between two points, with no mention of 1D length. In any such sequence, the dimensionality at which “filling in” the space to give a field is necessarily arbitrary and is used to define the Lie algebras for that manifold. Focusing solely on connectivity at the chosen dimensionality, such as 2-volume for 2-spheres and 3-volume for human-scale objects, makes hole-preserving morphing transformations between shapes easier. For example, locking in at 3-volume allows one to make the classic tutorial assertion that donuts and coffee cups are identical in topology.

The catch is that to implement all of these transformations, you must also maintain information that uniquely identifies all of those otherwise identical parts. For example, to create a square out of four lines, you must also label the lines (e.g., A, B, C, D) and then link them (e.g., A to B, B to C, C to D, D to A). However, the very act of labeling the parts also unavoidably adds one or more orthogonal information dimensions to the system. Our label-focused primate cognition systems tend to discard such word-valued dimensions without even realizing it. However, that is unacceptably sloppy for real-world or real-computer representation of, for example, creating a square out of lines ends up creating at least a 2D space. After properly including labeling axes, the result ends up having the same dimensionality, at least, as an embedding space. It has to be that way since any lesser number of labeling axes collapses the parts on top of each other like a 2D projection of a 3D cube.

I should emphasize that if that is what you want to do, there is absolutely nothing wrong with focusing on connectivity and angles to define the topology of our universe. However, to be mathematically formal when creating such confections, one must explicitly discard the naming coordinates of the implicit embedding space, not just sloppily brush them away as if they do not matter.

They do matter, especially for physics. For example, if you color-code all six of the squares used to create a cube, you discover that each nominally unique connectivity combination results in two distinct cubes. In both full-labeling space and 3-space, they are mirror images of each other. In real physics, that’s important since it is equivalent to the difference between matter and antimatter!

(See 2 of 2 below)

37. (2 of 2)

So: If the main problem with connectivity-only topologies is that they are exceedingly sloppy about discarding implicit labeling dimensions, why did I begin this by focusing instead on the concept of thickness and inside-outside surfaces?

That’s an interesting one. If you scrupulously avoid dimensional sloppiness and embed your manifold properly within a labels-compatible embedding space, it turns out that all manifolds have thickness. For example, and a bit more bluntly, there is no such thing as a 2-sphere, no matter how many Lie algebras you define on top of it. The best you can do is a two-surface balloon that gets sloppy-merged at some unreachable infinitesimal limit. The fancy name for this sloppy, physically unreachable limit is “duality.”

I say “at some unreachable limit” because all representations of manifolds, whether in your head or a computer, use finite-precision labels (often called numbers) to represent the points that define the manifolds. Equations tell you how to generate these points, but folks tend to forget that until someone uses equations to generate actual points, the equations are nothing more than programs that use up resources to generate the points of the actual manifold. As long as the generated numbers have finite precision in the embedding space, they also have width in the embedding space. The two concepts — finite precision in calculation space and finite width in the embedding space — cannot be separated.

But does all this nitpicking about numbers make any significant difference to either math or physics? Yes, it does. For one thing, consistent explicit use of the always-there embedding spaces makes navigating higher spaces a lot easier.

Here’s an example: How many kinds of holes are there in 5-dimensional space? If you said “three,” give yourself a Fields medal, since as best I’ve been able to tell, someone once got one more-or-less for figuring out that 4-dimensional space has two types of holes. That gives you some idea of how challenging higher dimensional holes are to navigate if you force yourself to use only the sloppy and decidedly non-intuitive connectivity-only approach to manifolds that gained popularity over the past half-century.

The pattern for how many unique hole types there are in higher spaces is almost absurdly simple: 1 in 3-space, 2 in 4-space, 3 in 5-space… you get the picture. Even better, those two holes in 4-space also exist in 3-space, just with slightly different properties. We call them “rings” and “balloons” [1]. When folks talk about “four-dimensional toruses,” they almost always mean an ordinary 3-space ring that has been “fattened up” to exist in 4-space. The difference is that in 4-space, you can pass an entire infinitely large plane through the same hole that in 3-space only allowed strings to pass through. Similarly, fattening up a balloon to exist in 4-space allows its interior to pass strings (but not planes). This much less well-known type of 4D hole is the most like the beads-on-a-string one we know in 3-space.

String theory loves to talk about holes. Not once do they get any of this right.

----------
[1] Bollinger, Terry. Terry’s Quick Guide to Holes in Hyperspace. LinkedIn, July 17, 2019. https://www.linkedin.com/pulse/terrys-quick-guide-holes-hyperspace-terry-bollinger/

38. I think that the term 'expands into' is in general misleading. It would be better to say:
- "The space-time of the universe develops" or
- "goes through an own process of evolution"
Whereas no one really knows what the result will be in the end.

See Wikipedia --> "List of unsolved problems in physics" --> "Future of the universe"

39. The notion that an embedding space is needed to define a space (if indeed that is what anyone is claiming) would, it seems to me, lead to the infamous "infinite regress". Along with Zeno and Democritus, I distrust infinities. They just don't seem practical to me. (Maybe it is because I only have a finite number of neurons.)

40. “Do we expand with the universe? No, we don’t. Indeed, it’s not only that we don’t expand, but galaxies don’t expand either. It’s because they are held together by their own gravitational pull. They are “gravitationally bound”, as physicists say. The pull that comes from the expansion is just too weak. The same goes for solar systems and planet. And atoms are held together by much stronger forces, so atoms in intergalactic space also don’t expand. It’s only the space between them that expands.”

“The pull from the expansion is just too weak.” Two questions to it:

1) Does the expansion really cause a pull? Is it not in contrast so that everything moves with the space so that no force is occurring? In that case the stars and the galaxies should attract each other like in the case where no expansion happens.
2) What is about fields? Do they expand with the field? In that case at least constructs which are built by multipole fields like the ones forming a molecule should also be getting extended.

In the latter case also our bodies will expand. And in this case, would we have a chance to notice this expansion?

The general problem with considerations about space is that it cannot be handled in an objective way, meaning that we cannot measure space physically. Because, how could that work? A geodesic person on earth would fix reference objects to the ground to perform a measurement. For space, such objects would similarly have to be fixed to “the space”. How should that be done? It is simply not possible. So anything quantitative what we state about space is either philosophy or interpretation. – Is this a really good and meaningful physical practice?

41. JimV, A 'datum' kinetic state as a 'local background' system is essential to derive ANY 'propagation speed'! There may must then be hierarchical, as found in space. As Einstein put it, (after Minkowski) in '52 there are 'endlessly'('infinitely' HM) many spaces in motion within spaces. As he said; that's 'logically unavoidable' but "not yet part of scientific thought". He was ignored so it still isn't! HM also wrote "everywhere and everywhen there is substance".
It also reminds me of Lawrence Krause's book "A universe from nothing" even BETTER that his excellent climate change book!
P.S. Sabine; No comments on that topic?? Was the comment box missed off?

42. Sorry, but the FLRW coordinates we are usually using, ds^2 = dtau^2 - a^2(tau) dl^2, are those where distances are shrinking but everything remains on its place.

43. OK I'm a little late to this but I would like to point out a couple of things.

One, it is not at all a matter of convention to say expansion ~ shrinking modulo coordinates. One can imagine a physical field that acts on say the electron mass, which ultimately defines the size of things. The radius of the hydrogen atom goes like the inverse of the electron mass. So such a (scalar) field, if it changed the rest mass of the electron in the same way everywhere, would produce the same shrinking ratio everywhere. Now, it is quite easy to see that this would have an immediately observable consequence - Hubble's linear law! Objects would appear more distant in direct proportion to their distance, and we would impute, should we assume that we are in fact maintaining our size, that distant objects were receding with a radial speed proportional to their distance. Now - this is an enormous change in perspective - one has gone from imputing a dynamic event, the Big Bang, from the expansion, to a simple kinematical explanation based on the action of a local field. It is not at all a symmetrical situation, but an imputation of a dynamical cause which may in fact be absent - because Riemannian geometry involves an arbitary, global length scale (parallel transport does not change the length of vectors), then as you say, there is no way to decide between local shrinking vs. global expansion. In almost every way, the explanation based on local field action is more logically compelling (in the Occam sense) because we need only make the assumption that the acting field is a scalar.

This issue leads one to consider that Riemannian geometry is just an inadequate framework for fundamental theory, and to consider alternatives (Weyl geometry) which can bring in other field quantities to tighten the ropes.

1. "One, it is not at all a matter of convention to say expansion ~ shrinking modulo coordinates. One can imagine"

I suggest you don't "imagine" but do the math. There are many ways to "imagine" coordinate transformations wrongly.

2. C'mon Sabine - I've been doing such things for a very long time now. Again - Riemmanian geometry has an arbitrary global length standard. That is not a matter of coordinates at all, it is a matter of structure. This shrinking vs. expanding issue is just a demonstration of that. To make it concrete, one needs a way of breaking that global symmetry. Weyl geometry (among others) works. (It doesn't work as physics in 4d, but that's another "matter" hehe.)

-drl

3. drl,

You can of course introduce a length by defining a scale, but there's no reason to use any of those scales, and there is still no reason to use any particular coordinate system in GR. Look, the only thing I am saying is that you can take FRW in the standard comoving coordinates, make a coordinate transformation, and get a non-orthogonal metric with flat subspaces. That's weird, alright, but it's perfectly possible.

44. It expands into the same thing that it expanded out of, the Quantum Void. But since the Quantum Void is non-physical, the universe expands into nothing.

45. I think your 10-dimensional embedding manifold is a bit higher than needed. I recall from my GR courses that the flat embedding space for an n-dimensional manifold requires at most 2n dimensions; see https://en.wikipedia.org/wiki/Whitney_embedding_theorem.

1. (a) Please see the info below the video.
(b) Depends on the exact assumptions.

46. Dr. Sabine
I read that if you could ride a photon
you would see past, present, and future simultaenously
Is that so?
Could you relate that to the block universe and superdeterminism?

1. If you could ride a photon you'd be dead and wouldn't see anything.

47. Hi, shouldn't universe expansion have a "visible" effect on subatomic particles arround us? Theoretically their wave function fills the universe so it will strech with it in some places even with speeds FTL.

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