## Sunday, August 26, 2018

### Dear Dr B: What does the universe expand into?

“When the universe expands, into what is it expanding? In what medium is it expanding? Is the universe like a bubble in a higher dimension something?
[Anonymous], Indiana, USA”
 [image: luftballon-profi.at]

This is a very good question and one, I should add, I get frequently. It is, I believe, to no small part caused by the common illustrations of a curved universe: it’s a rubber-sheet with a bowling-ball on it, it’s an inflating balloon, or – in the rarer case that someone tries to illustrate negative curvature, it’s a potato chip (because really I have no idea what a saddle looks like).

But in each of these cases what the illustration actually shows is a two-dimensional surface embedded in a non-curved (“flat”) three-dimensional space. That’s good because you can draw it, but it’s bad because it raises the impression that to speak of curvature you need to put the surface into a larger space. That, however, isn’t so: Curvature is a property of the surface itself.

To get an idea of how this works, consider the simplest example of a curved surface, a ball. On the ball’s surface the angles of triangles will not add up to 180 degrees. You can calculate the curvature from measuring all the angles in all triangles that you could draw onto the ball. This is a measurement which can be done entirely on the surface itself. Or by ants crawling on the surface, if you wish, to use another common analogy.

Curvature, hence, is an intrinsic property of the surface – you do not need the embedding space to define it and to measure it. Also note that the curvature is a local property; it can change from one place to the next, just that a ball has constant curvature.

General relativity uses the same notion of local, intrinsic curvature, just that in this case we aren’t dealing with two dimensions of space and ants crawling on it, but with three dimensions of space, one dimension of time, and humans crawling around in it. So the math is more complicated and all the properties of space-time are collected in something called the curvature-tensor, but that is still an entirely internal construct. We can measure it by tracking the motion of particles, and it’s this curvature that creates the effect we usually refer to as gravity.

Now, what cosmologists mean when they speak of the expansion of the universe is a trend of certain measurement results that, using Einstein’s equations, can be interpreted as being due to an increasing distance between galaxies. Again, this expansion is an entirely internal notion. It is defined and measured in our universe. You do not have to embed this four dimensional space-time into anything else to quantify it. You do not need a medium and you do not need a larger space. Einstein’s theory is entirely self-contained with a four-dimensional, internally curved space-time.

While you do not have to embed space-time in a higher-dimensional flat space, you can. Indeed it can be mathematically proved that you can embed any curved four dimensional space-time into a ten dimensional flat space-time. The reason physicists don’t normally do this is that these additional dimensions are superfluous and they don’t aid the math either.

 Black hole embedding diagram. Only the surface itself has physical meaning. The surrounding space is for visual purposes. [Image source: Quora]
We do, however, on occasion use what is called an “embedding diagram”, which
can be useful to visualize the extrinsic curvature of certain slices of space-time. This is, for example, what gives rise to the idea that when matter collapses to a black hole, space develops a long throat with a bubble that eventually pinches off. But please keep in mind that these are merely visual aids. They have their uses as such, but one has to be very careful in interpreting them because they depend on the chosen embedding.

Now you ask what does the universe expand into? It doesn’t expand into anything, it just expands. That the universe expands is a statement about what happens inside the universe, supported by measurements inside the universe. It’s an entirely internal notion that does not require us to speak of an outside of the universe or a medium into which it is embedded.

Thanks for an interesting question!

1. Thanks for the blog!
Thats a very interesting view, so you are saying because we can not comprehend negative curvature - akin to maybe a tall hill on earth that artifically "expands" the travel distance between two places, although in 3 dimensions the actual distance stays the same, which isnt the case in what we observe through redshift but for the idea of visualising negative curvature it might be just fine - its actually not needed to think about where the universe expands into, as it is only spacetime, an entirely 4D concept that our brain can not visualise, that is expanding?

Where does negative curvature come from though? We have not yet found anything with negative mass, so how could this even work?

2. This comment has been removed by the author.

3. Thank you for this. It confirms my prior beliefs and so makes perfect sense to me.

What I don't understand is when people talk about collisions between universes leaving imprints on the CMB. I find it hard to think about this without envisioning the universes embedded in and expanding into a higher dimension space. My preference is to assume that there is not a multiverse so I don't need to contemplate this possibility, which seems consistent with all empirical data but may mean that I'm burying my head in the sand.

4. derdotte,

I don't know what makes you say "we cannot comprehend negative curvature". We can. Or some of us can, anyway. As I wrote, the saddle/potato chip is the common example. If the generic potato chip is the source of confusion, I originally wrote Pringles, but then thought it might look like advertisement.

5. Mike,

Yes, indeed, I have the same problem. But you can calculate it, and that works for me. Really I often find visualization attempts more confusing than enlightening.

6. Curvature, hence, is an intrinsic property of the surface – you do not need the embedding space to define it and to measure it. Four LIGO-observed black hole mergers are 2(+ ε)-dimensional bubbles joining. The surface is all observables. No "internal" singularity exists. BH evaporation reveals nothing.

QM and GR can be selectively defective by their own rules, yet consistent with all prior observations. Chemistry uses an unmeasurable observable physics denies exists. Observe the black swan (DOI:10.1002/anie.201704221 with a different molecule).

7. My best quess is that the universe expands into space and develops into a new universe.

8. Dr B, you said, " It doesn’t expand into anything, it just expands". While I agree with the notion of intrinsic curvature as you presented it, isn't that just talking about our ability to explain things in a particular way with math. It does NOT prove your statement above. So, wouldn't it be better to answer with, "our current knowledge is uncertain in these regards"?

9. Can you explain why a four-dimensional curved spacetime needs ten dimensions to be embedded in a flat spacetime? Why aren't, say, five dimensions of flat spacetime enough?

10. Can the observed expansion of space=time in the observed universe be the relaxation of the gravity well of the universe as it's density is reduced? If so, can it be considered as a local distortion of a greater (infinite?) space-time?

11. I follow your answer and it is very elegantly stated not to mention it is very informative. I enjoyed reading it. But with all due respect doesn't it still beg the question? Some would answer by "We are part of a multiverse ! Alas we are so far removed that it matters not." Other's like Hawking would say "It's like asking what's north of the North Pole", meaning that there is nothing beyond the beyond for anything to expand into anyway. You are getting around the problem by saying the question is not necessary because the properties of space time allow me to expand in such a manner that what is "outside " of me isn't needed. Clever I like it but we both know it does not answer the readers question.

12. Thanks for clarification. But as far as I know our notion is based on gravity as mentioned in general relativity. Tomorrow If the gravity is unified with electromagnetism perhaps we should review and rewrite all these arguments which are related to the gravity. ..

13. hiwa,

Doesn't matter what happens tomorrow, GR will remain an excellent theory.

14. sedumjoy,

The multiverse too isn't embedded in anything.

15. Leibniz,

10 dimensions will always do, but sometimes fewer will be sufficient. Depends on the number of symmetries. I can't recall the details but I believe it just comes from the number of independent components in the metric tensor.

16. Evan,

You don't need it, as a matter of fact, according to all current observations. GR works well without an embedding. You can always postulate the existence of things that our observations don't necessitate. If that's what you meant to say, that's correct.

17. Obviously, it expands into Heaven.

18. Would the balloon+ants analogy be more useful if the ants were crawling in the inside of the balloon, and for whom the universe ends at the rubber surface?

19. "Can you explain why a four-dimensional curved spacetime needs ten dimensions to be embedded in a flat spacetime? Why aren't, say, five dimensions of flat spacetime enough?"

An m-dimensional space needs, in general, at least n dimensions to be embedded in, where n = m + (m-1) + (m-2) + ... + 1. So, for m = 2, n = 2 + 1 = 3 and for m=4, n = 4 + 3 + 2 + 1 = 10.

Basically, to describe the curvature, you need an m x m matrix. The diagonal has length m. Due to symmetry, the upper and lower off-diagonal elements are equivalent, so there are only n independent quantities.

20. Sabine,
thanks for your post, and also for the blog as a whole. I enjoy having occasional reads on your blog.

Your daily-life analogy of ants on the ball was illuminating. Analogy, is used correctly, can impart the right insight and help avoid the incorrect one---like you did in this case. My question is, can you please also provide an apt daily-life analogy of something that expands, but does not necessarily expand into anything?

Many thanks.

21. Since as a non-physicist I only have to worry about seeing empirical evidence in a way that’s visually sensible and consistent with observations, I dismiss the two dimensional curvature diagrams as being physically viewable. Instead I merely look at space-time curvature as being the scale for which space-time is relative to another observer. It may be wrong, but for an amateur just wanting consistent visual dynamics that work for SR too, it makes a good approximation that seemingly adheres to evidence and observations, and allows for some predictive approximations for relative views of space-time too.

22. Dr. Hossenfelder,
Does finite vs infinite space affect whether it is embedded?
In a 2D analog, a sphere can be finite, when it expands, it doesn’t seem to need embedding. An infinite sheet can expand without embedding as well. A finite sheet can expand, but it seems to be expanding into something.
Thanks

23. @ Phillip Helbig: Look up Whitney embedding theorem. The strong form where for an n dimensional space it can be smoothly embedded in 2n dimensions such as R^{2n}.

There are ways of defining curvature. Riemannian curvature is defined entirely by the parallel translation of vectors within a manifold. It does not have any definition with respect to an embedding space of larger dimension. This can be generalized with sectional curvatures defined on a two dimensional tangent plane on a manifold. If this tangent manifold changes under translation so one of the vectors is no longer defined on the manifold then there is a reference to an embedding space. We can think of one of these vectors transforming into some ancillary dimension. Conversely if there is a vector locally normal to a tangent plane to a manifold that under translation becomes linearly dependent with a local basis on the manifold there is then an extrinsic curvature.

Spacetime is most often not defined with any reference to an embedding manifold. We may however think of spacetime as 3-space plus time so spatial surfaces foliate the spacetime. Different spatial surfaces are related by some homeomorphism (topological isomorphism) that is differential or smooth. So for a Euclidean R^3 space we can imagine a spatial surface related to another by having the distance between points changed, and that distance change is a linear function of their initial distance. This is in effect what the Hubble relation v = Hd, H = 74km/sec-Mpc the Hubble constant and d the distance between galaxies. These galaxies are then frame dragged with this expansion of space or the increasing distance between points of space. As such very distant points with a distance d > c/H move apart faster than light speed. They have redshift z > 1 and we are able to witness them from a prior time when their comoving velocity was less than c.

This does mean that extremely distant points are frame dragged out at extreme velocity. If the redshift is larger than the local cosmological horizon distance d = sqrt{3/Λ} then these regions are utterly unobservable. Here Λ is the cosmological constant, which for the observable universe is a curvature factor Λ ~ 10^{52}m^{-2}. Even if we observe a galaxy with red shift z > 1 we will never be able to send a signal to it, nor can we ever observe it as it is on the Hubble frame.

The de Sitter and anti-de Sitter spacetimes are defined according to an embedding.in 5 dimensions. The 5 dimensional Euclidean space R^5 can have the metric

s^2 = t^2 - u^2 - x^2 - y^2 - z^2

where a constant metric distance s defines a hyperboloid that has the de Sitter metric. For de Sitter spacetimes the cosmological constant Λ ≥ 0. A 3 dimensional space has no intrinsic curvature, but the curvature of spacetime = 3-space plus time defines the way this space is embedded and in a sense “evolves” by exponentially expanding. If the additional coordinate direction u as a time meaning then

s^2 = t^2 + u^2 - x^2 - y^2 - z^2

and a constant s defines two hyperboloids and these are anti-de Sitter spacetimes. Here the cosmological constant Λ < 0 is negative. These spacetimes are quotient geometries defined by the quotient of orthogonal Lie groups. In effect this means these spacetimes have a principal bundle on them that reflects this embedding.

The cosmos is often portrayed as a flat Euclidean space that evolves in time on a Hubble frame. This description has a funny frame dependency, which gets us into some subtle issues of energy and time. The manifold does not have to be a flat space. It can be a 3 dimensional sphere, or a 3 dimensional hyperboloid.

24. Sabine,

I someone had 6 superconducting balls arranged in a hexagon, with current traveling through these balls, would the current of each ball look like it is accelerating from every viewpoint within each ball? If so would that be an analogy of why our universe looks like it is accelerating?

25. Babak, no, clearly not! Everything that we see expanding, like balloons or bellies after brunch, can be seen as expanding into space. We like to see it that way, because we live here. But now we're trying to answer the question, "What is the expansion of space like?" This is not so mundane.

The best analogy is maybe to the math of two dimensional surfaces, like balloons with ants on them, or lady bugs. JQ, it doesn't matter if the ants are inside or outside - the ants are imagined to have no height, and the balloon we're concerned with is just the surface. If you consider the balloon embedded in R^3, it has an interior volume, but that's a property of the embedding, not of the balloon itself.

The history of surface geometry started with looking at everything embedded in three dimensional space, because that is where we look at things, as people. And it is Gauss's Theorema Egregium that showed that all the curvature properties could be reproduced purely locally with on-surface measurements, without any reference to some embedding. This is a potent theorem, and it essentially keeps working in higher dimensions.

Personally, this caused me a lot of frustration when I took an algebraic geometry course. About two weeks I finally spilled my confusion onto the professor. "But these properties, they're all dependent on an embedding. Why does anyone care about any of this?" It turns out there may be some reasons, but general relativity isn't really one of them.

26. Hallelujah! It took Google three months but finally comment notification is working again!

27. Babak,

Everything inside our universe is embedded in that universe. Hence, there can't be a real-life example.

28. JQ,

In the balloon analogy, the universe is only the surface of the balloon. As any analogy, it has its shortcomings. The ants should actually not crawl on the surface, but they should be two-dimensional ("flat world") ants crawling in the surface. The side of the surface, hence, isn't relevant in this analogy.

29. I made an error in my last post here. I wrote " curvature factor Λ ~ 10^{52}m^{-2}," when in fact the curvature factor or cosmological constant is very small Λ ~ 10^{-52}m^{-2}.

I have to further write that Bee has written a reasonable account of how the observable universe expands. The one problem I have noted with physics is how it shocks our intuitive idea of the world, and this case is one example. Further, I have to say it is very unlikely the expanding universe has a boundary. This would imply a sort of auxiliary set of conditions. This is part of the Hartle-Hawking result.

30. I just remembered these animations by "Zogg of Betelgeuse" on how the universe has no edge. These are pretty good and funny in places.

31. A 4-dimensional curved spacetime needs a 90-dimensional flat spacetime with 87 space dimensions and 3 time dimensions to be embedded in. Source:

Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian manifolds," Proc. Roy. Soc. A314 (1970) 417-428

My gut feeling is that a quantum mechanical spacetime curvature theory actually needs those 90 dimensions!!

32. For the embedding one needs an isometric emmbedding, so Nash theorem is the one to look at. And better the one, if there is one, about the lorentzian case.

33. Observe an ongoing irresolvable debate of which math models are appropriate to the task. Nobody wonders whether postulates are appropriate. Newton postulated (outright or tacit) wrong values for lightspeed (infinite), Planck's and Boltzmann's constants (zero). Newton opened Principia postulating the Equivalence Principle. All but the last are empirically wrong. The EP has only been tested to validate (via "accepted theory"), never to falsify.

Maths succeed but physics fails. Both QM and the EP are vulnerable to a day of analytical chemistry. Denial cannot heal physics. (Organic chemistry was gobstruck by palladium. The name reactions are Asian – who did not know it was impossible.)

34. I viewed the "Zogg of Betelgeuse" explainer on "Shape of Universe" suggested by Lawrence Crowell ... nicely done video series, BTW!

Zogg explains you can have a torus-shaped surface and still be euclidean & flat.

But that doesn't seem right. Imagine a 2D torus surface and a flat-world physicist living inside the surface.

Suppose flat-physicist is on the "outer" radius of the torus surface, and attempts to determine if parallel lines exist. Two separate laser rays are sent out, suppose initially the rays are 1 meter apart.

If the rays are pointed toward the torus's smaller inner radius, the distance between the 2 rays will be different than 1m at the torus's inner radius, or at intermediate points along the rays' journey. Depending on the curvature & distances involved, the 2 rays might even cross.

Some rays would remain parallel at same distance apart, but other rays might have variable separations depending on the specific path they take.

I am not a mathematics geometer ... but in general I don't see how a torus can always be "flat", although maybe there are peculiar versions that are flat.

-- TomH

35. " Now you ask what does the universe expand into? It doesn’t expand into anything, it just expands "

This is a very profound statement. It implies that all reality is contained within our particular universe. And that there are no other universes or anything else for that matter.

36. The main reason I think curved spacetime needs to be embedded in a higher dimensional flat spacetime in a quantum gravity theory, is because you ultimately want to calculate the path integral of the curved spacetime. You cannot trace the path of an n-dimensional object in an n-dimensional universe. You need higher dimensions to draw it into. You cannot trace the path of a point particle inside a point space, or a string inside 1-dimensional space.

Furthermore, I think if you want to know if one end of the universe joins the other end, you have to embed this universe in higher dimensions, to be able to track where the universe intersects itself.

It turns out that for a particular spacetime such as the black hole solution multiple ways are possible to embed it into.

37. The expansion without expanding into anything doesn't seem that weird to me. Imagine an infinite 'rubber' sheet that is being stretched by internal forces (e.g., it was initially compressed ['dark energy']).

Like any analogy this one has its pitfalls, but it seems to deal with the 'expanding into' issue.There always plenty of room at infinity.

38. Is the 4-->10 embedding related to why string theory 'likes' 10 dimensions?

39. Hello Sabine and thank you for taking time to answer laymen questions!

However, as a philosophical layman i feel like i need more clarity.

What I think I got from your answer is that we can know (assuming that the best explanation of observed phenomena is space expansion of course) that there is sapce expansion, and we can describe it and measure it without assuming that the expansion is 'into' something.

Take the ball example: the ball is in fact expanding into something, and conceptually or at least physically must be expanding into something, even if the expansion could be described and measured without mentioning something which it expands into. So as you acknowledge and point out, the fact that you can internally measure space expansion, and thus describe it mathematically without reference to any external stuff does not mean there isn't this external. Why wouldn't it be, just like in the ball example, the most straightforward and natural thing to assume the universe is also expanding into something?

it seems like there are philosophical assumptions here that you are not making explicit enough and not defending. you are assuming that if something isn't needed to describe a phenomenon mathematically, although it is as far as most can tell, conceptually needed to describe it, then by some sort of okhamian-positivist principle, it is really not needed, and you can expand without expanding into something. Many people would claim what you just described is, while maybe not a logical impossibility, certainly a conceptual one. And to say "I can conceptualize expansion without expansion-into-something"- I can claim that i can coneptualize a triangle that isn't a triangle, that doesn't make me right. We have to take you on blind faith that you can actually have this conception, but the burden is on you to convince us and make us see that it can actually be conceptualized, not just said.

i would suggest the burden is on you to generally support or justify this principle of adherence to the mathematics over conceptual clarity, and more specifically, show how you're still justified in using the word 'expansion' and not just inventing some nonsensical use for that word.

40. FWIW, Hawking's idea that "there's nothing north of the North Pole" never worked for me.

It fails for me as a metaphor, because of course you can go north of the North Pole -- you just need a rocket ship. Our sense of "north" does not end on Earth, but continues into space. The context changes quite a lot, but the direction of north still applies.

It also fails (for me) because, even though Hawking intended to show how our spacetime "ends" at the North Pole because the Earth does, it also shows that something does continue in the "north" direction.

So it doesn't come close to answering the question: What existed "outside" the Big Bang?

For surely there was some context, some set of physical laws, that allowed the BB to occur.

(This connects back to the discussion of time a few posts previous. If the BB was an event that occurred, then some sort of time seems even more fundamental than the BB-created universe itself.)

Unless one posits that everything, enabling law and result, all sprang into existence at once. For no reason, governed by no physical law, occurring outside of any kind of time or space.

Turtles all the way down. :)

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