I disagree with that last sentence. The fact that a space has curvature does *not* mean it is imbedded in a higher-dimensional space. The curvature *can* simply be intrinsic to the space.
I have come across your pov several times, always in the context of physics and the curved nature of space time. Intuitively to me, at least having curved space time implies an embedding in a higher dimensional euclidian space. It's hard to imagine a curved space in the absence of such an n dimensional ambient space in which it is embedded. But I'm sure you can mathematically make a notion like that rigorous. In any case, we can say such a space is homeomorphic to an actual similarly dimensional space embedded in some higher dimensional(probably n+1) space.
You're right! Hopefully, my ending was not too confusing, but I think this is what I am saying in my second to last sentence. In general relativity, we do not need to consider any embedding of spacetime. My last sentence is just to suggest the possibility of higher dimensions. But of course, these dimensions have never been found experimentally.
@@dekippiesip Indeed, we are comfortable thinking about curves (or manifolds, more generally) as being embedded in some ambient space. However, because of the intrinsic nature of these objects, the modern tendency is to study manifolds independently of any embedding. For example, when specifying a "Riemannian manifold," one only needs to give a pair (M, g) where M is a smooth manifold (fundamentally, a special kind of topological space) and g is an inner product. We can do this all without specifying an ambient space! How cool. If you would like, however, the Whitney embedding theorem guarantees that one can always embed an n-dimensional manifold (conditions: smooth and real) in a 2n-dimensional space. Just as a random example, a Klein bottle is a 2-D manifold (i.e., a surface) that needs 4 dimensions for an embedding.
@@feedyourhead-mq9fu yeah it's crazy. But it removes a lot of overhead of having to deal with that higher dimensional space. Concepts like derivatives and tangent lines/planes/... must be much more abstract then too though. Because normally you picture them as kissing the curve on 1 point and being out of the manifold for most others.
Great video! Now to share a bunch of tangential philosophical asides that have always bothered me: How could flat-landers possibly “meausure” lengths and angles if all their instrumentation was already curved? If you think about this, it makes no sense. If you lived on the surface of a sphere, you wouldn’t be able to construct the tangent plane which is necessary for determining the angle between two adjoining curves. We can measure angles on the surface of the earth by walking in “straight lines” then turning, but these straight lines are characterized through projections and connections of the tangent planes. Which would mean it would be impossible to determine what the “straightest path” was on a curved surface without first imbedding it in some higher dimension, or at least defining a way to construct a tangent surface. Likewise in GR curvature is measured via the acceleration of nearby free-falling bodies. But this acceleration is measured relative to idealized rigid bodies-these rigid bodies form the idea of the “tangent” space which defines the connections which allows for determining curvature. If you truly “lived” in a curved surface, you could never know it, because all your measuring instruments would be warping too. Hence, some notion of euclidean geometry must always be prior to any non-euclidean geometry. That’s my take at least!
What happens with an expanding surface? Imagine living on an expanding sphere with a finite speed of light. The path light takes follows a path through the bulk while simultaneously tracing an apparent path along the surface. Things will never appear to be quite where they were when the light was emitted or where they are when it is received.
Your 3rd sentence assumes there *is* a "bulk", rather than the surface simply being curved and expanding. Nothing requires such a surface to be embedded in a larger space.
@@jursamaj True, but this whole video is about a surface embedded in a higher dimensional space and my question is specifically about expansion due to there being an additional dimension. There's a philosophical debate about whether the bulk "exists" in some sense (presentism vs eternalism) but I've always held the opinion that it's a red herring that boils down to a linguistic dispute over incorrect use of tense. The extra dimension is time in this case and it makes no sense to say the surface at some point in the past or future exists **now**. It existed and will exist, however.
@protocol6 @ yea thats where the measurement, redshift, ontics vs dynamics, and conservation problems come in. great video i really needs part of this variational analysis this for some idea i wanted to roughly evaluate. the LLM was confused about it. but now , so we can run simulations from "outside the box" to get this answers.. we can calculate "ontics " of any pov in the box if we feel we need do. Also consider redshift, time dilation, soliton interaction, fast fluctuations, adn sucn.. holographic theory, (maybe) so I think we would not know without making some new simplifying assumptions. Int topology use cases ( it works,m you dont need 3d+1) or complex tensors.. if reversible, combined with thermodynam and entropic consideration, where time becomes unmeasurable and its metric assumed to be invariariant, So Noether suggested somewhere, you can put your growing s2 sphere in a box.. represent mases as a nonlinear wave function bump with a soliton solution, or a circle in a box with time . with an 1D + 1 soliton travelling pulse if you expand and contract the circle to that it conserves ( time invariance) its cyclical. big bang not to s singularity/ bing crunch , to a limit. if seriously interested.. d'hooft, Noether Einstein on www.jstor.org/stable/1968902?read-now=1&seq=14#page_scan_tab_contents then if you follow the rabbit you will get to the soliton quantum gravity scalar field theory dynamics.. dont !
I was a bit confused by a points towards the end of the video about general relativity. Does the Gaussian curvature of a point in space depend on the matter in its vicinity or is it constant? If the former, when scientists say they're trying to measure the curvature of the universe, are they referring to the underlying curvature of space (which is distinct from the local curvature of a point)?
At the end of the video, I tried to make some generalizations beyond surfaces to a general manifold. In general relativity, we do not consider a 2-dimensional surface but rather a 4-dimensional manifold (3 for space, 1 for time). The appropriate notion here is Riemannian curvature (which reduces to Gaussian curvature in the 2-D case). The curvature of spacetime is certainly not constant--matter and energy cause it to curve. To your last point, I believe physicists are talking about the large-scale shape of the universe as opposed to the local shape. One way to answer this is to ask what happens to parallel lines as they extend to infinity. If they eventually intersect, that indicates positive curvature. If they diverge, that indicates negative curvature. If they remain equidistant, the universe is "flat." As far as I know, much experiential data points to the universe being flat, though this is still an unsolved problem in physics. Hope that helps!
@@feedyourhead-mq9fu It computerizes nicely but, computers aside, I always had the (undereducated) feeling that it could be a stepping stone to differential geometry.
