Curvature or Warpage?

>>7710221
If wood warps what does it do?

>>7710221
What the fuck is a "war page"?

>>7710221
curvature is exactly what you think, except it is impossible to imagine in 3d.
I believe Warpage is when 2 very distant point in space can be connected with a short path through a worm hole.
the later is scifi bullshit.

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>>7710245
Curves?

>>7710247
pic related?

>>7710221
This is a board for science and math. Videogames and "woah space" youtube videos aren't science.

>>7710221
The difference is kind of subtle, but still important. Curvature in the physics sense that I think you're talking about is a continuous deformation of C^n (normally C^2 or C^3 in any undergraduate calculations) a la classic homeomorphisms from intro topology, while warpages admit homotopy as well (think along the lines of how homotopy allows us to evaluate complex line integrals of functions holomorphic on domains with contractible regions on which the function might not be holomorphic). So, in that sense, every curvature of spacetime is indeed warpage, but every warpage is not necessarily analagous to a type of curvature. This will manifest itself in how you'll learn to deal with essential singularities in a minowski space via the higher dimensional version of Caserati-Weierstrass (think Hodge theory).

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>>7710259
>>7710268
Well, the reason I ask is because Stephen Hawking seems to think space-time "warpage" has a vital part in explaining inflation in the early universe. I assumed he meant curvature, as predicted by general relativity, but he seems to use the terms as distinct things. I don't know my shit, and I'm finding piss all when I google it.

>>7710280
Too bad. "Googling it" with no physics knowledge is going to serve absolutely no purpose. Go get the prereqs (learn actual introductory physics first, classical mechanics, E&M, then modern physics) and then you'll know.

>>7710278
I applaud thee. Thank you.

>>7710221
Yes, they're exactly the same. Physicists sometimes rarely use "warping" of spacetime to describe specifically curvature caused by some effect, but there's no real difference and "curvature" is the standard term for everything.

>>7710278
>homotopy
If complex line integrals involve complex numbers then they don't exist realistically.

>In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.

>In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces

If warping is homotopythic (HOMOTOPIC)

>if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

I think I see what you're saying though; it's homotopical because warping would require the CONTINUED bending of spacetime. On the other hand, all of space is homotopical considering the effects of gravity on spacetime.

>>7710278
>physics noob
>deliberately answers in advanced mathematics

Less correct but more layman-friendly:

>Homeomorphism mean you convert one thing into another thing by continuous bending, stretching, and compressing transformations, but only in a way that lets you continuously deform it right back if you want to - "continuous" meaning you never get to do anything like cutting it apart and pasting it back together. So a sphere is homeomorphic to a cube, and a coffee mug is homeomorphic to a donut.

>Homotopy is a slightly broader category that also includes continuous deformations that don't allow you to easily get back to the original shape - for instance, a solid sphere is homotopically equivalent to a point, since you can squish it inwards to nothing, but it's not homeomorphically equivalent since you can't simply stretch a single point out to make a disk.

So "curvature" is equivalent to deforming your space-time into something homeomorphically similar, and "warpage" is deforming it into something homotopically similar. The reason you'd care about "warpage" specifically in the early universe is that the Big Bang did, in fact, involve the universe starting out as a point and stretching out to a sphere.

So they express a very subtle mathematical difference, but are otherwise the exact same thing.

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>>7710330
>>7710331
>>7710335
>Right?

>>7710326
"continuously" and "continued" are not at all the same thing. "continuously" just means that there aren't any "discontinuities" - you get from shape 1 to shape 2 only by smoothly passing through other valid shapes. You never have something like:

>Step 1: Start with a sphere
>Step 2: ???
>Step 3: And now you have a donut.

So if you start with, for instance, a big mass at Point 1, and move it to Point 2, spacetime is deformed homeomorphically - because the shape of spacetime when the object was at 1 turned into the shape when the object was at 2 got to that point by being continuously bent as the object moved through all the points in between 1 and 2.

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>>7710342
Based anon delivers! Thank you!

>>7710280
Cosmologists use curvature to mean the global curvature of spacetime. It's a specific term to them but I think you have it mixed up. Inflation is vital in explaining the flat curvature of the universe.

My dream is to write a book titled "On the kurwature of Polish spaces", but I haven't had time to check if curvature can in any sensible way even be defined in them.

>>7710278
>Google results for warpage homotopy don't turn up anything relevant
>this post is the second result

>>7710326
So while it is true that spacetime as a minowski space is homotopical to any subspace with the same genus, it's not necessarily true that all subsets of minowski spaces are homotopic to the entire space. For this to make sense, consider a simple example: the annulus. A function holomorphic over a curve, gamma, running through the annulus does not allow us to immediately conclude that a line integrals of the function over gamma will be the same as a line integral of the function over a curve, eta, if eta isn't homothetic to gamma (ie homotopic with the same endpoints). This becomes interesting if we consider a function that's otherwise entire except for the region at the center of the annulus. In this case, we must consider that, at the center of the annulus, we may have a singularity, which would prevent the standard argument for "homotopically collapsing" the removed disk in the annulus to a point and using the strait line homotopy between gamma and eta. This is part of where we actually get then math for black holes or wormholes, which would be the case where we have either a pole or an essential singularity respectively lying in the disincluded region of homomorphicity. Probably the most interesting caveat this leads to is the notion that one is indeed restricted to simple fibrations like homeomorphisms and not more powerful fibrations, such as the hopf fibration, when one considers the line integrals of a holomorphic function over a region of nonzero genius along nonhomothetic curves. Essentially, this is exactly what >>7710342 is referencing.

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>>7710533
LOL
It's the first result for me.

>>7710386
What have I mixed it up with? You'll have to correct me, but this is how I--a complete idiot in this subject--understand it. Inflation in the early universe was postulated to have happened, because if it didn't happen there wouldn't be enough time in the history of the universe for the temperature of the universe to be as uniform as it is. (I think the uniformity that is adhered to is that shown by the CMBR, which almost, but not quite, completely uniform.) Inflation takes aspects of both general relativity and quantum mechanics into account, which makes it a bit dubious at the moment. But it predicts that there would be differences in temperature of the CMBR between different directions (?). And such differences were discovered by the COBE satellite 1992, and the WMAP satellite 2001 (?).

Anyway, inflation requires some weird improbable initial condition. Such a condition, or a better condition, or something (?), was proposed by Hawking and Hartle 1983 in their article "Wave Function of the Universe" (you can find it in google scholar), called the no-boundary condition. I have read it, but I didn't understand it. I have however read--like the noob I am--some popular science of him (The Grand Desing), and looked at some lectures. And I take it that something having to do with space-time warpage is needed to infer the no-boundary condition.

Is the above really ignorant?

>>7710623
>What have I mixed it up with?
You've mixed up "warpage" (a term I have never heard used before) with curvature and mixed up the relation. Flatness needs inflation to be explained, not the other way around (as it wouldn't make any sense).

>Inflation takes aspects of both general relativity and quantum mechanics into account, which makes it a bit dubious at the moment.
It doesn't really. It's pure GR. A consequence of inflation is that quantum fluctuations in the early universe would be blown up to be the seeds of galaxies. However this isn't a part of pure inflation, just a consequence.

>But it predicts that there would be differences in temperature of the CMBR between different directions
Any big bang model requires that not just inflation.

Inflation is very well accepted in the cosmological community. It's even less controversial than standard cosmology. I'm not sure why you think the initial condition is improbable.

Quantum cosmology and the "wave function of the universe" really isn't very useful.

>>7710674
>I'm not sure why you think the initial condition is improbable.

I took it on Stephen Hawking's authority. Why would he prefer the no-boundary condition?

>>7710692
I don't know, I'm not sure what you're referring to. Inflation however is very well accosted.

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>>7710722
I'm not sure either.

Inflation isn't important for the initial question though. I brought it up because, as I understood it, it had something to do with Hawking and Hartle's no-boundary condition. Wikipedia summarizes the conditions like the following:

"artle and Hawking suggest that if we could travel backward in time toward the beginning of the universe, we would note that quite near what might have otherwise been the beginning, time gives way to space such that at first there is only space and no time. "

Here is the technical explanation:

"The Hartle–Hawking state is the wave function of the Universe—a notion meant to figure out how the Universe started—that is calculated from Feynman's path integral.

More precisely, it is a hypothetical vector in the Hilbert space of a theory of quantum gravity that describes this wave function.

It is a functional of the metric tensor defined at a (D − 1)-dimensional compact surface, the Universe, where D is the spacetime dimension. The precise form of the Hartle–Hawking state is the path integral over all D-dimensional geometries that have the required induced metric on their boundary. According to the theory time diverged from three state dimension—as we know the time now[clarification needed]—after the universe was at the age of the Planck time.[2]

Such a wave function of the Universe can be shown to satisfy the Wheeler–DeWitt equation."

Stephen Hawking, in the Grand Design, and in various lectures, stated for time to behave like space, there had to be gravity present to warp space-time. Or something along those lines. What made me pose the initial question was the fact that he used "warp" and "warpage" and so forth, as he explained it, and not curvature, which was the term he had used before.