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Some corrections to hopefully solve some misunderstandings. > A Lorentizian manifold looks locally hyperbolic, so we need to imagine a light cone. Now the curv
by pa7x1 6y ago
Some corrections to hopefully solve some misunderstandings.
> A Lorentizian manifold looks locally hyperbolic, so we need to imagine a light cone. Now the curvature of this manifold tells us how we need to distort light cones at different points to build up the full space.
A Lorentzian manifold is not necessarily locally hyperbolic and certainly the existence of light-cones do not follow from its hyperbolic nature. A Lorentzian manifold may be flat or curved, if it's flat we usually call it also a Minkowski space. Curved spaces come in many shapes of forms depending on the curvature which is a complicated geometric object. One particular case of curved space is a hyperbolic space which is one with constant negative curvature. A hyperbolic (constant negative curvature) Lorentzian (space+time) manifold is also called an Anti-de Sitter (AdS) space and plays a very important role in physics (see AdS/CFT correspondence).
> How the fuck do I visualize this? Is there some way to embed this in low dimension to play with it? For example, to understand regular curvature, you can play around with a sphere and a cone to get a feel for curvature ~= angle deficit ~= loopy deficit ~= holonomy.
Well, we humans suck at visualizing dimensions greater than 3 and projecting higher dimensional spaces into lower dimensional spaces (like we usually do representing a hypercube) is very hard to grasp unless you already know what you should be seeing. So your best bet is to choose problems with high degree of symmetry and visualizing only the interesting dimensions. For example, a stationary (i.e. doesn't change with time -> 1 dimension less to visualize) and spherical solution (2 more symmetries -> 2 dimensions less to visualize) can be easily visualized, in fact only has one interesting dimension! This solution is usually called Schwarzschild's solution. And because it has so high degree of symmetry it can be embedded a low dimensional space we can see, we usually call that Flamm's paraboloid and you can find a pretty picture here: https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s...
> I don't know how to visualize the curvature of space-time, and I'm completely unconvinced I actually understand it beyond squiggles on paper. I'd love a way to embed and visualize this. Please tell me if you know how!
Unless you develop the ability the visualize higher dimensional spaces your best bet is to select highly symmetrical problems and visualize the interesting dimensions of those, the way we generalize to higher dimensions is through math not through the visual cortex. The brain is a highly plastic structure so maybe with training you could develop the ability to work visually with higher dimensional spaces but I also wouldn't be surprised if the visual cortex has been evolved and finely tuned to work with the macroscopic dimensions it had available and this exercise would turn out to be futile. After all, the ability to generalize to arbitrary high dimensions may have some cost and animals would have benefited very little from it.
- codethief 6y ago>> A Lorentizian manifold looks locally hyperbolic, so we need to imagine a light cone. Now the curvature of this manifold tells us how we need to distort light cones at different points to build up the full space. > A Lorentzian manifold is not necessarily locally hyperbolic Maybe OP was thinking of some local notion of global hyperbolicity[0] (which, of course, has nothing to do with hyperbolic Riemannian manifolds)? [0] https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold
- pa7x1 6y agoI think you are right on the source of confusion, I have tried to explain the relation between hyperbolic space and Lorentz transformations in my latest answer. See here: https://news.ycombinator.com/item?id=23549361 https://news.ycombinator.com/item?id=23549361
- codethief 6y agoThis is a great answer! Thanks so much for notifying me!
- cinquemb 6y agoIt seems like something in our way of understanding gravity is also confounded that our perception of it is skewed by our assumptions of time orientability in any given region of spacetime, extends to all regions of spacetime, even those outside of our light cone or those of a light cone trapped within a event horizon.
- bollu 6y ago> A Lorentzian manifold is not necessarily locally hyperbolic and certainly the existence of light-cones do not follow from its hyperbolic nature. A Lorentzian manifold may be flat or curved, if it's flat we usually call it also a Minkowski space. Curved spaces come in many shapes of forms depending on the curvature which is a complicated geometric object. One particular case of curved space is a hyperbolic space which is one with constant negative curvature. A hyperbolic (constant negative curvature) Lorentzian (space+time) manifold is also called an Anti-de Sitter (AdS) space and plays a very important role in physics (see AdS/CFT correspondence). I am confused. Let me write down the definition I know, and let's error-correct to a point of agreement? - A Lorentzian manifold is a pseudo-Riemannian manifold with a metric tensor that can be positive or negative definite, but not zero. - A neighbourhood of a Lorentzian manifold is flat if one can assign a coordinate system where we have the minowski metric ds^2 = dx1^2 dx2^2 + ... + dx(n-1)^2 - dxn^2. So it's hyperbolic space of dimension n, Hn. This space has constant negative curvature; You can check by computing the curvature tensor. - Thanks for that particular model of Schwarzschild's solution! I wasn't aware of it. - Do you have more examples of a similar flavour? I don't care if the solution obeys the einstein field equations. I'm mostly just interested in examples of Lorentzian manifolds with curvature. - Thanks for the advice regarding visualizing higher dimensions. I am quite comfortable with the algebra. However, it is often the case that people have great insight into the structure of hyperbolic space, which I am trying to build. Consider Thurston and his famous insight into the structure of 3-manifolds and hyperbolic space.