Comment by bollu | Hacker News Reader

Comment by bollu | Hacker News Reader

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> 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.

> I am confused. Let me write down the definition I know, and let's error-correct to a point of agreement?

Fixing the definitions is a good idea because they may vary slightly in the literature.

> - A Lorentzian manifold is a pseudo-Riemannian manifold with a metric tensor that can be positive or negative definite, but not zero.

A bit unconventional definition as it places some of the characteristics of a pseudo-Riemannian manifold into Lorentzian territory. Usually the Lorentzian manifold restricts the signature of the metric to (1, n-1) or (n-1, 1). That is to say, restricts the pseudo-Riemannian manifold to one with only 1 time dimension.

> - 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.

If the space is flat (Riemann curvature tensor = 0) how can it have constant negative curvature? Hope you see with this question that something is off. Here is a more detailed explanation:

There is a tight link between Hyperbolic space and the Lorentz transformations. The proper (i.e. no space inversion) ortochronous (i.e. no time inversion) Lorentz group of transformations SO^+(1,n) is an isometry group of H^n. To be noted here the dimensionality difference, this means that a hyperbolic space H^n can be embedded in R^(1,n) and that the action of Lorentz transformations leaves H^n invariant. You can see a neat visualization in the wiki: https://en.wikipedia.org/wiki/Lorentz_group#orthochronous, the Lorentz transformations leave invariant the hyperboloids shown in the picture. The action of a Lorentz transformation in SO^+(1,n) on any vector in one those hyperboloids H^n produces another vector in the same hyperboloid. But H^n is not Minkowski space, it's just one "shell" embedded in a Minkowski space of one additional dimension that is left invariant by SO^+(1,n). These shells have constant negative curvature but the space they are embedded into (Minkowski) is flat.

This is equivalent to the relation between Euclidean space and spheres. The group of rotations in n-dimensional Euclidean space E^n leaves invariant the spheres S^(n-1). The spheres have positive constant curvature and are embedded in a higher dimensional flat space.

> - 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.

Not that I can think of, to visualize a curved space you always have to embed it in a higher dimensional flat space. To "see" one curved dimension, you need (at least) 2 flat ones in which to embed it. This is what Flamm's paraboloid does, it shows the curved radial dimension of Schwarzschild solution embedded in 2D so we can see how distances change along the way. The problem is that we run very quickly out of "real estate" to show the curvature. If you are a highly visual person I could recommend you to get a copy of Gravitation by Misner, Thorne, Wheeler. It is a masterpiece and comes with huge amounts of visual intuition.

> If the space is flat (Riemann curvature tensor = 0) how can it have constant negative curvature?

You are wrong. Locally our space is _hyperbolic_. So "flat" means "flat space-time" means "constant negative curvature", not "zero curvature". This is unlike the Riemannian case where "flat = zero curvature". You seem to be confusing the Riemannian and Lorentzian cases?

> "flat" means "flat space-time" means "constant negative curvature", not "zero curvature"

So Minkowski space is not flat according to your definition? This way of defining flatness seems quite unusual to me both from a math and a physics perspective. Could you provide a source?

> Locally our space is _hyperbolic_.

How would this even be possible? A hyperbolic manifold is a Riemannian manifold by definition[0] but here we're dealing with a Lorentzian one!

[0] https://en.wikipedia.org/wiki/Hyperbolic_manifold

So I think I understand the confusion here. - In Riemannian manifolds, we take R^n as the building block - Every hyperbolic space can be locally represented by R^n. If you do this, the hyperbolic manifold will have negative curvature. Hence, hyperbolic space is a Riemannian manifold. - On the other hand, one can take Minkowski space as a given, and then build spaces where we have space that is locally hyperbolic. So we are trying to complete the analogy:

    Locally R^n :: Riemannian
    Locally Minkowski [signature (n-1,1) :: ???

The answer to `???` is _Lorentzian/Psuedo Riemannian_. You can check the wiki link for more(https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold#Pro...), to quote:

> Just as Euclidean space can be thought of as the model Riemannian manifold, Minkowski space with the flat Minkowski metric is the model Lorentzian manifold.

Notice the use of the word flat here: Just as flatness in the Riemannian case means "globally like R^n", flatness in the psuedo-Riemannian case means "globally like Minkowski".

You can also notice that the tangent space structure of the psuedo-Riemannian manifold is different from the Riemannian case:

> Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. [Once again from the wiki page]

As for _why_ we do this, my understanding is that we want to generalize SR(special relativity). For SR, we use a single space of signature (3,1). For GR, we want to 'put together' many such spaces, which are then distorted by gravity (this is curvature of space-time). Hence, we take the local space to be (3, 1), and the global space connects versions of (3, 1).

Does this make sense? Please tell me if this is incoherent; This is my current understanding of the state of things, I could well be wrong.

> In Riemannian manifolds, we take ℝ^n as the building block

We take ℝ^n as the building block for any smooth manifold, not just Riemannian ones. Note that there is a difference between ℝ^n and Euclidean space. ℝ^n is a smooth manifold without a metric structure, so it is not a Riemannian manifold automatically. In contrast, Euclidean space is ℝ^n with a metric structure, namely the one given by the standard Euclidean metric δ. So Euclidean space is a Riemannian manifold and it is often denoted by the tuple (ℝ^n, δ).

Finally, if we denote by η the (n-1, 1) Minkowski metric on ℝ^n, then (ℝ^n, η) is just Minkowski space (in particular, it is a Lorentzian / pseudo-Riemannian manifold).

> Just as flatness in the Riemannian case means "globally like ℝ^n"

This is not correct. There are many flat Riemannian manifolds that are not globally isometric (or even diffeomorphic) to Euclidean space (ℝ^n, δ). Take e.g. the torus T^n = S¹ × … × S¹ of any dimension.

> On the other hand, one can take Minkowski space as a given, and then build spaces where we have space that is locally hyperbolic

Again, I think you're confusing the terms "Lorentzian" and "hyperbolic" here. As I said (and you then yourself said), a hyperbolic manifold is a Riemannian manifold, not a Lorentzian one. A Lorentzian manifold is locally Lorentzian, not locally hyperbolic. You might want to read pa7x1's excellent explanation again.

Also note that the quotes you gave from Wikipedia don't contradict what I (or pa7x1) said in the slightest.

> As for _why_ we do this, my understanding is that we want to generalize SR(special relativity). For SR, we use a single space of signature (3,1). For GR, we want to 'put together' many such spaces, which are then distorted by gravity (this is curvature of space-time). Hence, we take the local space to be (3, 1), and the global space connects versions of (3, 1).

This is certainly one of way of thinking about it, yes: Locally, we see Minkowski space but globally the geometry (and topology) might be more complicated and this is what GR allows us to implement through a general Lorentzian manifold.

Thank you, It does appear I am hopefully confused between hyperbolic and Lorentzian :) Thanks for patiently showing me where I am going wrong. For my selfish benefit, could you please collect the definitions of (i) Hyperbolic space (ii) Minkowski space (iii) Lorentzian manifold (iv) Pseudo Riemannian manifold?

I'm glad I could help!

> could you please collect the definitions of (i) Hyperbolic space (ii) Minkowski space (iii) Lorentzian manifold (iv) Pseudo Riemannian manifold?

Sure:

A Riemannian manifold is a smooth manifold M together with a positive-definite metric g on M.

A hyperbolic manifold is a Riemannian manifold M of constant negative sectional curvature. It is often required in addition that M be a complete manifold.

A pseudo-Riemannian manifold is a smooth manifold M together with a non-degenerate metric g on M (of arbitrary signature).

A Lorentzian manifold of dimension n is a pseudo-Riemannian manifold (of dimension n) whose metric has signature (1, n-1) or, equivalently, (n-1, 1).

(n+1)-dimensional Minkowski space is the Lorentzian manifold (ℝ^(n+1), η), where η = - (dx⁰)² + (dx¹)² + … + (dx^n)² with respect to the standard coordinates (x⁰, x¹, …, x^n) on ℝ^(n+1).

And what is the precise relationship between hyperbolic space and Minkowski space? am I right in understanding that one can _embed_ a model of hyperbolic space (the hyperboloid moidel, obeying the equation `-x0^2 + x1^2 + ... xn^2 = 0` as a subspace of the Minkowski space? But Minkowski space is far larger than hyperbolic space?

pa7x1 already answered that question: https://news.ycombinator.com/item?id=23549361

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