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Sure, I can explain how to visualise it (I think :) )

First off, a note: lots of people think they know how to visualise GR, but actually don't. For example, explaining with marbles falling down curved rubber sheets doesn't help, because you're using gravity (the marble being pulled down) to explain gravity. And if you think you can just explain it with geodesics along a curved rubber sheet with a heavy weight in the middle, then think about how to explain why a stationary object falls towards the heavy weight.

Anyway, here is how to visualise it:

Imagine you are driving a car along some surface. We require that each wheel rotates at exactly the same speed with respect to the surface. Now imagine there is a decreasing curvature gradient from left to right across the path of your car - in other words the surface below the left wheel is curving faster than under the right wheel. Seen from above, the left wheel has extra distance to travel in the vertical direction than the right wheel, to achieve a given horizontal distance. This causes the direction of the car to turn slowly to the left - in other words the car is 'gravitationally attracted' to the left - i.e. towards the region of greater curvature, i.e. towards the attracting mass.

To explain why static objects are gravitationally attracted to a mass, you just have to realise that according to Einstein, there are no such things as stationary objects (in some sense). Rather, even a 'stationary' object is travelling rapidly through spacetime. To go back to the car metaphor - the car is not stationary, rather it is always driving forwards (through spacetime).

Some notes on this - think about what a bizarre, weirdly-shaped surface the car must be driving over.

You can also think of the car front axle as being a wavefront of light.

And finally - if you think this explanation is too bizarre, don't blame me, blame Einstein. Personally I don't actually like GR as a theory due to how weird it is.

> For example, explaining with marbles falling down curved rubber sheets doesn't help, because you're using gravity (the marble being pulled down) to explain gravity.

Exactly, and it drives me nuts. There are many similar examples in SR and GR that maybe help a layperson intuitively, but creates confusion and frustration when you actually try to learn the theory. I can think of 2:

The "escape velocity" explanation of black hole event horizon. Yes, the number comes out correct. No, it does not make sense and does not help me understand how black hole works.

"Mass increases when object moves". No it doesn't. And no it doesn't explain why massive particles can't reach speed of light, because in the frame of the object, its speed is zero.

I can't tell you how much time I've spent searching for the answers, and drowned by these answers that seemingly make sense.

For the latter concern, you may find this Physics Today article from Okun helpful:

http://www.hysafe.org/science/KareemChin/PhysicsToday_v42_p3...

isn't it still gravity that keeps the car in contact with the road in your example, meaning it's also using gravity as part of its explanation of gravity?

In this case, the mechanism holding the car to the road is not relevant. The important part of the visualization is the implication of all wheel rotating at the same speed on a curved surface. Any visualization on Earth would have this problem, because gravity effects the possible configurations you can think of.

If it makes you feel better, move the car and road into space and hold them together via magnatism.

Test particles are super-useful.

Synge in his textbook [1] discusses a five-point curvature detector, which is the formal treatment for a small cloud of non-interacting dust that reveals the Riemann curvature tensor.

Baez does this succinctly and less formally at [2], using a decomposition of the Riemann curvature tensor into the Weyl tensor and the Ricci tensor.

One could substitute ticking clocks for coffee grounds: in e.g. exterior Schwarzschild, when the flat-spacetime round-ball is instead a Weyl-curved spacetime American-football, the long axis of the football is radial to M, so the clock closest to M ticks slower than the others (cf. the Riemann tensor and especially the last sentence at [3]).

One may compare with [4] (and the focusing theorem on the same page).

- --

[1] Synge, J.L., "Gravitation: The general theory". (1964) Ch. XI, §8

[2] http://www.math.ucr.edu/home/baez/gr/ricci.weyl.html

[3] https://en.wikipedia.org/wiki/Schwarzschild_metric#Curvature...

[4] https://en.wikipedia.org/wiki/Raychaudhuri_equation#Intuitiv...

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...

> 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.

>> 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

I 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

This is a great answer! Thanks so much for notifying me!

It 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.

> 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