General relativity and Higher dimensions

lightseeker
Hi everyone , this is my first time here :)
My question is simple , if gravitation is basically a space distortion so it's fair to say that at least a fourth dimension exists , since a distortion must occur in higher dimension than the one of the concerned space .
Is it correct putting it like that ?

Altabeh
Hi everyone , this is my first time here :)
My question is simple , if gravitation is basically a space distortion so it's fair to say that at least a fourth dimension exists , since a distortion must occur in higher dimension than the one of the concerned space .
Is it correct putting it like that ?

Gravitation only can't exist in spaces with 1 and 2 dimensions. The reason why a two dimensional space can't contain gravitation in it is that the Weyl tensor in 2D spaces always vanishes. For spaces of higher dimensions than 2, gravitation can be defined as in the theory of GR which lives in the 3 dimensional spaces with an extra time dimension.

AB

Homework Helper
Welcome to PF!

Hi lightseeker! Welcome to PF!
if gravitation is basically a space distortion so it's fair to say that at least a fourth dimension exists , since a distortion must occur in higher dimension than the one of the concerned space .
Is it correct putting it like that ?

No.

A space can be entirely self-contained, and still have what you call "distortion".

It does not need to be embedded in a higher-dimensional space.

From http://en.wikipedia.org/wiki/Curvature" [Broken] …
There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold.

What you are describing is extrinsic curvature, but mathematicians and physicists are usually talking about intrinsic curvature.

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Altabeh

Hi lightseeker! Welcome to PF!

No.

A space can be entirely self-contained, and still have what you call "distortion".

I think the OP does not ask whether the space can be distorted if there is no gravitational field or matter. (E.g. the surface of a 3-sphere.) The question is essentially about the dimentional aspects of spaces for which distortion chalks up to the presence of gravitation.

AB

Homework Helper
I think the OP does not ask whether the space can be distorted if there is no gravitational field or matter. (E.g. the surface of a 3-sphere.) The question is essentially about the dimentional aspects of spaces for which distortion chalks up to the presence of gravitation.

AB

Yes, and I've said that curvature (or "distortion") can be intrinsic, meaning that eg a 3D space have have curvature without being part of a 4D space.

lightseeker
I am not a mathematician but trying to figure this out from an intuitive view point , I see extrinsic curvature is dealing with space itself , and intrinsic curvature dealing with structure of something inside space , but as I said it's just an intuition, so would you please explain this self-contained distortion ?

Staff Emeritus
Gold Member
I am not a mathematician but trying to figure this out from an intuitive view point , I see extrinsic curvature is dealing with space itself , and intrinsic curvature dealing with structure of something inside space , but as I said it's just an intuition, so would you please explain this self-contained distortion ?

Intrinsic curvature just means that you can measure it without having to see or even hypothesize extra dimensions. For example, if the circumference of a circle is not exactly equal to $2\pi r$, then that's an indication of curvature.

lightseeker
Intrinsic curvature just means that you can measure it without having to see or even hypothesize extra dimensions. For example, if the circumference of a circle is not exactly equal to $2\pi r$, then that's an indication of curvature.

When you say a circle or an other defined shape it becomes something in space , since I am initially talking about general space defined with N Dimensions.

Homework Helper
Hi lightseeker!
… I see extrinsic curvature is dealing with space itself , and intrinsic curvature dealing with structure of something inside space …

Yes, except it's the other way round …

intrinsic curvature is in itself, extrinsic curvature is in something outside.

lightseeker
intrinsic curvature is in itself, extrinsic curvature is in something outside.

Would you give me a concrete example of an intrinsic curvature and show how it applies to space of certain dimensions using parallelism .

If you draw a triangle on a flat piece of paper, and measure the angles, their sum is 180 degrees. If you draw a triangle on the surface of the earth, the angles will not be 180 degrees, indicating that the surface of the earth is curved. Drawing a triangle on the surface of the earth is an operation confined to the surface that allows us to determine if that surface is curved or not - in this way, the curvature of the earth's surface is intrinsically 2D, even though the earth is embedded in 3D space.

In normal space, squared length are positive. The difference with spacetime is that some "squared lengths" are negative. Also, spacetime is 4D. To draw a triangle in spacetime, you can use light rays and freely falling particles.

Altabeh
Would you give me a concrete example of an intrinsic curvature and show how it applies to space of certain dimensions using parallelism .

Another example of an intrinsic curvature is http://mathworld.wolfram.com/GaussianCurvature.html" [Broken] in GR is built on an isotropic and homogeneous line-element which again includes the Gaussian curvature $$k$$, and when $$k=1$$ or $$-1$$ the metric always has an instrinsic uniform curvature which constructs an elliptical or hyperbolic spacetime, respectively. But on the other hand, you can have a space(time) that is not intrinsically curved as I can name the "cylinder" which can be easily embedded in an Euclidean 3-space. You can find a change of coordinates which brings the local coordinates of the cylinder to those of Euclidean 3-space and thus retrieving a flat space through a coordinate transformation.

AB

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Homework Helper
Would you give me a concrete example of an intrinsic curvature and show how it applies to space of certain dimensions using parallelism .

The Poincaré disc … an ordinary 2D disc with a curved (or "distorted") geometry … is an example of intrinsic curvature.

See http://en.wikipedia.org/wiki/Poincaré_disc_model" [Broken] for details.

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lightseeker
If you draw a triangle on a flat piece of paper, and measure the angles, their sum is 180 degrees. If you draw a triangle on the surface of the earth, the angles will not be 180 degrees, indicating that the surface of the earth is curved.

Let us consider your example which is a little bit intelligible , a triangle in an undisturbed flat piece of paper is an element of 2D space , and when it's seen from 3D view it's confined in a planar 2D space , but once you draw it on the surface of the earth the curvature becomes evident not only from 2D view (measuring angles ) but also in third dimension since the triangle don't keep confined in the planar 2D space .

Altabeh
Let us consider your example which is a little bit intelligible , a triangle in an undisturbed flat piece of paper is an element of 2D space , and when it's seen from 3D view it's confined in a planar 2D space , but once you draw it on the surface of the earth the curvature becomes evident not only from 2D view (measuring angles ) but also in third dimension since the triangle don't keep confined in the planar 2D space .

No! The metric of the surface of the Earth is dependent on two elements, basically the Earth radius $$r$$ and the latitude $$\theta$$:

$$ds^2=r^2d\theta^2+r^2\sin^2(\theta)d\phi^2.$$

This metric describes the geometry of the Earth surface and you can see it is completely 2D since the Earth is embedded in the 3D Euclidean space and thus its surface is a 2D submanifold! I think you are misunderstanding two implications of space and manifold by looking at the surface of the Earth as a space! It is a submanifold which has 2 dimensions or it can be described by two local coordinates, giving rise to it being 2D!

AB