Slightly confused about embedding picture and imagining spacetime

[jmz34]

I'm slightly confused as to how we can use the picture of a 2D surface embedded in 3D space as an analogue to understand (maybe not picture!) 4D spacetime. My initial thinking was that trying to imagine a 4D spacetime isn't really possible, it's just a mathematical concept which one should not try to picture. Am I right in thinking this?

But to be a small being/insect on a 2D cylinder, say, on embedded in 3D space the intrinsic geometry would be Euclidean, if the insect traversed the whole cylinder and arrived it at the same point then it could infer some information about the topology of the surface. I'm fine with all this. Problem arises when I try to take these concepts for a 3D manifold embedded in 4D spacetime.

If this 3D manifold is curved, then by taking measurements locally we should (like the insect) be able to deduce information about the intrinsic geometry of the space. Does this mean that, if we were near a massive spherical body say, and were able to fly from the north pole, towards the equator, fly to the left (without changing orientation), then fly backwards up to the north pole- the orientation of the spacecraft would differ from the initial? Is this a symptom of 3D space being curved?

 

[pervect]

I haven't calculated the specific example you cite, but in general, when you parallel transport a vector around a closed curve, in a curved space-time its orientation changes, and in a flat space-time it doesn't.

If you represent the notion of the front-back of the spaceship with a vector, and you interpret "keeping the direction the same" as parallel transport (which is a reasonable interpetation) this is equivalent to what you asked.

 

[A.T.]

But to be a small being/insect on a 2D cylinder, say, on embedded in 3D space the intrinsic geometry would be Euclidean ... Problem arises when I try to take these concepts for a 3D manifold embedded in 4D spacetime.

I think you are confusing two meanings of "embedding" here:

1) A non-Euclidean manifold can be embedded in a higher dimensional Euclidean manifold.

- For example the surface of a ball is a non-Euclidean 2D manifold embedded in a 3D Euclidean manifold.
- Or your 2D cylinder surface could be deformed to be non-Euclidean too and then embedded in a Euclidean 3D manifold for visualization, like done here:
http://www.relativitet.se/spacetime1.html
http://www.adamtoons.de/physics/gravitation.swf

2) The non-Euclidean 3D space is a subspace of the 4D space-time which is also non-Euclidean.

This is not usually called embedding. To visualize the non-Euclidean 3D space or the non-Euclidean 4D space-time using embedding you would need even higher dimensional Euclidean spaces.

If this 3D manifold is curved, then by taking measurements locally we should (like the insect) be able to deduce information about the intrinsic geometry of the space. Does this mean that, if we were near a massive spherical body say, and were able to fly from the north pole, towards the equator, fly to the left (without changing orientation), then fly backwards up to the north pole- the orientation of the spacecraft would differ from the initial? Is this a symptom of 3D space being curved?

Yes, you could use a gyroscope for that. A gyroscope that orbits a massive body on a circular path will arrive with a different orientation at it's initial position.