# Does rotation of a 3d volume about a 2d plane create gravity?

1. Dec 3, 2012

### Sunfire

Hello,

If a rotation of a 2-d disk about a 1-d line can produce centrifugal gravity, is it right to infer

that

rotation of a 3-d volume about a 2-d plane would create gravity, observable in the 3-d volume?

It is confusing to think about the direction of this gravity, if it exists

Thank you.

2. Dec 3, 2012

### Staff: Mentor

How do you rotate about a plane?

3. Dec 3, 2012

### Staff: Mentor

Centrifugal force isn't gravity, and the axis of rotation of a three-dimensional object is still a 1-dimensional line (imagine the earth rotating about its axis - the axis is a line that passes through both North and South poles). But with that said, yes, there will be a centrifugal force, its magnitude will depend on the distance between the point at which the force is being measured and the axis, and its direction will be perpendicular to the axis.

It's worth noting that the magnitude of the force is necessarily zero at the poles, the point where the axis of rotation insects the surface.

4. Dec 3, 2012

### Sunfire

The only example I have of such rotation is that of a hypercube about a 2-d plane

http://en.wikipedia.org/wiki/Tesseract

One needs to scroll down to the animated gif showing the rotation

"A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom"

5. Dec 3, 2012

### Sunfire

Nugatory,

the rotation of the 2-d disk about a 1-d line takes place in 3-d space;
my question is about a rotation of a 3-d volume about a 2-d plane in 4-dimensional space (sorry for being unclear on that)

6. Dec 3, 2012

### Staff: Mentor

Actually, a "rotation" in a general n-dimensional space is described by an antisymmetric tensor, with two indexes corresponding to the plane of rotation. The reason we can describe rotations in 3 dimensions as rotations "about an axis" is that in 3 dimensions, there is a one-to-one mapping between antisymmetric tensors and vectors (strictly speaking, the mapping is to pseudovectors, which behave differently under parity transformations than ordinary vectors do).

So in 3-D, we can equally well describe the same rotation as being in the x-y plane, or being about the z axis. But in higher dimensions, the only way to describe a rotation is by its plane, since there is no way to define a single "axis" for it because there is no way to map antisymmetric tensors one-to-one to vectors (the number of components is different, 6 for a 4-D antisymmetric tensor but only 4 for a 4-vector).

7. Dec 3, 2012

### Staff: Mentor

Interesting. So then there is no difference between rotation about an axis and rotation in a plane in 3D, but in higher dimensions there is only rotation in a plane.

So for Sunfire's question, the dimensionality of the object is not relevant, objects only rotate in planes regardless of the dimension.

8. Dec 3, 2012

### Staff: Mentor

Yes. As far as whether rotation "creates gravity", I would say yes, but I haven't seen any real treatment of the details in more than 3 spatial dimensions.

9. Dec 3, 2012

### Sunfire

This would mean that the simple rotation of a tesseract about a plane (shown on the left figure here http://en.wikipedia.org/wiki/Tesseract) still produces a centrifugal field. It is probably not visible in 3-d due to the manner it was projected in 3-d.

It is easier to see the in-plane rotation on the RHS figure, where the tesseract is rotating about 2 orthogonal planes; then its 3-d projection still shows rotation and it is intuitive to accept there is induced c.f. gravity.