Can Two Cylinder Space-Time Maps Fully Cover De Sitter's Hyperboloid?

[Zserdman]

Hello,

I am in a college seminar on Einstein and his theories. One of the topics we have recently been discussing is space-time geometry, particularly that of De Sitter. A question arose the other day about whether two of Einstein's Cylinder space-time maps could be added together in order to compose all of De Sitter's space-time map given by a hyperboloid. I understand that a cylinder space-time map corresponds to a 'wedge' in De Sitter's space time map because one point must represent where the conversion factor g44=0. So, if you took another cylinder and rotated it 90 degrees so that the point where the conversion factor g44=0 was still located on the same cross-section, it would correspond to a new wedge in the De Sitter Space-Time geometry. It is the claim of some of my peers that if you added these two wedges together (setting them on top of each other) they would cover all of the space enclosed by a hyperboloid of one sheet. It is my opinion that these two wedges would in fact not cover all of the space enclosed by a hyperboloid of one sheet and would instead leave space uncovered in the center at the top.

A way of looking at this problem without needing background knowledge is asking this: You take a circle enclosed by a hyperboloid of one sheet at the waist and note the volume of it when you rotate it along an axis located at the waist of the hyperboloid of one sheet and is exactly parallel with the waist, up 45 degrees and down 45 degrees. The circle expands itself upon each rotation so that it reaches the sides of hyperboloid of one sheet. You then rotate the axis 45 degrees so that the new axis of rotation is exactly perpendicular to the previous and once again rotate the circle up 45 degrees and down 45 degrees having it reach the sides of the hyperboloid of one sheet. If these two new volumes were added together would they compose the entire volume enclosed by the hyperboloid of one sheet or would there be some space that was not covered by the addition of these two volumes of rotation.

 

[Ambitwistor]

Thank you for bringing up this interesting question about the composition of De Sitter's space-time map. I can offer some insights into this topic.

Firstly, it is important to understand that Einstein's Cylinder space-time map and De Sitter's space-time map are two different models that represent different aspects of space-time geometry. While the cylinder map is a simple and intuitive representation of space-time, De Sitter's map is a more complex and accurate depiction of the curvature of space-time.

In order to answer your question, we need to consider the underlying principles of space-time geometry. In De Sitter's map, the conversion factor g44=0 represents a singularity, where the curvature of space-time becomes infinite. This singularity is a crucial aspect of De Sitter's map and cannot be ignored.

Now, let's imagine adding two cylinder space-time maps together. As you correctly pointed out, the two wedges would not cover all of the space enclosed by a hyperboloid of one sheet. This is because the singularity, represented by the point where g44=0, would still be present in the center of the composition. In other words, the two wedges would not be able to cover the space where the curvature of space-time is infinite.

To answer your second question about rotating a circle enclosed by a hyperboloid of one sheet, the same principle applies. When you rotate the circle along an axis parallel to the waist of the hyperboloid, the volume increases but it does not cover the singularity at the center. And when you rotate the circle along an axis perpendicular to the waist, the volume increases but it still does not cover the singularity.

In conclusion, the addition of two cylinder space-time maps cannot fully compose De Sitter's space-time map because it would leave out the crucial singularity at the center. I hope this explanation helps clarify the concept of space-time geometry and its representation in different models. Thank you for your question and continued interest in Einstein's theories.