Shapes of Space-Time

Question

Ok let’s get the legwork done first.

A common way to visualize our current space-time is as a 3D grid. The lines of this grid are perfectly parallel until a significant amount of matter causes them to bend inward toward the center of mass, thus causing gravity. In this case (since it appears to describe the world we live in) we know how time, light, acceleration, gravity, etc all behave in this arrangement of space-time.

My question is how a different arrangement of space-time might affect things. Of course that’s a rather general statement so let’s narrow the focus a bit.

• This specific space-time is a 3D spiral instead of a grid.
• All constants that would not be affected by this change are exactly the same as in our 3 dimensions.
• Each of the 3 new dimensions are spirals, so the new X, Y, and Z “lines” of space-time are now each spirals.

Would this arrangement of space-time change the “flow” of time? For instance, near a gravity well in our universe, time moves forward and down toward the center of mass. Around black holes, time does this but also dilates to the point that time might not even pass at all within the singularity itself, due to the fact that escape velocity now exceeds the speed of light. Would this naturally spiraling space-time cause alterations in the flow of time like this, in a completely different way, at all?

Any insights on how this might change the flow of time, as well as how it might change the behavior of things like light, gravity, the expansion of the universe, etc… as well as how such a dimension might look and generally operate are also appreciated. Also if you just happen to know how other “shapes of space-time” (angled, looping, knotted, or what have you) might affect these things that would be cool to know too, otherwise I may have to make this a series.

Edit 1: To clarify what I mean by each dimension is a spiral, think of an xyz coordinate graph. Instead of the x, y, and z being straight lines, each of them are intersecting spirals. So if we view our space-time as a grid of intersecting xyz lines where all of x is parallel, all of y is parallel, all of z is parallel, and they are all at right angles to each other, this new space-time is laid out the same, but instead of straight lines the xyz lines are now spirals. As far as the scale goes, I guess this would be on the macro side. However, the inspiration behind this question came from reading about how, in string theory, spatial dimensions could be curled up or compacted. This got me wondering how a universe might behave if space-time were twisted into different shapes, much how gravity does currently.

Answer 1

Before delving into the maths I don't quite understand what you want to achieve with your alternate space time? If you have an answer to that we may be able to give you ideas on how to modify it to achieve that result.

For space time you need to distinguish between local and global effects.

Locally in a small neighborhood our space time looks like a 3 dimensional rectangular grid. If there are no major masses nearby the grid is exactly rectangular and the grid lines are parallel. Major masses make the grid lines somewhat bend.

This is completely independent of what the global topology looks like. A regular piece of paper is flat, the analogy of a 2 dimensional grid with no masses. You can put the paper on the table and it will have 4 edges (or, if you want, be infinite in two directions). You can also glue together 2 opposing edges to form a cylinder. The paper now has only two edges (or is infinite in one direction). Mathematically you could even glue together the two remaining edges. You would then have a torus with no edges but with finite total area. You can't do that with paper in real life 3-d but you could do that with a 2-d regular flat paper in a 4 dimensional space.

Now putting the dimensions of a 3 dimensional space time into spiral shape doesn't change anything locally. In a small neighborhood it is still approximately flat. It also doesn't change anything globally in the topology, each dimension is still in infinite line. So all this does is putting your space time into some higher dimensionmal space in a complicated way, it doesn't change anything for the beings inside the space time.