Simulating bendy space

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amylizzle
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Simulating bendy space

Postby amylizzle » Mon Jun 17, 2013 9:01 am UTC

In practically every simulation I've ever built, space has been either a pixelated 3D grid or a smooth plane. It's always been fixed, and the objects always moved through it at velocity directly proportional to their momentum. This worked fine for those simulations, but space isn't really like that. Space can bend in our universe, and objects can distort space, as well as time.

I have no idea how I'd go about modelling that. I thought maybe a pixel grid in which you could add further pixels in between other pixels to represent space "stretching", but that doesn't really deal with time dilation, and doesn't work in reverse(ie, condensing space). It occurred to me that if anybody would know, they'd probably be on this board, so I figured I'd ask. This is just a thought experiment, I'm simply curious and I don't really have any intention of implementing this right now, though I might in the future.

DeGuerre
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Re: Simulating bendy space

Postby DeGuerre » Tue Jun 18, 2013 4:31 am UTC

The language of "bendy space" is differential geometry. You can understand differential geometry as the study of geometric objects on which we can do calculus. Calculus in high-dimensional bendy space looks different from the calculus in Euclidean space that you might be used to, but it's calculus nonetheless.

We can "discretise" Euclidean space in many ways: into regularly-spaced grids, or simplicial meshes (a "simplex" is a triangle, tetrahedron or higher-dimensional equivalent), or other finite element approaches. The equivalent for more general space falls under the general heading of discrete differential geometry. It's a relatively new field, and an active research area. That page should give you more than enough to get you started.

Oh, it wouldn't hurt brushing up on tensor analysis if you haven't seen it or are a bit rusty.

Good luck!

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amylizzle
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Re: Simulating bendy space

Postby amylizzle » Tue Jun 18, 2013 11:20 am UTC

Thanks for the informative answer! I'll do some reading an inevitably come back to the thread with more questions.

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eternauta3k
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Re: Simulating bendy space

Postby eternauta3k » Fri Jun 21, 2013 5:15 pm UTC

You can store the position of objects in a cartesian grid, and apply a transformation to the grid. For example, (u,v,w) = T(x,y,z) = (x, y, z^2). Then, using the derivatives of T you can know how volumes, areas and arc lengths change under the transformation. However, it's not clear to me what you want to do with this.
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amylizzle
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Re: Simulating bendy space

Postby amylizzle » Tue Jun 25, 2013 6:43 pm UTC

Not specifically intending to do anything, I was just wondering if it'd be ridiculously complex to design a simulation using general relativity instead of newtonian mechanics.

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PM 2Ring
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Re: Simulating bendy space

Postby PM 2Ring » Wed Jun 26, 2013 9:34 am UTC

amylizzle wrote:Not specifically intending to do anything, I was just wondering if it'd be ridiculously complex to design a simulation using general relativity instead of newtonian mechanics.


It'd certainly be more complex. But luckily, GR says that any sufficiently small region of spacetime is locally flat. So you don't need to simulate the bendiness at the 4D spacetime pixel level, you can cut your large region into small flat regions and use the Metric tensor to glue those small regions together, similar to how we can map the curved surface of the Earth onto an atlas with flat pages. And when you do need to deal with scenarios where the spacetime curvature is large, eg near neutron stars and small black holes, so the locally flat regions are very small, there will generally be some radial symmetry that you can take advantage of by an appropriate choice of coordinate system.


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