Wednesday, January 9, 2008

The Omnipresence / Eternalness Riddle - B :
Dimension of Space Time.

Note : This is a follow up to this post. Please don't bother about the precise meaning of mathematical terms I use. Trust me, they have the nicest intuitive meaning ! Just read on.

A one dimensional world cannot be observed 'simultaneously' while being 'inside' the world. Only a being of 'higher' dimensions can have such a global view. A two dimensional being can observe one dimensional world without being part of it, and can also 'intervene' in it without being restrained by the topology of it. Similarly to be able to intervene in our usual (3,1) dimensional space-time, a being must necessarily inhabit a world which embeds our world as a part of it. Such a world must be at least of dimension 5. Hence an omnipresent / eternal God must inhabit this 5-D world which straddles our world. So for such a God to exist, this Embedding postulate (i,e our world being embedded inside a higher dimensional space) must hold. Please note that truth of embedding postulate does not imply existence of God, but its falsity forbids it.

Suppose such a God exists, then we can contemplate of a God of even higher dimensions. So let us denote and grade such Gods as G(m,n), where m denotes the spatial dimension and n the temporal dimension of the world it inhabits. We also assume that m and n are the maximal dimensions that G(m,n) inhabits e.g, G(4,1) can possibly 'rule' our world since it has the ability to 'intervene' in our (3,1)-D world, but would be 'powerless' against G(5,1) or G(5,2) and so on. As of now we have scant evidence of any higher dimension inside which we are embedded. But to allow this postulate we have these possibilities -

  1. If our world is exactly minkowskian of dim (3, -1) - it means it has 3 spatial and one temporal dimension and is geometrically flat throughout - then there doesn't seem to be any way to detect its embedding inside a higher dimensional world. In this case the embedding postulate can be possibly verified only when divine intervention occurs and gives us proof of His existence, which hasn't happened indubitably so far. Thankfully for God, our world is known to be non-minkowskian. So the God hypothesis makes a dashing comeback !
  2. It makes much more sense to pose this 'embedding' riddle for a non-minkowskian space, which holds for our natural world. However, mathematically, it is possible to embed it in a huge number of ways in some higher dimension. Let us demand the embedding to be a smooth topological embedding (I mean that it is not an immersion, plz don't bother about these terms, if it doesn't make sense to you). Further we distinguish two embeddings only upto isometries of the ambient space, which is a very reasonable assumption (in future, we may possibly have to distinguish between orientation preserving and orientation reversing cases). We also demand that the ambient space be minkowskian. This is the nicest possible case.
    So to illustrate : since a Klein bottle can be embedded in 4D space, so if our universe were a Klein bottle (which is 2 dimensional), its ambient space would need minimum 4 dimensions to properly embed it. I highlight this example merely to point out that the ambient space need not be of just one dimension higher.

  3. The third case is when the ambient space is not minkowskian, but embedded in a higher dimensional space, which in turn is again embedded higher up, until we reach a minkowskian space. Let us assume that this embedding stops after a finite number of such embeddings.
  4. For the fourth case, we assume that embeddings never stops. By this we mean that never in the chain we get a minkowskian space. Please note from case 1 that a minkowskian space effectively neutralizes the embedding postulate and no need to further investigate. Getting a minkowskian space is tantamount to stopping of this chain.
  5. Fifth, we assume that the chain stops but the final space is still non-minkowskian. This final minkowskian space has a stand alone spooky existence not needing any ambient space to exist. I find this case the strangest.
Note : The reason I am discounting non trivial isometric immersions is that, then the topology of the space would not be the same as that of its ambient space (actually strictly finer), forbidding the God to intervene. I mean the intervention is also assumed to be topologically smooth.

We will continue ...

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