- Joined
- Jun 21, 2003
Any comments?
Is the Pope Catholic.
There are a couple of points that need to be clarified with regard to the hypersphere model. First, the three-dimensional hypersphere is not compact (closed). This means that the topology of space-time at the big bang is quite strangely singular, with the consequence that the entirety of space-time is not just a four-dimensional hypersphere in this model.
Again, it is our mental pictures that are leading us astray. For a (positively curved) sphere, the cone on a 3-sphere is indeed toplogically equivalent (but not geometrically equivalent -- Steven Hawking proved this in his PdD dissertation under Roger Penrose) to an ordinary four dimensional disc. This is why the Hawking-Hartle Theory of quantum cosmology works so nicely in the case of a sphere, but is problematic in the case of negatively curved spaces.
The spherical version of this model (called deSitter space, or Einstein-deSitter Space) has been around for almost a century. Without trying to predict the results of future data-gathering, I think that this model has pretty much been ruled out by experiement.
In my opinion, a more tractable model (and one that matches better with our mental pictures), would start like this. Take the ordinary three-dimensional hypersphere, with the Poincare metric. Take the quotient of this space by the action of a group of isometries. The resulting model has the same local curvature properties as the original hypersphere, but provides a more satisfactory model (in my opinion), sacrificing only simple conectivity.
I do not agree that scientists are puzzled by this. Einstein's great insight was not that curvature causes gravitational acceleration. It is that gravity and curvature are different names for the same thing. In general relativity, there is no such thing as the "force" of gravity. Gravity is not a force at all. What we interpret as acceleration due to gravity is just objects following their natural geodesics in curved space.
That is why -- in my opinion -- we will never achieve a theory of quantum gravity. (I eagerly await being proved wrong!)
By the way, this also emphasizes why "intrinsic" curvatue (curvature of the metric tensor) is the right way to approach this topic, rather than "extrinsic" (rubber sheet) curvature. Gauss proved in his famous Theorem Egregium that the two kinds of curvature are formally equivalent. But if we define a geodesic as the "shortest" (or in the case of the space-time interval, the longest) distance between two points, then first we need to define distance.
True, but the "expandimg anything" hypothesis serves just as well. There is no reason why the expanding space-like submnifolds should be hyperspheres rather than any other kind of three-dimensional space.
This cannot be determined by theorizing. It must be verified by hard evidence. Perhaps surprizingly, there is a lot of it, especially from studies of the cosminc background radiation. As much as I wish it were otherwse, so far the Euclidean model is prevailing -- fairly dramatically, actually -- over both the positively curved model, like the sphere, and the miriad negatively curved models, like the hypersphere.
Even if this is right, however, there are still many interesting models that are local Euclidean but have interesting global topological features. Toroidal models, for instance. Again, in principle, all these things can be determined experimentally.
In order to make these ideas into a genuine "hypothesis," this is what we must do.
(1) Descirbe a metric on the hyperspheres (or whatever 3-fold we wish to study), such that this metric
(a) is symmetric and isotropic (and also satifies some other technical conditions called "energy conditions," which basically say that the contibution of matter dominates the "stress" in the fabric of empty space), and
(b) satisfies the Einstein field equations for a distribution of matter that matches what we see in the real universe.
(2) Devise some experiements that will turn out one way if your hypothesis is correct, a different way if a competing hypethesis is correct instead.
If you are seriously interested in investigating these ideas, a good place to get started is The Large Scale Structure of Space-time by Hawking and Ellis (Cambridge Monographs on Mathematical Physics). Start reading in chapter five (exact solutions) and you will see the hypersphere model discussed in context will several others. (Do not read pop science paperbacks, lol.)
Then, if you really want to jump in, go to
http://grtensor.phy.queensu.ca/
and download (for free) a software product called GRTensor. This allows you to create your own universe. Just enter a metric tensor (either one of the standard ones or one of your own invention), hit the button, and GRTensor calculates all the curvature tensors and displays the physical properties of your universe. This is totally cool for amateur dabblers and serious professionals alike.
Beep-beep, this GRTensor product is a pedagogical tool that can be used to introduce exceptionally well-motivated students to the mathematics of general relativity without getting lost in all the calculations.
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