Q & A: big extra dimensions

According to M-theory there are 11 dimensions. 4 large dimension (3 space, 1 time), and 7 tiny dimensions. My question: If large masses "bends" space-time(gravity), doesn't that imply another large space-time dimension? If the definition of a dimension is directions in which something can move, and space-time is being bent, then wouldn't there have to be another large dimension? It also seems to me that if you view space-time as two dimensional (3 large spacial dimensions and 2 large time dimensions), you could explain things like velocity time dilation. If you have two objects accelerating from each other, both will observe the others clocks as slowing down. Could this be because you have two objects observing each other relative to their perspective from their "positions" in time?
- James Carlson (age 34)
Houston, Tx

You don't really need any extra dimensions to talk about curvature of the ones you've got. Remember that the idea of mass bending spacetime was already built into General Relativity, which has no extra dimensions. All the curvature is defined within the existing dimensions. For example, in a 2D space in which the circumference of a circle isn't 2π*radius, there's curvature, quite aside from any ideas about how to embed that space in some other space. So the large-scale curvature of our geometry doesn't have any direct information on whether the hypothetical remaining dimensions are all curled up on very small scales or not.

I wasn't able to understand your second question well enough to answer it.

Mike W.

(published on 05/26/2011)

I guess the point of my question lies in the fact that in order to bend 2D space...you need 3 dimensions. In other words, if a trampoline is representative of 2 dimensional space, and you put a bowling ball on it, regardless of how the 2D beings perceive the bending of their space...you have 3 functional dimensions in play (2 spacial dimensions + 1 to account for the dip in the trampoline). The 2D beings wouldn't be able to perceive the third dimension directly, but they would perceive the effects it had on their 2D world. Most likely, they would be pulled towards the dip much like gravity. I carried that thought over to spacetime. The curvature of space from your answer would be like the perceived effects of a 5th large dimension on 4 dimensional creatures. I'm not suggesting general relativity is wrong, I'm questioning the interpretation of the results. Am I way off base? If so...what am I missing?
- James Carlson (age 34)
Houston, Tx

Your intuition is just what we all start with as we view these questions pictorially. Our mental picturing apparatus seems to have some hard-wired limitations, specifically to assume a Euclidean structure. Kant even claimed that we were not able to think of space in other terms, an unfortunate choice given the important role non-Euclidean geometries was to play in physics. In formal math courses, we learn some other ways of thinking about curvature, defined purely in terms of distances between points in a single space, without reference to embedding in some bigger Euclidean space. That's not to say that curved spaces can't be embedded in higher-dimensional flat space, just that the only real point of discussing that would be if it had some observational implications. Otherwise, you're just tacking some hypothetical dimensions onto the observed world.

Your 2D critters can measure various effects of the curvature of their space. If it's very mild, they can say they live in a flat space with some gravity. If it's stronger and they want a reasonably compact description, they'll say they live in a curved space. If all the curvature fits a theory in which the causes are already observable in the 2D space, they gain nothing by hypothesizing bowling balls or even higher dimensions. If there are features (e.g. analogous to our accelerating cosmic expansion) that have no observed source in the 2D space, they may want to explore ideas about higher-dimensions to find a theory that explains these phenomena too.

I know this is hard to wrap your mind around, and I have no illusions of having explained it well, so feel free to keep following up.

Mike W.

(published on 06/02/11)