We are to think of the universe as the skin of a balloon as it inflates: there is no point on the surface which could be considered the “center”, as the Big Bang occurred everywhere.
Is it at all possible to think of the center of the interior of the balloon as the “past” and the outside to which the balloon expands the “future”?
And what if the universe contracts into a Big Crunch…would we begin to see time run backwards, since the skin is once more moving into the “past”?
This is a perfectly understandable misunderstanding in basic differential geometry that I’m forced to unteach to students year after year after year.
In the expanding balloon model, there is nothing in the physics corresponding to the space outside the balloon. Yes, it’s there in the model but it’s just as mistaken to think it has anything to do with anything as it would be to use cereal pieces to teach counting to your child and then ask whether the number 2 is crunchy.
Everything in the model of importance is the surface of the balloon. All the geometry is intrinsic to that surface and nothing depends on its extrinsic properties (how it happens to be embedded into a Euclidean space). For example: a torus is curved as we usually see it, but “really” (intrinsically) it’s as flat as a sheet of paper.
Thus: there’s no “interior of the balloon”.
In answer to your second question, there are several different “arrows of time” (or definitions of past and future) in physics. There’s the cosmological arrow of time, “The future is the time when the Universe is bigger” and the thermodynamic arrow of time, “The future is the time when there’s more entropy”. There’s also a particle physics arrow of time, but it’s very subtle, and seems only to be relevant to a few subatomic particles such as the neutral K meson.
In any event, most of the human concepts of time (i.e., “The past is the time I can remember, and the future is the time I can’t”, or “A cause is in the past, while an effect is in the future”) are based on the thermodynamic arrow of time. And there’s no reason to suppose any connection between the cosmological arrow of time and athe thermodynamic arrow. Even if the Universe turned around and started contracting (which is not currently expected to ever happen, incidentally), it would not change the direction our memories work, or that cause precedes effect.
The point of the balloon analogy is that the galaxies get further away from each other without moving. They stay where they are and space expands. This excellent SciAm article explains that even astronomers interpret the analogy incorrectly.
In three dimensions, the analogy becomes that of an expanding plum pudding (with the raisins being the galaxies-at-rest), but a plum pudding with no “outside”. In this sense, yes, a less expanded universe is the past, a more expanded universe is the future: The expansion of the universe is the “time” of the universe. At the Big Bang (I hate that phrase - why did we allow such a misleading insult anyway?) the universe is as unexpanded as it can be - one cannot get any further back in “time”.
Also, keep in mind another theory- the universe really only wears black to hide the fact that it’s expanding.
(Not only is this not about the OP, but I stole it from Conan)
I’m not sure that the OP’s analogy is that bad, actually. If you actually write out the metric for a four-dimensional closed FRW universe, it’s conformally equivalent to the metric for four-dimensional space in spherical coordinates, as long as you take the “radial” coordinate to be time. There are then two sources of curvature in the total spacetime: the intrinsic curvature of the spatial surfaces (the 3-spheres), and the extrinsic curvature describing how these surfaces are “bending” with respect to the remaining dimension.
Not to step on Mathochist’s toes, but it is possible to embed a curved space(time) in a flat spacetime of a higher dimension. This is basically what ES is suggesting. (It may require more than one extra dimension, but it never requires more than some maximum: twice the original dimension, maybe? I forget.) That said, it’s still true that this additional dimension is completely unnecessary from the math point of view, and irrelevant from the physics pov, since you can never get there.
Sure, why not?
You’re in good company with this suggestion. Stephen Hawking at one point suggested that this might happen. He has since changed his mind, though.
Of course it’s possible. It’s possible to embed any differentiable manifold of dimension n into Euclidean space of dimension 2n. I think you might need 2n+1 to get an isometry for a given metric.
That’s beside the point, though. The curvature of the embedded submanifold splits (not naturally (“natural” is a technical term)) into an intrinsic and an extrinsic part, and the equations of general relativity are only concerned with intrinsic curvature.
True, but it is the case that both the intrinsic and extrinsic curvature of the spacelike hypersurfaces in a chosen foliation are important in GR. (This is the most obvious in the 3+1 decompostion of Einstein’s equation.) And in the usual “balloon” analogy, the “balloon” represents a spacelike three-surface at a given time, so its extrinsic curvature matters as well.
The more I think about this, the more I feel like it’s a pretty good analogy.
Okay, I didn’t want to split this hair too finely.
There are two different extrinsic curvatures in the balloon analogy: the extrinsic curvature of the 3-manifold the balloon traces out as it expands in the embedding 4-dimensional space (3-space + time). This is what I was saying doesn’t matter. The other is the extrinsic curvature of a given 2-dimensional slice – the balloon at a given moment – within the 3-manifold the balloon sweeps out. Yes, that extrinsic curvature does matter, but it still is strictly confined to the balloon and not the air around it.
Sorry to keep hijacking the thread like this… but it’s essentially what the OP was asking about, and it’s an interesting debate to a GR geek like myself.
So what you’re saying (unless I miss my guess) is that there are two different “non-balloon” coordinates you’re considering: time and a radial coordinate. Our three-dimensional spacetime, which is topologically S[sup]2[/sup] x R, is then embedded in a four-dimensional space, R[sup]4[/sup]. If that’s what you’re saying, then yes, I would agree with you that there’s some part of the extrinsic curvature that won’t be physically important.
What I (and, I think, the OP) was saying is: why do you need two different non-balloon coordinates? Why not just call the radial coordinate, which we don’t have too hard of a time visualizing, “time”? Then we have our three-dimensional spacetime, which is topologically S[sup]2[/sup] x R (if we remove the origin, which corresponds to the Big Bang singularity anyways), and there are no irrelevant curvatures – both the intrinsic curvature of the two-surfaces (the curvature of “space”) and their extrinsic curvature (how “space” is embedded in spacetime) are physically relevant.
The only part of this that you have to be careful with, I think, is that the “radial” coordinate is really a time coordinate and so comes into the metric with a flipped sign. But maybe there’s some problem with this analogy that I haven’t thought of.
What I read the OP as saying was to see the fixed point in the center of the balloon (which traces out a copy of R) as the point where things exploded from. Since at every value of time this is outside the balloon it has no counterpart in the physics.
The other problem with the balloon analogy is that it implies a positive curvature to space. While this is not an absurd possibility, it’s equally not-absurd to suppose that the Universe is flat or negatively curved, and the current body of evidence favors flat. Flat space can be accomodated by the raisin bread analogy (so long as you can picture the raisin bread being infinite), but I don’t know of any familiar analogy for an expanding negatively-curved space. And the raisin bread analogy and presumably any negatively curved analogy one might construct doesn’t have any radial coordinate to associate with time, so such identification is of strictly limited usefulness.