At the end, a physicist opines that only 35 decimal places of pi are necessary to calculate the circumference of the known universe to the accuracy of the diameter of a hydrogen atom. His reasoning is based on a calculation of the size of the universe derived by multiplying the age of the universe (20 billion years) by the speed of light.
My question - is this valid? My amateur understanding of some theories of physics is that in the extremely early stages of the life of the universe, inflation occurred in which the universe expanded enormously (relative to its initial size). My understanding has been (and I seek guidance here) that this inflation was not limited by the speed of light, because the speed of light limit applies to things moving within space, not to the expansion of the fabric of space itself.
If my understanding is correct, then the abovementioned calculation of the size of the universe may be invalid. Of course, it may also be that the point I make is true but trivial, in that it wouldn’t make much of a difference in the calculations.
Well, I was browsing A Short History of Nearly Everything and the author claimed that the universe was billions of miles across a second after the big bang which isn’t possible if c is the rate of expansion. I’m not sure of Bill Bryson’s science backround, so I’m not sure if that’s accurate, I don’t think it is.
Bryson bailed college after one term and read about eighteen bajillion books. He’s highly erudite but prone to vector bad info now and then, simply because he’s such a generalist.
The person who did the calculation was using “universe” to mean “observable universe” which is one of the ways the word is used. Since there is, as I understand it, neither any way to be certain the size of the actual totality of the space-time continuum, nor any generalized agreement on the size of all of the universe, observable or not, it’s as reasonable a use of the word as any, I guess.
(Nods). Yeah, makes sense. Upon reflection, the inherent epistemological ambiguity in the expression “known” universe is only capable of being meaningfully resolved by converting it to a strictly empirical expression like “observable”, in which case the problem disappears. Thanks DS. Nice to see a lawyer’s mindset has application in fields outside courtrooms.
I do like Bryson, though I’ve read a few criticisms of some of the “facts” in his books. But in the intro to that particular one, he says he got each chapter checked thoroughly by the relevant scientists; I wonder how it slipped through the net?
The distances used to talk about the universe are often confusing. Popular usage and strict scientific usage overlap, but also sometimes contradict.
The nutshell version. We talk about the universe being 13.7 light years in radius because that’s how long we think it took the light from the farthest objects to reach us. That does not mean that those objects would be 13.7 billion light years away if we could magically see them today. They are much farther than that because of the continued expansion of the universe.
The question about using pi to measure the universe doesn’t really change, however. It’s obvious that each additional decimal point adds a factor of ten in accuracy. But a radius of 78 is only four times a radius of 20, so no additional decimal point would be needed. As an approximation it’s within the margin the error. 35 decimal places is all you need, regardless.
Bryson’s A Short History of Nearly Everythingis purely for entertainment purposes and any concordance with scientific fact is strictly coincidenal.
The rate of expansion of the universe depends on the distance between two points per Hubble’s Law, and at a boundary (the cosmic event horizon) the expansion rate is c; beyond that, we can’t observe anything, so that is the size of the observable universe. The current radius of the observable universe is somewhere around 45 Bly, and at the time the light from stars currently seen at that distance was emitted they were somewhere around 40 Mly away, given the assumption of a constant Hubble expansion rate.
Real space, described by Riemannian geometry, is non-Euclidian, i.e. not flat. The essential consequence of this is that parallel lines are not “straight” but instead are curved (when projected onto a planar topology). This changes the value of pi from being a constant to a function of local curvature. Fortunately for us, space is locally flat enough that we can use the Euclidian value for pi in everyday calculations, and do so even in regions which are curved by using hyperbolic functions to map warped space onto a Cartesian plenum. Since most curvatures that we deal with are nearly-spherical in shape, this is a simple transformation, though when you get two more more regions of warped space together the math gets very, very messy and we have to use certain tricks or numerical solutions to get a credible representation. However, in areas where space is extremely warped (very massive objects, or objects with very high angular momentum) or over literally cosmic distances the value of pi changes dramatically, and may depend instantaneously upon your own orientation or momentum.
The claim that the first 35 digits of pi can accurately represent the scale of the universe doesn’t (and isn’t intended to) discuss the underlying non-Euclidian geometry of the universe, but rather to illustrate that we are capable of making perfectly valid mathematical models and statements that are far more precise than our ability to observe in nature; it’s just a simplification for the purposes of illustration and amusement. Heck, for most phenomena that engineers deal with on a regular basis, the first six digits are more than adequate.
To be fair, it’s not really pi that’s changing, but the circumference-to-radius ratio of a circle and other geometric quantities. After all, no one would say 1 + e[sup]i pi[/sup] = 0 is false near a black hole.
Because it is not an error. The universe can expand faster than the speed of light. The expansion of the universe is really the expansion of nothing. That is, it is the distance between points that is expanding. No object is actually moving faster than the speed of light.
It is misleading, though, to say that the expansion of the Universe was faster than light at a particular time. The speed of the expansion of the Universe depends on how far apart the points you’re comparing are. At any given time, there’s a cosmological horizon at some distance, beyond which the Universe is expanding faster than light. The difference between now and, say, the inflationary era is that back then, the cosmological horizon was very, very close.
Chronos has a good point. If some region is moving away from you faster than the speed of light, then light from that region can not reach you, and by definition it is beyond the horizon. Of course, at some later time the expansion rate can slow, and light from that region, emitted during the time of fast expansion, can reach you.
For some reason, the concept of faster than light expansion of the universe is easier to grasp than that of faster than light contraction, for me.
