The last bit of evidence from physics which I’ll discuss is the “no-boundary” proposal of Jim Hartle and Stephen Hawking (and some related ideas).\u00a0 The Hartle-Hawking proposal was described in Hawking’s well known pop book, A Brief History of Time<\/a>.\u00a0 This is an excellent pop description of Science, which also doubles as a somewhat dubious resource for the history of religious cosmology, as for example in this off-handed comment:<\/p>\n [The Ptolemaic Model of Astronomy]<\/em> was adopted by the Christian church as the picture of the universe that was in accordance with Scripture, for it had the great advantage that it left lots of room outside the sphere of fixed stars for heaven and hell.$$^{[citation\\,needed!]}$$ Carroll, after making some metaphysical comments about how outdated Aristotelian metaphysics is<\/em>, and how the only things you really need in a physical model are mathematical consistency and fitting the data\u2014this is Carroll’s main<\/em> point, well worthy of discussion, but not the subject of this post\u2014goes on to comment on the Hartle-Hawking state in this way:<\/p>\n Can I build a model where the universe had a beginning but did not have a cause? The answer is yes. It\u2019s been done. Thirty years ago, very famously, Stephen Hawking and Jim Hartle presented the no-boundary quantum cosmology model. The point about this model is not that it\u2019s the right model, I don\u2019t think that we\u2019re anywhere near the right model yet. The point is that it\u2019s completely self-contained. It is an entire history of the universe that does not rely on anything outside. It just is like that.<\/p><\/blockquote>\n Temporarily setting aside Carroll’s comment that he doesn’t actually think this specific model is true\u2014we’ll see some possible reasons for this later\u2014the first thing to clear up about this is that the Hartle-Hawking model <\/em>doesn’t actually have a beginning!\u00a0 <\/strong>At least, it probably<\/em> doesn’t have a beginning, not in the traditional sense of the word.\u00a0 To the extent that we can reliably extract predictions from it at all, one typically obtains an eternal universe, something like a de Sitter spacetime<\/a>.\u00a0 This is an eternal spacetime which contracts down to a minimum size and then expands: as we’ve already discussed in the context of the Aguirre-Gratton model<\/a>.<\/p>\n This is because the Hartle-Hawking idea involves performing a “trick”, which is often done in mathematical physics, although in this case the physical meaning is not entirely clear.\u00a0 The trick is called Wick rotation, and involves going to imaginary values <\/em>of the time parameter $$t$$.\u00a0 The supposed “beginning of time” actually occurs at values of the time parameter that are imaginary!\u00a0 If you only think about values of $$t$$ which are real, most calculations seem to indicate that with high probability you get a universe which is eternal in both directions.<\/p>\n Now why is the Hartle-Hawking model so revolutionary?\u00a0 In order to make predictions in physics you need to specify two different things: (1) the “initial conditions” for how a particular system (or the universe) starts out at some moment of time, and (2) the “dynamics<\/a>“, i.e. the rule for how the universe changes as time passes.<\/p>\n Most of the time, we try to find beautiful theories concerning (2), but for (1) we often just have to look at the real world.\u00a0 In cosmology, the effective initial conditions we see are fairly simple but have various features which haven’t yet been explained.\u00a0 What’s interesting about the Hartle-Hawking proposal is that is a rather elegant<\/a> proposal for (1), the actual initial state of a closed universe.<\/p>\n One reason that the Hartle-Hawking proposal is so elegant is that the rule for the initial condition is, in a certain sense, almost the exact same rule as the rule for the dynamics, except that it uses imaginary values of the time $$t$$ instead of real values.<\/em>\u00a0 Thus, in some sense the proposal, if true, unifies <\/em>the description of (1) and (2).\u00a0 However, the proposal is far from inevitable, since there is no particularly good reason (*) to think that this special state is the only <\/em>allowed state of a closed universe in a theory of quantum gravity.\u00a0 There are lots of others, and if God wanted to create the universe in one of those other states, so far as I can see nothing in that choice would be inconsistent with the dynamical Laws of Nature in (2).<\/p>\n (Hawking has a paragraph in his book asserting that the proposal leaves no room for a Creator, but I’ll put my comments on that into a later post!)<\/p>\n In the context of a gravitational theory, imaginary time means that instead of thinking about metrics whose signature<\/a> is $$(-, +, +, +)$$, as normal for special<\/a> or general<\/a> relativity, we think about “Euclidean” (or “Riemannian”) signature metrics whose signature is $$(+, +, +, +)$$.\u00a0 So we have a 4 dimensional curved space (no longer spacetime).<\/p>\n The assumption is that time has an imaginary “beginning”, in the sense that it is finite when extended into the imaginary time direction.\u00a0 However, because there is no notion of “past” or “future” when the signature of spacetime, it’s arbitrary which point you call the “beginning”.\u00a0 What’s more, unlike the case of the Big Bang singularity in real time, there’s nothing which blows up to infinity or becomes unsmooth at any of the points.<\/p>\n All possible such metrics are considered, but they are weighted with a probability factor which is calculated using the imaginary time dynamics.\u00a0 However, there are some rather hand-waving arguments that the most <\/em>probable Euclidean spacetime looks like a uniform spherical geometry. The spherical geometry is approximately classical, but there are also quantum fluctuations around it.\u00a0 When you convert it back to real time, a sphere looks like de Sitter space: hence the Hartle-Hawking state predicts that the universe should look have an initial condition that looks roughly like de Sitter space, plus some quantum fluctuations.<\/p>\n I say handwaving, because first of all nobody really knows how to do quantum gravity.\u00a0 The Hartle-Hawking approach involves writing down what’s called a functional integral <\/em>over the space of all possible metrics for the imaginary-time goemetry.\u00a0 There are\u00a0<\/em>an infinite-dimensional space of these metrics, and in this case nobody knows how to make sense of it.\u00a0 Even if we did know how to make sense of it, nobody has actually proven that there isn’t a classical geometry that isn’t even more probable than the sphere.\u00a0 Worst of all,\u00a0 it appears that for some of the directions in this infinite dimensional space, the classical geometries are a minimum <\/em>of the probability density rather than a maximum!\u00a0 This gives rise to instabilities, which if interpreted naively give you a “probability” distribution which is unnormalizable, meaning that there’s no way to get the probabilities to add up to 1.<\/p>\n So Hartle and Hawking do <\/em>what’s called formal <\/em>calculations, which is when you take a bunch of equations that don’t really make sense, manipulate them algebraically as if they did<\/em> make sense, cross your fingers and hope for the best.\u00a0 In theoretical physics, sometimes this works surprisingly well, and sometimes you fall flat on your face.<\/p>\n Unfortunately, it appears that the predictions of the Hartle-Hawking state, interpreted in this way, are also wrong when you use the laws of physics in the real universe!\u00a0 The trouble is that there are two periods of time when the universe looks approximately like a tiny de Sitter space, (a) in the very early universe during inflation, and (b) at very late times, when the acceleration of the universe makes it look like a very big de Sitter space.\u00a0 Unfortunately, the Hartle-Hawking state seems to predict that the odds the universe should begin in a big de Sitter space is about $$10^{120}$$ times greater than the odds that it begins in the little one.\u00a0 That’s a shame because if it began in the little one, you would plausibly get a history of the universe which looks roughly like our own.\u00a0 Whereas the big one is rather boring: since it has maximum <\/em>generalized entropy<\/a>, nothing interesting happens (except for thermal fluctuations<\/a>).\u00a0 St. Don Page has a nice article explaining this problem<\/a>, and suggesting some possible solutions which even he believes are implausible.<\/p>\n Alex Vilenkin has suggested a different “tunnelling” proposal, in which the universe quantum fluctuates out of “nothing” in real time rather than imaginary time.\u00a0 This proposal doesn’t actually explain how to get rid of the initial singularity, and requires at least as much handwaving as the Hartle-Hawking proposal, but it has the advantage that it favors a small de Sitter space over a big one.\u00a0 From the perspective of agreeing with observation, this proposal seems better.\u00a0 And<\/em> it has an actual beginning in real time, something which (despite all the press to the contrary) isn’t true for Hartle-Hawking.<\/p>\n (*) There is however at least one bad<\/em> reason to think this, based on a naive interpretation of the putative “Holographic Principle” of quantum gravity, in which the information in the universe is stored on the boundary.\u00a0 A closed universe has<\/em> no boundary, and therefore one might think it has no information, meaning that it has only one allowed state!\u00a0 (The argument here is similar to the one saying the energy is zero<\/a>.)\u00a0 At one time I took this idea seriously, but I now believe that such a strong version of the Holographic Principle has to be wrong.\u00a0\u00a0 There are lots of other contexts where this “naive” version of the Holographic Principle gets the wrong answer for the information content of regions, and actual calculations<\/a> of the information content of de Sitter-like spacetimes give a nonzero answer.\u00a0 So I’m pretty sure this isn’t actually true.<\/p>\n","protected":false},"excerpt":{"rendered":" The last bit of evidence from physics which I’ll discuss is the “no-boundary” proposal of Jim Hartle and Stephen Hawking (and some related ideas).\u00a0 The Hartle-Hawking proposal was described in Hawking’s well known pop book, A Brief History of Time.\u00a0 … Continue reading
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