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It would certainly change the stability analysis, yes, but the universe does not have a significant angular momentum. That would leave a characteristic signature on the CMB, a preferred direction, and it's been hunted, with each new and improved dataset, and we still don't have it. The hunt has turned up other interesting anomalies such as the appallingly-named "axis of evil", but those signatures are also (probably) not due to an intrinsic net angular momentum. That doesn't say there isn't one, just that it's not particularly significant and is, as yet, unobservable. (In case you're interested, a universe with a net angular momentum could be described by one of the Bianchi models rather than a (Frieman-Lemaitre-)Robertson-Walker. A simpler case that might interest people is the Goedel universe, notorious because it contains closed timelike loops, which allow for the possibility of time travel. The Goedel universe wouldn't form a realistic model of the universe but it's an interesting spacetime.)

Well, it violates the weak energy condition, although it does satisfy (just) the strong energy condition. The weak energy condition says that density + three times pressure is greater than zero, and that's pretty easy to violate if you've got a scalar field. The strong energy condition says that density + pressure is greater than zero, which is far harder to violate; dark energy is right on the border of it, and a cosmological constant *is* the border (pressure = - density).

Of course, you're totally right - none of this means that they violate conservation of energy. That's built in to the theory...

"What they've proved mathematically as that at the event horizon of a black hole the math fails. It falls apart and no longer makes any sense because the numbers get too large on one side of the equation."

Not so. The maths dies at the singularity at the centre of the hole, but it doesn't at the event horizon except in a badly-chosen coordinate system. Alas, the usual coordinate system we'd present the Schwarzschild solution in is indeed badly-chosen and has an apparent singularity at the horizon, but this is not an actual singularity, as can be seen quickly by calculating a scalar curvature invariant - the Ricci scalar is the immediate choice, it's basically a 4d generalisation of the more-familiar Gaussian curvature - and seeing that it's entirely well-behaved except at the centre of the hole. So we look for a coordinate system well-behaved at the horizon and quickly come across Painleve-Gullstrand coordinates, in which spacetime is locally flat and perfectly behaved at the horizon. The implication is the poor sod wouldn't be able to tell that he'd got to the horizon, except through tidal forces (which depend on the size of the hole), and then he'd struggle to navigate before slamming into a singularity.

Even more confusingly, for a *realistic* hole, the insides are rather different. A Schwarzschild hole has a singularity inevitably in the future - all future-directed paths one can travel on, or light can travel on, end at the singularity. That's a bit of a bummer if you happen to be in a Schwarzschild hole. But a Schwarzschild hole is not physical; it is a non-rotating, uncharged hole, and that's not a realistic setup. In a charged (Reisser-Noerdstrom) or a rotating (Kerr) or, come to that, a charged rotating (Kerr-Newman) hole the singularity is "spacelike" -- there exist paths on which we could, in principle, travel, that avoid the singularity. In the case of a Kerr(-Newman) hole it's even smeared out into the edge of a disc. In reality, good luck navigating in there, but the singularity is not inevitably in the future in there.

A bit closer to the point, you're right that speed doesn't really have anything to do with it. Instead it's the type of path you can travel on, and where *they* go. An event horizon can be defined as the surface on which "null" geodesics, on which light travels, remain equidistant from the hole. If you travel, as massive particles do, on a "timelike" geodesic then you're fucked; you're never going to be able to accelerate enough that you even travel on a null geodesic, let alone a "spacelike" geodesic along which you can basically access anywhere. On a spacelike geodesic you could get out of a hole no problem. You could also travel in time, and you could break causality fifteen times before breakfast. I'd like to travel on a spacelike geodesic - it would be fun. Though managing to get back to a timelike geodesic might be significantly less so.

"Another obvious but often overlooked theory is that our universe IS a black hole inside a larger universe."

That's an extraordinarily strong statement. Our universe might be indistinguishable from a black hole from the outside, yes, but there's a big "might" in there, and an "outside" that doesn't necessarily make much sense either. It all depends on the setup you're assuming. Sure, we could end up finding that the universe is "inside" a black hole for a given definition of "universe", "inside" and "black hole", or we might find that that statement does not make any extent. I wouldn't want to say anything stronger than that, frankly, not least as I'm aware of models of cosmology that are observationally indistinguishable from a standard, infinitely-extended, flat universe, which are also flat, but which have finite extent. One way to do so is to simply put the universe into a toroidal topology. Since GR is a local theory it says nothing about topology, and it would be hard to argue that a universe extended on a torus would look like a black hole from the "outside", since that would be the entire extent of spacetime.

Also, more to the direct point, in the article we've kind of moved away from discussing is this snippet [p5]:

He [Einstein] notes that some theoretical attempts have already been made to explain the new observations: “Several investigators have attempted to account for the new facts by means of a spherical space, whose radius P is variable over time”. Once again, no specific citations are made, so we can only presume that Einstein is referring to works such as those by Lemaître, Eddington, de Sitter and Tolman (Lemaître 1927; Eddington 1930; de Sitter 1930a,b; Tolman 1929, 1930). Indeed, the only specific reference in the entire paper is to Alexander Friedmann’s model of 1922: “The first to try this approach, uninfluenced by observations, was A. Friedman, on whose calculations I base the following remarks”

This not only provides a few references to papers that Einstein -- in 1931, writing before the derivation of the Einstein-de Sitter model -- may well have based his work on, including the Friedman and Lemaitre models that were proven in the 30s by Robertson and Walker (working independently) to be the unique dynamical, homogeneous and isotropic models, but also shows that Einstein was aware of the Friedman model. (If you're interested, Tolman's work was on spherically-symmetric models that are isotropic but not homogeneous, and are subsets of what are now known as the Lemaitre-Tolman-Bondi models. The de Sitter papers, I think, established de Sitter and anti-de Sitter space, and I wasn't aware of Eddington's paper before but reading through the article here it turns out to have been on the instability of the Einstein static universe.)

The EdS model is not "the workhorse of modern cosmology", no matter what the author of this summary wants you to think. If any model could be described thus it would be the Friedman-Lemaitre-Robertson-Walker model, which was already known (thanks to Friedman and Lemaitre who developed it in the 20s) by 1931. The EdS model is a specialisation of the FLRW to a universe containing pure pressureless matter, and an expansion is necessarily decelerating. As such not only can it not describe the early universe, when the existence of the CMB and the expansion of the universe together imply a period where the universe was instead dominated by radiation, nor the late universe, where observations imply that expansion is instead accelerating. EdS was used as an approximation to the late time universe until the 90s when it was obvious that it was in conflict with observation. It's sometimes still used for rough approximations thanks to the simple solutions one can find for linear perturbations, but those are only valid up to redshifts of approximately 1, and no later.

Similar - adblock, noscript and flashget. 4gig RAM and Firefox routinely taking about 900meg since the update to 29. I tend to only have three or four tabs open, and usage was around 500meg in 28.

Where do *you* live? That's a genuine question, because here in the UK, South America is that big southern lump featuring the likes of Argentina and Brazil, Central America is the thin wiggly bit featuring the world-famous Panama Canal which is based in Panama, and North America is the big lump on top which Mexico, the USA and Canada call home.

Actually, sorry, you're wrong about that. The geography skills of the guy who wrote the summary are deeply questionable (Mexico seemed to pull off the trick of joining South America remarkably quietly), but "maths" is the abbreviation of "mathematics" used in the UK. It's not a plural of "math" -- and, after all, we don't call it "mathematic". So I think we've pinned down where "thephydes" is from - he or she is either British, or learned British English.

So I think the main thing we can learn from the summary is that despite the characterisation of Americans has having no knowledge of geography outside their own country, the rest of the English-speaking world is just as bad and should probably stop feeling so damned smug and look at the fucking great log in their own eyes.

I've had to remote Firefox too, chiefly to access papers that are behind paywalls my university has access to but I have no access to at a university I'm visiting, or at home. You're making Firefox over X sound a lot better than it actually is.

Well, I wasn't a particular fan of that comment either...:) But your comments about the conservation of energy are basically wrong -- like I say, the expansion of the universe is governed basically by two equations:

1) The Friedman equation, which is the "Hamiltonian constraint" and which can be interpreted as an energy constraint equation 2) The matter continuity equations, which arise directly from conservation of equation of matter

(Technically, and apologies for the ugly LaTeX notation here, matter is described in relativity by "stress-energy" or "energy-momentum" tensors, which bundle together the classical concepts of energy, momentum flux, energy flow, and momentum flux density. The gradient of the more familiar Momentum flux density produces momentum conservation, while the time derivative and gradient of the energy give, well, energy conservation otherwise known as matter continuity. In terms of the stress-energy tensors, if energy is conserved then the covariant gradient of these is conserved -- written T^{\mu \nu}_{; \nu} = 0 where the \mu and \nu describe the four spacetime coordinates, and the repeated \nu means we sum across them. In Euclidean spacetime this immediately reduces to normal energy and momentum conservation. That semi-colon means "covariant derivative", meaning it has contributions coming from interactions with the spacetime geometry; if instead you insist on writing the conservation equation as a partial derivative (as in familiar classical mechanics and fluid dynamics) then you get T^{\mu \nu}_{, \nu} = [gravitational terms]^\mu and if you were unaware of the form of a covariant derivative this would look like a violation of energy and momentum conservation rather than what it is -- an interaction of the stress-energy of matter with the geometry of spacetime.

(The energy density, with respect to some chosen time coordinate which can be described as the vector n^\mu normal to a purely spatial three-dimensional surface, is found by the contraction \rho = n^\mu n^\nu T_{\mu \nu} which basically just means the scalar product of the stress-energy tensor with the time coordinate, or phrased differently, the stress-energy tensor projected along the time coordinate. This energy is conserved. Similarly, if we take the full Einstein equations they can be written as G^\mu_\nu = 8\pi G T^\mu_\nu where the G^\mu_\nu is the "Einstein tensor" that basically characterises the local gravitational field, and the G is newton's constant which infuriatingly has the same god damned symbol. If we project these equations along the time coordinate then we get out an energy on the right-hand side, and something that therefore can be interpreted as embodying the "gravitational energy" on the left-hand side. The result of this is the Hamiltonian constraint, and in the specific case of a cosmological spacetime the result is the Friedmann equation. Both these equations therefore express energy conservation -- one for the gravitational field, and the other for the matter.)

Said a bit less technically, the second equation is simply what you expect -- for normal matter we except the density changes with scale factor as rho = rho_0 / a^3. That is, the density decays as the volume of the universe increases. For photons we would expect instead rho=rho_0 / a^4, where we've got the normal volume dilution (1/a^3) and, since a photon's energy is proportional to length and the expansion of the universe will stretch that length, we should have an extra decay in the energy of 1/a. So rho = rho_0 / a^4. To a good approximation the same can be said for neutrinos (although to be exact we should make these massive, albeit extremely light.) Put these together and you have rho = rho_matter / a^3 + rho_photons / a^4. This is nothing but what happens when space expands.

That's well and good but it doesn't answer your issue with energy conservation, because it seems to violate it. That's where the other equation comes in, because these equations are only *half* the story. In GR (and other related, geometric theories of gravity) gravity also has an associated energy. It's hard to describe where the Friedmann equation comes from in this picture without the full theory, but it's obvious that if the energy density of photons is being diluted there must be some impact on the geometry that underpins what we call "gravity". In cosmology there is basically one quantity that determines that geometry -- the same scale factor, a. So the energy equation should determine how a evolves in time. Given that we can characterise it with the Hubble rate, which is the change in a with time, divided by a. This doesn't have the units of energy which (since we've set the speed of sound equal to 1, meaning we measure seconds and centimetres with the same stick) has units of 1/time^2. So let's square the Hubble rate to get something in the right units to be an energy. That gives us a crude estimate of the gravitational "energy conservation" as H^2 = ?. First guess for the right hand side has to be to slap in something proportional to the matter energy. This then gives us H^2 = \kappa \rho. This isn't a particularly convincing way to justify the Friedmann equation (which is what that is), but it's better than the usual approach which is to assume an expanding sphere of pure dust and which misses the point of gravitational energy completely.

The point of all of that, both technical and ineptly non-technical, is that while if you just look at the matter sector alone and forget that you have to take the geometry into account, you conclude that energy is not being conserved. However, remembering that in the theory cosmology is based on we have to include interactions with gravity (which can also be described as the "energy" of the gravtiational field though I'd advise caution in taking such interpretations too strongly) everything suddenly falls into place -- the apparent loss of energy from matter is balanced by the gravitational field.

Jesus. Sorry for the length of that one, you didn't deserve that. Oh well, hopefully it interested you -- ultimately I'd agree with the statement "physics has problems too" because if it didn't there'd be one hell of a lot of unemployed physicists mooching about looking unhappy.

It's just how gravity works - if you fill a universe with not quite enough matter to make it collapse it will expand. Meaning if the universe was a bit denser it would collapse, but since it's not it's expanding. Then, as Neil Boekend says, due to something we're currently calling "dark energy" for want of a better term, with the amount of matter we've got we would expect the universe to be expanding but slowing whereas it's actually accelerating.

Sorry I can't give a better answer but really no-one can -- just that given the theory we've got, and the starting point we assume (an initial expansion coming from the point at which the theory simply breaks down), that's how it goes.

That's like a description of high energy physics in general, and it's a very good one for M theory, particularly with the elephant's shadow given the AdS/CFT duality....

Well, we can have a very good bash at it by taking the science we have at the minute, extrapolating back, noting every place where the story seems to get shaky and the edge cases that can arise, and see when things die. Surprisingly, the story of the last five billion years is a hell of a lot shakier than the story from, say, the fifth second up to the ten billionth year. Very early cosmology, yes, I totally agree -- it's speculation, and in particular (the slight overreaction to the BICEP2 results aside) I don't think anyone actually working on inflationary theories would pretend that they are dealing with anything that isn't, strictly speaking, phenomenology. I think most theorists believe something that *behaved* like inflation had to occur, and I would agree with that, but I don't think anyone has made any particularly strong claims for any particular inflationary theory, since they're all by their nature nothing but speculation about how higher-energy theories might act viewed with our current techniques.

But by the time of big bang nucleosynthesis in the first seconds to the first minutes, the science is really well understood -- we have direct evidence from accelerators for how matter behaves at such energy scales, and the very fact that the predictions of the theory based on this match so closely to observation at every point up to (and even beyond) the ten billionth year of the galaxy can give us a hell of a lot of faith that the fundamental picture is (at least phenomenologically) correct. I say this as someone who has chopped away at cosmology's fundamentals for over a decade -- the broad idea of inflation is probably accurate; something that closely resembles a universe composed of "baryonic" (ie normal) matter (about 5% of critical density), cold dark matter (about 25% critical density), photons, and three species of slightly massive neutrinos, *had* to hold between the first few seconds and roughly around the 10th or 12th billionth year, at which point that model breaks down unless one additionally adds in something that looks like a cosmological constant or a dark energy. Whether the physical identifications are actually precise or not is a totally different question -- and they're not, with dark energy on particularly shaky grounds but "dark matter" certainly hiding a world of complication beneath its single boring equation (w = 0) -- but the actual model is far too successful to simply be ignored. And I wouldn't consider using a different model for the period from just before BBN up until around a redshift of, say, 2 or 3, because there would be little to be acheived. Precisely how I generate the different components of course is model dependent, but it will look and act a lot like Lambda CDM.

For lower redshifts, meh, many bets are still off.