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Archive for April, 2011

Today’s arxiv find is a historical fact that I really should have known by now. From “The linear redshift-distance relationship: Lemaître beats Hubble by two years“:

The facts are simple: Friedman (1922) was the first to publish non- static solutions to Einstein’s field equations. However, he did not extend that into a cosmological model built on astronomical observations. In 1927 Lemaître rediscovered these dynamical solutions. In the same publication he extracted (on theoretical grounds) the linear velocity–distance relationship v=Hr. Combining redshifts published by Stromberg (1925) (who relied mostly on redshifts from Slipher (e.g. Slipher 1917)) and Hubble’s distances via magnitudes (Hubble 1926), he calculated for the “Hubble constant” two values, 575 and 670 km/sec/Mpc depending on how the data is grouped. For Lemaître these results showed that the Universe was expanding. Two years later Hubble found the same velocity–distance relationship v=Hr on observational grounds from practically the same observations that Lemaître used in 1927. However, Hubble does not credit anyone for the redshifts, most of which again came from Slipher.

Why is this not more widely known? In 1931, Lemaître’s paper was translated into English with the help of Eddington, but  “the two pages from the 1927 paper that contain Lemaître’s estimates of the Hubble Constant are not in the 1931 MNRAS paper”.

The standard story is that Friedman discovered the equations. Lemaître rediscovered the equations and promoted them. Hubble used Slipher’s observations to show that the expansion predicted by the equations was actually true. It seems that Lemaître did it all! And, according to Wikipedia, was one of the inventors of the Fast Fourier Transform.

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More Sweet Nothings …

I’ve had an interesting set of responses to my post “Of Nothing”, so I can’t resist a sequel. (I’m not really taking a position on who “won” the debate. I don’t really care. I’m just commenting on some of the issues raised.) Here, I’ll respond to a few more of Krauss’ comments on nothing and cosmology, this time from this article in the Wall Street Journal:

There are remarkable, testable arguments that provide firmer empirical evidence of the possibility that our universe arose from nothing. … the existence of dark energy and a flat universe has profound implications for those of us who suspected the universe might arise from nothing.  Why? Because if you add up the total energy of a flat universe, the result is precisely zero. How can this be? When you include the effects of gravity, energy comes in two forms. Mass corresponds to positive energy, but the gravitational attraction between massive objects can correspond to negative energy. If the positive energy and the negative gravitational energy of the universe cancel out, we end up in a flat universe.  Think about it: If our universe arose spontaneously from nothing at all, one might predict that its total energy should be zero. And when we measure the total energy of the universe, which could have been anything, the answer turns out to be the only one consistent with this possibility.  Coincidence? Maybe. But data like this coming in from our revolutionary new tools promise to turn much of what is now metaphysics into physics.

If you read the whole article, you quickly discover that Krauss simply doesn’t understand the question “why is there something rather than nothing?” – see my last post for more on that. Here, I will raise two issues with the quote above.

The Energy of the Universe

It is not true that if the universe is flat then it has zero total energy. This idea relies on a Newtonian calculation of the gravitational potential energy of an expanding universe. However, such a calculation simply cannot be performed in general relativity (and even the Newtonian calculation is suspect because the Poisson equation has no solution for a uniform unbounded fluid. See Rindler 1977, pg 199). There is simply no way to calculate the total mass-energy of a general spacetime, and the energy of the gravitational field cannot be calculated in a coordinate invariant way. John Baez and Sean Carroll have excellent discussions of this point, as does Hartle in his GR textbook: “Conserved quantities … cannot be expected in a general spacetime that has no special symmetries”. Note that energy conservation is a consequence of time-translation symmetry, which an expanding universe (and its Robertson-Walker spacetime) does not possess. Here’s Gravitation by Misner, Thorne and Wheeler (MTW):

When evaluating the 4-momentum and angular momentum of a localised system, one must apply the flux integrals only in asymptotically Minkowskian coordinates. If such coordinates do not exist (spacetime is not flat at infinity), one must completely abandon the flux integrals, and any quantities that rely on them for definition: the total mass, momentum, and angular momentum of the gravitating source … ‘Total mass-energy’ is a limited concept, useful only when one adopts a limited viewpoint that ignores cosmology. (Pg 463)

The Robertson-Walker metric is an example of a spacetime that is not asymptotically flat, and thus attempting to calculate its total mass-energy will “easily and unavoidably produce nonsense” (MTW). (Don’t confuse the two different uses of the word flat above. Krauss is referring to the space curvature of an expanding universe, while MTW are referring to spacetime curvature. A spatially-flat expanding universe is not spacetime-flat.)

The point of the above passage is that we cannot in general add up the local contributions of the mass-energy of the universe to produce a total mass-energy. When it comes to gravitational energy, it gets even worse – we can’t even define it locally: (more…)

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Congratulations to India for winning the cricket World Cup. They clearly deserved it. Attention now will turn to the next World Cup, and the controversial plan to cut the number of teams to 10. Allow me to give my opinion, as an avid cricket watcher. (Alas, playing cricket is rather difficult in Switzerland).

At least 2 associate teams must be included. The performance of Ireland and the Netherlands was not only impressive, it was exciting. Every world cup has featured at least one “minnow” that created an upset. Such upsets are what makes sport interesting.

At the same time, the games involving Canada and Kenya held little interest. It was quite clear early on that no upset was on the cards. They were easily beaten by the test-playing nations, and no-one really cared when they played each other.

The inclusion of the minnows meant that a lot of the group games were very one-sided. There were 42 group games, but each test playing nation played a minnow as often as they played another test playing nation. What we really come to see in the world cup are the top teams playing each other. But once we passed the group stage, the tournament was over within 7 sudden death matches.

So, here is what we want from an ideal world cup:

  • We want minnow vs. big-gun games only when the minnow has a reasonable chance of an upset.
  • We want to maximise the number of big-gun vs big-gun games. When we’ve eliminated the rest, we want more than 7 games of the best.
  • This world cup transitioned very quickly from the “plenty of second chances” group stage to the sudden-death stage. This isn’t ideal.

Can this be done? Here is my idea.

Seed the groups. Pre-tournament, the top 12 teams in the world (incl. 2 associate teams) are seeded by their ICC ranking.

  • Group A: Teams 1,2,3,4
  • Group B: Teams 5,6,7,8
  • Group C: Teams 9,10,11,12

Group stage: Each team plays each member of its group once. 18 games. At the end of the group stage, teams are ranked: A1, A2, A3, A4, B1, B2 … C3, C4.

  • C3 and C4 are eliminated.
  • A1 and A2 go through to the final 8.

Final 8 playoff: The middle 8 teams playoff in 2 tiers. 4 games

  • Alpha playoffs: A3 vs B2, A4 vs. B1.
  • Beta playoffs: B3 vs C2, B4 vs. C1.
  • After these playoffs, we have a ranking: Alpha 1, Alpha 2, … Beta4
  • Beta 3 and Beta 4 are eliminated.

Final 8: The ranking in the final 8 is very important – a higher ranking makes it harder to be eliminated. The ranking order is:

A1 : A2 : Alpha1 : Alpha2 : Alpha3 : Alpha4 : Beta1 : Beta2

We can now do more than the 7 game, sudden-death quarter finals. At the very least, we could use the McIntyre final-8 system. My preferred system is as follows – it ensures that a final 8 team must lose at least 2 games to be eliminated. Each win puts you in a better situation – there is no incentive to lose.

  • Rank the final 8: 1,2,3 … 8.
  • Round 1: 1 vs. 8, 2 vs. 7, 3 vs. 6, 4 vs. 5. Rank the winners W1 … W4, losers L1 … L4
  • Round 2: W1 vs W4, W2 vs. W3. L1 vs. L4, L2 vs. L3. Losers in the L games are eliminated. Winners of the W games receive a bye.
  • Round 3: Play the 4 teams (winners in the L games, losers in the W games) against each other. Losers are eliminated.
  • Semi-finals
  • Finals – best of three.

This is 15 games, including the three finals. The entire tournament has 37 games – 12 less than this World Cup. Minnows that play big-guns have earned their place, are in form and are facing a big-gun that is coming off losses – a recipe for an upset. There are as many minnow vs minnow games as in this world cup, fewer minnow vs big-gun and many more big-gun vs. big-gun. It would be easy to play minnow vs. minnow (group C) games on the same day as Group A games.

Tell the ICC, won’t you … I’ll just wait here for the Barnes system to catch on …

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I’ve just finished listening to a debate between philosopher William Lane Craig and cosmologist Lawrence Krauss on the debate topic “Is there evidence for God?”. I have a load of these on my iPod – some are very good (Craig vs Austin Dacey is probably the best), while some represent 2 hours of my life that I’ll never get back. They get a bit repetitive after a while. The debate with Krauss was somewhere in the middle. Craig was polished and concise, presenting the same 5 arguments (contingency, Kalam, fine-tuning, moral, resurrection of Jesus) he’s presented for decades. Krauss was less organised and much less focussed. I’ve responded to some of Craig’s claims elsewhere. I’ll focus on some of what Krauss said.

First and foremost, I’m getting really rather sick of cosmologists talking about universes being created out of nothing. Krauss repeatedly talked about universes coming out of nothing, particles coming out of nothing, different types of nothing, nothing being unstable. This is nonsense. The word nothing is often used loosely – I have nothing in my hand, there’s nothing in the fridge etc. But the proper definition of nothing is “not anything”. Nothing is not a type of something, not a kind of thing. It is the absence of anything.

Some of the best examples of the fallacy of equivocation involve treating the word nothing as if it were a type of something:

  • Margarine is better than nothing.
  • Nothing is better than butter.
  • Thus, margarine is better than butter.

We can uncover the fallacy by simply rephrasing the premises, avoiding the word nothing:

  • It is better to have margarine than to not have anything.
  • There does not exist anything that is better than butter.

The conclusion (margarine is better than butter) does not follow from these premises.

Now let’s look at Krauss’ claims again. Does it make sense to say that there are different types of not anything? That not anything is not stable? This is bollocks. What Krauss is really talking about is the quantum vacuum. The quantum vacuum is a type of something. It has properties. It has energy, it fluctuates, it can cause the expansion of the universe to accelerate, it obeys the (highly non-trivial) equations of quantum field theory. We can describe it. We can calculate, predict and falsify its properties. The quantum vacuum is not nothing. (more…)

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