Preposterous Universe

Monday, May 31, 2004
Energy and intelligence

I should tie up some loose ends (read: "potentially misleading intemperate statements") in my post below about the anti-Big-Bang petition.

First, an actual physics point: Does Einstein's general relativity really say that energy is not conserved? You will be unsurprised to hear that the answer depends on what you mean by "energy." and what you mean by "conserved." Before general relativity came along, when spacetime was thought of as a fixed, static background on which all the rest of physics played itself out, the answer was unambiguous; at any moment in time, there was a number we could compute (for any closed system) called the "energy," and that number would be the same as at any other moment in time. (One way to derive this statement is as a consequence of the word "static"; time-translation invariance implies energy conservation.) Most often, we were lucky enough that the energy came in the form of an energy density defined at each point in space, which we could add up over the entire system to get the total energy.

GR changes the rules of the game. Spacetime is a dynamical object whose geometry responds to the presence of matter fields. We can now ask two separate questions: Is energy conserved for the matter fields in a given spacetime background? and Is the total energy of the universe, including matter and gravity (as manifested in spacetime curvature) conserved?

The answer to each question still depends on what you mean. Consider matter evolving in some background spacetime (so we ignore the possible energy of the gravitational field, whatever that may be). There is now no number we can calculate for a closed system that corresponds to "energy" and is conserved. This shouldn't be a surprise, since we have violated time-translation invariance; the background geometry could be expanding or contracting, for example. In cosmology, there is no "total energy of the universe" which is supposed to be conserved -- that's why Lerner's statement was so silly. On the other hand, there is a rule for "covariant" conservation of the local energy-momentum tensor (for experts, it's DaTab=0). This rule can be thought of as telling us exactly how the energy changes in response to changes in the background geometry, and it is what replaces the flat-spacetime notion of energy conservation. So the number of rules is the same in flat or curved spacetime; it's not as if anything goes. But we can't, once again, define a conserved total energy in any reasonable way.

So we should just include the energy of the gravitational field, obviously, right? The problem is there's no good way to do that. If we blindly follow the rules for calculating the energy and apply them to general relativity, we find that they don't give us an energy density at each point in space, but rather a boundary contribution defined solely at infinity. In other words, there is no local definition of energy density in general relativity. In the weak-field limit we can come up with good approximate notions of a gravitational energy density, and these are useful e.g. when calculating the energy lost through gravitational radiation in orbiting bodies. So perhaps we could give up on locality and stick with just a global definition. Given appropriate boundary conditions (typically that spacetime is flat at infinity) this makes sense, and we can define different notions of energy (the ADM energy, the Bondi energy) appropriate to different circumstances. But in cosmology the universe is not flat at infinity, so these circumstances don't apply -- there is generally, once again, no such thing as the conserved total energy. (In a closed universe there is -- and it's always exactly zero, for all the good it does us.)

The situation is thus a little ambiguous; whether energy is conserved in GR depends on the situation you are talking about, and what you would qualify as energy conservation. Don't get me wrong: nobody who understands what's going on has any disagreement about the equations or their solutions, it's just that there are different words we can reasonably apply to them. This is actually a good example of what Thomas Kuhn talked about in The Structure of Scientific Revolutions, where he discusses how words mean different things before and after a paradigm shift. The notion of "energy" is very useful, but its status in GR is different than it is in flat-spacetime physics. The one thing we can all agree on is that background energy density that remains constant as the universe expands, and whose integral over space therefore grows, is perfectly consistent with everything we know about physics.

The other thing I wanted to revisit is my defaming remark that Big-Bang opponents aren't very smart. Peter Woit points out a counterexample: Irving Segal, a well-known mathematical physicist who developed "chronometric cosmology" as an alternative to the Big Bang. Of course there are other counterexamples, notable among whom we should mention Sir Fred Hoyle, who did extremely important work in stellar nucleosynthesis, and later became well-known as a supporter of the Steady State model. (It was Hoyle who actually coined the term "Big Bang," as a derogatory term to belittle the model we now know to be correct.) (Update: Another unfair slander! See the comments.)

I shouldn't have given such a blanket indictment of the intellectual prowess of the anti-Bang folks. For all I know, Eric Lerner is a grandmaster at chess, a gourmet cook, and a crossword-puzzle wizard. What I should have simply said is that the criticisms leveled by these folks at the Big Bang are just not very smart. I can certainly imagine intelligent and reasonable arguments given against almost any scientific position; but the anti-Banging is generally done from a position of deep philosophical conviction, which tends to result in rather weak argumentation. Segal, for example, had a theory in which the apparent velocity of a distant galaxy should be proportional to its distance squared (rather than simply the distance, as in the conventional theory). He would insist that modern statistical techniques reinforced his result; typically, these techniques would involve tossing out the data that manifestly disagreed with his theory. He was extremely bright in some ways, but about this he was blind.

The point really is that the anti-Big-Bang crowd are not visionary mavericks being unfairly undermined by a narrow-minded scientific establishment; they are just crackpots. The difference can be quite subtle and even subjective, but in this case it's pretty clear.

Ideas on culture, science, politics.
Sean Carroll

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