Spinor Info

The National Ignition Facility has achieved a net power gain in its experimental fusion reactor. This is heralded as a major breakthrough.

Does this mean that in 50 years, we will have practical nuclear fusion power our world?

Oh wait. We were told exactly that some 50 years ago:

At the beginning of the 1950s, it seemed that success is not far away. But later, difficulties arose one after another […]
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Unfortunately today there are still gigantic difficulties in the path towards utilizing this fabulously rich supply of energy […]
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In fourteen countries of the world, more than two thousand engineers and scientists are laboring on working out different types of fusion devices.
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To date, more than a hundred different models have been devised […]
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Let us introduce only one group of these: the Soviet Tokamak devices, because around the world, these are the ones in which researchers have the most faith, viewing them as prototypes of future fusion power plants.
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A year and a half ago, in an experiment carried out in collaboration between Soviet and English physicists, they directly measured the temperature and density of the plasma of Tokamak-3, and it became clear that the results were even better than indicated by prior measurements. To date, no other device could produce plasma of such quality.
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When will the first fusion power plants be realized, when will the investigation of controlled nuclear fusion exit the constraints of laboratory experiments? According to Professor Igor Golovin, the world-renowned expert on thermonuclear research, it will be possible to develop Tokamak devices into electricity-producing equipment by the last decade of our century. L. Hirsch, one of the leading physicists of the American Atomic Energy Commission is a little more cautious. According to him the path from the first experiments to the worldwide spread of fusion power plants is longer, and we’re lucky if they will enter the world’s energy production market in fifty years.

These are all quotes (my translations) from a 1972 Hungarian-language educational children’s publication, “Boys’ Almanac 1973”.

As I express my (probably uninformed) skepticism concerning practical fusion power generation, I note that in the deep interior of the Sun, under gravitational confinement due to the combined mass of more than 300,000 Earths, fusion progresses at the leisurely rate of a few hundred watts per cubic meter. (The power output of a well-maintained industrial compost pile.) For practical power generation, we need something that is at least a million times that, a few hundred megawatts per cubic meter… and we don’t have 300,000 Earths for gravitational confinement.

Of course I’d be delighted if they proved me wrong.

A few days ago I had a silly thought about the metric tensor of general relativity.

This tensor is usually assumed to be symmetric, on account of the fact that even if it has an antisymmetric part, \(g_{[\mu\nu]}dx^\mu dx^\nu\) will be identically zero anyway.

But then, nothing constrains \(g_{\mu\nu}\) to be symmetric. Such a constraint should normally appear, in the Lagrangian formalism of the theory, as a Lagrange-multiplier. What if we add just such a Lagrange-multiplier to the Einstein-Hilbert Lagrangian of general relativity?

That is, let’s write the action of general relativity in the form,

$$S_{\rm G} = \int~d^4x\sqrt{-g}(R – 2\Lambda + \lambda^{[\mu\nu]}g_{\mu\nu}),$$

where we introduced the Lagrange-multiplier \(\lambda^{[\mu\nu]}\) in the form of a fully antisymmetric tensor. We know that

$$\lambda^{[\mu\nu]}g_{\mu\nu}=\lambda^{[\mu\nu]}(g_{(\mu\nu)}+g_{[\mu\nu]})=\lambda^{[\mu\nu]}g_{[\mu\nu]},$$

since the product of an antisymmetric and a symmetric tensor is identically zero. Therefore, variation with respect to \(\lambda^{[\mu\nu]}\) yields \(g_{[\mu\nu]}=0,\) which is what we want.

But what about variation with respect to \(g_{\mu\nu}?\) The Lagrange-multipliers represent new (non-dynamic) degrees of freedom. Indeed, in the corresponding Euler-Lagrange equation, we end up with new terms:

$$\frac{\partial}{\partial g_{\alpha\beta}}(\sqrt{-g}\lambda^{[\mu\nu]}g_{[\mu\nu]})=
\frac{1}{2}g^{\alpha\beta}\sqrt{-g}\lambda^{[\mu\nu]}g_{[\mu\nu]}+\sqrt{-g}\lambda^{[\mu\nu]}(\delta^\alpha_\mu\delta^\beta_\nu-\delta^\alpha_\nu\delta^\beta_\mu)=2\sqrt{-g}\lambda^{[\mu\nu]}=0.$$

But this just leads to the trivial equation, \(\lambda^{[\mu\nu]}=0,\) for the Lagrange-multipliers. In other words, we get back General Relativity, just the way we were supposed to.

So in the end, we gain nothing. My silly thought was just that, a silly exercise in pedantry that added nothing to the theory, just showed what we already knew, namely that the antisymmetric part of the metric tensor contributes nothing.

Now if we were to add a dynamical term involving the antisymmetric part, that would be different of course. Then we’d end up with either Einstein’s attempt at a unified field theory (with the antisymmetric part corresponding to electromagnetism) or Moffat’s nonsymmetric gravitational theory. But that’s a whole different game.