Spinor Info

I don’t always agree with Sabine Hossenfelder but every once in a while, she hits the nail spot on.

Case in point: Her article, published in The Guardian on September 26, about the state of particle physics.

Imagine going to a zoology conference, she says, where a researcher discusses a hypothesis (complete with a computer-generated 3D model) of a 12-legged purple spider living in the Arctic. Probably doesn’t exist but still, how about proposing a mission to the Arctic to search for one? After all, a null result also contains valuable data. Or how about a flying earthworm that lives in caves? Martian octopuses, anyone?

Zoology conferences do not usually discuss such imaginary monsters but, Sabine argues (and she is spot on) this is pretty much what particle physics conferences are like: “invent new particles for which there is no evidence, publish papers about them, write more papers about these particles’ properties, and demand the hypothesis be experimentally tested”. Worse yet, real money is being spent (wasted might be a better word) on carrying out such experiments.

She points out that while it is true that good science is falsifiable, the opposite isn’t always the case: Just because something is falsifiable does not make it good science.

And not just particle physics, I hasten to add. How about cosmology and gravitation? Discussions about what may or may not have happened during the Planck epoch? Exploring exotic spacetime topologies, often in dimensions other than four? And let me not even mention quantum computing or fusion energy…

Perhaps I am a born skeptic lacking imagination, but to me, these are all 12-legged purple Arctic spiders. The science we actually know and have the ability to confirm are general relativity in a spacetime that is by and large the perturbed Minkowski metric; and the Standard Model of particle physics, extended with a neutrino mass mixing matrix. These are the things that work. Not perfectly, mind you. General relativity needs “dark matter” (name aside, we don’t know what it is except that it has a dust equation of state) and “dark energy” (again, it has a name but beyond that, we don’t know what it is beyond its equation of state) to account for galaxy dynamics and cosmic evolution. The Hubble tension, the discrepancy between values of the Hubble parameter measured using different methods, is real. Observations by the James Webb space telescope suggest that we do not understand the “dark ages”, the first few hundred million years after the surface of last scattering (i.e., the epoch when the cosmic microwave background radiation was produced), well. Massive neutrinos invite the question about the apparent absence right-handed neutrinos.

And yes, we are very much in the blind concerning these issues. Nature has not yet provided hints and we are not smart enough to figure out the answers entirely on our own. But how is that an excuse for inventing 12-legged spiders?

I think it isn’t.

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.