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.