Lorentz scalar Lagrangian in curved spacetime

Question

This question might be very simple but I guess I'm missing something. We know that Lagrangian has to be a Lorentz scalar. I can see why that should be the case when dealing with inertial frames of reference as we want to ensure Lorentz invariance. However, I don't understand why the Lagrangian has to be a Lorentz scalar even in curved spacetime. For example for a scalar field, $\phi$, in curved spacetime, the Lagrangian contains the term $g^{\mu\nu}\phi_{\mu}\phi_{\nu}$ to keep it a Lorentz scalar. Why does the Lagrangian have to be a Lorentz scalar in curved spacetime when the very notion of Lorentz invariance is not well defined in that type of spacetime?

Answer 1

First, the main answer to your question is that in general relativity, we want to be able to express the action in a manner that is invariant under changes of coordinates. There are fancy ways to express this (like "diffeomorphism symmetry is the gauge symmetry of GR"), but at a basic level we cannot make assumptions about the form of the coordinates when dealing with a general manifold, so we have to allow for any choice of coordinates on the manifold. For example, like you are getting at, we cannot assume there is set of coordinates that look like inertial coordinates from special relativity and are defined everywhere on the manifold. However, the action principle, plus locality, suggests that the dynamics of the theory should be derivable from an action that is an integral of a Lagrangian scalar field that is a function of the fields and metric at each point on the manifold. So we generalize the notion of a Lorentz scalar, to a scalar field on a manifold, and require that the Lagrangian be a scalar in the latter sense. One possible source of confusion here is the principle of minimal coupling, which suggests that to go from special relativity to general relativity you simply take a Lorentz scalar like $\eta^{\mu\nu} \partial_\mu \phi \partial_\nu \phi$, and promote $\eta_{\mu\nu} \rightarrow g_{\mu\nu}$ and $\partial_\mu \rightarrow \nabla_\mu$, so you would end up with a term like $g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi$. This procedure for guessing generalizations of interactions in special relativity might make you think the symmetries are the same, but in the former case the symmetry is Lorentz invariance, and in the latter case it is general coordinate invariance, and this guessing procedure is not unique since you can add terms proportional to the curvature which will vanish in a flat spacetime.

Second, as a technical point, a manifold, to get a coordinate-independent integral, you should integrate a density as the integrand. The true scalar density that is integrated in a curved spacetime is $\sqrt{-g}\mathcal{L}$, where $g$ is the determinant of the metric. In other words, the action $S$ is calculated as $$ S = \int d^4 x \sqrt{-g} \mathcal{L} $$ so that $S$ does not depend on the choice of coordinates on the manifold. The factor of $\sqrt{-g}$ guarantees that even though $\mathcal{L}$ is a scalar field on the manifold, the action $S$ is independent of the choice of coordinates.

Answer 2

Andrew's answer is (as usual) correct and complete, so this is mostly a complement.

In curved spacetime, the Lagrangian should actually be a scalar under general transformations of coordinates, not only Lorentz transformations. For example, under integration by parts we can convert $$\partial_\mu \phi \partial^\mu \phi \to - \phi \partial_\mu \partial^\mu \phi.$$ In curved spacetime, this calculation is incorrect (and the last term is not what occurs in the Lagrangian). Instead, the correct one is $$\partial_\mu \phi \partial^\mu \phi = \nabla_\mu \phi \nabla^\mu \phi \to - \phi \nabla_\mu \nabla^\mu \phi.$$ This is invariant under general transformations of coordinates.

This comment actually also holds in Minkowski spacetime. The Lagrangian is always a scalar under general transformations of coordinates, not only Lorentz transformations, but in flat spacetime this is only relevant if you plan on using curvilinear coordinates.