General relativity: Principle of minimal coupling computations

Question

I have a question about computations in general relativity and transition from a Lorentz frame to a general fame just by substituting the flat metric with a general one and ordinary derivatives with covariant ones, I'm a little confused by this. For example if I have to compute the d'alambertian of the square of a scalar field I'd do the following:

First compute it in a Lorentz frame: $$ \Box f^2 = \partial_\mu \partial^\mu f^2 = \partial_\mu(2 f \partial^\mu f) = 2 \partial_\mu f \, \partial^\mu f + 2f \partial_\mu \partial^\mu f $$

Now I go to a general frame replacing ordinary derivative with covariant derivatives obtaining

$$ 2(\nabla_\mu f \, \nabla^\mu f + f \, \nabla_\mu \, \nabla^\mu f) = 2 (\partial_\mu f \, \partial^\mu f + f \, \Box f) $$

Where the in the last equality the I used the fact that the covariant derivative of a scalar field is an ordinary derivative and $ \nabla_\mu \, \nabla^\mu = \Box$.

Is this the correct way to procede in these situations?

Can I apply this method everytime?

As far as I understand everything I've done here is right but I'm not completely sure about it, are there any other smart and right ways to do these computations?

Answer 1

There is no universal answer here; the method is really an encouragement to try out ideas in a certain direction, but there is no substitute for thinking as you go along.

For every equation involving $\partial_a$ in flat space there is at least one equation involving $\nabla_a$ in a general space which will reduce to the former in the right circumstances. The method called minimal coupling can help you find such an equation, but it will not be unique. Also, the mere replacement of $\partial_a$ by $\nabla_a$ is not always mathematically correct, because there may be some assumption that required zero connection coefficients in the result involving $\partial_a$.

Finally, even if the equation involving $\nabla_a$ is mathematically consistent, there is no guarantee that it correctly captures any physical phenomenon, because in curved spacetime there may be some aspect of the physics that goes beyond minimal coupling---a term involving curvature in a way above and beyond what is captured by $\partial_a \rightarrow \nabla_a$.

However, to our marvel and delight, it often happens that the natural world is as simple as it could reasonably be, in its fundamental equations, so the method of minimal coupling is in practice a very fruitful line of inquiry in fundamental physics.