So suppose we have a black hole with hair, that is a background solution in our field theory that describes a black hole spacetime and in which a field coupled to gravity has a non zero configuration.

For the purpose of the discussion suppose that the metric is Schwartzschild, and without loss of generality let us choose the Schwartzschild coordinates. So the metric and its inverse have divergent components at the horizon:$$g^{00}=g_{rr}=\frac{1}{f(r)}=\frac{1}{1-\frac{a}{r}}$$ Now when we consider the quantum fluctuations around this background, don't we have problems defining the perturbative theory near the horizon?

My impression is that if we write the operators of the theory for fluctuations some of them will apriori diverge.

Indeed we write the operators for the fluctuations by substituting some (from 1 to all) background fields with a fluctuation of that field in the monomials of the original Lagrangian. Even if these monomials will be bounded (this will usually be the case if there are no naked singularities), when we substitute with a fluctuation one of the fields that in these monomials approach zero at the horizon (and that tame the divergence of another field in the same monomial, on the background solution), we get a fluctuation operator which has fluctuation times divergent term. Therefore this operator at the horizon has an infinite strong coupling with respect to some of the others, in particular with respect to the kinetic term for fluctuations of the field that we are coupling to gravity.

This disparity will be present even among operators with the same number of fluctuations: already by substituting one background field with one fluctuation we can get operators which diverge or which go to zero at the horizon (depending if we substitute the zero or the divergent background field).

Another problem of this setting is that now there are operators with many fluctuations that can diverge and therefore be more important with respect to the ones of order one with less fluctuations (for instance the kinetic terms).

So in conclusion it would seem that the presence of the horizon causes problems in defining quantum fluctuations around it, unless one imposes conditions on these fluctuations: going like the respective background for instance. Is this a meaningful requirement?

Of course I am doing everything in the sad Schwartzschild coordinate system. But the Lagrangian of the fluctuations is a scalar and the various operators should be too if we define the fluctuations of the fields to be tensors (always possible I would say). So I would expect that these divergences are indeed independent from the coordinates.

Maybe I presented the question in a somewhat involved way (I tried my best not to do so); in particular I am not sure I need some other field in addition to gravity to get this puzzle. I hope somebody already debunked this! Thanks for any input.


I think I found the answer by myself: there are actually some conditions on the fluctuations.

The fact is that independently on the coordinate system we choose, we want the fluctuations to be bounded or at least to be way smaller than the respective background fields. If we do so, then when we go to the schwartzschild coordinates the fluctuations will be zero. If they were non zero, then by changing to smooth (Kruskal) coordinates, they would become infinite.

So the strong coupling in schwartzschild coordinates is actually due to having infinite fluctuations in other coordinates.

Therefore in the schwartzschild coordinates the fluctuations will have to go like their respective fields at the horizon.


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