We note that if $X=0$ and $k=l=0$ (namely, $\mathsf{\Phi}$ is scalar), then
Corollary 5.12 holds with the optimal $\kappa$ (like in [16, 17, 28]) for a
certain range of $\sigma$. In particular, here we can take $\mathcal{C}_{0}$
in Corollary 5.12, and the smallest allowed value of $\kappa$ is given by

$\displaystyle\kappa=\begin{cases}\frac{n-2}{2}+\sqrt{\frac{n^{2}}{4}-\sigma}\text{,}&\quad\sigma\leq\frac{n^{2}-1}{4}\text{,}\\\
\frac{n-1}{2}\text{,}&\quad\sigma\geq\frac{n^{2}-1}{4}\text{.}\end{cases}$

###  5.3. Proof of Theorem 5.11

Throughout this subsection, we assume the hypotheses of Theorem 5.11. The
proof is, to a large extent, analogous to that of the Carleman estimate from
[28, Theorem 5.11]. As a result, we will omit details from the parts that are
close to [28] and focus our attention primarily on the arguments that differ.
262626The main differences with [28] are that we use a different function
$\eta$ in our definition of $f$, and that we use a different Riemannian metric
$\mathfrak{h}$ to measure the sizes of our vertical tensor fields.

By [28, Corollary 3.5] and a partition of unity argument, there is a $\zeta\in
C^{2}(\bar{\mathcal{D}})$ such that

$(\mathfrak{D}^{2}\eta-\eta\,\bar{\mathfrak{g}}-\zeta\,\mathfrak{g})(\mathfrak{X},\mathfrak{X})>c\eta\,\mathfrak{h}(\mathfrak{X},\mathfrak{X})$
(5.18)

for any tangent vector $\mathfrak{X}\in T\mathcal{D}$. We then define
$\pi_{\zeta}$ and $w_{\zeta}$ as in Definition 2.19, using the above $\zeta$.
In addition, we let $N$ be as in Definition 2.14, and we set the following:

  * •

We define the vector field $S$ and the real-valued function $v_{\zeta}$ by
272727Here, $\operatorname{div}_{g}$ denotes the divergence with respect to
$g$.

$S:=f^{n-3}\nabla^{\sharp}f\text{,}\qquad
v_{\zeta}:=f^{n-3}w_{\zeta}+\frac{1}{2}\operatorname{div}_{g}S\text{.}$ (5.19)

  * •

We define the auxiliary quantities

$F:=\kappa\log f+\lambda
p^{-1}f^{p}\text{,}\qquad\mathsf{\Psi}:=e^{-F}\mathsf{\Phi}\text{.}$ (5.20)

  * •

Throughout the proof, we will view $F:=F(f)$ as a function of $f$. In
particular, we write $F^{\prime},F^{\prime\prime},\dots$ to denote derivatives
of $F$ with respect to $f$.

  * •

We also define the following differential operators:

$\mathcal{L}:=e^{-F}(\breve{\Box}_{g}+\sigma)e^{F}\text{,}\qquad\mathcal{L}^{\dagger}:=e^{-F}(\breve{\Box}_{g}+\sigma+\rho^{2}\breve{\nabla}_{X})e^{F}\text{,}\qquad\breve{S}_{\zeta}:=\breve{\nabla}_{S}+v_{\zeta}\text{.}$
(5.21)

Finally, we assume that for all $\mathcal{O}$-notations, the constants can
depend on $\mathsf{g}$, $\mathcal{D}$, $X$, $k$, $l$.

####  5.3.1. Preliminary Computations

Here, we collect a number of computations and asymptotic properties that we
will require later in the proof.

######  Lemma 5.14 (Asymptotics for $S$ and $v_{\zeta}$).

The following hold for $S$:

$S=[f^{n-2}+\mathcal{O}(f^{n})]N\text{,}\qquad\operatorname{div}_{g}S=-2f^{n-2}+\mathcal{O}(f^{n})\text{.}$
(5.22)

Moreover, the following hold for $v_{\zeta}$:

$\displaystyle v_{\zeta}=\mathcal{O}(f^{n})\text{,}$
$\displaystyle\qquad\nabla_{\rho}v_{\zeta}=\mathcal{O}(\rho^{-1}f^{n})\text{,}$
(5.23)
$\displaystyle\mathsf{D}v_{\zeta}=\mathcal{O}(\rho^{-1}f^{n+1})\text{,}$
$\displaystyle\qquad\Box_{g}v_{\zeta}=\mathcal{O}(f^{n})\text{.}$

###### Proof.

The first two parts of (5.22) follow immediately from Lemma 2.11, (2.19), and
(5.19). For the last part of (5.22), we first apply Lemmas 2.11 and 2.13 to
obtain

$\displaystyle\operatorname{div}_{g}S$
$\displaystyle=(n-3)f^{n-4}\,g(\nabla^{\sharp}f,\nabla^{\sharp})+f^{n-3}\Box_{g}f$
(5.24)
$\displaystyle=-2f^{n-2}+(n-1)f^{n}\,g(\mathsf{D}^{\sharp}\eta,\mathsf{D}^{\sharp}\eta)-\rho
f^{n-1}\operatorname{tr}_{\mathsf{g}}\mathsf{D}^{2}\eta+\frac{1}{2}\rho
f^{n-2}\operatorname{tr}_{\mathsf{g}}\mathcal{L}_{\rho}\mathsf{g}\text{.}$

The asymptotics for $\operatorname{div}_{g}S$ then follow from the above,
along with the asymptotics

$\mathsf{D}\eta=\mathcal{O}(1)\text{,}\qquad\mathsf{D}^{2}\eta=\mathcal{O}(1)\text{,}\qquad\mathcal{L}_{\rho}\mathsf{g}=\mathcal{O}(\rho)\text{,}$

and the observation that $\rho\lesssim f$.

Next, from (2.27), (5.19), and (5.22), we compute

$\displaystyle v_{\zeta}$ $\displaystyle=\rho
f^{n-1}\zeta+\frac{1}{2}(n-1)f^{n}\,\mathsf{g}(\mathsf{D}^{\sharp}\eta,\mathsf{D}^{\sharp}\eta)-\frac{1}{2}\rho
f^{n-1}\operatorname{tr}_{\mathsf{g}}\mathsf{D}^{2}\eta+\frac{1}{4}\rho
f^{n-2}\operatorname{tr}_{\mathsf{g}}\mathcal{L}_{\rho}\mathsf{g}\text{.}$
(5.25)

The first part of (5.23) follows immediately from the above. The remaining
parts of (5.23) then follow from differentiating (5.25) as needed, along with
some tedious computations; see [28, Lemma 5.23] for further details from an
analogous computation. ∎

######  Lemma 5.15 (Asymptotics for $\smash{\breve{R}}$).

Let $\mathsf{A}$ be a vertical tensor field of rank $(k,l)$. In addition, let
$\mathfrak{E}$ be a vector field on $\mathcal{D}$, and let
$E=\mathcal{P}\mathfrak{E}$. 282828See Definition 2.15 for the definition of
$\mathcal{P}$. Then,

$|\breve{R}_{NE}[\mathsf{A}]|_{\mathfrak{h}}\lesssim(k+l)\rho^{2}f\,|\mathfrak{E}|_{\mathfrak{h}}|\mathsf{A}|_{\mathfrak{h}}\text{.}$
(5.26)

###### Proof.

Fix a compact coordinate system $(U,\varphi)$ on $\mathcal{I}$. The main
observations are the bounds

$|\breve{R}_{ab}[\mathsf{A}]|_{0,\varphi}\lesssim_{\varphi}(k+l)\,|\mathsf{A}|_{0,\varphi}\text{,}\qquad|\breve{R}_{\rho
a}[\mathsf{A}]|_{0,\varphi}\lesssim_{\varphi}(k+l)\rho\,|\mathsf{A}|_{0,\varphi}\text{,}$
(5.27)

stated in terms of $\varphi$-coordinates; for their derivations, see the proof
of [28, Lemma 5.24], which is directly applicable to our current setting.
292929The essential formulas for expanding $\breve{R}[\mathsf{A}]$ can be
found in [28, Proposition 2.32]. Now, by (2.19) and (2.20),

$\breve{R}_{NE}[\mathsf{A}]=\breve{R}[\mathsf{A}](\mathcal{O}(\rho)\,\partial_{\rho}+\mathcal{O}(\rho
f)\,\mathsf{D}^{\sharp}\eta,\,\rho
f(\mathfrak{E}\eta)\,\partial_{\rho}+\rho\mathfrak{E})\text{.}$

The bound (5.26) now follows from (5.27) and the above. ∎

######  Lemma 5.16 (Asymptotics for $\mathfrak{h}$).

Let $\mathfrak{E}$ be a vector field on $\mathcal{D}$, and let
$E=\mathcal{P}\mathfrak{E}$. Then,

$\displaystyle\breve{\nabla}_{\rho}\mathfrak{h}=\mathcal{O}(\rho)\text{,}$
$\displaystyle\qquad\breve{\nabla}_{N}\mathfrak{h}=\mathcal{O}(\rho
f)\text{,}\qquad\breve{\nabla}_{E}\mathfrak{h}=\mathcal{O}(\rho)(\mathfrak{E})\text{,}\qquad\breve{\Box}_{g}\mathfrak{h}=\mathcal{O}(\rho^{2})\text{.}$
(5.28)

###### Proof.

First, we clearly have that

$\mathfrak{h}=\mathcal{O}(1)\text{,}\qquad\breve{\mathsf{D}}\mathfrak{h}=\mathcal{O}(1)\text{,}\qquad\breve{\mathsf{D}}^{2}\mathfrak{h}=\mathcal{O}(1)\text{.}$
(5.29)

For convenience, in the upcoming computations, we will state quantities with
respect to a compact coordinate system $(U,\varphi)$ on $\mathcal{I}$. Since
$\mathcal{L}_{\rho}\mathfrak{h}=0$ by definition, we derive that

$\breve{\nabla}_{\rho}\mathfrak{h}_{ab}=-\frac{1}{2}\mathsf{g}^{dc}\mathcal{L}_{\rho}\mathsf{g}_{ac}\mathfrak{h}_{db}-\frac{1}{2}\mathsf{g}^{dc}\mathcal{L}_{\rho}\mathsf{g}_{bc}\mathfrak{h}_{ad}\text{.}$
(5.30)

The first part of (5.28) follows from Lemma 2.8 and the above. Next, by (2.19)
and (2.20),

$\breve{\nabla}_{N}\mathfrak{h}=\mathcal{O}(\rho)\,\breve{\nabla}_{\rho}\mathfrak{h}+\mathcal{O}(\rho
f)\,\mathsf{D}_{\mathsf{D}^{\sharp}\eta}\mathfrak{h}\text{,}\qquad\breve{\nabla}_{E}\mathfrak{h}=\rho
f(\mathfrak{E}\eta)\,\breve{\nabla}_{\rho}\mathfrak{h}+\rho\mathsf{D}_{\mathfrak{E}}\mathfrak{h}\text{.}$

The second and third parts of (5.28) now follow from the first part and from
the above.

Now, we unwind the definitions of $\smash{\breve{\nabla}}^{2}$ and
$\mathsf{D}^{2}$ (see Definitions 5.6 and 5.8) to obtain

$\displaystyle\breve{\nabla}^{2}_{ab}\mathfrak{h}$
$\displaystyle=\mathsf{D}_{ab}\mathfrak{h}-\breve{\mathsf{D}}_{\nabla_{a}\partial_{b}-\mathsf{D}_{a}\partial_{b}}\mathfrak{h}$
(5.31)
$\displaystyle=\mathsf{D}_{ab}\mathfrak{h}+\left(\frac{1}{2}\mathcal{L}_{\rho}\mathsf{g}_{ab}-\rho^{-1}\mathsf{g}_{ab}\right)\breve{\mathsf{D}}_{\rho}\mathfrak{h}$
$\displaystyle=\mathcal{O}(1)_{ab}\text{,}$

where we also used the first part of (5.28) and (5.29). Moreover,
differentiating (5.30) yields

$\breve{\mathsf{D}}_{\rho}(\breve{\mathsf{D}}_{\rho}\mathfrak{h})_{ab}=-\frac{1}{2}\mathsf{g}^{dc}\,\breve{\mathsf{D}}_{\rho}(\mathcal{L}_{\rho}\mathsf{g}_{ac}\mathfrak{h}_{db}+\mathcal{L}_{\rho}\mathsf{g}_{bc}\mathfrak{h}_{ad})=\mathcal{O}(1)_{ab}\text{,}$
(5.32)

where we again recalled Lemma 2.8 and the first part of (5.28). Unwinding the
definitions of $\smash{\breve{\nabla}}^{2}$ and
$(\smash{\breve{\mathsf{D}}}_{\rho})^{2}$, and then applying the first part of
(5.28) and (5.32), we obtain

$\displaystyle\breve{\nabla}^{2}_{\rho\rho}\mathfrak{h}$
$\displaystyle=\breve{\mathsf{D}}_{\rho}(\breve{\mathsf{D}}_{\rho}\mathfrak{h})-\breve{\nabla}_{\nabla_{\rho}\partial_{\rho}}\mathfrak{h}$
(5.33)
$\displaystyle=\breve{\mathsf{D}}_{\rho}(\breve{\mathsf{D}}_{\rho}\mathfrak{h})+\rho^{-1}\breve{\nabla}_{\rho}\mathfrak{h}$
$\displaystyle=\mathcal{O}(1)$

Finally, combining Lemma 2.8, (5.31), and (5.33) yields

$\breve{\Box}_{g}\mathfrak{h}=g^{\rho\rho}\breve{\nabla}_{\rho\rho}\mathfrak{h}+g^{ab}\breve{\nabla}_{ab}\mathfrak{h}=\mathcal{O}(\rho^{2})\text{,}$

which is precisely the last part of (5.28). ∎

######  Remark 5.17.

Lemma 5.16 represents one part of the proof of Theorem 5.11 that differs from
[28]. This is due to our use of a different Riemannian metric to measure
vertical tensor fields. However, our metric $\mathfrak{h}$ satisfies the same
asymptotic estimates as its counterpart $\mathsf{h}$ in [28], so this does not
significantly affect other portions of the proof compared to [28].

######  Lemma 5.18 (Expansion of $\mathcal{L}$).

The following formula holds for $\mathcal{L}$:

$\displaystyle\mathcal{L}$
$\displaystyle=\breve{\Box}_{g}+2F^{\prime}f^{-(n-3)}\breve{\nabla}_{S}+\mathcal{A}\text{,}$
(5.34) $\displaystyle\mathcal{A}$
$\displaystyle=[(F^{\prime})^{2}+F^{\prime\prime}]\,g(\nabla^{\sharp}f,\nabla^{\sharp}f)+F^{\prime}\Box_{g}f+\sigma\text{.}$

In addition, we have the following asymptotic formulas:

$\displaystyle\mathcal{A}$
$\displaystyle=(\kappa^{2}-n\kappa+\sigma)+(2\kappa-n+p)\lambda
f^{p}+\lambda^{2}f^{2p}+\lambda^{2}\mathcal{O}(f^{2})\text{,}$ (5.35)
$\displaystyle-\frac{1}{2}\operatorname{div}_{g}(\mathcal{A}S)$
$\displaystyle=(\kappa^{2}-n\kappa+\sigma)f^{n-2}+\left(1-\frac{p}{2}\right)(2\kappa-n+p)\lambda
f^{n-2+p}$
$\displaystyle\qquad+(1-p)\lambda^{2}f^{n-2+2p}+\lambda^{2}\mathcal{O}(f^{n})\text{.}$

###### Proof.

The formulas (5.34) follow from direct computations and the definitions
(5.19)–(5.21). The asymptotic relations (5.35) are also direct computations;
these formulas are the same as [28, Lemma 5.26], and their derivations are
identical, hence we omit the details here. ∎

####  5.3.2. The GNCC and Pseudoconvexity

Next, we establish the crucial (and new compared to [28]) estimate in the
proof of Theorem 5.11—that the GNCC implies the level sets of $f$ are
pseudoconvex. This is captured in the following lemma, which shows that the
modified deformation tensor $\pi_{\zeta}$ is positive-definite in the
directions tangent to the level sets of $f$:

######  Lemma 5.19 (Lower bound for $\pi$).

Given a spacetime tangent vector $Z\in T\mathcal{M}$, we write 303030Note that
$\mathcal{P}$ is invertible, so we can define
$\mathfrak{E}:=\mathcal{P}^{-1}E$ in the above.

$Z:=Z^{N}N+E\text{,}\qquad E:=\mathcal{P}\mathfrak{E}\text{,}$ (5.36)

where $E$ is normal to $N$ (and hence tangent to level sets of
$f$).313131Observe that $N$ is spacelike when $f$ is sufficiently small. Then,
there exists a constant $K>0$ such that the following inequality holds on
$\mathcal{D}(f_{\star})$:

$\pi(Z,Z)\geq-[(n-1)f^{n-2}+\mathcal{O}(f^{n})]\,(Z^{N})^{2}+K\rho^{2}f^{n-2}\,\mathfrak{h}(\mathfrak{E},\mathfrak{E})\text{.}$
(5.37)

###### Proof.

First, we recall Lemma 2.20 and (5.36), which imply

$\displaystyle\pi(Z,Z)$
$\displaystyle=(Z^{N})^{2}\,\pi(N,N)+2Z^{N}\,\pi(N,E)+\pi(E,E)$ (5.38)
$\displaystyle=-[(n-1)f^{n-2}+\mathcal{O}(f^{n})](Z^{N})^{2}+\mathcal{O}(\rho
f^{n})(\mathfrak{E})\,Z^{N}$ $\displaystyle\qquad+\rho
f^{n-1}(\mathfrak{D}^{2}\eta-\eta\bar{\mathfrak{g}}-\zeta\mathfrak{g})(\mathfrak{E},\mathfrak{E})+\mathcal{O}(\rho^{2}f^{n-1})(\mathfrak{E},\mathfrak{E})\text{.}$

We apply (5.18) to treat the third term on the right-hand side of (5.38):

$\displaystyle\rho
f^{n-1}(\mathfrak{D}^{2}\eta-\eta\bar{\mathfrak{g}}-\zeta\mathfrak{g})(\mathfrak{E},\mathfrak{E})$
$\displaystyle\geq c\rho f^{n-1}\eta\,\mathfrak{h}(\mathfrak{E},\mathfrak{E})$
(5.39)
$\displaystyle=c\rho^{2}f^{n-2}\,\mathfrak{h}(\mathfrak{E},\mathfrak{E})\text{.}$

Moreover, for the second term, the Cauchy inequality implies

$\mathcal{O}(\rho
f^{n})(\mathfrak{E})\,Z^{N}\gtrsim\mathcal{O}(f^{n})\,(Z^{N})^{2}+\mathcal{O}(\rho^{2}f^{n})(\mathfrak{E},\mathfrak{E})\text{.}$

Combining (5.38) and (5.39) with the above, we obtain

$\pi(Z,Z)\geq-[(n-1)f^{n-2}+\mathcal{O}(f^{n})](Z^{N})^{2}+c\rho^{2}f^{n-2}\,\mathfrak{h}(\mathfrak{E},\mathfrak{E})+\mathcal{O}(\rho^{2}f^{n-1})(\mathfrak{E},\mathfrak{E})\text{,}$

and the desired (5.37) follows from the above by taking $f_{\ast}$ small
enough. ∎

####  5.3.3. Pointwise Estimate for $\mathsf{\Psi}$

We now establish a pointwise estimate for the conjugated operator
$\mathcal{L}^{\dagger}$. This will serve as a precursor for our main Carleman
estimate (5.14). The first step is to derive an estimate comparing the
operators $\mathcal{L}$ and $\mathcal{L}^{\dagger}$:

######  Lemma 5.20 (Relation between $\mathcal{L}$ and
$\mathcal{L}^{\dagger}$).

Let $\Psi$ be as in (5.20), and define

$J(\mathsf{\Psi}):=\frac{1}{2}\langle\mathcal{L}(\mathfrak{h}^{\star}\mathsf{\Psi}),\breve{S}_{\zeta}(\mathsf{\Psi})\rangle+\frac{1}{2}\langle\mathcal{L}(\mathsf{\Psi}),\breve{S}_{\zeta}(\mathfrak{h}^{\star}\mathsf{\Psi})\rangle\text{.}$
(5.40)

Then, there exists $\mathcal{C}_{0}>0$ (depending only on $\mathsf{g}$,
$\mathcal{D}$, $X$, $k$, $l$) such that on $\mathcal{D}(f_{\star})$,

$\displaystyle|J(\mathsf{\Psi})|$
$\displaystyle\leq\lambda^{-1}f^{n-2-p}|\mathcal{L}^{\dagger}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+f^{n-2}|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\left(\mathcal{C}_{0}+\frac{1}{2}\lambda
f^{p}\right)$ (5.41)
$\displaystyle\qquad+\frac{1}{4}K\rho^{2}f^{n-2}\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\lambda\,\mathcal{O}(f^{n-1})(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$

where the constant $K>0$ is as in the statement of Lemma 5.19, and where
$E_{A}:=\mathcal{P}\mathfrak{E}_{A}$ ($1\leq A\leq n$) is a local orthonormal
frame on the level sets of $f$ satisfying

$|\mathfrak{E}_{A}|_{\mathfrak{h}}\simeq 1\text{,}\qquad 1\leq A\leq
n\text{.}$

Furthermore, if $X=0$ and $k=l=0$, then (5.41) holds with $\mathcal{C}_{0}=0$.

###### Proof.

First, we obtain from (5.21) the relation

$\mathcal{L}^{\dagger}=\mathcal{L}+\breve{\nabla}_{Y}+F^{\prime}(Yf)\text{,}\qquad
Y:=\rho^{2}X\text{.}$ (5.42)

Expanding (5.40) using the product rule and the above, we obtain 323232The
notation $\smash{\breve{\mathfrak{h}}}$ in $J_{4}$ was defined in (5.1). Note
that we can think of $\smash{\breve{\mathfrak{h}}}$ as a rank $(2l,2k)$
vertical tensor field, hence we can make sense of the mixed derivatives of
$\breve{\mathfrak{h}}$ in (5.43).

$\displaystyle J(\mathsf{\Psi})$
$\displaystyle=\breve{\mathfrak{h}}(\mathcal{L}^{\dagger}\mathsf{\Psi},\breve{S}_{\zeta}\mathsf{\Psi})-F^{\prime}(Yf)\,\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{S}_{\zeta}\mathsf{\Psi})-\breve{\mathfrak{h}}(\breve{\nabla}_{Y}\mathsf{\Psi},\breve{S}_{\zeta}\mathsf{\Psi})$
(5.43)
$\displaystyle\qquad+g^{\mu\nu}\breve{\nabla}_{\mu}\breve{\mathfrak{h}}(\breve{\nabla}_{\nu}\mathsf{\Psi},\breve{S}_{\zeta}\mathsf{\Psi})+F^{\prime}f^{-(n-3)}\,\breve{\nabla}_{S}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{S}_{\zeta}\mathsf{\Psi})+\frac{1}{2}\breve{\Box}_{g}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{S}_{\zeta}\mathsf{\Psi})$
$\displaystyle\qquad+\frac{1}{2}\breve{\nabla}_{S}\breve{\mathfrak{h}}(\mathcal{L}^{\dagger}\mathsf{\Psi},\mathsf{\Psi})-\frac{1}{2}\breve{\nabla}_{S}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{\nabla}_{Y}\mathsf{\Psi})-\frac{1}{2}F^{\prime}(Yf)\,\breve{\nabla}_{S}\breve{\mathfrak{h}}(\mathsf{\Psi},\mathsf{\Psi})$
$\displaystyle:=J_{1}+\dots+J_{9}\text{,}$

where the indices $\mu$, $\nu$ in $J_{4}$ are with respect to an arbitrary
coordinate system on $\mathcal{M}$.

The next step is to estimate each of the $J_{i}$’s in (5.43), using Lemmas
5.14 and 5.16. While this is a lengthy process, the estimates obtained are
analogous to those found in the proof of [28, Lemma 5.27]. 333333Strictly
speaking, the estimates here and in [28] are not quite identical, as different
Riemannian metrics are used. However, the estimates for $\mathfrak{h}$ in
Lemma 5.16 are identical to those in [28, Lemma 5.25], so the same proof holds
in both settings. Moreover, note our frames $\\{E_{A}\\}$ correspond to the
frames $\\{V,E_{A}\\}$ in [28]. Hence, here we omit the details and list the
resulting bounds:

$\displaystyle|J_{1}|$
$\displaystyle\leq\lambda^{-1}\left[\frac{1}{2}f^{n-2-p}+\mathcal{O}(f^{n})\right]|\mathcal{L}^{\dagger}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\frac{1}{2}\lambda
f^{n-2+p}|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+\lambda\,\mathcal{O}(f^{n})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{2}|$ $\displaystyle\leq\lambda\,\mathcal{O}(\rho
f^{n-2})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{3}|$
$\displaystyle\leq(|\mathfrak{X}^{\rho}|^{2}+|\mathfrak{X}|_{\mathfrak{h}}^{2})\,\mathcal{O}(f^{n-2})|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\frac{1}{16}K\rho^{2}f^{n-2}\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+\mathcal{O}(f^{n-1})(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{4}|$
$\displaystyle\leq(k+l)^{2}\mathcal{O}(f^{n-2})\,|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\frac{1}{16}K\rho^{2}f^{n-2}\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+\mathcal{O}(f^{n})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{5}|$ $\displaystyle\leq\lambda\,\mathcal{O}(\rho
f^{n-1})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{6}|$
$\displaystyle\leq\mathcal{O}(\rho^{2}f^{n-2})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{7}|$ $\displaystyle\leq\mathcal{O}(\rho
f^{n-1})\,(|\mathcal{L}^{\dagger}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|J_{8}|$
$\displaystyle\leq\mathcal{O}(\rho^{3}f^{n-1})\left(\|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)+\mathcal{O}(\rho
f^{n-1})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$ $\displaystyle|J_{9}|$
$\displaystyle\leq\lambda\mathcal{O}(\rho^{2}f^{n-1})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

Combining (5.43) and the above, while taking $f_{\star}$ sufficiently small,
yields (5.41), with

$\mathcal{C}_{0}\lesssim_{\mathsf{g},\mathcal{D},X,k,l}(k+l)^{2}+|\mathfrak{X}^{\rho}|^{2}+|\mathfrak{X}|_{\mathfrak{h}}^{2}\text{.}$

In particular, $\mathcal{C}_{0}=0$ whenever $X=0$ and $k=l=0$. ∎

From now on, we set $\mathcal{C}_{0}$ as in Lemma 5.20, so that (5.12) holds
with this choice of $\mathcal{C}_{0}$.

######  Lemma 5.21 (Pointwise estimate for $\mathcal{L}^{\dagger}$).

There exists a constant $\mathcal{C}>0$ (depending on $\mathsf{g}$,
$\mathcal{D}$, $X$, $k$, $l$) such that the following bound holds on
$\mathcal{D}(f_{\star})$:

$\displaystyle\lambda^{-1}f^{n-2-p}|\mathcal{L}^{\dagger}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\geq\mathcal{C}\rho^{2}f^{n-2}\left(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)$
(5.44)
$\displaystyle\qquad+\frac{1}{4}\lambda^{2}f^{n-2+2p}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\operatorname{div}_{g}P\text{,}$

where $E_{A}$, $1\leq A\leq n$, denotes the same local orthonormal frames as
in Lemma 5.20, and where $P$ is the spacetime 1-form given, with respect to
any coordinate system on $\mathcal{M}$, by

$\displaystyle P_{\beta}$
$\displaystyle:=\frac{1}{2}[\langle\breve{\nabla}_{S}(\mathfrak{h}^{\star}\mathsf{\Psi}),\breve{\nabla}_{\beta}\mathsf{\Psi}\rangle+\langle\breve{\nabla}_{S}\mathsf{\Psi},\breve{\nabla}_{\beta}(\mathfrak{h}^{\star}\mathsf{\Psi})\rangle-
g_{\alpha\beta}g^{\mu\nu}S^{\beta}\langle\breve{\nabla}_{\mu}(\mathfrak{h}^{\star}\mathsf{\Psi}),\breve{\nabla}_{\nu}\mathsf{\Psi}\rangle]$
(5.45)
$\displaystyle\qquad+\frac{1}{2}v_{\zeta}[\langle\breve{\nabla}_{\beta}(\mathfrak{h}^{\star}\mathsf{\Psi}),\mathsf{\Psi}\rangle+\langle\breve{\nabla}_{\beta}\mathsf{\Psi},\mathfrak{h}^{\star}\mathsf{\Psi}\rangle]-\frac{1}{2}\nabla_{\beta}v_{\zeta}\,|\mathsf{\Psi}|_{\breve{\mathsf{h}}(\rho)}^{2}+\frac{1}{2}g_{\alpha\beta}S^{\beta}\mathcal{A}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+(2\kappa-n+1-\mathcal{C}_{0})f^{n-3}\nabla_{\beta}f\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\frac{1}{2}(2-p)\lambda
f^{n-3+p}\nabla_{\beta}f\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

Furthermore, there exists $\mathcal{C}_{b}>0$ (depending on $\mathsf{g}$,
$\mathcal{D}$, $X$, $k$, $l$) such that

$P(\rho\partial_{\rho})\leq\mathcal{C}_{b}f^{n-2}\rho^{2}\,(|\breve{\mathsf{D}}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\breve{\mathsf{D}}\mathsf{\Psi}|_{\mathfrak{h}}^{2})+\mathcal{C}_{b}\lambda^{2}f^{n-2}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$
(5.46)

###### Proof.

Let $J(\Psi)$ be as in (5.40). We then expand $J(\Psi)$ using (5.19) and Lemma
5.18. After some extensive computations (we omit the details here, but this
mirrors the process shown within the proof of [28, Lemma 5.28]), we obtain the
following identity for $\Psi$,

$\displaystyle J(\mathsf{\Psi})$
$\displaystyle=\operatorname{div}_{g}(P^{S}+P^{Q})+2F^{\prime}f^{-(n-3)}\,|\breve{\nabla}_{S}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+g^{\alpha\mu}g^{\beta\nu}(\pi_{\zeta})_{\alpha\beta}\,\breve{\mathfrak{h}}(\breve{\nabla}_{\mu}\mathsf{\Psi},\breve{\nabla}_{\nu}\mathsf{\Psi})$
(5.47)
$\displaystyle\qquad+\left[\mathcal{A}v_{\zeta}-\frac{1}{2}\operatorname{div}_{g}(\mathcal{A}S)+\frac{1}{2}\breve{\Box}_{g}v_{\zeta}\right]\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+I_{S}+I_{\zeta}+I_{R}+I_{\pi}\text{,}$

where the (spacetime) $1$-forms $P^{S}$, $P^{Q}$ and the scalar quantities
$I_{S}$, $I_{\zeta}$, $I_{R}$, $I_{\pi}$ are given by

$\displaystyle P_{\nu}^{S}$
$\displaystyle:=\frac{1}{2}g_{\nu\mu}S^{\mu}\mathcal{A}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$
(5.48) $\displaystyle P_{\nu}^{Q}$
$\displaystyle:=Q_{\mu\nu}S^{\mu}+\frac{1}{2}v_{\zeta}[\langle\breve{\nabla}_{\nu}(\mathfrak{h}^{\star}\mathsf{\Psi}),\mathsf{\Psi}\rangle+\langle\breve{\nabla}_{\nu}\mathsf{\Psi},\mathfrak{h}^{\star}\mathsf{\Psi}\rangle]-\frac{1}{2}\nabla_{\nu}v_{\zeta}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$
$\displaystyle Q_{\mu\nu}$
$\displaystyle:=\frac{1}{2}[\langle\breve{\nabla}_{\mu}(\mathfrak{h}^{\star}\mathsf{\Psi}),\breve{\nabla}_{\nu}\mathsf{\Psi}\rangle+\langle\breve{\nabla}_{\mu}\mathsf{\Psi},\breve{\nabla}_{\nu}(\mathfrak{h}^{\star}\mathsf{\Psi})\rangle-
g_{\mu\nu}g^{\alpha\beta}\langle\nabla_{\alpha}(\mathfrak{h}^{\star}\mathsf{\Psi}),\breve{\nabla}_{\beta}\mathsf{\Psi}\rangle]\text{,}$
$\displaystyle I_{S}$
$\displaystyle:=2F^{\prime}f^{-(n-3)}\,\breve{\nabla}_{S}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{\nabla}_{S}\mathsf{\Psi})\text{,}$
$\displaystyle I_{\zeta}$
$\displaystyle:=Ff^{-(n-3)}v_{\zeta}\,[\langle\breve{\nabla}_{S}(\mathfrak{h}^{\star}\mathsf{\Psi}),\mathsf{\Psi}\rangle+\langle\breve{\nabla}_{S}\mathsf{\Psi},\mathfrak{h}^{\star}\mathsf{\Psi}\rangle]\text{,}$
$\displaystyle I_{R}$
$\displaystyle:=\frac{1}{2}g^{\alpha\beta}S^{\nu}\,[\langle\breve{R}_{\nu\alpha}[\mathfrak{h}^{\star}\mathsf{\Psi}],\breve{\nabla}_{\beta}\mathsf{\Psi}\rangle+\langle\breve{R}_{\nu\alpha}[\mathsf{\Psi}],\breve{\nabla}_{\beta}(\mathfrak{h}^{\star}\mathsf{\Psi})]\text{,}$
$\displaystyle I_{\pi}$
$\displaystyle:=g^{\alpha\mu}g^{\beta\nu}(\pi_{\zeta})_{\alpha\beta}\,\breve{\nabla}_{\mu}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{\nabla}_{\nu}\mathsf{\Psi}).$

Next, we obtain asymptotic formulas for various terms in (5.47) and (5.48).
Again, this process mirrors that in the proof of [28, Lemma 5.28], so the
reader is referred there for further details. 343434Again, while we use a
different Riemannian metric than in [28], both metrics satisfy the same
estimates. Furthermore, our frames $\\{E_{A}\\}$ here again correspond to the
frames $\\{V,E_{A}\\}$ in [28]. For the second term in (5.47), we use (5.20)
and (5.22) to infer

$2F^{\prime}f^{-(n-3)}|\breve{\nabla}_{S}\mathsf{\Psi}|_{\mathfrak{h}}^{2}=[2\kappa
f^{n-2}+2\lambda
f^{n-2+p}+\mathcal{O}(f^{n})]\,|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$
(5.49)

For the fourth term in (5.47), we use (5.23) and (5.35) to obtain

$\displaystyle\mathcal{A}v_{\zeta}-\frac{1}{2}\operatorname{div}_{g}(\mathcal{A}S)+\frac{1}{2}\breve{\Box}_{g}v_{\zeta}$
$\displaystyle=(\kappa^{2}-n\kappa+\sigma)f^{n-2}+\left(1-\frac{p}{2}\right)(2\kappa-n+p)\lambda
f^{n-2+p}$ (5.50)
$\displaystyle\qquad+(1-p)\lambda^{2}f^{n-2+2p}+\lambda^{2}\,\mathcal{O}(f^{n})\text{.}$

For the third term in (5.47), we expand the $g$-contractions in terms of the
frame $\\{N,E_{A}\\}$:

$g^{\alpha\mu}g^{\beta\nu}(\pi_{\zeta})_{\alpha\beta}\,\breve{\mathfrak{h}}(\breve{\nabla}_{\mu}\mathsf{\Psi},\breve{\nabla}_{\nu}\mathsf{\Psi})=\sum_{Z_{1},Z_{2}\in\\{N,E_{A}\\}}\mu_{Z_{1}}\mu_{Z_{2}}\,\pi_{\zeta}(Z_{1},Z_{2})\,\breve{\mathfrak{h}}(\breve{\nabla}_{Z_{1}}\mathsf{\Psi},\breve{\nabla}_{Z_{2}}\mathsf{\Psi})\text{,}$

where each $\mu_{Z_{i}}=\pm 1$, depending on whether $Z_{i}$ is timelike or
spacelike. From the above, we fully expand the factors
$\smash{\breve{\mathfrak{h}}(\breve{\nabla}_{Z_{1}}\mathsf{\Psi},\breve{\nabla}_{Z_{2}}\mathsf{\Psi})}$
using an $\mathfrak{h}$-orthonormal frame, we apply Lemma 5.19, and we then
recall the assumptions $|\mathfrak{E}_{A}|_{\mathfrak{h}}\simeq 1$ in order to
deduce

$\displaystyle
g^{\alpha\mu}g^{\beta\nu}(\pi_{\zeta})_{\alpha\beta}\,\breve{\mathfrak{h}}(\breve{\nabla}_{\mu}\mathsf{\Psi},\breve{\nabla}_{\nu}\mathsf{\Psi})$
$\displaystyle\geq-[(n-1)f^{n-2}+\mathcal{O}(f^{n})]\,|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
(5.51)
$\displaystyle\qquad+K\rho^{2}f^{n-2}\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$

where $K>0$ is as in the statement of Lemma 5.19.

The error terms $I_{S}$, $I_{\zeta}$, $I_{R}$ and $I_{\pi}$ contain no
leading-order terms and are controlled using Lemmas 5.14–5.16 by following the
process in the proof of [28, Lemma 5.28]: 353535There are no $N$-derivatives
of $\mathsf{\Psi}$ in the estimate for $I_{R}$, since
$\smash{\breve{R}}_{NN}[\mathsf{\Psi}]=0$.

$\displaystyle|I_{S}|$ $\displaystyle\leq\lambda\,\mathcal{O}(\rho
f^{n-1})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
(5.52) $\displaystyle|I_{\zeta}|$
$\displaystyle\leq\lambda\,\mathcal{O}(f^{n})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})\text{,}$
$\displaystyle|I_{R}|$
$\displaystyle\leq\mathcal{O}(\rho^{3}f^{n-1})\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\mathcal{O}(\rho
f^{n-1})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$ $\displaystyle|I_{\pi}|$
$\displaystyle\leq\mathcal{O}(\rho
f^{n-1})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})+\mathcal{O}(\rho^{3}f^{n-1})\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

Combining (5.47)–(5.52) with the estimate (5.41) for $J(\mathsf{\Psi})$, we
obtain

$\displaystyle\lambda^{-1}f^{n-2-p}|\mathcal{L}^{\dagger}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\geq\left[(2\kappa-n+1-\mathcal{C}_{0})f^{n-2}+\frac{3}{2}\lambda
f^{n-2+p}+\lambda\,\mathcal{O}(f^{n-1})\right]\,|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
(5.53)
$\displaystyle\qquad+\frac{1}{4}K\rho^{2}f^{n-2}\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+(\kappa^{2}-n\kappa+\sigma)f^{n-2}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+\left[\left(1-\frac{p}{2}\right)(2\kappa-n+p)\lambda
f^{n-2+p}+(1-p)\lambda^{2}f^{n-2+2p}\right]\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+\lambda^{2}\mathcal{O}(f^{n-1})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\operatorname{div}_{g}(P^{S}+P^{Q})\text{.}$

Now, for any $q\in\mathbb{R}$, we expand the inequality

$0\leq
f^{q-2}\left|\breve{\nabla}_{\nabla^{\sharp}f}\mathsf{\Psi}+\frac{1}{2}(q-n)f\,\mathsf{\Psi}\right|_{\mathfrak{h}}^{2}\text{,}$

and we recall the asymptotics from (2.14), (2.17), and Lemma 5.16 to obtain
363636See the derivation of [28, Equations (5.78)–(5.79)] for details.

$\displaystyle f^{q}\,|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\geq\frac{1}{4}(q-n)^{2}f^{q}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+(1+q^{2})\,\mathcal{O}(f^{q+2})\,(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+|\mathsf{\Psi}|_{\mathfrak{h}}^{2})$
(5.54) $\displaystyle\qquad+(q-n)\,\mathcal{O}(\rho
f^{q+1})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\operatorname{div}_{g}P^{H,q}\text{,}$

where $P^{H,q}$ is defined as

$P^{H,q}:=-\frac{1}{2}(q-n)f^{q-1}\nabla^{\sharp}f\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$
(5.55)

We then apply the inequality (5.54) twice, with $q:=n-2$ and $q:=n-2+p$, to
the estimate (5.53). Noting in particular that
$2\kappa-n+1-\mathcal{C}_{0}\geq 0$ by (5.12), we then obtain

$\displaystyle\lambda^{-1}f^{n-2-p}|\mathcal{L}^{\dagger}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\geq\left[\frac{1}{2}\lambda
f^{n-2+p}+\lambda\,\mathcal{O}(f^{n-1})\right]|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\frac{1}{4}K\rho^{2}f^{n-2}\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
(5.56)
$\displaystyle\qquad+(\kappa^{2}-(n-2)\kappa+\sigma-(n-1)-\mathcal{C}_{0})f^{n-2}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+\left(1-\frac{p}{2}\right)(2\kappa-n+1+\frac{p}{2})\lambda
f^{n-2+p}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\qquad+[(1-p)\lambda^{2}f^{n-2+2p}+\lambda^{2}\,\mathcal{O}(f^{n-1})]\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\operatorname{div}_{g}P\text{,}$

where we have set

$P:=P^{S}+P^{Q}+\lambda\,P^{H,n-2+p}+(2\kappa-n+1-\mathcal{C}_{0})\,P^{H,n-2}\text{.}$
(5.57)

Observe that (5.56) immediately implies the desired (5.44), once we set
$f_{\star}$ sufficiently small and $\lambda$ sufficiently large as in (5.13).
Furthermore, by the definitions (5.48) and (5.55), we see that the quantity
$P$ defined in (5.57) precisely matches that of (5.45).

It remains to prove the bound (5.46) for $P(\rho\partial_{\rho})$; as before,
the steps mirror those in the end of the proof of [28, Lemma 5.28]. First, by
Lemma 2.10, (2.14), (5.19), and the crude estimate
$\mathcal{A}=\lambda^{2}\,\mathcal{O}(1)$ from Lemma 5.18, we obtain

$\displaystyle|P^{S}(\rho\partial_{\rho})|$
$\displaystyle\leq\lambda^{2}\,\mathcal{O}(f^{n-2})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$
(5.58)
$\displaystyle|\lambda\,P^{H,n-2+p}(\rho\partial_{\rho})+(2\kappa-n+1-\mathcal{C}_{0})\,P^{H,n-2}(\rho\partial_{\rho})|$
$\displaystyle\leq\lambda\,\mathcal{O}(f^{n-2})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

The bound $P^{Q}$ is similar, but there are more terms involved. We begin by
expanding

$\displaystyle P^{Q}(\rho\partial_{\rho})$
$\displaystyle=\frac{1}{2}\rho[2\,\breve{\mathfrak{h}}(\breve{\nabla}_{\rho}\mathsf{\Psi},\breve{\nabla}_{S}\mathsf{\Psi})+\breve{\nabla}_{\rho}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{\nabla}_{S}\mathsf{\Psi})+\breve{\nabla}_{S}\breve{\mathfrak{h}}(\breve{\nabla}_{\rho}\mathsf{\Psi},\mathsf{\Psi})]$
(5.59) $\displaystyle\qquad-\frac{1}{2}\rho
g(\partial_{\rho},S)\,g^{\mu\nu}[\breve{\mathfrak{h}}(\breve{\nabla}_{\mu}\mathsf{\Psi},\breve{\nabla}_{\nu}\mathsf{\Psi})+\breve{\nabla}_{\mu}\breve{\mathfrak{h}}(\mathsf{\Psi},\breve{\nabla}_{\nu}\mathsf{\Psi})]$
$\displaystyle\qquad+\frac{1}{2}\rho
v_{\zeta}[2\,\breve{\mathfrak{h}}(\breve{\nabla}_{\rho}\mathsf{\Psi},\mathsf{\Psi})+\breve{\nabla}_{\rho}\breve{\mathfrak{h}}(\mathsf{\Psi},\mathsf{\Psi})]-\frac{1}{2}\rho\nabla_{\rho}v_{\zeta}\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle:=B_{1}+B_{2}+B_{3}+B_{4}\text{.}$

Following the proof of [28, Lemma 5.28], by applying (2.14) and Lemma 5.16, we
estimate

$\displaystyle|B_{1}|$
$\displaystyle\leq\mathcal{O}(\rho^{2}f^{n-2})\,|\breve{\mathsf{D}}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\mathcal{O}(\rho^{2}f^{n})\,|\mathsf{D}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\mathcal{O}(\rho^{2}f^{n})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$
(5.60) $\displaystyle|B_{3}+B_{4}|$
$\displaystyle\leq\mathcal{O}(\rho^{2}f^{n})\,|\breve{\mathsf{D}}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\mathcal{O}(f^{n})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

For $B_{2}$, we apply Lemma 2.8, (2.12), and Lemma 5.16, which yields 373737In
contrast to [28, Lemma 5.28], it is easier to expand in terms of coordinates
rather than frames.

$\displaystyle B_{2}$
$\displaystyle\leq\frac{1}{2}f^{n-2}(-\rho^{2}|\breve{\mathsf{D}}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\rho^{2}|\mathsf{D}\mathsf{\Psi}|_{\mathfrak{h}^{2}})+\mathcal{O}(f^{n-2})\,(\rho^{3}\,|\breve{\nabla}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}+\rho^{2}\,|\mathsf{D}\mathsf{\Psi}|_{\mathfrak{h}})\,|\mathsf{\Psi}|_{\mathfrak{h}}$
(5.61)
$\displaystyle\leq\mathcal{O}(\rho^{2}f^{n})\,|\breve{\mathsf{D}}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\mathcal{O}(\rho^{2}f^{n-2})\,|\mathsf{D}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\mathcal{O}(\rho^{2}f^{n-2})\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

Finally, combining (5.58)–(5.61) yields the desired estimate (5.46) for
$P(\rho\partial_{\rho})$. ∎

####  5.3.4. Pointwise Estimate for $\mathsf{\Phi}$

We now convert Lemma 5.21 into a pointwise estimate for the wave operator
$\smash{\breve{\Box}_{g}+\sigma+\rho^{2}\breve{\nabla}_{X}}$ and for the
original unknown $\mathsf{\Phi}$.

######  Lemma 5.22 (Main pointwise estimate).

There exists a constant $\mathcal{C}>0$ (depending on $\mathsf{g}$,
$\mathcal{D}$, $X$, $k$, $l$) such that the following inequality holds on
$\mathcal{D}(f_{\star})$,

$\displaystyle\lambda^{-1}\mathcal{E}f^{-p}\,|(\breve{\Box}_{g}+\sigma+\rho^{2}\breve{\nabla}_{X})\mathsf{\Phi}|_{\mathfrak{h}}^{2}$
$\displaystyle\geq\mathcal{C}\mathcal{E}\rho^{4}\,(|\breve{\mathsf{D}}_{\rho}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+|\mathsf{D}\mathsf{\Phi}|_{\mathfrak{h}}^{2})$
(5.62)
$\displaystyle\qquad+\frac{1}{8}\lambda^{2}\mathcal{E}f^{2p}\,|\mathsf{\Phi}|_{\mathfrak{h}}^{2}+\operatorname{div}_{g}P\text{,}$

where $P$ is as in Lemma 5.21, and where the weight $\mathcal{E}$ is given by

$\mathcal{E}:=e^{-2F}f^{n-2}=e^{-2\lambda p^{-1}f^{p}}f^{n-2-2\kappa}\text{.}$
(5.63)

Furthermore, there exists $\mathcal{C}_{b}>0$ (depending on $\mathsf{g}$,
$\mathcal{D}$, $X$, $k$, $l$) such that

$\rho^{-n}\,P(\rho\partial_{\rho})\leq\mathcal{C}_{b}\,[|\breve{\mathsf{D}}_{\rho}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+|\mathsf{D}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+\lambda^{2}\,|\rho^{-\kappa-1}\mathsf{\Phi}|_{\mathfrak{h}}^{2}]\text{.}$
(5.64)

###### Proof.

The proof is analogous to that of [28, Lemma 5.29], so we again omit many of
the common details. First, applying the definitions (5.20) and (5.21) for
$\Psi$ and $\mathcal{L}^{\dagger}$ to Lemma 5.21, we see that there exists
$\mathcal{C}>0$ (depending on $\mathsf{g}$, $\mathcal{D}$, $X$, $k$, $l$) with

$\displaystyle\lambda^{-1}\mathcal{E}f^{-p}\,|(\breve{\Box}_{g}+\sigma+\rho^{2}\breve{\nabla}_{X})\mathsf{\Phi}|_{\mathfrak{h}}^{2}$
$\displaystyle\geq\mathcal{C}\rho^{2}f^{n-2}\left(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)$
(5.65)
$\displaystyle\qquad+\frac{1}{4}\lambda^{2}\mathcal{E}f^{2p}\,|\mathsf{\Phi}|_{\mathfrak{h}}^{2}+\operatorname{div}_{g}P\text{,}$

where the frames $\\{N,E_{A}\\}$ are as in Lemma 5.21.

From (2.19), (2.20), and the definition of the $E_{A}$’s, we can estimate

$\displaystyle\rho^{4}|\breve{\mathsf{D}}_{\rho}\mathsf{\Phi}|_{\mathfrak{h}}^{2}$
$\displaystyle\leq\mathcal{O}(\rho^{2})\,|\breve{\nabla}_{N}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+\mathcal{O}(f^{2}\rho^{2})\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Phi}|_{\mathfrak{h}}^{2}\text{,}$
(5.66) $\displaystyle\rho^{4}|\mathsf{D}\mathsf{\Phi}|_{\mathfrak{h}}^{2}$
$\displaystyle\leq\mathcal{O}(f^{2}\rho^{2})\,|\breve{\nabla}_{N}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+\mathcal{O}(\rho^{2})\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Phi}|_{\mathfrak{h}}^{2}\text{.}$

Moreover, by (2.14), (2.19), Lemma 2.16, (5.12), and (5.20),

$e^{-2F}|\breve{\nabla}_{N}\mathsf{\Phi}|_{\mathfrak{h}}^{2}\leq|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\lambda^{2}\,\mathcal{O}(1)\,|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}\qquad
e^{-2F}|\breve{\nabla}_{E_{A}}\mathsf{\Phi}|_{\mathfrak{h}}^{2}=|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$
(5.67)

From (5.66) and (5.67), we then conclude

$\mathcal{E}\rho^{4}(|\breve{\mathsf{D}}_{\rho}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+|\mathsf{D}\mathsf{\Phi}|_{\mathfrak{h}}^{2})\leq\mathcal{O}(\rho^{2}f^{n-2})\,\left(|\breve{\nabla}_{N}\mathsf{\Psi}|_{\mathfrak{h}}^{2}+\sum_{A=1}^{n}|\breve{\nabla}_{E_{A}}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)+\mathcal{O}(f^{2})\,\lambda^{2}\mathcal{E}|\mathsf{\Phi}|_{\mathfrak{h}}^{2}\text{.}$

Combining (5.65) and the above results in the estimate (5.62).

It remains to show (5.64). For this, we apply (2.12) and (5.20), and recall
that $\rho\leq f$ (by (2.11)) and that $2\kappa\geq n-1$ (by (5.12)). Putting
all this together, we obtain

$\displaystyle f^{n-2}\rho^{-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\leq|\rho^{-\kappa-1}\mathsf{\Phi}|_{\mathfrak{h}}^{2}\text{,}$
$\displaystyle
f^{n-2}\rho^{-n+2}|\breve{\mathsf{D}}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\leq|\breve{\mathsf{D}}_{\rho}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+\lambda^{2}\,\mathcal{O}(1)\,|\rho^{-\kappa-1}\mathsf{\Phi}|_{\mathfrak{h}}^{2}\text{,}$
$\displaystyle f^{n-2}\rho^{-n+2}|\mathsf{D}\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\leq|\mathsf{D}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+\lambda^{2}\,\mathcal{O}(1)\,|\rho^{-\kappa-1}\mathsf{\Phi}|_{\mathfrak{h}}^{2}\text{.}$

The bound (5.64) now follows immediately from (5.46) and the above. ∎

####  5.3.5. Completion of the Proof

The final step is to integrate our pointwise inequality over
$\mathcal{D}(f_{\star})$. 383838The steps are analogous to [28, Section
5.3.4]; the current situation is in fact easier, since our version of $f$ is
smooth, while [28] also had to deal with discontinuities in higher derivatives
of $f$. Since $\mathcal{D}(f_{\star})$ has infinite volume, we first apply an
additional cutoff to $\mathcal{D}(f_{\star})$:

$\mathcal{D}(f_{\star},\rho_{\star}):=\mathcal{D}(f_{\star})\cap\\{\rho>\rho_{\star}\\}\text{,}\qquad
0<\rho_{\star}\ll f_{\star}\text{.}$ (5.68)

Integrating (5.62) over $\mathcal{D}(f_{\star},\rho_{\star})$ then yields

$\displaystyle\int_{\mathcal{D}(f_{\star},\rho_{\star})}\mathcal{E}f^{-p}|(\breve{\Box}_{g}+\sigma+\rho^{2}\breve{\nabla}_{X})\mathsf{\Phi}|_{\mathfrak{h}}^{2}\,d\mu_{g}-\lambda\int_{\mathcal{D}(f_{\star},\rho_{\star})}\operatorname{div}_{g}P\,d\mu_{g}$
(5.69)
$\displaystyle\quad\geq\lambda\mathcal{C}\int_{\mathcal{D}(f_{\star},\rho_{\star})}\mathcal{E}\rho^{4}(|\breve{\mathsf{D}}_{\rho}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+|\mathsf{D}\mathsf{\Phi}|_{\mathfrak{h}}^{2})\,d\mu_{g}+\frac{1}{8}\lambda^{3}\int_{\mathcal{D}(f_{\star},\rho_{\star})}\mathcal{E}f^{2p}|\mathsf{\Phi}|_{\mathfrak{h}}^{2}\,d\mu_{g}\text{,}$

where $\mathcal{C}$ and $\mathcal{E}$ are as in the statement of Lemma 5.22.

Observe that $\partial\mathcal{D}(f_{\star},\rho_{\star})$ consists two
components:

  * •

The first is $\\{f=f_{\star}\\}$, on which both $\mathsf{\Phi}$ and
$\breve{\nabla}\mathsf{\Phi}$ are assumed to vanish.

  * •

The second is $\mathcal{D}(f_{\star})\cap\\{\rho=\rho_{\star}\\}$; observe
that $\rho^{-n}\,d\mu_{\mathsf{g}}$ is the induced volume measure, and that
$-\rho\partial_{\rho}$ is the outward-facing unit normal.

Thus, applying the divergence theorem and recalling (5.64) yields

$\displaystyle-\int_{\mathcal{D}(f_{\star},\rho_{\star})}\operatorname{div}_{g}P\,d\mu_{g}$
$\displaystyle=\int_{\mathcal{D}(f_{\star})\cap\\{\rho=\rho_{\star}\\}}\rho^{-n}\,P(\rho\partial_{\rho})\,d\mu_{\mathsf{g}}$
$\displaystyle\quad\leq\mathcal{C}_{b}\lambda^{2}\int_{\mathcal{D}(f_{\star})\cap\\{\rho=\rho_{\star}\\}}[|\breve{\mathsf{D}}_{\rho}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+|\mathsf{D}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+|\rho^{-\kappa-1}\mathsf{\Phi}|_{\mathfrak{h}}^{2}]\,d\mu_{\mathsf{g}}\text{.}$

Finally, the Carleman estimate (5.14) follows from combining (5.69) with the
above, taking $\rho_{\star}\searrow 0$, and then applying the monotone
convergence theorem.

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# Geometry and thermodynamics of coherent quantum black holes

Roberto Casadioab

aDipartimento di Fisica e Astronomia, Università di Bologna
via Irnerio 46, 40126 Bologna, Italy

bI.N.F.N., Sezione di Bologna, I.S. FLAG
viale B. Pichat 6/2, 40127 Bologna, Italy E-mail<EMAIL_ADDRESS>
###### Abstract

We present a quantum description of black holes given by coherent states of
gravitons sourced by a matter core. The expected behaviour in the weak-field
region outside the horizon is recovered, with arbitrarily good approximation,
but the classical central singularity is not resolved because the coherent
states may not contain modes of arbitrarily short wavelength and the matter
core must therefore have finite size. Ensuing quantum corrections both in the
interior and exterior are also estimated by assuming the mean-field
approximation holds everywhere. These deviations from the classical black hole
geometry can be viewed as quantum hair and lead to a quantum corrected horizon
radius and thermodynamics.

##  1 Introduction and motivation

The gravitational collapse of compact objects generates geodesically
incomplete spacetimes in general relativity if a trapping surface appears [1]
and eternal point-like sources are not mathematically compatible with
Einstein’s field equations [2]. We expect the quantum theory will fix this
inconsistent classical picture of the gravitational interaction, in the same
way that quantum mechanics explains the stability of atoms by not admitting
quantum states corresponding to the classical ultraviolet catastrophe.
Although the fundamental dynamics remains unaffected, the condition for the
existence of a quantum state can greatly modify the effective description of
any systems. At the same time, the quantum state of a macroscopic black hole
must be such that the effective description reproduces the phenomenology of
spacetime we detect experimentally [3, 4].

There are many quantum models of black holes in the literature (for a very
partial list, see Refs. [5, 6, 7]). In particular, the corpuscular picture [5]
belongs to the class of approaches for which geometry should only emerge at
suitable (macroscopic) scales from the underlying (microscopic) quantum field
theory of gravitons [8, 9]. The key idea is that the constituents of black
holes are soft gravitons marginally bound in their own potential and forming a
condensate [5] with characteristic Compton-de Broglie wavelength $\lambda_{\rm
G}\sim R_{\rm H}$, where the gravitational (or Schwarzschild) radius of the
black hole of Arnowitt-Deser-Misner (ADM) mass [11] $M$ is given by 111Units
with $c=1$ are used throughout and the Newton constant $G_{\rm N}=\ell_{\rm
p}/m_{\rm p}$, where $\ell_{\rm p}$ is the Planck length and $m_{\rm p}$ the
Planck mass, so that $\hbar=\ell_{\rm p}\,m_{\rm p}$.

$\displaystyle R_{\rm H}=2\,G_{\rm N}\,M\ .$ (1.1)

The energy scale of the gravitons is correspondingly given by $\epsilon_{\rm
G}\sim\hbar/\lambda_{\rm G}$ and, if one assumes that the total mass of the
black hole $M\simeq N_{\rm G}\,\epsilon_{\rm G}$, there immediately follows
the scaling relation for the total number of gravitons

$\displaystyle N_{\rm G}\sim\frac{M^{2}}{m_{\rm p}^{2}}\sim\frac{R_{\rm
H}^{2}}{\ell_{\rm p}^{2}}\ ,$ (1.2)

which reproduces Bekenstein’s conjecture for the horizon area quantisation
[12].

The nonlinearity of the gravitational interaction plays a crucial role in this
picture. This can be seen by considering that the (negative) gravitational
energy of a source of mass $M$ localised inside a sphere of radius $R_{\rm s}$
is given by $U_{\rm N}\sim M\,V_{\rm N}(R_{\rm s})$, where

$\displaystyle{V}_{\rm N}=-\frac{G_{\rm N}\,M}{r}$ (1.3)

here is the Newtonian potential. This potential can be obtained as the
expectation value of a scalar field on a coherent state, whose normalisation
then yields the graviton number (1.2) for any values of $R_{\rm s}\gtrsim
R_{\rm H}$ [13, 16, 14, 15]. In addition to that, assuming most gravitons have
the same wavelength $\lambda_{\rm G}$, the binding energy of each graviton is
given by

$\displaystyle\epsilon_{\rm G}\sim\frac{U_{\rm N}}{N_{\rm
G}}\sim-\frac{\ell_{\rm p}\,m_{\rm p}}{R_{\rm s}}\ ,$ (1.4)

which yields the typical Compton-de Broglie length $\lambda_{\rm G}\sim R_{\rm
s}$. The graviton self-interaction energy hence reproduces the (positive)
post-Newtonian energy,

$\displaystyle U_{\rm GG}\sim N_{\rm G}\,\epsilon_{\rm G}\,V_{\rm N}(R_{\rm
s})\sim\frac{G_{\rm N}^{2}\,M^{3}}{R_{\rm s}^{2}}\ ,$ (1.5)

and the fact that gravitons in a black hole are marginally bound [5], that is
$U_{\rm N}+U_{\rm GG}\simeq 0$, finally yields the scaling $\lambda_{\rm
G}\sim R_{\rm s}\simeq R_{\rm H}$ [13, 16].

A key feature of the above scenario is that, like for all bound states in
quantum physics, it does not contain modes of arbitrarily small wavelengths
and the classical central singularity is therefore not realised. 222This
assumption can be viewed as a manifestation of the classicalization of gravity
[17]. Independent arguments in support of this condition, based on the quantum
nature of the source, were also given in Refs. [18, 19]. For other similar
considerations, see Refs. [21]. However, viewing a black hole as a quantum
state made of only gravitons with one typical wavelength $\lambda_{\rm G}$
cannot reproduce the gravitational field in the accessible outer spacetime,
even in the simple Newtonian approximation. Of course, general relativity is a
metric theory and a complete description of quantum gravity remains beyond our
scope, but this (conceptually and phenomenologically) important issue can be
addressed in a simplified form for static and spherically symmetric systems
[13, 22, 24, 23]. In particular, we limit the quantum description to the
potential function (1.3) appearing in the Schwarzschild metric

$\displaystyle\mbox{${\rm d}$}s^{2}=-\left(1+2\,V_{\rm N}\right)\mbox{${\rm
d}$}t^{2}+\frac{\mbox{${\rm d}$}r^{2}}{1+2\,V_{\rm N}}+r^{2}\,\mbox{${\rm
d}$}\Omega^{2}\ .$ (1.6)

In fact, the function $V_{\rm N}$ plays the role of a potential in the
geodesic equation of radial motion,

$\displaystyle\frac{\mbox{${\rm d}$}^{2}r}{\mbox{${\rm
d}$}\tau^{2}}\equiv\ddot{r}=-\frac{G_{\rm N}\,M}{r^{2}}=-V_{\rm N}^{\prime}\
,$ (1.7)

where the geodesic $r=r(\tau)$ is parameterised by the proper time $\tau$,
333One should also notice that the general relativistic function $V_{\rm N}$
depends on the areal radius $r$, whereas the post-Newtonian viewpoint in Eqs.
(1.3)-(1.5) makes use of the harmonic radius [25, 19, 20]. and this equation
is the starting point to derive a discrete spectrum for a collapsing dust core
[19].

The first important step in a field theoretic description of gravity is to
identify the quantum vacuum $\ket{0}$, which we assume is the realisation of a
universe where no modes (of matter or gravity) are excited. We could then
associate the Minkowski metric $\eta_{\mu\nu}$ to such an absolute vacuum,
since this is the metric used to describe linearised gravity and to define
both matter and gravitational excitations in this regime. The linearised
theory should provide a reliable description for small matter sources (say
with total energy $M\ll m_{\rm p}$), for which it allows one to recover the
Newtonian potential from simple tree-level graviton exchanges in the non-
relativistic limit. In this context, the potential $V_{\rm N}$ is not a
fundamental scalar, but it emerges from the (non-propagating) longitudinal
polarisation of virtual gravitons [8]. For large sources (that is, with $M\gg
m_{\rm p}$), one can in principle reconstruct the complete classical dynamics
[9], 444See Ref. [10] for a critical discussion of this conjecture. but to
obtain the proper quantum state from the excitations of the linearised theory
appears rather hopeless in this highly non-linear regime. In order to
circumvent this issue and, at the same time, to compare with experimental
data, one usually assumes that there exists a classical background geometry to
replace $\eta_{\mu\nu}$ with a suitable solution $g_{\mu\nu}$ of the
corresponding classical Einstein’s equations. However, in our perspective, the
Einstein equations should emerge as effectively describing the macroscopic
dynamics and a mean-field metric should correspond to a suitable quantum
state. Since we are interested in static and spherically symmetric
configurations representing a black hole, we just require that the relevant
quantum state of gravity effectively reproduces (as closely as possible) the
expected Schwarzschild geometry (1.6) like it was done in Ref. [22]. This
approach is thus akin to quantising the longitudinal mode of gravity [8, 26]
and can be obtained by employing coherent states in a suitable Fock space
built upon the Minkowski vacuum. The use of coherent states is generically
motivated by their property of minimising the quantum uncertainty, and is
further supported by studies of electrodynamics [14, 27], linearised gravity
[15] and the de Sitter spacetime [28].

In the original corpuscular picture, as briefly reviewed above, baryonic
matter sourcing the gravitational field and triggering the gravitational
collapse is argued to become essentially irrelevant after the black hole forms
[5]. In Refs. [19, 22], we instead found that a finite size $R_{\rm s}$ of the
matter source is both a consequence of the quantum nature of collapsing matter
and a necessary condition for the existence of a proper quantum state which
(approximately) reproduces the outer Schwarzschild geometry. 555In a fully
quantum picture, one expects that $R_{\rm s}$ be at best the expectation value
of some emergent operator in the relevant state of matter fields. For more
considerations on the scale $R_{\rm s}$ for quantum black holes, see the
concluding remarks and Refs. [19, 30]. This supports the point of view of
Refs. [13, 16, 18, 29], according to which matter inside the black hole still
plays a very significant role in defining the structure of the interior of
astrophysical black holes (and possibly microscopic ones) and leads to the
existence of quantum hair [31]. In fact, the quantum version of $V_{\rm N}$
can be employed in order to reconstruct a quantum corrected complete metric
[22]. In this work, we analyse in details the quantum corrected black hole
geometry and thermodynamics, thus complementing the study of the entropy of
the matter core of Ref. [32].

In the next section, we will briefly review how coherent states of a massless
scalar field on a reference flat spacetime can be used to reproduce a
classical static field configuration, in general; a consistent quantum state
for the metric function (1.3) is then introduced in Section 3, where the
corresponding geometry and thermodynamics will be analysed in details; final
remarks, outlook and connections with other works will be given in Section 4.

##  2 Quantum coherent states for classical static configurations

We will first review how to describe a generic static $V=V(r)$ as the mean
field of the coherent state of a free massless scalar field (for more details,
see, e.g. Ref. [14, 16]). It is important to remark again that this
description is not fundamental and the use of a scalar field to represent the
(non-perturbative) behaviour of the true degrees of freedom of Einstein
gravity in the (rather unphysical) static limit is a simplification supported
by more refined analyses [15, 28].

We first rescale the dimensionless $V$ so as to obtain a canonically
normalised real scalar field $\Phi=\sqrt{{m_{\rm p}}/{\ell_{\rm p}}}\,V$, and
then quantise $\Phi$ as a massless field satisfying the free wave equation

$\displaystyle\left[-\frac{\partial^{2}}{\partial
t^{2}}+\frac{1}{r^{2}}\,\frac{\partial}{\partial
r}\left(r^{2}\,\frac{\partial}{\partial
r}\right)\right]\Phi(t,r)\equiv\left(-\partial_{t}^{2}+\triangle\right)\Phi=0\
.$ (2.1)

Solutions to Eq. (2.1) can be conveniently written as

$\displaystyle u_{k}(t,r)=e^{-i\,k\,t}\,j_{0}(k\,r)\ ,$ (2.2)

where $k>0$ and $j_{0}={\sin(k\,r)}/{k\,r}$ are spherical Bessel functions
satisfying the orthogonality relation

$\displaystyle
4\,\pi\int_{0}^{\infty}r^{2}\,{\mathrm{d}}r\,j_{0}(k\,r)\,j_{0}(p\,r)=\frac{2\,\pi^{2}}{k^{2}}\,\delta(k-p)\
.$ (2.3)

The quantum field operator and its conjugate momentum read

$\displaystyle\hat{\Phi}(t,r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,\sqrt{\frac{\hbar}{2\,k}}\left[\hat{a}_{k}\,u_{k}(t,r)+\hat{a}^{\dagger}_{k}\,u^{*}_{k}(t,r)\right]$
(2.4) $\displaystyle\hat{\Pi}(t,r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!i\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,\sqrt{\frac{\hbar\,k}{2}}\left[\hat{a}_{k}\,u_{k}(t,r)-\hat{a}^{\dagger}_{k}\,u^{*}_{k}(t,r)\right]\
,$ (2.5)

which satisfy the equal time commutation relations,

$\displaystyle\left[\hat{\Phi}(t,r),\hat{\Pi}(t,s)\right]=\frac{i\,\hbar}{4\,\pi\,r^{2}}\,\delta(r-s)\
,$ (2.6)

provided the creation and annihilation operators obey the commutation rules

$\displaystyle\left[\hat{a}_{k},\hat{a}^{\dagger}_{p}\right]=\frac{2\,\pi^{2}}{k^{2}}\,\delta(k-p)\
.$ (2.7)

The Fock space of quantum states is then built from the vacuum defined by
$\hat{a}_{k}\ket{0}=0$ for all $k>0$. The choice of the flat Minkowski metric
in Eq. (2.1) follows from the expectation that this vacuum $\ket{0}$ is meant
to describe a completely empty spacetime devoid of any matter source and
without excitations of the gravitational field, as we argued in the
introductory Section 1.

Classical configurations of the scalar field that can be realised in the
quantum theory must correspond to suitable states in this Fock space, and a
natural choice is given by coherent states $\ket{g}$ such that

$\displaystyle\hat{a}_{k}\ket{g}=g_{k}\,e^{i\,\gamma_{k}(t)}\ket{g}\ .$ (2.8)

In particular, we are interested in those $\ket{g}$ for which the expectation
value of the quantum field $\hat{\Phi}$ reproduces the classical potential,
namely

$\displaystyle\sqrt{\frac{\ell_{\rm p}}{m_{\rm
p}}}\bra{g}\hat{\Phi}(t,r)\ket{g}=V(r)\ .$ (2.9)

From the expansion (2.4), we obtain

$\displaystyle\bra{g}\hat{\Phi}(t,r)\ket{g}=\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,\sqrt{\frac{2\,\ell_{\rm
p}\,m_{\rm p}}{k}}\,g_{k}\,\cos[\gamma_{k}(t)-k\,t]\,j_{0}(k\,r)\ .$ (2.10)

If we now write

$\displaystyle
V=\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,\tilde{V}(k)\,j_{0}(k\,r)\
,$ (2.11)

we immediately obtain 666This formal choice of phases eliminates the time
dependence from the normal modes (2.2) and allows for reproducing static
configurations. From a physical point of view, one can consider such a
limiting approximation holds for time intervals shorter than $\Delta t\sim
k^{-1}$.

$\displaystyle\gamma_{k}=k\,t$ (2.12)

and

$\displaystyle g_{k}=\sqrt{\frac{k}{2}}\,\frac{\tilde{V}(k)}{\ell_{\rm p}}\ .$
(2.13)

The coherent state finally reads

$\displaystyle\ket{g}=e^{-N_{\rm
G}/2}\,\exp\left\\{\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,g_{k}\,\hat{a}^{\dagger}_{k}\right\\}\ket{0}\
,$ (2.14)

where

$\displaystyle{N_{\rm
G}}=\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,g_{k}^{2}$
(2.15)

is the total occupation number. We note in particular that the value of
${N_{\rm G}}$ measures the “distance” in the Fock Space of $\ket{g}$ from the
vacuum $\ket{0}$ corresponding to ${N_{\rm G}}=0$. Another quantity of
interest is given by

$\displaystyle\mbox{$\langle\,k\,\rangle$}=\int_{0}^{\infty}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,k\,g_{k}^{2}\
,$ (2.16)

from which one obtains the “average” wavelength $\lambda_{\rm G}={N_{\rm
G}}/\mbox{$\langle\,k\,\rangle$}$.

##  3 Quantum Schwarzschild black holes

We will now apply results from the previous section to the Schwarzschild
metric (1.6). This geometry contains only the function $V_{\rm N}=\sqrt{G_{\rm
N}}\,\Phi$, and all of the relevant expressions introduced can be explicitly
computed from the coefficients $g_{k}$ representing the occupation numbers of
the modes $u_{k}$.

In particular, by inverting Eq. (2.11), we find

$\displaystyle\tilde{V}_{\rm N}=-4\,\pi\,G_{\rm N}\,\frac{M}{k^{2}}$ (3.1)

and the coefficients

$\displaystyle g_{k}=-\frac{4\,\pi\,M}{\sqrt{2\,k^{3}}\,m_{\rm p}}\ .$ (3.2)

For such a coherent state, we obtain

$\displaystyle N_{\rm G}=\frac{4\,M^{2}}{m_{\rm
p}^{2}}\int_{0}^{\infty}\frac{{\mathrm{d}}k}{k}$ (3.3)

and

$\displaystyle\mbox{$\langle\,k\,\rangle$}=\frac{4\,M^{2}}{m_{\rm
p}^{2}}\int_{0}^{\infty}{\mathrm{d}}k\ .$ (3.4)

The number of quanta $N_{\rm G}$ contains a logarithmic divergence both in the
infrared (IR) and the ultraviolet (UV), whereas $\langle\,k\,\rangle$ only
diverges (linearly) in the UV.

The meaning of such divergences was already explored in details in previous
works [16, 18, 22]. In particular, we recall that the UV divergences are due
to the vanishing size of the source, hence they would not be present if the
density were regular. Instead of smoothing out the source, the UV divergences
can be formally regularised by introducing a cut-off $k_{\rm UV}\sim 1/R_{\rm
s}$, where $R_{\rm s}$ can be interpreted as the finite radius of a would-be-
regular matter source. Such a cut-off is just a mathematically simple way of
describing the fact that the very existence of a proper quantum state
$\ket{g}$ requires the coefficients $g_{k}$ to depart from their purely
classical expression (3.2) for $k\to 0$ and $k\to\infty$. Of course, for a
black hole spacetime, we must have $R_{\rm s}<R_{\rm H}$. Likewise, we
introduce a IR cut-off $k_{\rm IR}=1/{R_{\infty}}$ to account for the
necessarily finite life-time $\tau\sim{R_{\infty}}$ of a realistic source [16]
777Any disturbance in the source will propagate (at most) at the speed of
light. Alternatively, we could take for $R_{\infty}$ the size of the
observable Universe as an upper bound [24]. In any case, modes with wavelength
$k^{-1}$ many orders of magnitude larger than $R_{\rm H}$ do not contribute
significantly to the determination of $V=V(r)$ in Eq. (2.9) (for the details,
see Ref. [18, 33]) and we shall assume $k_{\rm IR}\to 0$ whenever possible.
and rewrite

$\displaystyle N_{\rm G}=\frac{4\,M^{2}}{m_{\rm p}^{2}}\int_{k_{\rm
IR}}^{k_{\rm UV}}\frac{{\mathrm{d}}k}{k}=4\,\frac{M^{2}}{m_{\rm
p}^{2}}\,\ln\left(\frac{R_{\infty}}{R_{\rm s}}\right)$ (3.5)

and

$\displaystyle\mbox{$\langle\,k\,\rangle$}=\frac{4\,M^{2}}{m_{\rm
p}^{2}}\int_{k_{\rm IR}}^{k_{\rm UV}}{\mathrm{d}}k=4\,\frac{M^{2}}{m_{\rm
p}^{2}}\left(\frac{1}{R_{\rm s}}-\frac{1}{R_{\infty}}\right)\ .$ (3.6)

The corpuscular scaling (1.2) for the number $N_{\rm G}$ with the square of
the energy $M$ of the system already appears at this stage, whereas the second
crucial result

$\displaystyle\lambda_{\rm G}=\frac{N_{\rm
G}}{\mbox{$\langle\,k\,\rangle$}}\sim\ell_{\rm p}\,\frac{M}{m_{\rm p}}\ ,$
(3.7)

is obtained from Eqs. (3.5) and (3.6) only provided the cut-offs satisfy

$\displaystyle\ln\left(\frac{R_{\infty}}{R_{\rm s}}\right)\simeq\frac{R_{\rm
H}}{R_{\rm s}}\ .$ (3.8)

Assuming $R_{\rm s}\lesssim R_{\rm H}\ll R_{\infty}$, the above yields (see
Fig. 1 for a graphical comparison with the exact solution)

$\displaystyle R_{\rm s}\simeq\frac{R_{\rm H}}{\ln\left({R_{\infty}}/{R_{\rm
H}}\right)}\ ,$ (3.9)

so that the size of the inner source and the radius of the outer region
containing a gravitational field appear connected in the quantum description.
We will further comment about possible consequences of this result in the
concluding Section 4.

Figure 1: Exact solution to Eq. (3.8) (solid line) compared to its
approximation (3.9) (dashed line).

###  3.1 Consistent black hole states

We started from the condition in Eq. (2.9), which demands that the coherent
state $\ket{g}$ reproduces the classical potential everywhere. We then found
that acceptable occupation numbers $g_{k}$ do not exist which satisfy this
requirement for $k\to 0$ and $k\to\infty$. We next observe that, for a black
hole, the mean field needs only reproduce the classical $V_{\rm N}$ with
sufficient accuracy to comply with experimental bounds at most in the region
of outer communication outside the horizon. This means that the coherent state
$\ket{g_{\rm BH}}$ representing a black hole must give

$\displaystyle\sqrt{\frac{\ell_{\rm p}}{m_{\rm p}}}\bra{g_{\rm
BH}}\hat{\Phi}(t,r)\ket{g_{\rm BH}}\simeq V_{\rm N}(r)\qquad{\rm for}\
r\gtrsim R_{\rm H}\ ,$ (3.10)

where we recall that $V_{\rm N}(R_{\rm H})=-1/2$ and the approximate equality
is subject to experimental precision. In practice, this weaker condition means
that $\ket{g_{\rm BH}}$ does not need to contain the modes of infinitely short
wavelength that are necessary to resolve the classical singularity at $r=0$.

In fact, Eq. (3.10) can be satisfied by building the coherent state
$\ket{g_{\rm BH}}$ according to Eq. (2.14) with modes of wavelength $k^{-1}$
larger than some fraction of the size of the gravitational radius $R_{\rm H}$
of the source, which we can further identify with the UV cut-off $R_{\rm s}$.
By momentarily considering also the IR scale $k_{\rm IR}$, we thus have that
only the modes $k$ satisfying

$\displaystyle R_{\infty}^{-1}\sim k_{\rm IR}\lesssim k\lesssim k_{\rm UV}\sim
R_{\rm s}^{-1}$ (3.11)

are significantly populated in the quantum state $\ket{g_{\rm BH}}$. This
yields an effective quantum potential

$\displaystyle V_{\rm QN}$ $\displaystyle\\!\\!\simeq\\!\\!$
$\displaystyle\int_{k_{\rm IR}}^{k_{\rm
UV}}\frac{k^{2}\,{\mathrm{d}}k}{2\,\pi^{2}}\,\tilde{V}_{\rm
N}(k)\,j_{0}(k\,r)$ (3.12) $\displaystyle\\!\\!\simeq\\!\\!$
$\displaystyle-\frac{2\,\ell_{\rm p}\,M}{\pi\,m_{\rm p}\,r}\int^{r/R_{\rm
s}}_{0}{\mathrm{d}}z\,\frac{\sin z}{z}\ ,$

where we defined $z=k\,r$ and let $k_{\rm IR}=1/R_{\infty}\to 0$ as mentioned
above. We thus find

$\displaystyle V_{\rm QN}$ $\displaystyle\\!\\!\simeq\\!\\!$
$\displaystyle-\frac{2\,G_{\rm N}\,M}{\pi\,r}\,{\rm Si}\left(\frac{r}{R_{\rm
s}}\right)$ (3.13) $\displaystyle\\!\\!\simeq\\!\\!$ $\displaystyle V_{\rm
N}\left\\{1-\left[1-\frac{2}{\pi}\,{\rm Si}\left(\frac{r}{R_{\rm
s}}\right)\right]\right\\}\ ,$

where ${\rm Si}$ denotes the sine integral function (see Fig. 2 for an
example).

Figure 2: Quantum metric function $V_{\rm QN}$ in Eq. (3.13) (solid line)
compared to $V_{\rm N}$ (dashed line) for $R_{\rm s}=R_{\rm H}/20$. The
horizontal thin line marks the location of the horizon for $V=-1/2$.

It is important to remark that different shapes for the deviation from the
classical potential $V_{\rm N}$ would be obtained if one employed a different
UV cut-off in the integral (3.12). In particular, one could consider a smooth
window function rather than the hard cut-off $k<R_{\rm s}^{-1}$. Assuming such
a smooth function is physically related to the distribution of matter in the
black hole interior, one could in principle detect different matter profiles
from analysing (test particle motion in) the black hole exterior, whose
geometry is going to be described in details next. However, we shall not
endeavour in the survey of other possibilities in the present work, as very
little can be accomplished analytically.

###  3.2 Geometry

Figure 3: Oscillations of the quantum potential $V_{\rm QN}$ in Eq. (3.13)
around the Schwarzschild expression $V_{\rm N}$ for $R_{\rm s}=G_{\rm
N}\,M=R_{\rm H}/2$ (solid line) and $R_{\rm s}=R_{\rm H}/20$ (dashed line) in
the region outside the horizon $R_{\rm H}=2\,G_{\rm N}\,M$.

The potential (3.13) can be used to reconstruct the full spacetime metric
straightforwardly as [22]

$\displaystyle\mbox{${\rm d}$}s^{2}=-\left(1+2\,V_{\rm QN}\right)\mbox{${\rm
d}$}t^{2}+\frac{\mbox{${\rm d}$}r^{2}}{1+2\,V_{\rm QN}}+r^{2}\,\mbox{${\rm
d}$}\Omega^{2}\ ,$ (3.14)

where the dependence of $V_{\rm QN}$ on $R_{\rm s}$ therefore results in a
quantum violation of the no-hair theorem [31]. Near the origin, this quantum
corrected metric function reads

$\displaystyle V_{\rm QN}\simeq-\frac{2\,G_{\rm N}\,M}{\pi\,R_{\rm
s}}\left[1-\frac{\pi\,r^{2}}{18\,R_{\rm s}^{2}}\right]\ ,$ (3.15)

so that it is bounded and its derivative vanishes for $r=0$ (see Fig. 2). This
suggests that gravitational forces remain finite, as we shall show in more
details next.

In the classical Schwarzschild spacetime (1.6), the Kretschmann scalar
$R_{\alpha\beta\mu\nu}\,R^{\alpha\beta\mu\nu}\sim R^{2}\sim r^{-6}$ for $r\to
0$, whereas for the above quantum corrected metric we have

$\displaystyle R_{\alpha\beta\mu\nu}\,R^{\alpha\beta\mu\nu}\simeq
R^{2}\simeq\frac{64\,G_{\rm N}^{2}\,M^{2}}{\pi\,R_{\rm s}^{2}\,r^{4}}\ .$
(3.16)

This ensures that tidal forces remain finite all the way to the centre, as can
be seen more explicitly from the relative acceleration of radial geodesics
approaching $r=0$, to wit

$\displaystyle\frac{\ddot{\delta r}}{\delta r}=-R^{1}_{\
010}\simeq\frac{8\,G_{\rm N}^{2}\,M^{2}}{9\,\pi^{2}\,R_{\rm
s}^{4}}\left(1-\frac{\pi\,R_{\rm s}}{4\,G_{\rm N}\,M}\right)\ ,$ (3.17)

where $\delta r$ is the separation between two nearby radial geodesics and a
dot denotes again the derivative with respect to the proper time, like in Eq.
(1.7). We recall that, in the Schwarzschild spacetime, ${\ddot{\delta
r}}/{\delta r}\sim r^{-4}$, which causes the so-called “spaghettification” of
matter approaching the central singularity. The point $r=0$ in the quantum
corrected geometry can instead be seen as an integrable singularity [34],
where some geometric invariants still diverge but no harmful effects occur to
matter.

One can further compute the effective energy-momentum tensor $T_{\mu\nu}$ from
the Einstein tensor $G_{\mu\nu}$ of the metric (3.13) and find the effective
energy density

$\displaystyle\rho=-\frac{G^{0}_{\ 0}}{8\,\pi\,G_{\rm
N}}=\frac{M}{2\,\pi^{2}\,r^{3}}\,\sin\\!\left(\frac{r}{R_{\rm s}}\right)\ ,$
(3.18)

the effective radial pressure

$\displaystyle p_{r}=\frac{G^{1}_{\ 1}}{8\,\pi\,G_{\rm N}}=-\rho$ (3.19)

and the effective tension

$\displaystyle p_{t}\ =\frac{G^{2}_{\ 2}}{8\,\pi\,G_{\rm
N}}=\frac{M}{4\,\pi^{2}\,r^{3}}\left[\sin\\!\left(\frac{r}{R_{\rm
s}}\right)-\frac{r}{R_{\rm s}}\,\cos\\!\left(\frac{r}{R_{\rm
s}}\right)\right]\ .$ (3.20)

The integrals of these quantities over space are finite and, in particular,
one obtains

$\displaystyle 4\,\pi\,\int_{0}^{\infty}r^{2}\,\rho(r)\,\mbox{${\rm
d}$}r=-4\,\pi\,\int_{0}^{\infty}r^{2}\,p_{r}(r)\,\mbox{${\rm d}$}r=M$ (3.21)

and

$\displaystyle 4\,\pi\,\int_{0}^{\infty}r^{2}\,p_{t}(r)\,\mbox{${\rm
d}$}r=\frac{M}{2}\ .$ (3.22)

Another quality of this quantum corrected geometry is that it does not contain
a (inner) Cauchy horizon (whenever there exists the outer event horizon), a
property which generalises to electrically charged black holes [23]. This
excludes potentially serious casual issues which are often present in regular
black hole candidates (see, e.g. Refs. [35] and references therein).

In light of the above results, the quantum geometry given by coherent states
is consistent with a inner matter source that does not collapse to a
singularity [19], similarly to what is found in the bootstrapped Newtonian
approach [36, 37]. The latter could therefore provide a compatible effective
description of the quantum black hole interior including (some of the)
nonlinearities. Moreover, the term inside square brackets in Eq. (3.13)
induces oscillations around the classical $V_{\rm N}$ whose effects on test
bodies could be observed at $r>R_{\rm H}$. Such oscillations are determined by
the (quantum) size $R_{\rm s}$ of the matter source and another important
observation is that the amplitude of these fluctuations around $V_{\rm N}$ in
the outer region (for $r>R_{\rm H}$) decreases for decreasing values of
$R_{\rm s}/R_{\rm H}$ (see Fig. 3). Therefore, one can always choose (finite)
values of $R_{\rm s}/R_{\rm H}$ so that the oscillations are too small to be
measured by a distant observer. On the other hand, one could interpret this
effect as a damping of transients as $R_{\rm s}$ shrinks and the collapse
proceeds inside the horizon.

Figure 4: Radius $r_{\rm H}$ of the quantum corrected horizon (solid line)
compared to the classical horizon $R_{\rm H}=2\,G_{\rm N}\,M$ (dotted line)
for different values of $R_{\rm s}$. Shaded regions cover points with $r_{\rm
H}<R_{\rm s}$ and do not correspond to black holes. Figure 5: Temperature
$T_{\rm Q}$ of the quantum corrected black hole (solid line) compared to the
Hawking result $T_{\rm H}=\hbar/4\,\pi\,R_{\rm H}$ (dotted line) for different
values of $R_{\rm s}$. Shaded regions cover points with $r_{\rm H}<R_{\rm s}$
and do not correspond to black holes.

As can be seen from Fig. 2, the quantum corrected metric still contains a
horizon when the cut-off scale $R_{\rm s}\lesssim R_{\rm H}$. The horizon
radius $r=r_{\rm H}$ is given by the solution of $2\,V_{\rm QN}=-1$, which can
be computed numerically for different values of $R_{\rm s}$ (see Fig. 4). In
particular, one can see that $r_{\rm H}=R_{\rm H}$ for very specific values of
$R_{\rm s}$, the largest one being $R_{\rm s}\simeq 0.53\,R_{\rm H}$, which is
very close to the preferred case obtained in Ref. [19]. A remarkable result is
that the horizon radius $r_{\rm H}$ shrinks for $R_{\rm s}$ approaching
$R_{\rm H}$. In fact, $r_{\rm H}=R_{\rm s}$ for $R_{\rm s}\simeq 0.6\,R_{\rm
H}$ and $r_{\rm H}$ would further vanish for $R_{\rm s}\simeq 0.7\,R_{\rm H}$.
This means that the material core cannot be too close in size to the classical
gravitational radius $R_{\rm H}$ in order for it to lie inside the actual
horizon. The shaded regions in Fig. 4 correspond to values of $r_{\rm
H}<R_{\rm s}$, which are therefore not black holes.

###  3.3 Thermodynamics

The above departure of $r_{\rm H}$ from the Schwarzschild radius $R_{\rm
H}=2\,G_{\rm N}\,M$ will also give rise to corrections for the horizon area
$\mathcal{A}_{\rm H}$ and Bekenstein-Hawking entropy [12],

$\displaystyle S_{\rm QBH}=\frac{\mathcal{A}_{\rm H}}{4\,\ell_{\rm
p}^{2}}=\frac{\pi\,r_{\rm H}^{2}}{\ell_{\rm p}^{2}}$ (3.23)

and for the black hole temperature [38]

$\displaystyle T_{\rm
Q}=\frac{\hbar\,\kappa}{2\,\pi}=\frac{\hbar}{2\,\pi}\left.\frac{\partial
V_{\rm Q}}{\partial r}\right|_{r=r_{\rm H}}\ ,$ (3.24)

where $\kappa$ is the surface gravity at the horizon. In particular, Fig. 4
shows that the quantum corrected horizon radius $r_{\rm H}\simeq R_{\rm H}$
for $R_{\rm s}\lesssim R_{\rm H}/2=G_{\rm N}\,M$. Therefore, the quantum
corrected gravitational entropy (3.23) remains very close to its classical
Schwarzschild value $S_{\rm BH}=4\,\pi\,M^{2}/\ell_{\rm p}^{2}$ as long as the
matter core is sufficiently smaller than $R_{\rm H}$.

The quantum corrected Hawking temperature can also be computed only
numerically and is displayed in Fig. 5. One can then see that quantum
corrected black holes are colder then their classical counterpart, that is
$T_{\rm Q}<T_{\rm H}$, if $R_{\rm s}\gtrsim R_{\rm H}/2$. In fact, $T_{\rm Q}$
would vanish for $R_{\rm s}\simeq 0.7\,R_{\rm H}$, which precisely correspond
to $r_{\rm H}=0$ but falls outside the range $r_{\rm H}>R_{\rm s}$
representing proper black holes. One could argue that the shaded regions in
Fig. 4 still represent some intermediate stage in the gravitational collapse
or will perhaps become relevant at the end of the Hawking evaporation. 888For
the late stage of the Hawking evaporation in the corpuscular picture, see e.g.
[29] and references therein.

##  4 Conclusions and outlook

In this work we started from the description of the static and spherically
symmetric Schwarzschild geometry in terms of the coherent state of a massless
scalar field on a reference flat spacetime, as previously employed, e.g. in
Refs. [13, 14, 16, 18]. We recall that, in order to grant the coherent state a
proper normalisation, we must relax the condition that its mean field
reproduces the classical behaviour everywhere inside the horizon of a black
hole [22]. In fact, it remains questionable whether the mean field
approximation should make sense at all in the interior of a black hole as a
(macroscopic) quantum object [6, 7, 19, 30], but here we took the less drastic
viewpoint that a mean field exists, although it cannot reproduce the expected
classical solution.

The weaker requirement in Eq. (3.10) that the coherent state yields the
classical behaviour in the exterior can be achieved by removing modes of
wavelength shorter than a UV cut-off $R_{\rm s}\lesssim R_{\rm H}$, where
$R_{\rm s}$ can be physically interpreted as the size of the (quantum) matter
source [19]. The expectation value of the gravitational potential then
displays oscillations around the classical $V_{\rm N}$, whose amplitudes
decrease for shrinking $R_{\rm s}$. We already commented that the specific
shape of such deviations will be affected by the form of the UV cut-off, but
the qualitative dependence on the scale $R_{\rm s}$ should be fairly well
captured by the simplest choice of a sharp cut-off in Eq. (3.12). One could
then interpret this effect as a decay in time due to the relative collapse of
the inner material core of size $R_{\rm s}$ with respect to the outer boundary
$R_{\infty}$ of the Newtonian region. Eq. (3.8) then tells us that a core that
shrinks down to $R_{\rm s}\simeq R_{\rm H}/65$ roughly corresponds to an
increase by 26 orders of magnitude in the outer Newtonian region of size
$R_{\infty}$. Assuming the last signals generated by the black hole formation
(when $R_{\rm s}$ crosses $R_{\rm H}$) are approximately located at the edge
of the outer Newtonian potential, and $R_{\infty}$ expands at the speed of
light, for a solar mass black hole with $R_{\rm H}\sim 1\,$km, this means a
time of around $10^{10}$ years, the present age of the Universe. Such a
consideration could be relevant for scenarios which predict a semiclassical
bounce [39]. It is also tempting to push the model towards smaller and smaller
black hole masses and conjecture that the oscillations shown in Fig. 3 grow
significantly and totally disrupt the classical picture for $R_{\rm s}\sim
R_{\rm H}\sim\ell_{\rm p}$. This would exclude the existence of stable
remnants of Planckian size at the end of the Hawking evaporation [38], in
agreement with other approaches among those in Refs. [7].

The simple analysis we presented in this work neglects the precise quantum
nature of the matter source and a better understanding of the interior might
be obtained from the bootstrapped Newtonian picture [16, 36, 37, 20] or other
modifications of the gravity-matter coupling [40]. For a quantum source, the
size $R_{\rm s}$ needs not be sharply defined and one should include the
effects of its uncertainty onto the gravitational state, like it was done by
considering the bootstrapped Newtonian potential generated by a source of
finite size in Ref. [18], or like in Ref. [19], where a bound on the
compactness was obtained from the quantisation of the geodesic motion of the
surface of a sphere of dust. The latter result in particular provides a clear
motivation for including the UV scale $R_{\rm s}\sim G_{\rm N}\,M$ in the
definition (3.10) of the coherent state of gravitons, although the use of a
sharp cut-off in Eq. (3.12) is justified mostly by the simplicity of the
calculation. Moreover, we note that the natural time evolution of quantum
states would imply that the oscillations around the expected classical
potential could make the region around the horizon rather fuzzy [41, 7], with
possibly relevant implications for the causal structure of astrophysical black
holes and their interaction with infalling matter.

We conclude by remarking that the modified thermodynamics described in Section
3.3 complements the results from Ref. [32], in which different forms of
information entropy where computed for the matter core of Ref. [19]. In
particular, it was found that matter entropy grows with the ADM mass $M$ like
$\ln(M^{\alpha})$, where the power $0.4<\alpha<0.9$. Since the quantum
corrected gravitational entropy (3.23) is still proportional to the horizon
area and grows like $M^{2}$ for sufficiently small matter cores, the entropy
of quantum black holes is expected to be dominated by the gravitational
contribution, like in their classical counterparts. It will be interesting to
investigate whether this conclusion is also supported by a direct calculation
of the information entropy for the coherent states of the geometry [42].

## Acknowledgments

I would like to thank A. Giusti and A. Platania for useful discussions. I am
partially supported by the INFN grant FLAG and my work has also been carried
out in the framework of activities of the National Group of Mathematical
Physics (GNFM, INdAM).

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