# Analytical computation of boundary integrals for the Helmholtz equation in
three dimensions

Nail A. Gumerov and Ramani Duraiswami
Perceptual Interfaces and Reality Lab, UMIACS,
University of Maryland, College Park, MD

###### Abstract

A key issue in the solution of partial differential equations via integral
equation methods is the evaluation of possibly singular integrals involving
the Green’s function and its derivatives multiplied by simple functions over
discretized representations of the boundary. For the Helmholtz equation, while
many authors use numerical quadrature to evaluate these boundary integrals, we
present analytical expressions for such integrals over flat polygons in the
form of infinite series. These can be efficiently truncated based on the
accurate error bounds, which is key to their integration in methods such as
the Fast Multipole Method.

##  1 Introduction

Boundary integral methods can be used to solve the Helmholtz equation either
in the direct form, e.g., [GD09], or the indirect form e.g., [GAD13]. The
boundary in these approaches was discretized with flat triangles or
quadrilaterals, and the discrete integrals evaluated. In this brief note, we
provide expressions for computation of boundary integrals for the Helmholtz
equation in three dimensions. Such integrals can be used both in the direct
and indirect boundary element methods (BEM). Particularly, in our paper
[GAD13] we did not provide the expressions for the integrals due to the lack
of space for short communications

Given surface $S$, the problem is to compute the single and double layer
potentials and their derivatives,

$\displaystyle L_{k}\left[\sigma\right]\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\int_{S}\sigma\left(\mathbf{r}^{\prime}\right)G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)dS\left(\mathbf{r}^{\prime}\right),$
(1) $\displaystyle M_{k}\left[\mu\right]\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\int_{S}\mu\left(\mathbf{r}^{\prime}\right)\frac{\partial
G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)}{\partial
n\left(\mathbf{r}^{\prime}\right)}dS\left(\mathbf{r}^{\prime}\right),$
$\displaystyle\mathbf{L}_{k}^{\prime}\left[\sigma\right]\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\nabla\int_{S}\sigma\left(\mathbf{r}^{\prime}\right)G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)dS\left(\mathbf{r}^{\prime}\right),$
$\displaystyle\mathbf{M}_{k}^{\prime}\left[\mu\right]\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\nabla\int_{S}\mu\left(\mathbf{r}^{\prime}\right)\frac{\partial
G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)}{\partial
n\left(\mathbf{r}^{\prime}\right)}dS\left(\mathbf{r}^{\prime}\right).$

where $\sigma$ and $\mu$ are the single and double layer densities,
$\mathbf{n}$ is the normal to the surface, and
$G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)$ is the free-space Green
function for given wavenumber $k$,

$G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\frac{e^{ik\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{4\pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}.$
(2)

Figure 1: Boundary integrals over flat polygons: notation. The polygon is
defined by an ordered tuple of points $\left({\bf r}_{1},{\bf
r}_{2},\ldots,{\bf r}_{n},{\bf r}_{1}\right)$, which are arranged according to
the right hand rule so that the normal n points outward. The approach to
computing the integrals uses the Gauss’ divergence theorem and reduces the
surface integral to a contour integral which is evaluated over the line
segments.

###  1.1 Computation of boundary integrals

In center panel collocation with the boundary discretized by triangles, the
integrals are approximated by $N$ integrals over flat patches (e.g., triangles
$\Delta_{l}$) centered at $\mathbf{r}_{l}^{(c)}$,

$\displaystyle L_{k}\left[\sigma\right]\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\sum_{l^{\prime}=1}^{N}\int_{\Delta_{l^{\prime}}}\sigma\left(\mathbf{r}^{\prime}\right)G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)dS\left(\mathbf{r}^{\prime}\right)\approx\sum_{l^{\prime}=1}^{N}L_{ll^{\prime}}\sigma_{l^{\prime}},$
(3) $\displaystyle\sigma_{l^{\prime}}$ $\displaystyle=$
$\displaystyle\sigma\left(\mathbf{r}_{l^{\prime}}^{(c)}\right),\quad
L_{ll^{\prime}}=\int_{\Delta_{l^{\prime}}}G_{k}\left(\mathbf{r},\mathbf{r}^{\prime}\right)dS\left(\mathbf{r}^{\prime}\right).$

Similar expressions can be written for other boundary integrals. For a flat
patch $\Delta$ with vertices ($\mathbf{r}_{1},\mathbf{...},\mathbf{r}_{n})$
defining the polygon as a right-hand oriented contour $C$ with normal
$\mathbf{n}$ (see Fig. 1), we need to compute the integrals

$\displaystyle L_{k\Delta}\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\int_{\Delta}G_{k}\left(\mathbf{r,r}^{\prime}\right)dS\left(\mathbf{r}^{\prime}\right),\quad
M_{k\Delta}\left(\mathbf{r}\right)=\int_{\Delta}\mathbf{n}\cdot\nabla_{\mathbf{r}^{\prime}}G_{k}\left(\mathbf{r}^{\prime}\mathbf{,r}\right)dS\left(\mathbf{r}^{\prime}\right),$
(4) $\displaystyle\mathbf{L}_{k\Delta}^{\prime}\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\nabla_{\mathbf{r}}\int_{\Delta}G_{k}\left(\mathbf{r}^{\prime}\mathbf{,r}\right)dS\left(\mathbf{r}^{\prime}\right),\quad\mathbf{M}_{k\Delta}^{\prime}\left(\mathbf{r}\right)=\nabla_{\mathbf{r}}\int_{\Delta}\mathbf{n}\cdot\nabla_{\mathbf{r}^{\prime}}G_{k}\left(\mathbf{r}^{\prime}\mathbf{,r}\right)dS\left(\mathbf{r}^{\prime}\right).$

Note that

$\displaystyle
G_{k}\left(\mathbf{r,r}^{\prime}\right)=\nabla_{s}^{\prime}\cdot\left[\mathbf{\rho}f_{k}\left(\rho;h\right)\right],\quad
f_{k}\left(\rho;h\right)=\frac{e^{ikr}-e^{ik\left|h\right|}}{4\pi
ik\rho^{2}},$ (5) $\displaystyle h=\mathbf{n\cdot
r,\quad\rho=r}^{\prime}-\mathbf{r+n}h,\quad
r=\left|\mathbf{r}-\mathbf{r}^{\prime}\right|=\sqrt{\rho^{2}+h^{2}},\quad\rho=\left|\mathbf{\rho}\right|,$

where $\nabla_{s}^{\prime}$ is the surface gradient operator with respect to
$\mathbf{r}^{\prime}$ and $h$ is the projection of $\mathbf{r}$ to the normal.
Using the Gauss divergence theorem, we obtain

$L_{k\Delta}\left(\mathbf{r}\right)=\sum_{j=1}^{n}I_{j},\quad
I_{j}=\int_{C_{j}}\left(\mathbf{n}_{j}^{\prime}\cdot\mathbf{\rho}\right)f_{k}\left(\rho;h\right)dC,$
(6)

where $C_{j}$ are line segments constituting the triangle sides and
$\mathbf{n}_{j}^{\prime}\mathbf{\ }$are the normals to these segments outward
to the triangle. Expressions for $I_{j}$ can be simplified to

$\displaystyle I_{j}$ $\displaystyle=$ $\displaystyle
H\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-H\left(-x^{\prime},y^{\prime},z^{\prime}\right),\quad
j=1,2,3,$ (7) $\displaystyle H\left(x,y^{\prime},z^{\prime}\right)$
$\displaystyle=$ $\displaystyle-z^{\prime}\int
f\left(\rho;y^{\prime}\right)dx=\frac{-z^{\prime}}{4\pi
ik}\int\frac{e^{ikr}-e^{iky^{\prime}}}{x^{2}+z^{\prime 2}}dx,\quad
r=\sqrt{x^{2}+y^{\prime 2}+z^{\prime 2}},\quad y^{\prime}=\left|h\right|,$

where $l_{j}$ is the length of segment $C_{j}$,
$\left(x^{\prime},y^{\prime},z^{\prime}\right)$ are the coordinates of the
evaluation point $\mathbf{r}$ in the local right-hand oriented Euclidean
reference frame centered at the triangle vertex, which $x$-axis (basis vector
$\mathbf{i}_{x}^{\prime}$) is directed along the segment, and the other basis
vectors are $\mathbf{i}_{y}^{\prime}=\mathbf{n}$,
$\mathbf{i}_{z}^{\prime}=\mathbf{n}_{j}^{\prime}$. Expressions for the surface
integrals can be obtained by summation of the line integrals:

$\displaystyle L_{k\Delta}\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}I_{j}\left(x_{j}^{\prime},h,z_{j}^{\prime}\right),\quad
I_{j}\left(x^{\prime},y^{\prime},z^{\prime}\right)=H\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-H\left(-x^{\prime},y^{\prime},z^{\prime}\right),$
(8) $\displaystyle\mathbf{L}_{k\Delta}^{\prime}\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\mathbf{I}_{j}^{\prime}\left(x_{j}^{\prime},h,z_{j}^{\prime}\right),\quad\mathbf{I}_{j}^{\prime}\left(x^{\prime},y^{\prime},z^{\prime}\right)=\mathbf{H}^{\prime}\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-\mathbf{H}^{\prime}\left(-x^{\prime},y^{\prime},z^{\prime}\right),\quad$
$\displaystyle M_{k\Delta}\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}J_{j}\left(x_{j}^{\prime},h,z_{j}^{\prime}\right),\quad
J_{j}\left(x^{\prime},y^{\prime},z^{\prime}\right)=K\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-K\left(-x^{\prime},y^{\prime},z^{\prime}\right),$
$\displaystyle\mathbf{M}_{k\Delta}^{\prime}\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\mathbf{J}_{j}^{\prime}\left(x_{j}^{\prime},h,z_{j}^{\prime}\right),\quad\mathbf{J}_{j}^{\prime}\left(x^{\prime},y^{\prime},z^{\prime}\right)=\mathbf{K}^{\prime}\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-\mathbf{K}^{\prime}\left(-x^{\prime},y^{\prime},z^{\prime}\right),$

where

$\mathbf{H}^{\prime}=-\mathbf{i}_{x}^{\prime}\frac{\partial H}{\partial
x}+\mathbf{i}_{y}^{\prime}\frac{\partial H}{\partial
y^{\prime}}+\mathbf{i}_{z}^{\prime}\frac{\partial H}{\partial
z^{\prime}}\mathbf{,\quad}K=-\frac{\partial H}{\partial
y^{\prime}},\quad\mathbf{K}^{\prime}=\mathbf{i}_{x}^{\prime}\frac{\partial^{2}H}{\partial
x\partial y^{\prime}}-\mathbf{i}_{y}^{\prime}\frac{\partial^{2}H}{\partial
y^{\prime 2}}-\mathbf{i}_{z}^{\prime}\frac{\partial^{2}H}{\partial
z^{\prime}\partial y^{\prime}}.$ (9)

The above expressions are obtained for $h\geqslant 0$. For negative $h$ the
following symmetry relations can be applied

$\displaystyle I_{j}\left(x^{\prime},-y^{\prime},z^{\prime}\right)$
$\displaystyle=$ $\displaystyle
H\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-H\left(-x^{\prime},y^{\prime},z^{\prime}\right),$
(10)
$\displaystyle\mathbf{I}_{j}^{\prime}\left(x^{\prime},-y^{\prime},z^{\prime}\right)$
$\displaystyle=$
$\displaystyle\widehat{\mathbf{H}}^{\prime}\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-\widehat{\mathbf{H}}^{\prime}\left(-x^{\prime},y^{\prime},z^{\prime}\right),\quad$
$\displaystyle J_{j}\left(x^{\prime},-y^{\prime},z^{\prime}\right)$
$\displaystyle=$
$\displaystyle\widehat{K}\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-\widehat{K}\left(-x^{\prime},y^{\prime},z^{\prime}\right),\quad$
$\displaystyle\mathbf{J}_{j}^{\prime}\left(x^{\prime},-y^{\prime},z^{\prime}\right)$
$\displaystyle=$
$\displaystyle\widehat{\mathbf{K}}^{\prime}\left(l_{j}-x^{\prime},y^{\prime},z^{\prime}\right)-\widehat{\mathbf{K}}^{\prime}\left(-x^{\prime},y^{\prime},z^{\prime}\right),\quad$

where

$\widehat{\mathbf{H}}^{\prime}=-\mathbf{i}_{x}^{\prime}\frac{\partial
H}{\partial x}-\mathbf{i}_{y}^{\prime}\frac{\partial H}{\partial
y^{\prime}}+\mathbf{i}_{z}^{\prime}\frac{\partial H}{\partial
z^{\prime}}\mathbf{,\quad}\widehat{K}=\frac{\partial H}{\partial
y^{\prime}},\quad\widehat{\mathbf{K}}^{\prime}=-\mathbf{i}_{x}^{\prime}\frac{\partial^{2}H}{\partial
x\partial y^{\prime}}-\mathbf{i}_{y}^{\prime}\frac{\partial^{2}H}{\partial
y^{\prime 2}}+\mathbf{i}_{z}^{\prime}\frac{\partial^{2}H}{\partial
z^{\prime}\partial y^{\prime}}.$ (11)

####  1.1.1 Primitives

The use of the BEM with center panel collocation presumes that the triangle
size is much smaller than the wavelength, so
$k\left|\mathbf{r}-\mathbf{r}_{j}\right|\ll\pi.$ The triangle inequality then
shows

$k\left|r-r_{0}\right|=k\left|\left|\mathbf{y}-\mathbf{x}\right|-\left|\mathbf{y}-\mathbf{x}_{j}\right|\right|\leqslant
k\left|\mathbf{x}-\mathbf{x}_{j}\right|\ll\pi,\quad r_{0}=\sqrt{x^{\prime
2}+y^{\prime 2}+z^{\prime 2}}.$ (12)

For any integration point, we can use the following truncated expansion of the
exponent

$e^{ikr}=e^{ikr_{0}}e^{ik\left(r-r_{0}\right)}=e^{ikr_{0}}\sum_{m=0}^{p-1}\frac{\left(ik\right)^{m}}{m!}\left(r-r_{0}\right)^{m}+O\left(\frac{1}{p!}\max\left(k^{p}\left|r-r_{0}\right|^{p}\right)\right).$
(13)

Using the binomial formula for $\left(r-r_{0}\right)^{m}$, substituting
expansion into Eq. (7), and neglecting the truncation error term, we obtain

$H\left(x,y^{\prime},z^{\prime}\right)=\frac{1}{4\pi
ik}\left[e^{iky^{\prime}}k_{0}\left(x,y^{\prime},z^{\prime}\right)-e^{ikr_{0}}\sum_{l=0}^{p-1}A_{l}^{(p)}k_{l}\left(x,y^{\prime},z^{\prime}\right)\right],$
(14)
$A_{l}^{(p)}=\frac{\left(ik\right)^{l}}{l!}a_{p-l}\left(-ikr_{0}\right),\quad
a_{l}\left(\xi\right)=\sum_{m=0}^{l-1}\frac{\xi^{m}}{m!}.\quad l=0,...,p-1,$
(15)

and the functions

$k_{m}\left(x,y^{\prime},z^{\prime}\right)=z^{\prime}\int\frac{r^{m}}{x^{2}+z^{\prime
2}}dx,\quad r=\sqrt{x^{2}+y^{\prime 2}+z^{\prime 2}},\quad m=0,\pm 1,\pm
2,...,$ (16)

can be recursively computed as described in the Appendix.

Computation of the derivatives of the single and double layer potentials
requires derivatives of the primitives with respect to any of their three
arguments. This can be done using a similar method, with the only notice,
that, first, the derivative should be computed using integral form (7) and
then truncated expansion (13) should be inserted there. Dropping these
derivations we present only the final results, which can be written as follows
(for all functions arguments are $\left(x,y^{\prime},z^{\prime}\right)$).

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{ik}\left(R-S\right),\quad\frac{\partial H}{\partial
x}=-z^{\prime}T,\quad\frac{\partial H}{\partial
y^{\prime}}=R-V,\quad\frac{\partial H}{\partial z^{\prime}}=xT-U,$ (17)
$\displaystyle\frac{\partial^{2}H}{\partial x\partial y^{\prime}}$
$\displaystyle=$ $\displaystyle-z^{\prime}P,\quad\frac{\partial^{2}H}{\partial
y^{\prime 2}}=-k^{2}H+z^{\prime}Q,\quad\frac{\partial^{2}H}{\partial
z^{\prime}\partial y^{\prime}}=xP-y^{\prime}Q,$ $\displaystyle P$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi}\frac{1}{x^{2}+z^{\prime
2}}\left(\frac{y^{\prime}}{r}e^{ikr}-e^{iky^{\prime}}\right),\quad
R=\frac{1}{4\pi}e^{iky^{\prime}}k_{0},\quad T=\frac{1}{4\pi
ik}\frac{e^{ikr}-e^{iky^{\prime}}}{x^{2}+z^{\prime 2}},$ (18) $\displaystyle
S$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}e^{ikr_{0}}\sum_{l=0}^{p-1}A_{l}^{(p)}k_{l},\quad
U=\frac{1}{4\pi}e^{ikr_{0}}\sum_{l=0}^{p-1}A_{l}^{(p)}i_{l-1},\quad
V=\frac{y^{\prime}}{4\pi}e^{ikr_{0}}\sum_{l=0}^{p-1}A_{l}^{(p)}k_{l-1},$
$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}e^{ikr_{0}}\sum_{l=0}^{p-1}A_{l}^{(p)}\left(iki_{l-2}-i_{l-3}\right),\quad$

where functions

$i_{m}\left(x;a\right)=\int r^{m}dx,\quad r=\sqrt{x^{2}+a^{2}},\quad
a^{2}=y^{\prime 2}+z^{\prime 2},\quad m=0,\pm 1,\pm 2,...,$ (19)

can be recursively computed as described in the Appendix.

We note now that in computation of primitives some cases with removable or
non-removable (strong) singularities can appear. Strong singularities appear
only for the cases when the evaluation point belongs to contour $C$, in which
case, indeed the hypersingular integral blows up (in fact, cancellation of
these singularities happens when summing up integrals over neighbor elements;
such situation is typical for evaluation of integrals which exist in a sense
of principal Cauchy value, but blow up if the integration bounds coincide with
singular points). However, this case is never encountered in the center panel
collocation method, as the center of the element never belongs to its edge.
Field points also are non-singular since they do not belong to the boundary.
However, such singularities should be avoided in evaluation of spatial
gradients of the double layer potential (e.g. evaluation point should not
approach the boundary closer than some small distance consistent with the flat
constant panel discretization errors.

Note that all singularities occur only if $z^{\prime}=0$. In this case,
however,

$H=\frac{\partial H}{\partial x}=\frac{\partial H}{\partial
y^{\prime}}=\frac{\partial^{2}H}{\partial x\partial
y^{\prime}}=\frac{\partial^{2}H}{\partial y^{\prime 2}}=0,$ (20)

so these functions should be set to zero immediately. Other derivatives are
regular and computable if $x\neq 0$ and $y^{\prime}\neq 0.$ Strong singular
case corresponds to $y^{\prime}=0$ and $x=0$, which should be avoided as the
program should avoid computation of integrals at edge evaluation points where
$y^{\prime}=0$ and $x=0$ appears on the integration path. Formulae for
limiting cases $x=0$ and $y\neq 0$ and $x\neq 0$ and $y=0$ can be found
straightforward from the above analytical expressions.

In case $k\rightarrow 0$ the Green function of the Helmholtz equation smoothly
transits to the Green function of the Laplace equation. In this case the
series related toe the expansions of the exponents disappear and all
primitives can be expressed via elementary functions:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{y^{\prime}k_{0}+k_{1}}{4\pi},\quad\frac{\partial
H}{\partial x}=-\frac{z^{\prime}}{4\pi}\frac{r-y^{\prime}}{x^{2}+z^{\prime
2}},\quad\frac{\partial H}{\partial
y^{\prime}}=\frac{k_{0}-y^{\prime}k_{-1}}{4\pi},$ (21)
$\displaystyle\frac{\partial H}{\partial z^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{x}{4\pi}\frac{r-y^{\prime}}{x^{2}+z^{\prime
2}}-\frac{i_{-1}}{4\pi},\quad\frac{\partial^{2}H}{\partial x\partial
y^{\prime}}=-\frac{z^{\prime}}{4\pi}\frac{1}{\left(x^{2}+z^{\prime
2}\right)}\left(\frac{y^{\prime}}{r}-1\right),$
$\displaystyle\frac{\partial^{2}H}{\partial y^{\prime 2}}$ $\displaystyle=$
$\displaystyle-\frac{z^{\prime}i_{-3}}{4\pi},\quad\frac{\partial^{2}H}{\partial
z^{\prime}\partial y^{\prime}}=\frac{x}{4\pi}\frac{1}{\left(x^{2}+z^{\prime
2}\right)}\left(\frac{y^{\prime}}{r}-1\right)+\frac{y^{\prime}i_{-3}}{4\pi}.$

These expressions are also useful for understanding of the singularities in
general case $k\neq 0$, as these singularities are caused by singularities of
the Laplacian kernel can be efficiently computed as described in the Appendix.

##  Appendix A Elementary integral computations

###  A.1 Computation of integrals $i_{m}$

In this appendix we show how integrals

$i_{m}\left(x;a\right)=\int r^{m}dx,\quad r=\sqrt{x^{2}+a^{2}},\quad
a^{2}=y^{\prime 2}+z^{\prime 2},\quad m=0,\pm 1,\pm 2,...$ (22)

can be computed analytically. For small values of even and odd $m$ these
integrals can be computed

$\displaystyle i_{0}\left(x;a\right)$ $\displaystyle=$ $\displaystyle\int
dx=x,\quad i_{-1}\left(x;a\right)=\int r^{-1}dx=\ln\left|r+x\right|,$ (23)
$\displaystyle i_{-2}\left(x;a\right)$ $\displaystyle=$ $\displaystyle\int
r^{-2}dx=\frac{1}{a}\arctan\frac{x}{a},\quad i_{-3}\left(x;a\right)=\int
r^{-3}dx=\frac{x}{a^{2}r}.$

We have the recurrence:

$\displaystyle i_{m+2}$ $\displaystyle=$ $\displaystyle\int
r^{m+2}dx=xr^{m+2}-\left(m+2\right)\int x^{2}r^{m}dx$ $\displaystyle=$
$\displaystyle xr^{m+2}-\left(m+2\right)i_{m+2}+(m+2)a^{2}i_{m}.$
$i_{m+2}=\frac{xr^{m+2}}{m+3}+\frac{m+2}{m+3}a^{2}i_{m},\quad m\neq-3.$ (25)

The special case $m=-3$ is not so important, since we know for
$i_{-1}\left(x;a\right).$ We need integral values for $m\geqslant-3$, and have
explicit expressions for all required non-positive $m$ and we can recursively
find all positive $m$ starting the recurrence from $m=0$ for even $m$ and from
$m=-1$ for odd $m$. For even $m=2n$ we can evaluate the integrals as series
explicitly using the binomial expansion

$\\!\\!\\!\\!\\!\\!\\!\\!i_{2n}\left(x;a\right)=\int
r^{2n}dx=\int\left(x^{2}+a^{2}\right)^{n}dx=\sum_{l=0}^{n}\frac{n!}{l!(n-l)!}a^{2(n-l)}\int
x^{2l}dx=\sum_{l=0}^{n}\frac{n!a^{2(n-l)}x^{2l+1}}{l!(n-l)!\left(2l+1\right)}.$
(26)

For odd $m=2n+1$ we have the recurrence

$i_{2n+1}=\frac{1}{2n+2}\left[xr^{2n+1}+\left(2n+1\right)a^{2}i_{2n-1}\right].$
(27)

Expanding this down to $i_{-1}$ we obtain

$i_{2n+1}\left(x;a\right)=x\sum_{l=0}^{n}\frac{2^{2l}\left(2n+2\right)!(l!)^{2}}{2^{2n+2}\left(2l+1\right)!\left[\left(n+1\right)!\right]^{2}}r^{2l+1}a^{2(n-l)}+\frac{\left(2n+2\right)!}{2^{2n+2}\left[\left(n+1\right)!\right]^{2}}a^{2n+2}\ln\left|r+x\right|.$
(28)

Note that primitives $i_{m}\left(x;a\right)$ may have singularities only if
$a=0.$ In such cases we have

$i_{m}\left(x;0\right)=\frac{x\left|x\right|^{m}}{m+1},\quad m\neq-1,\quad
i_{-1}\left(x;0\right)=\mbox{sgn}(x)\ln\left|x\right|.$ (29)

This shows that all functions with negative index are singular at $x=0$ and
this should be accounted for in formulae involving them.

###  A.2 Computation of integrals $k_{m}$

In this appendix we show how integrals

$k_{m}\left(x;y^{\prime},z^{\prime}\right)=z^{\prime}\int\frac{r^{m}}{x^{2}+z^{\prime
2}}dx,\quad r=\sqrt{x^{2}+y^{\prime 2}+z^{\prime 2}},\quad m=0,\pm 1,\pm
2,...$ (30)

can be computed analytically. Note, that for $y^{\prime}=0$ these reduce to
the integrals $i_{m}$:

$k_{m}\left(x;0,z^{\prime}\right)=z^{\prime}\int\frac{r^{m}}{r^{2}}dx=z^{\prime}i_{m-2}\left(x;\left|z\right|\right).$
(31)

So we only need consider the case $y^{\prime}\neq 0.$ We derive recurrences
for all needed $k_{m}$ and find initial values to use them. We have

$\displaystyle k_{m+2}$ $\displaystyle=$ $\displaystyle
z^{\prime}\int\frac{r^{m+2}}{x^{2}+z^{\prime
2}}dx=z^{\prime}\int\frac{\left(x^{2}+z^{\prime 2}\right)+y^{\prime
2}}{x^{2}+z^{\prime 2}}\rho^{m}dx$ $\displaystyle=$ $\displaystyle
z^{\prime}i_{m}\left(x;\sqrt{y^{\prime 2}+z^{\prime 2}}\right)+y^{\prime
2}k_{m}.$

With known expressions for the initial values (odd and even), all integrals
can be computed recursively. For this we have

$\displaystyle k_{0}\left(x;y^{\prime},z^{\prime}\right)$ $\displaystyle=$
$\displaystyle\frac{z^{\prime}}{\left|z^{\prime}\right|}\arctan\frac{x}{\left|z^{\prime}\right|}=\mbox{sgn}(z^{\prime})\arctan\frac{x}{\left|z^{\prime}\right|}$
(33) $\displaystyle k_{1}\left(x;y^{\prime},z^{\prime}\right)$
$\displaystyle=$
$\displaystyle\frac{y^{\prime}z^{\prime}}{\left|z^{\prime}\right|}\arctan\frac{y^{\prime}x}{\left|z^{\prime}\right|r}+z^{\prime}\ln\left|r+x\right|=y^{\prime}\mbox{sgn}(z^{\prime})\arctan\frac{y^{\prime}x}{\left|z^{\prime}\right|r}+z^{\prime}\ln\left|r+x\right|.$

For the boundary integrals we need only one negative value $m=-1.$ We can find
this value from the same recurrence and known $k_{1}:$

$\displaystyle k_{-1}\left(x;y^{\prime},z^{\prime}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{y^{\prime
2}}\left[k_{1}\left(x;y^{\prime},z^{\prime}\right)-z^{\prime}i_{-1}\left(x;\sqrt{y^{\prime
2}+z^{\prime 2}}\right)\right]$ $\displaystyle=$
$\displaystyle\frac{z^{\prime}}{y^{\prime}\left|z^{\prime}\right|}\arctan\frac{y^{\prime}x}{\left|z^{\prime}\right|r}=\frac{\mbox{sgn}(z^{\prime})}{y^{\prime}}\arctan\frac{y^{\prime}x}{\left|z^{\prime}\right|r}.$

Note that $y^{\prime}k_{-1}\left(x;y^{\prime},z^{\prime}\right)$ entering the
primitive expressions are not singular even when $z^{\prime}$ and $y^{\prime}$
approaches zero, while depend on the path (ratio $y^{\prime}/z^{\prime}$).

## References

  * [GAD13] N Gumerov, R Adelman, and R Duraiswami.  Fast multipole accelerated indirect boundary elements for the Helmholtz equation.  Proc. Meet. Acoust., 2013.
  * [GD09] Nail A Gumerov and Ramani Duraiswami.  A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation.  J. Acoust. Soc. Am., 125(1):191–205, January 2009.


# Overspinning a rotating black hole in semiclassical gravity with type-A
trace anomaly

Jie Jiang<EMAIL_ADDRESS>College of Education for the Future,
Beijing Normal University, Zhuhai 519087, China  Ming Zhang
<EMAIL_ADDRESS>(Corresponding author) Department of Physics, Jiangxi
Normal University, Nanchang 330022, China

###### Abstract

Recently, Fernandes discovered an analytic solution for rotating black holes
in semiclassical gravity induced by the trace anomaly. These solutions exhibit
some distinctive characteristics, including a non-spherically symmetric event
horizon and violations of the Kerr bound. As a crucial assumption to uphold
causality in spacetime, we investigate the validity of the weak cosmic
censorship conjecture (WCCC) within this class of solutions with type-A trace
anomaly by introducing a test particle on the equatorial plane. Our study
reveals three distinct mechanisms that can potentially destroy the event
horizon, leading to a violation of the WCCC. Our findings indicate that, with
the exception of extremal Kerr, static extremal, and static singular black
holes, the WCCC may be violated under the first-order perturbation of the test
particle. These results suggest the need for further exploration of
modifications to the behavior of the test particle under quantum effects in
order to address the violation of the WCCC in this system.

##  I Introduction

The Hawking-Penrose singularity theorem states that gravitational collapse
inevitably ends up in a spacetime singularity Penrose:1964wq ; Hawking:1970zqf
. However, these singularities in gravitational theories can lead to
unpredictable results, making it difficult for us to understand the rules of
the universe. Penrose suggested a solution known as the weak cosmic censorship
conjecture (WCCC) Penrose:1969pc , which proposes that these singularities
must be hidden by an event horizon, maintaining the predictability of
gravitational theories. Resolving this issue is crucial for understanding
classical gravitational theory and could offer significant insights into the
nature of the universe.

The WCCC has been tested in various ways, such as through numerical
simulations with collapsing matter fields and disturbed black holes, and in
simulations of merging black holes in higher dimensions Christodoulou:1984mz ;
Ori:1987hg ; Shapiro:1991zza ; Lemos:1991uz ; Choptuik:1992jv ;
Corelli:2021ikv ; Eperon:2019viw ; Crisford:2017zpi ; Figueras:2017zwa ;
Figueras:2015hkb ; Lehner:2010pn ; Hertog:2003zs ; Andrade:2020dgc ;
Andrade:2019edf ; Andrade:2018yqu ; Sperhake:2009jz . In 1974, Wald designed a
gedanken experiment Wald:1974 , which demonstrated that an extremal Kerr-
Newman black hole could resist destruction from a test particle under the
first-order approximation from the particle perturbation. Then, Hubeny
expanded it to consider near-extremal black holes and second-order
perturbations, suggesting these black holes could potentially be destroyed
Hubeny:1998ga . Many follow-up studies agreed with this finding
deFelice:2001wj ; Hod:2002pm ; Jacobson:2010iu ; Chirco:2010rq ; Saa:2011wq ;
Gao:2012ca . However, as Hubeny Hubeny:1998ga  discussed, to confirm whether
black holes actually disintegrate, all second-order effects must be taken into
account. In 2017, a more complex version of the gedanken experiment was
proposed by Sorce and Wald based on the Noether charge method, which
considered the full dynamics of spacetime and perturbation matters
Sorce:2017dst  and showed that a near-extremal Kerr-Newman black hole cannot
be destroyed under second-order perturbation when the matters satisfy the null
energy condition. Additionally, field scattering is another method used to
examine the WCCC across different gravitational systems Semiz:2005gs ;
Gwak:2018akg ; Gwak:2021tcl ; Liang:2020hjz ; Natario:2016bay ;
Goncalves:2020ccm ; Gwak:2019rcz ; Yang:2020iat ; Yang:2020czk ; Feng:2020tyc
; Yang:2022yvq

The exploration of quantum phenomena offers us a deeper understanding of the
universal laws of physics. Trace anomaly is one such phenomenon that emerges
in a fundamentally conformally invariant classical theory due to the breaking
of conformal symmetry by one-loop quantum corrections Capper:1974ic ;
Duff:1993wm . This results in the renormalization of the stress-energy tensor,
leading to a non-zero trace. Interestingly, this trace is independent of the
quantum state of the quantum fields and solely depends on the local curvature
of spacetime, marking it as a general characteristic of quantum theories in
gravitational fields Anderson:2007eu ; Mottola:2022tcn ; Mottola:2006ew ;
Mottola:2016mpl .

In a four-dimensional spacetime, the trace anomaly can be expressed in terms
of the square of the Weyl tensor $C$ and the Gauss-Bonnet scalar
$\mathcal{G}$, which are commonly referred to as type-A and type-B anomalies
respectively Deser:1993yx . This can be captured by the following equation,

$\displaystyle g^{\mu\nu}\left\langle
T_{\mu\nu}\right\rangle=\frac{\beta}{2}C^{2}-\frac{\alpha}{2}\mathcal{G}\,.$
(1)

When contemplating modifications to General Relativity, the contributions of
the trace anomaly are critical. They are anticipated to produce observable
macroscopic effects Mottola:2022tcn ; Mottola:2006ew ; Mottola:2016mpl ,
thereby necessitating their inclusion in the low-energy effective field theory
of gravity.

Figure 1:  The $\left(\alpha/M^{2},a/M\right)$ diagram illustrating the domain
of existence for black hole solutions. The shaded region represents the
parameter space where black hole solutions exist. The boundary of this region
is determined by three distinct boundaries. The type-I boundary (red) is
defined by $\Delta_{\text{equator}}^{\text{min}}=0$, the type-II boundary
(blue) is defined by $\Delta_{\text{poles}}^{\text{min}}=0$, and the type-III
boundary (red) corresponds to the singular black hole solutions defined by
$\Delta_{\text{equator}}^{r_{s}}=0$.

Utilizing the semi-classical approach allows us to account for the
backreaction of quantum fields and their influence on spacetime geometry. This
process transforms the Einstein equation into

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G\left\langle
T_{\mu\nu}\right\rangle\,.$ (2)

A notable challenge when studying the backreaction of quantum fields lies in
the typically unknown expectation value of the renormalized stress-energy
tensor even in the static and spherically symmetric system Christensen:1977jc
; Ho:2018fwq ; Abedi:2015yga . To overcome this challenge, Ref. Cai:2009ua
introduces an additional condition: the geometry should rely solely on a
single free function, which effectively establishes an additional equation of
state for the stress-energy tensor. Through the adoption of this methodology,
and focusing solely on the type-A anomaly $(\beta=0)$, Ref. Cai:2009ua
manages to fully derive the renormalized stress-energy tensor and obtain an
analytic static and spherically symmetric black hole solution from the semi-
classical Einstein equations. Intriguingly, these solutions show a logarithmic
correction to their entropy, aligning with the expectation that primary
quantum corrections to black hole entropy should be logarithmic Page:2004xp ;
Sen:2012dw . Most recently, an analytic stationary and axially-symmetric black
hole solution to the semiclassical Einstein equations induced by the trace
anomaly has been found in Ref. Fernandes:2023vux . Unlike conventional
stationary black hole solutions, this new solution presents several distinct
features, including the violation of the Kerr bound and an event horizon that
lacks spherical symmetry, leading to a mismatch between the event horizon and
the Killing horizon.

As a fundamental assumption for ensuring causality in spacetime, a natural
question arises as to whether the WCCC still holds when considering quantum
effects such as the trace anomaly. Therefore, the aim of this paper is to
examine the WCCC in the rotating stationary solutions obtained in Ref.
Fernandes:2023vux . Specifically, we will investigate the possibility of
destroying the event horizon by dropping a test particle into the black hole,
thereby forming a naked singularity. The structure of this paper is as
follows. In Sec. II, we will introduce rotating black hole solutions in
semiclassical gravity with type-A trace anomaly and discuss their spacetime
structures. In Sec. III, we will first present the equations of motion for
test particles on the equatorial plane and derive the condition for test
particle to enter the black hole. We will then discuss whether the black hole
can be destroyed under the first-order approximation of the perturbation
caused by the test particle, thus violating the WCCC. Finally, in Sec. IV, we
will provide our conclusions and summarize the findings of our paper.

##  II Rotating black hole solutions in semiclassical gravity

In this paper, we focus exclusively on type-A anomalies (i.e., $\beta=0$) in
Einstein gravity. Most recently, Ref. Fernandes:2023vux  provides the solution
to the semiclassical Einstein equations with trace anomaly, which corresponds
to an asymptotically flat and rotating spacetime. In the ingoing Kerr-like
coordinates $x^{\mu}=(v,r,\theta,\varphi)$, the metric is given by

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\frac{2\mathcal{M}(r,\theta)r}{\Sigma}\right)\left(dv-a\sin^{2}\theta
d\varphi\right)^{2}$ (3) $\displaystyle+2\left(dv-a\sin^{2}\theta
d\varphi\right)\left(dr-a\sin^{2}\theta d\varphi\right)$
$\displaystyle+\Sigma\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right),$

where

$\displaystyle\mathcal{M}(r,\theta)$
$\displaystyle=\frac{2M}{1+\sqrt{1-{8\alpha r\xi M}/{\Sigma^{3}}}}\,,$ (4)
$\displaystyle\Sigma$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta\,,$
$\displaystyle\xi$ $\displaystyle=r^{2}-3a^{2}\cos^{2}\theta\,.$

Here, after assuming the spacetime is asymptotically flat, the symbol $M$
represents an integration constant. The mass and angular momentum of the
spacetime can be obtained by using the Komar integral, which yields $M$ and
$J=Ma$, respectively.

Figure 2: Diagrams illustrating the process of destroying an event horizon by
increasing the angular momentum in parameter-region I ($-1\leq\alpha/M^{2}\leq
0$). The profile shows the coordinate location of the inner horizon
(represented by the red line) and the event horizon (represented by the blue
line) as a function of the angular coordinate $\theta$. We consider a black
hole with $\alpha/M^{2}=-0.3$ and varying values of $a/M^{2}$. Here,
$a_{\text{ex}}/M=0.667538$ corresponds to the extremal value of $a/M$ for
$\alpha/M^{2}=-0.3$. The black dashed line indicates the position of the
curvature singularity.

Figure 3: Diagrams illustrating the process of destroying an event horizon by
increasing the angular momentum in parameter-region II
($(0<\alpha/M^{2}<6.2754)$). The profile shows the coordinate location of the
inner horizon (represented by the red line) and the event horizon (represented
by the blue line) as a function of the angular coordinate $\theta$. We
consider a black hole with $\alpha/M^{2}=2$ and varying values of $a/M^{2}$.
Here, $a_{\text{ex}}/M=0.890307$ corresponds to the extremal value of $a/M$
for $\alpha/M^{2}=2$. The black dashed line indicates the position of the
curvature singularity.

Figure 4: Diagrams illustrating the process of destroying an event horizon by
increasing the angular momentum in parameter-region III
($6.2754\leq\alpha/M^{2}\leq 8$). The profile shows the coordinate location of
the event horizon (represented by the blue line) and the singularity
(represented by the black dashed line) as a function of the angular coordinate
$\theta$. We consider a black hole with $\alpha/M^{2}=7$ and varying values of
$a/M$. Here, $a_{\text{sb}}/M=0.816227$ corresponds to the angular momentum of
the singular black hole when $\alpha/M^{2}=7$.

By analyzing the spacetime metric (3), we can identify the existence of two
singularities. The first singularity is the well-known ring singularity,
positioned at $\Sigma=0$, which corresponds to $r=0$ and $\theta=\pi/2$. The
second singularity arises due to the quantum effects of Gauss-Bonnet theory.
It is located at the point $r=r_{s}(\theta)$ where the expression inside the
square root in the mass function (4) becomes zero, i.e.,

$\displaystyle 1-\frac{8\alpha r\xi}{\Sigma^{3}}M=0\,.$ (5)

Solving this condition requires a numerical approach, except for
$\theta=\pi/2$, where we find

$\displaystyle r_{s}(\pi/2)=2(M\alpha)^{1/3}\,.$ (6)

In the context of describing a black hole spacetime, there exists an event
horizon located at $r=r_{H}(\theta)$ that conceals the singularity within it.
This event horizon satisfies the following differential equation

$\displaystyle\begin{aligned}
\left[\partial_{\theta}r_{H}(\theta)\right]^{2}+\left.\Delta\right|_{r=r_{H}(\theta)}=0\,,\end{aligned}$
(7)

where

$\Delta=r^{2}+a^{2}-2r\mathcal{M}(r,\theta).$ (8)

Considering the symmetries of the problem, we need to establish the boundary
conditions $\partial_{\theta}r_{H}(0)=\partial_{\theta}r_{H}(\pi/2)=0$.
Together with Eq. (7), these boundary conditions imply

$\displaystyle\begin{aligned}
\Delta_{\text{poles}}|_{r=r_{H}(0)}=0\,,\quad\text{and}\quad\Delta_{\text{equator}}|_{r=r_{H}(\pi/2)}=0\,,\end{aligned}$
(9)

in which we define

$\displaystyle\begin{aligned}
\Delta_{\text{poles}}(r)&\equiv\Delta(r,\theta=0,\pi)\,,\\\
\Delta_{\text{equator}}(r)&\equiv\Delta(r,\theta=\pi/2)\,.\end{aligned}$ (10)

To establish the presence of an event horizon, it is necessary for both
$\Delta_{\text{poles}}(r)$ and $\Delta_{\text{equator}}(r)$ to have at least
one root. Solving Eq. (7) using the pseudospectral method enables us to
determine the range of black hole existence, illustrated in Fig. 1. The figure
reveals three distinct boundary behaviors within this range.

First, in the parameter-region I $(-1\leq\alpha/M^{2}\leq 0)$, the boundary of
the black hole solution is represented by the red curve in Fig. 1. They
describe extremal black hole solutions where the event horizon and the inner
horizon coincide at the equator ($\theta=\pi/2$), see Fig. 2. In this case,
$\Delta_{\text{equator}}(r_{H}(\pi/2))=\partial_{r}\Delta_{\text{equator}}(r_{H}(\pi/2))=0$,
or the minimum value of $\Delta_{\text{equator}}(r)$, denoted as
$\Delta_{\text{equator}}^{\text{min}}$, is zero. Increasing the angular
momentum further results in the absence of positive roots for
$\Delta_{\text{equator}}(r)$ (i.e., $\Delta_{\text{equator}}^{\text{min}}>0$),
leading to the destruction of the event horizon and the exposure of the
singularity.

Second, in the parameter-region II $(0<\alpha/M^{2}<6.2754)$, the boundary of
the black hole solutions is represented by the blue curve in Fig. 1. They
describe extremal black hole solutions where the event horizon and the inner
horizon overlap at $\theta=0$ and $\pi$, see Fig. 3. In this case, the minimum
value of $\Delta_{\text{poles}}(r)$, denoted as
$\Delta_{\text{poles}}^{\text{min}}$, is zero. Further increasing the angular
momentum leads to $\Delta_{\text{poles}}^{\text{min}}<0$, resulting in the
destruction of the event horizon.

Finally, in the parameter-region III $(6.2754\leq\alpha/M^{2}\leq 8)$, we find
the boundary of the third type showed by the green curve in Fig. 1, where the
event horizon and the singularity overlap, see Fig. 4. Specifically, this
boundary is determined by the equation
$r_{H}(\pi/2)=r_{s}(\pi/2)=2(M\alpha)^{1/3}$, which also implies

$\displaystyle\begin{aligned}
\Delta_{\text{equator}}^{r_{s}}\equiv\Delta_{\text{equator}}|_{r=r_{s}(\pi/2)}=0\,.\end{aligned}$
(11)

##  III Destroy the event horizon by throwing a test particle

In this section, our goal is to investigate whether the event horizon of a
rotating black hole in semiclassical gravity with type-A anomaly can be
destroyed after absorbing a test charged particle. The equation of motion for
the test particle is given by

$\displaystyle\begin{aligned} U^{b}\nabla_{b}U^{a}=0,\end{aligned}$ (12)

where

$\displaystyle\begin{aligned}
U^{a}=\left(\frac{\partial}{\partial\tau}\right)^{a}\end{aligned}$ (13)

represents the four-velocity of the particle, and $\tau$ denotes the
particle’s proper time, ensuring $U^{a}U_{a}=-1$. The energy $E$ and angular
momentum $L$ of the test particle are defined as

$\displaystyle\begin{aligned} E&\equiv-mU_{a}\left(\frac{\partial}{\partial
v}\right)^{a}=-mU_{v}\,,\\\ L&\equiv
mU_{a}\left(\frac{\partial}{\partial\varphi}\right)^{a}=mU_{\varphi}\,.\end{aligned}$
(14)

For simplicity and without loss of generality in our subsequent analysis, we
set $m=1$. To treat the particle as a test body, we assume that its energy $E$
and angular momentum $L$ are small compared to those of the black hole, i.e.,

$\displaystyle E\ll M\,,\quad\text{and}\quad L\ll J\,.$ (15)

Considering the reflection symmetry of the black hole with respect to the
equatorial plane, we focus on the scenario where the particle moves on the
equatorial plane $(\theta=\pi/2)$ with some angular momentum, resulting in
zero components of velocity $U_{\theta}=0$. Using the normalized condition
$U^{a}U_{a}=-1$ and the definitions of energy and angular momentum, we can
obtain the following expressions,

$\displaystyle\begin{aligned}
\left(\frac{dr}{d\tau}\right)^{2}&=\left.\frac{a^{2}\left(E-\Omega_{H}L\right)^{2}-[r_{H}^{2}+(aE-L)^{2}]\Delta}{r_{H}^{4}\Omega_{H}^{2}}\right|_{\theta=\pi/2}\,,\\\
\frac{dv}{d\tau}&=\left.\frac{r_{H}^{2}(a^{2}+r_{H}^{2})+\mathcal{O}(E,L)}{2r_{H}^{2}(a^{2}E-aL+Er_{H}^{2})}\right|_{\theta=\pi/2}\,,\end{aligned}$
(16)

in which

$\displaystyle\begin{aligned}
\Omega_{H}\equiv\frac{a}{a^{2}+r_{H}^{2}}\end{aligned}$ (17)

is the angular velocity of the Killing horizon $r=r_{K}(\theta)$, determined
by $\Delta(r,\theta)=0$.

The condition for the test particle to enter the event horizon on the
equatorial plane requires its motion near the event horizon $r=r_{H}$ to be
timelike and future-directed, which implies

$\displaystyle E>\Omega_{H}^{\text{equator}}L$ (18)

with

$\displaystyle\begin{aligned}
\Omega_{H}^{\text{equator}}\equiv\Omega_{H}|_{\theta=\pi/2}\,.\end{aligned}$
(19)

Here we have used the assumption (15). The above inequality gives the lower
bound of $E/L>\Omega_{H}^{\text{equator}}$ such that the test particle can
enter the event horizon on the equatorial plane.

Next, we explore the conditions required to destroy the black hole.
Specifically, the test particle must be capable of entering the event horizon,
and the black hole should become overspun after absorbing the test particle.
These conditions establish a relationship between the energy $E$ and angular
momentum $J$ of the test particle. After the test particle is dropped into the
black hole, the parameters of the final state become

$\displaystyle\begin{aligned} M&\rightarrow M^{\prime}=M+\delta M,\\\
J&\rightarrow J^{\prime}=J+\delta J,\end{aligned}$ (20)

where $\delta M=E$ and $\delta J=L$. To examine the validity of the WCCC, we
need to determine whether the spacetime with mass $M^{\prime}$ and
$J^{\prime}$ still represents a black hole solution, i.e., whether
$(M^{\prime},J^{\prime})$ lies within the domain of existence of black holes
as shown in Fig. 1.

In the previous section, we showed that there are three different types of
parameter regions, each associated with distinct mechanisms for the
destruction of the event horizon. In the following, we will discuss the
conditions for the destruction of black holes in each parameter region and
analyze the possibility of black hole destruction when considering the
particle conditions mentioned earlier.

Figure 5:  $\partial_{M}\Delta_{\text{equator}}(r_{\text{min}},M,J)$ and
$\Delta\Omega_{\text{I}}$ of the extremal black hole solution as a function of
the coupling constant in the parameter-region I.

###  III.1 Type-I violation

Firstly, let’s consider the case where the coupling constant satisfies
$-1\leq\alpha/M^{2}\leq 0$, which corresponds to parameter-region I. In this
region, the boundary curve of the black hole solutions represents extremal
black holes where the event horizon and the inner horizon overlap at the
equator $\theta=\pi/2$. To illustrate the mechanism for destroying the event
horizon and exposing the singularity, in Fig. 2, we depict the profiles of the
inner horizon, outer horizon, and singularity in $(\theta,r/M)$ coordinates
with $\alpha/M^{2}=-0.3$ and different angular momenta near the boundary line.
From the figure, we observe that by increasing the angular momentum, the inner
and event horizons will tend to coincide at the equator $\theta=\pi/2$, and
ultimately, the event horizon is destroyed when the angular momentum exceeds
its extremal value $a=a_{\text{ex}}$. We refer to this phenomenon as a type-I
violation of the black hole in our study. Therefore, we can use
$\Delta_{\text{equator}}^{\text{min}}$ to determine whether the spacetime
describes a black hole in parameter-region I. Through numerical calculations,
we find that the boundary (red line in Fig. 1) is determined by
$\Delta_{\text{equator}}^{\text{min}}=0$,
$\Delta_{\text{equator}}^{\text{min}}<0$ in the red shaded region in Fig. 1,
and $\Delta_{\text{equator}}^{\text{min}}>0$ in the white region of Fig. 1.
Hence, the destruction of the black hole after dropping a test particle in
this parameter region implies that

$\Delta^{\prime}{}_{\text{equator}}^{\text{min}}>0,$ (21)

where $\Delta^{\prime}{}_{\text{equator}}^{\text{min}}$ is the minimal value
of

$\Delta_{\text{equator}}(r,M+\delta M,J+\delta J)\equiv\Delta(r,\pi/2,M+\delta
M,J+\delta J).$ (22)

Next, we consider only the first-order approximation from the test particle,
and the initial state of the spacetime is an extremal black hole, i.e., the
solution lies on the red boundary curve in Fig. 1. Let $r_{\text{min}}$ be the
point of minimal value for the function $\Delta_{\text{equator}}(r,M,J)$,
meaning that we have
$\Delta_{\text{equator}}^{\text{min}}=\Delta_{\text{equator}}(r_{\text{min}},M,J)$.
The condition for the initial state to be extremal implies

$\Delta_{\text{equator}}(r_{\text{min}},M,J)=0,$ (23)

which also implies that $r_{\text{min}}=r_{H}(\pi/2)$.

After dropping a test particle, the minimal point shifts infinitesimally to
$r_{\text{min}}+\delta r_{\text{min}}$. For infinitesimal changes, we have

$\displaystyle\Delta^{\prime}{}_{\text{equator}}^{\text{min}}$
$\displaystyle=\Delta_{\text{equator}}(r_{\text{min}}+\delta
r_{\text{min}},M+\delta M,J+\delta J)$ (24)
$\displaystyle=\frac{\partial\Delta_{\text{equator}}}{\partial M}\delta
M+\frac{\partial\Delta_{\text{equator}}}{\partial J}\delta J,$

under the first-order approximation of the particle perturbation. Here, we
have used the assumption (23) that the initial state is extremal and the
condition that $r_{\text{min}}$ is the point of minimal value of
$\Delta_{\text{equator}}(r,M,J)$, i.e.,

$\partial_{r}\Delta_{\text{equator}}(r_{\text{min}},M,J)=0.$ (25)

In the left panel of Fig. 5, we demonstrate that

$\partial_{M}\Delta_{\text{equator}}(r_{\text{min}},M,J)<0$ (26)

for the extremal black hole solutions in parameter-region I, where
$-1\leq\alpha/M^{2}\leq 0$. Then, the destruction condition (21), together
with Eq. (24), yields

$\delta M<\Omega_{\text{I}}\delta J,$ (27)

where we define

$\Omega_{\text{I}}=-\frac{\partial_{J}\Delta_{\text{equator}}(r_{\text{min}},M,J)}{\partial_{M}\Delta_{\text{equator}}(r_{\text{min}},M,J)}.$
(28)

This provides an upper bound for $E/L=\delta M/\delta J$. Together with the
condition (18) that the particle can be dropped into the black hole on the
equatorial plane, the allowed range of $E/L$ for destroying the black hole is

$\Omega_{H}^{\text{equator}}<E/L<\Omega_{\text{I}}.$ (29)

In the right panel of Fig. 5, we show the allowed length

$\Delta\Omega_{\text{I}}\equiv\Omega_{\text{I}}-\Omega_{H}^{\text{equator}}$
(30)

of $E/L$ as a function of the coupling constant $\alpha/M^{2}$ in parameter-
region I. Consequently, we observe that the allowed length $\Delta\Omega$ is
positive in the parameter region $-1<\alpha/M^{2}<0$, and $\Delta\Omega=0$ for
$\alpha/M^{2}=0$ and $\alpha/M^{2}=-1$. The above results indicate that,
except for the extremal Kerr limit ($\alpha/M^{2}=0$) and static limit
($\alpha/M^{2}=-1$) cases, the extremal black hole can be overspun by throwing
a test particle. In other words, the WCCC is violated in these cases.

Figure 6: $\partial_{M}\Delta_{\text{equator}}(r_{\text{min}},M,J)$ and
$\Delta\Omega_{\text{II}}$ of the extremal black hole solution as a function
of the coupling constant in the parameter-region II.

###  III.2 Type-II violation

Let’s proceed to analyze the situation when the coupling constant satisfies
$0<\alpha/M^{2}<6.2754$, which corresponds to the parameter-region II. In this
region, the boundary curve of the black hole solution represents extremal
black hole where the event horizon and the inner horizon overlap at the poles
$\theta=0,\pi$. To illustrate the mechanism for destroying the event horizon
and exposing the singularity, in Fig. 3, we present the profiles of the inner
horizon, outer horizon, and singularity in $(\theta,r/M)$ coordinates with
$\alpha/M^{2}=2$ and different angular momenta near the boundary line. From
this figure, we observe that by increasing the angular momentum, the inner and
event horizons tend to overlap at the poles, and eventually, the event horizon
is destroyed when the angular momentum exceeds its extremal value
$a=a_{\text{ex}}$. This scenario is referred to as a type-II violation of the
black hole in our study. In this case, whether the solutions describe a black
hole or a naked singularity is determined by the judging function
$\Delta_{\text{poles}}^{\text{min}}$, which represents the minimal value of
$\Delta_{\text{poles}}(r,M,J)$.

When $\Delta_{\text{poles}}^{\text{min}}>0$, the spacetime solution represents
a naked singularity, otherwise, it is a black hole. Therefore, the destruction
of the black hole after dropping a test particle in parameter-region II
requires

$\Delta^{\prime}{}_{\text{poles}}^{\text{min}}=\Delta_{\text{poles}}(\tilde{r}^{\prime}_{\text{min}},M+\delta
M,J+\delta J)>0,$ (31)

where $\tilde{r}^{\prime}_{\text{min}}$ is the minimal point of the function
$\Delta_{\text{poles}}(r,M+\delta M,J+\delta J)$. Let $\tilde{r}_{\text{min}}$
denote the minimal point of $\Delta_{\text{poles}}(r,M,J)$. Similar to the
previous subsection, we assume that the initial state of the spacetime is an
extremal black hole, i.e., the solution lies on the blue boundary curve in
Fig. 1. Therefore, we have $\tilde{r}_{\text{min}}=r_{H}(0)$, which implies

$\partial_{r}\Delta_{\text{poles}}(\tilde{r}_{\text{min}},M,J)=\Delta_{\text{poles}}(\tilde{r}_{\text{min}},M,J)=0.$
(32)

After dropping a test particle, the minimal point shifts infinitesimally to
$\tilde{r}^{\prime}_{\text{min}}=\tilde{r}_{\text{min}}+\delta\tilde{r}_{\text{min}}$.
Utilizing the above results, the destruction condition (31) implies

$\displaystyle\Delta_{\text{poles}}(\tilde{r}_{\text{min}}+\delta\tilde{r}_{\text{min}},M+\delta
M,J+\delta J)$ (33)
$\displaystyle=\frac{\partial\Delta_{\text{poles}}}{\partial M}\delta
M+\frac{\partial\Delta_{\text{poles}}}{\partial J}\delta J>0,$

under the first-order approximation of the particle perturbation. In the left
panel of Fig. 6, we demonstrate that

${\partial_{r}}\Delta_{\text{poles}}(\tilde{r}_{\text{min}},M,J)<0$ (34)

for the extremal black hole solutions in parameter-region II, where
$0<\alpha/M^{2}<6.2754$. Consequently, the destruction condition
(LABEL:descdII) gives

$\delta M<\Omega_{\text{II}}\delta J,$ (35)

where we define

$\Omega_{\text{II}}=-\frac{\partial_{J}\Delta_{\text{poles}}(\tilde{r}_{\text{min}},M,J)}{\partial_{M}\Delta_{\text{poles}}(\tilde{r}_{\text{min}},M,J)}.$
(36)

This provides an upper bound for $E/L$. Together with the condition (18) that
the particle can be dropped into the black hole on the equatorial plane, the
allowed range of $E/L$ for destroying the black hole is

$\Omega_{H}^{\text{equator}}<E/L<\Omega_{\text{II}}.$ (37)

In the right panel of Fig. 6, we illustrate the allowed length

$\Delta\Omega_{\text{II}}\equiv\Omega_{\text{II}}-\Omega_{H}^{\text{equator}}$
(38)

of $E/L$ as a function of the coupling constant $\alpha/M^{2}$ in parameter-
region II. As a result, we observe that the allowed length $\Delta\Omega$ is
positive in the parameter region $0<\alpha/M^{2}<6.2754$, indicating that an
extremal rotating black hole in these cases can be overspun by throwing a test
particle.

Figure 7: $\partial_{M}\Delta_{\text{equator}}(r_{\text{min}},M,J)$ and
$\Delta\Omega_{\text{III}}$ of the extremal black hole solution as a function
of the coupling constant in the parameter-region III.

###  III.3 Type-III violation

Lastly, let’s examine the scenario where the coupling constant satisfies
$6.2754\leq\alpha/M^{2}\leq 8$, which corresponds to parameter-region III. In
this region, the boundary curve of the black hole solutions represents
singular black holes where the event horizon and the singularity overlap at
the equator $\theta=\pi/2$. To illustrate the mechanism for destroying the
event horizon and exposing the singularity, in Fig. 4, we present the profiles
of the outer horizon and singularity in $(\theta,r/M)$ coordinates with
$\alpha/M^{2}=7$ and different angular momenta near the boundary line. From
this figure, we can observe that by increasing the angular momentum, the event
horizon and the singularity will tend to overlap at the equator, and
eventually, the event horizon is destroyed when the angular momentum exceeds
the critical value $a=a_{\text{sb}}$. This phenomenon is referred to as a
type-III violation of the black hole in our paper. To determine whether the
solutions describe a black hole or a naked singularity in parameter-region
III, we introduce the judging function

$\displaystyle\begin{aligned}
\Delta_{\text{equator}}^{r_{s}}(J,M)&=\Delta_{\text{equator}}(r_{s})\\\
&={J^{2}}/{M^{2}}-8M(\alpha M)^{1/3}+4(\alpha M)^{2/3}\,.\end{aligned}$ (39)

When $\Delta_{\text{equator}}^{r_{s}}>0$, the solution corresponds to a naked
singularity, otherwise, it describes a black hole. Therefore, the destruction
of the black hole after dropping a test particle in parameter-region III
requires that

$\Delta_{\text{equator}}^{r_{s}}(M+\delta M,J+\delta J)>0.$ (40)

Since we are considering only the first-order approximation under the
perturbation of the test particle, we focus on the scenario where the initial
state of the spacetime is a singular black hole, meaning the solution lies on
the green boundary curve in Fig. 1, and we have $r_{H}(\pi/2)=r_{s}$.
Consequently, the destruction condition (40) can be expressed as

$\frac{\partial\Delta_{\text{equator}}^{r_{s}}}{\partial M}\delta
M+\frac{\partial\Delta_{\text{equator}}^{r_{s}}}{\partial J}\delta J>0,$ (41)

within the first-order approximation of the particle perturbation. In the left
panel of Fig. 7, we demonstrate that

$\partial_{M}\Delta_{\text{equator}}^{r_{s}}(M,J)<0,$ (42)

for the singular black hole solutions in parameter-region III, i.e.,
$6.2754\leq\alpha/M^{2}\leq 8$. Based on Eq. (42), we obtain the destruction
condition

$\delta M<\Omega_{\text{III}}\delta J,$ (43)

where we define

$\Omega_{\text{III}}=-\frac{\partial_{J}\Delta_{\text{equator}}^{r_{s}}(M,J)}{\partial_{M}\Delta_{\text{equator}}^{r_{s}}(M,J)}.$
(44)

This provides an upper bound for $E/L$. Combining it with the condition (18)
that the particle can be dropped into the black hole on the equatorial plane,
we conclude that the allowed range of $E/L$ for destroying the black hole is

$\Omega_{H}^{\text{equator}}<E/L<\Omega_{\text{III}}.$ (45)

In the right panel of Fig. 7, we display the allowed length

$\Delta\Omega_{\text{III}}\equiv\Omega_{\text{III}}-\Omega_{H}^{\text{equator}}$
(46)

of $E/L$ as a function of the coupling constant $\alpha/M^{2}$ in parameter-
region III. Consequently, we observe that the allowed range
$\Delta\Omega_{\text{III}}$ is positive for $6.2754\leq\alpha/M^{2}<8$, while
$\Delta\Omega_{\text{III}}=0$ in the case of the static singular black hole
where $\alpha/M^{2}=8$ and $a=0$. This indicates that, except for the static
limit scenario at $\alpha/M^{2}=8$, the singular black hole can be overspun by
introducing a test particle, violating the WCCC.

##  IV Conclusion

In the present work, we have delved deeply into the WCCC for rotating black
holes in semiclassical gravity with a type-A trace anomaly. Specifically, we
have meticulously examined how a test particle alters the spacetime of
extremal or singular rotating black holes in such effective gravitational
theories, unveiling any potential violations of the WCCC. Through our
investigations, we found that depending on the three different ranges of the
coupling constant $\alpha/M^{2}$, we can observe three distinct boundary
characteristics of the black hole solutions, corresponding to three distinct
types of WCCC violations. The three types of violations of WCCC was explored
by launching a test particle towards the black hole on the equatorial plane.
With the exception of extremal Kerr, static extremal, and static singular
black holes (i.e., $\alpha/M^{2}=0,-1,8$), our results demonstrate that all
extremal black holes and singular black holes can be destroyed under the
first-order perturbation of the test particle, leading to a violation of the
WCCC.

For the non-static cases, the aforementioned analyses have clearly
demonstrated the possibility of breaching the WCCC. Taking the Type-I
violation as an example, even in near-extremal black hole scenarios, the
destruction conditions can still be satisfied as long as the initial state is
very close to extremal. Thus, if we can observe that the event horizon of
extremal (singular) black holes can be destroyed, then the possibility of
destruction in the near-extremal (singular) black holes also exists.
Therefore, we would only need to consider the near-extremal (singular) and the
second-order perturbation approximations in cases where the black hole cannot
be destroyed under the first-order perturbation, which precisely corresponds
to the $\alpha/M^{2}=-1,0,8$ cases in our model. This will be left for future
investigations. Furthermore, the equations of particle motion are governed by
the classical geodesic equation, with no considerations for the impact of
quantum corrections. As such, these findings suggest that the preservation of
the WCCC may necessitate the consideration of modifications to the test
particle behavior in the presence of quantum effects.

## Acknowledgements

JJ is supported by the National Natural Science Founda- tion of China with
Grant No. 12205014, the Guangdong Basic and Applied Research Foundation with
Grant No. 217200003 and the Talents Introduction Foundation of Beijing Normal
University with Grant No. 310432102. MZ is supported by the National Natural
Science Foundation of China with Grant No. 12005080.

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