* [32] Whitney, H.: Differentiable functions defined in closed sets. I.  Trans. Amer. Math. Soc. 36(2), 369–387 (1934).  URL https://doi.org/10.2307/1989844
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Further author information: (Send correspondence to Gilles Orban de Xivry)
G. Orban de Xivry: E-mail<EMAIL_ADDRESS>
# The VLT/ERIS vortex coronagraph: design, pointing control, and on-sky
performance

G. Orban de Xivry  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  O. Absil  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  R. J. De Rosa  European Southern Observatory, Alonso de Córdova 3107,
Vitacura, Santiago, Chile  M. J. Bonse  ETH Zurich, Institute of Particle
Physics and Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076
Tübingen, Germany  F. Dannert  ETH Zurich, Institute of Particle Physics and
Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland  J. Hayoz
ETH Zurich, Institute of Particle Physics and Astrophysics, Wolfgang-Pauli-
Strasse 27, 8093 Zurich, Switzerland  P. Grani  INAF – Osservatorio
Astrofisico di Arcetri, Largo E. Fermi 5., 50125 Firenze, Italy  A. Puglisi
INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5., 50125 Firenze,
Italy  A. Baruffolo  INAF – Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, 35122 Padova, Italy  B. Salasnich  INAF – Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy  R.
Davies  Max-Planck-Institut für extraterrestrische Physik, Postfach 1312,
85741 Garching, Germany  A. M. Glauser  ETH Zurich, Institute of Particle
Physics and Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
E. Huby  LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Université de Paris, 5 Place Janssen, 92195 Meudon, France  M.
Kenworthy  Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA
Leiden, The Netherlands  S. P. Quanz  ETH Zurich, Institute of Particle
Physics and Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland
W. Taylor  STFC UK ATC, Royal Observatory Edinburgh, Blackford Hill. Edinburgh
EH9 3HJ, UK  G. Zins  European Southern Observatory, Karl-Schwarzschildstr. 2,
85748 Garching, Germany

###### Abstract

The Enhanced Resolution Imager and Spectrograph (ERIS) is the new near-
infrared instrument at the VLT-UT4. ERIS replaces and extends the
observational capabilities formerly provided by SINFONI and NACO: integral
field spectroscopy at 1 - 2.5 $\mu$m, imaging at 1 - 5 $\mu$m with several
options for high-contrast imaging, and long-slit spectroscopy. In particular,
a vortex coronagraph is now available for high contrast observations at L and
M band. It is implemented using annular groove (or vortex) phase masks (one
for each of the L and M bands) in a focal plane, and a Lyot stop in a
downstream pupil plane. The vortex coronagraph has a discovery space starting
already at $\sim$1$\lambda$/D, and works well in broadbands. However, to reach
its optimal performance, it is critical to correct for slow pointing errors
onto the vortex phase mask, which mandates a dedicated pointing control
strategy. To do so, a control loop based on the QACITS algorithm has been
developed and commissioned for ERIS. Good pointing stability is now regularly
achieved with errors between 0.01 and 0.02 $\lambda$/D and a correction rate
of 0.2 Hz. In this contribution, we first review the design of the ERIS vortex
coronagraph. We then detail the implementation of the QACITS algorithm
describing the entire observing sequence, including the calibration steps, the
initial centering, and the stabilization during the observing template. We
then discuss performance based on commissioning data in terms of pointing
accuracy and stability. Finally, we present post-processed contrast curves
obtained during commissioning and compare them with NACO vortex data, showing
a significant improvement of about 1 mag at all separations.

###### keywords:

high-contrast imaging, coronagraph, mid-infrared imaging, observational

##  1 INTRODUCTION

The Enhanced Resolution Imager and Spectrograph for the VLT replaces NACO and
SINFONI in a single instrument and with improved diffraction limited and
spectroscopic capabilities[1]. The instrument consists of four main building
blocks: a calibration unit, the integral field spectrograph SPIFFIER, the
imager NIX, and the adaptive optics (AO) wavefront sensor.

NIX provides diffraction-limited imaging from 1 to 5 $\mu m$ and features a
comprehensive set of broadband and narrowband filters. Besides classical
imaging and long-slit spectroscopy, NIX supports several high-contrast imaging
modes:

  1. 1.

A pupil plane coronagraph based on a grating vector apodised phase plate
(gvAPP)[2]. This coronagraphic modes generates, for any star in the field, two
coronagraphic images with conjugated D-shaped dark holes, each with 10-30% of
the flux, depending on the filter[1]. When combined, the gvAPP provides almost
360$\degree$ discovery space from about 2 to 15 $\lambda/D$. Because the
separation of the two images scales with wavelength, the gvAPP is best suited
for narrow-band filters.

  2. 2.

Three sparse aperture masks provide non-redundant (or partially non-redundant)
apertures, yielding a resolution that is a factor of 2 better than the
diffraction limit of the telescope. The three masks offer three different
trade-offs between $u,v$ coverage, throughput, and non-redundancy.

  3. 3.

A focal plane coronagraph (FPC) using vortex phase masks, one for each of the
L and M bands, working well in broadband. It provides a clear 360$\degree$
discovery space starting already at $\sim 1\lambda/D$. To perform optimally,
however, it requires precise pointing control to keep the star centered on the
vortex phase mask.

This contribution focuses on the focal plane (vortex) coronagraphic mode,
detailing its design, optimization, and on-sky operation.

One significant enhancement of ERIS over NACO and SINFONI is its improved AO
system[3]. The ERIS AO system makes use of the VLT AO Facility including, most
notably, the Adaptive Secondary Mirror (ASM) on VLT-UT4. The system offers
several natural guide star and laser guide star modes, tailored to its various
science cases. The wavefront sensor is based on a Shack-Hartman with 40-by-40
subapertures, each with 6′′ $\times$6′′ field-of-view. The opto-mechanical
design employs only flat or slow optics to relay the telescope beam minimizing
and stabilizing non-common path aberrations (NCPAs). Indeed, during assembly,
integration, and verification, the NCPAs were calibrated, showing contribution
ranging from 22 to 32 nm rms wavefront error[3]. This low NCPA level is
attributed to the use of quasi-plano optics and the telecentric design,
enabling NGS acquisition with stages rather than a field steering mirror.

ERIS, with its extensive capabilities, is the new VLT workhorse instrument for
diffraction limited near-infrared observation of individual targets. Key
science cases[1] include monitoring stellar proper motions around Sgr A∗,
studying galaxy evolution at high redshifts ($z\sim 1-3$) to unveil the
physical processes shaping galaxies, and directly imaging self-luminous gas
giants and protoplanets as well as probing their atmospheres. ERIS saw its
first light in 2022, with the bulk of the commissioning happening that year. A
wide array of its modes has been offered since ESO Period 111 (for
observations beginning in April 2023) and the FPC mode is now offered since
ESO Period 113 (for observations beginning in April 2024).

In the context of direct imaging, the vector vortex coronagraph is an
attractive solution thanks to its high throughput, small inner working angle
down to $\sim 1\lambda/D$, its 360 degrees clear discovery space, and its
achromatic properties. Over the past decade, mid-infrared vortex coronagraphs
have been integrated into several instruments: VLT/VISIR[4], VLT/NACO[5],
LBT/LMIRCam[6], Keck/NIRC2[7], VLT/NEAR[8], and now VLT/ERIS. However, small
inner working angle focal-plane coronagraphs also mean a high sensitivity to
pointing drifts, and require specific control to perform optimally.

In this contribution, we review the principle and the design of the ERIS
vortex coronagraph. We then describe the pointing control algorithm
specifically tailored for the vortex, and the implementation of the FPC
operation into the observing templates. Finally, we present the obtained on-
sky results in terms of pointing accuracy, raw contrast, and post-processed
contrast.

##  2 VLT/ERIS VORTEX CORONAGRAPH

A vortex coronagraph features a vortex phase mask in its focal plane. The text
book effect is to move the light of an on-axis source outside the downstream
geometric pupil. A Lyot stop then blocks the diffracted light and provides
theoretically a perfect starlight cancellation for a circular entrance
aperture. However, realistic pupils featuring a central obscuration and
spiders lead to a leakage term not entirely blocked by the Lyot stop, i.e. the
vortex coronagraph (i.e. a vortex phase mask coupled with a downstream Lyot
stop) cannot be considered ‘perfect’ any longer.

The ERIS vortex focal plane coronagraph, similarly to all the mid-infrared
vortex coronagraphs so far, is implemented using an annular groove phase mask
(AGPM[9]), followed by its Lyot stop in the pupil plane. AGPMs are made up of
concentric subwavelength gratings etched onto diamond substrates. The
subwavelength grating acts locally as an achromatic half-wave plate and,
together with the concentric design, synthesizes a $4\pi$ helical (vortex)
phase ramp for the two orthogonal polarization states of light separately. For
improved transmission, an anti-reflective grating is etched on the backside of
the mask which significantly reduces the reflection from $\sim 17\%$ to $\sim
1.9\%$ (L band)[10].

ERIS has two AGPM masks, one for each of the L and M bands. The components
were developed and tested by Liège University and manufactured by Uppsala
University. The tests were performed on the VODCA test bench[11] where we
measured the component transmissions and rejection ratios. One of the two
components was initially under-performing and was subsequently re-etched and
tested. The final measurements performed on VODCA, also reported in Jolivet et
al. [11], are given in Tab. 1. Both components show excellent rejection
performance with ratio of 1773:1 (at 4.6$\mu m$) for the ERIS M band vortex,
and 1430:1 (measured on a bandpass of from 3575 to 4125 nm) for the ERIS L
band vortex.

Table 1: Measurements of the AGPM transmission and rejection ratio on the VODCA test bench of the two ERIS AGPM. See also Jolivet et al.[11] for more details. The L-band measurement are performed on a bandpass from 3.575$\mu m$ to 4.125$\mu m$.  Name | Transmission | Transmission | Rejection ratio | Rejection ratio
---|---|---|---|---
| (L- broadband) | (4.6$\mu$m) | (L broadband) | (4.6$\mu$m)
AGPM-L12r | $85\%\pm 0.7$ | $68.5\%\pm 2.5$ | $84\pm 5$ | $1773\pm 95$
AGPM-L15 | $83.5\%\pm 0.5$ | $66\%\pm 1$ | $1430\pm 82$ | $50\pm 3$

The downstream Lyot stop blocks the star light in the pupil plane. The stop is
typically undersized to block the light diffracted by the secondary
obscuration and the secondary support structure, which appears as a
monotonically decreasing flux moving away from these structures. The Lyot stop
therefore shrinks the perimeter of the entrance pupil in order to mask out the
scattered on-axis light. The trade-off consist in reducing the on-axis stellar
light leakage but maintaining a sufficient throughput for off-axis sources.

The ERIS Lyot mask itself is made from Molybdenum and blackened with Acktar
Fractal black coating. Two stops are installed: the regular Lyot stop, and a
Lyot stop coupled with a ND filter allowing the observation of very bright
stars. The final dimensions with respect to the VLT entrance pupil are given
in Fig. 1a. The outer diameter is 95.7% of the pupil diameter, the inner
diameter is 37% of the pupil diameter. The throughput of the Lyot stop is
74.77%. A composite image of the entrance pupil and the Lyot stop on the VLT-
UT4 is shown in Fig. 1b. As ERIS does not have a cold pupil stop upstream of
the vortex, the warm support structure of the VLT primary mirror and its
secondary is visible. With the Lyot stop slid in, we can verify the alignment
of the stop with respect to the entrance pupil.

Figure 1:  (Left) Vortex Lyot stop with its dimensions given with respect to
VLT M1. (Right) Composite pupil image obtained with NIX, showing the entrance
pupil including the warm support structure of M1 and M2, and the cold Lyot
stop with its thick spiders and oversized central obscuration. The four
central white areas are the portion of the pupil not blocked by the Lyot stop.

Figure 2: (Top) Theoretical coronagraphic transmission as a function of radial
distance to the vortex center, for a circular pupil (blue) and more
specifically for ERIS (green). The coronagraphic transmission reaches $\sim
50\%$ at 1 $\lambda/D$ which makes the vortex one of the most efficient in
terms of inner working angle, but also more prone to pointing errors (Bottom)
Illustration of the vortex coronagraphic PSFs for decentering from up to 0.5
$\lambda/D$ showing the rapid increase in transmission. The white circle
indicates the separation between the inner and outer estimator defined at 1.7
$\lambda/D$ radius.

The vortex components are mounted in the NIX aperture wheel, and the Lyot
stops are mounted in the NIX pupil wheel[12].

The combined throughput (vortex phase mask and Lyot stop) of the focal plane
coronagraph amounts to $\sim$62% in L band and $\sim$51% in M band. This
throughput does not take into account the Strehl ratio reduction due to the
Lyot stop, which is undersized compared to the entrance pupil.

##  3 VORTEX CENTERING WITH QACITS

To perform optimally, the star needs to be precisely centered on the vortex
phase mask, in which case it suppresses the central PSF significantly, leaving
only a ring of residual light. Indeed, for focal plane coronagraphs, small
inner working angles also mean high sensitivity to pointing drifts. This is
illustrated in Fig. 2 using Fourier-based simulations of the ERIS vortex
coronagraph. The transmission plot shows that at $\sim 1\lambda/D$, almost 50%
of the flux is transmitted. This is also visually illustrated in the image
sequence below with tilt going from 0.01 to 0.7 $\lambda/D$.

To keep the star centered on the mask during the observation, we use an
automated algorithm, specifically developed for the vortex coronagraph, called
Quadrant Analysis of Coronagraphic Images for Tip-Tilt Sensing (QACITS; Huby
et al.[13]). The QACITS technique is based on a quadrant analysis of the
vortex coronagraphic images. Because the relationship between the measured
shift and the tilt (or tip) is non-linear and generally not bijective when
considering the whole image, QACITS splits down the focal plane images in
several areas to build different estimators. The estimator providing the
largest linear range is the so-called ‘outer estimator’. It only considers the
quadrant flux asymmetry in an annulus going from 1.7 to 2.3 $\lambda/D$. The
outer estimator is robust to various effects, such as misalignment or low
order aberrations, but at the expense of a somewhat lower signal-to-noise
ratio (SNR).

###  3.1 On-sky calibration

To calibrate and validate our QACITS model, we observe a bright target on-sky,
manually center it on the vortex phase mask, and then incrementally offset the
star from the mask along one axis. We then reproduce this procedure along the
other axis. The offsets are injected by directly moving the AO stages.

Based on those measurements, we can produce the response curves of the QACITS
estimator which means defining the slope between measurement and tip-tilt. The
results, shown in Fig. 3, indicate a slope of 0.062. Comparing the on-sky
measurements to our simulations shows a relatively good match between the
response curves, validating our on-sky calibration. As shown by the curve, the
linear range of the outer estimator is approximately $1\lambda/D$ or $\sim
0.1^{\prime\prime}$.

Figure 3: Calibration of the QACITS response using the outer estimator. In
green, the $x$ and $y$ measurements obtained on-sky. In blue, the same
calibration performed in simulation (up to 1 $\lambda/D$ tilt).

###  3.2 Control aspects

The control sequence of QACITS is illustrated in Fig. 4 and works as follows:
QACITS measures the residual drift every 5 seconds based on the available
science images, then a total command is computed using a proportional-integral
(PI) controller and is sent to an asynchronous auxiliary AO loop with a
repetition rate of 3 seconds. The commands sent to the auxiliary AO loop
result in a movement of the AO wavefront sensor stages, which subsequently
causes a tip-tilt correction by the adaptive secondary mirror.

The control parameters of the PI controller were optimised on-sky by
monitoring the response to step perturbations. The perturbations were
introduced in AO closed-loop by moving one of AO stage by about 0.1′′ or 10
pixels on the NIX detector. Two examples of those tests are given in Fig. 4.
The measurements are compared with a toy model of the QACITS closed-loop. On
top of the two control parameters, the toy model is defined by the integration
time and the time delay between measurement and correction. The first is given
by the QACITS repetition rate, i.e. 5 seconds. The second is not precisely
known nor deterministic and was arbitrarily set in the toy model to 4 seconds
to best match the measurements. This time delay is the combined contributions
of computational delays, communication delays between the QACITS and the
asynchronous AO auxiliary loop, and the response time of the hardware itself.
Based on those tests, we select a proportional gain $K_{p}=0.25$ and an
integrator gain $K_{i}=0.5$ which seems a good compromise between response
speed and stability. With those parameters, the 0dB closed-loop error cut-off
frequency is estimated to $\sim$0.02Hz, meaning that minute-scale pointing
drift are efficiently rejected.

As a comparison, the VLT/NEAR and Keck/NIRC2 had respectively $\sim$30s and
$\sim$50s repetition rate. Therefore, the VLT/ERIS implementation provides a
5- to 10-fold improvement in terms of rejection bandwidth.

Figure 4: (Top) Control flow diagram. (Bottom) Step response curves to
perturbations of 10 pixels. The left figure shows the response obtained for
integrator and proportional gains $(K_{i},K_{p})=(1,0)$ and the right picture
shows the response after adjustment of the gain to $(K_{i},K_{p})=(0.5,0.25)$.
In blue, the on-sky measurements and in red, the simulated curves using our
toy model. The blue curve of the right plot changes to a dotted light blue at
around 20 seconds, indicating that the correction reached a software limit.
This is a specific experimental case and is not a concern for regular
operation.

###  3.3 Implementation in ERIS

The QACITS algorithm has been implemented in Python, and is fully integrated
to the ESO ERIS templates. A lot of the work has been dedicated to enable
effort-less and robust operations. The FPC operations are divided in two
specific templates, one for acquisition and one for observation.

After the regular telescope and AO target acquisitions, the FPC acquisition
template starts and executes the following steps:

  1. 1.

The target is first roughly centered onto the vortex mask (within $\sim
1\lambda/D$).

  2. 2.

The target is offset outside the field-of-view and a set of dark and sky
frames are acquired.

  3. 3.

Using a sky and dark frames with preset integration time, the vortex position
is measured by fitting the vortex center glow, see also Sec. 3.4.

  4. 4.

The target is moved in the field but far off-axis from the vortex center. An
unsaturated image of the star is acquired to provide a photometric reference
for QACITS, and the actual offset of the star is measured by fitting a
Gaussian profile.

  5. 5.

Based on the previous measurements, the star is moved onto the vortex. At the
end of this step, the star is centered onto the vortex mask within
approximately a 0.1 to 1 $\lambda/D$ accuracy (i.e. a few pixels).

  6. 6.

The centering is then improved by an optimization sequence consisting of a few
iterations of the QACITS loop, typically using shorter integration than the
scientific acquisitions. The optimization sequence performs three iterations
before moving to the observing template. The centering improvement is
illustrated in Fig. 6.

The acquisition time takes approximately 3.5 minutes. The acquisition can be
performed with either the regular Lyot stop or a Lyot stop coupled with a ND
filter (extinction $>1:60$) to avoid any saturation of the off-axis PSF on
very bright stars.

Figure 5: NIX sub-images during FPC acquisition. The large circle (left image)
represents the footprint of the vortex phase masks, equivalent to $\sim$16′′
diameter on-sky. The image features also cosmetics from the NIX detector, and
dust contamination of the vortex phase masks. The image and the zoom insets
display the diffraction-limited off-axis PSF and the vortex center glow, see
also Sec. 3.4. Figure 6: Improvement of the rejection of the on-axis star as
the centering is improved at the end of the QACITS acquisition sequence. The
images are sky subtracted and displayed with a square-root stretch. The left
image has an offset of about 1 $\lambda/D$ between the star and the vortex,
while the right image shows a well-centered star. The rejection of the stellar
light is about a factor 2 initially (left) and reaches more than a factor 100
at the end of the sequence (right). The images are 1.7′′ across; for reference
the FWHM in L band is 97 mas.

Once the acquisition template is completed, the FPC science observing template
starts. QACITS is enabled by default and measures pointing drift every 5
seconds based on the available data. One particularity is the monitoring of
the vortex center discussed in the next section: anytime the observer requests
a sky background, another background with a predefined integration time is
also acquired, from which the vortex position is computed using the vortex
center glow. This allows to keep track of any drift of the vortex position
onto the NIX camera.

During the observing template, a simple live display shows the QACITS
measurements. The QACITS measurements are also logged in the generic
instrument workstation logs. In the future, we plan to add quality metrics in
the data headers, and possibly provide a fits table with all the QACITS
measurements.

###  3.4 Vortex position measurement

Although the vortex mask is fully transparent, it typically produces a diffuse
emission on top of the thermal sky background. This emission, observed with
other instruments in the L, M, and N bands, is the result of thermal emission
from the warm telescope structure and the environment around the entrance
pupil. The vortex phase mask partially diffracts this thermal emission into
the pupil image, creating a diffuse bright spot in the downstream image
plane[14, 15]. If the AGPM was placed behind a cold pupil stop, we expect that
this emission would disappear and would be replaced by a dark spot.
Nevertheless, while this vortex center glow slightly increases photon noise
locally, it allows the position of the vortex to be easily identified and
monitored.

During commissioning, by monitoring off-line the vortex center glow, we
noticed a non negligible drift of the vortex position with respect to the NIX
detector, which is bound to directly impact the QACITS centering accuracy. To
mitigate any drift on pointing accuracy, we implemented an automatic vortex
center measurement on every sky frames, updating the target zero position of
QACITS. In Fig. 7, we illustrate the measured vortex position drift over a
$\sim$2-hr observing sequence. We can notice a total drift of about 20 mas, or
0.2 - 0.3 $\lambda/D$. Based on the fastest drift rates measured on-sky, we
need to re-measure the vortex position approximately every 5 min.

Figure 7: Representative vortex position drift in $x$ and $y$, as a function
of time.

##  4 ON-SKY RESULTS

###  4.1 Pointing control performance

We illustrate in Fig. 8 the QACITS pointing performance obtained on-sky based
on two $\sim$1hr observations performed during commissioning in May 2023, and
an entire night during Period 113 (Spring 2024). The first target is i Vel
with a magL=4.1 (‘bright case’) observed during 62-min where the pointing
precision (robust standard deviation) is $\sim$0.01$\lambda/D$. The second
target is GQ Lup with a magL=6.1 (‘faint case’) observed during 70-min where
the pointing accuracy is $\sim 0.02\lambda/D$. Finally, we illustrate an
entire night of focal plane coronagraphic observations of multiple targets
where the overall pointing precision is also $\sim 0.02\lambda/D$, testifying
the good robustness of QACITS.

Systematic shift, such as the one observed on i Vel of about 0.01$\lambda/D$,
can be explained by the star pointing drift. In Fig. 9, we plot the total
command of QACITS over the 1hr sequence. We can observe a drift of about 20
pixels/1hr or 4mas/min. Considering the closed-loop control and its inherent
delays, this is consistent with an observed shift of about 0.01 $\lambda/D$.
It goes without saying that without QACITS, this drift would strongly degrade
the vortex performance.

Figure 8: Residual pointing errors measured by QACITS. (Left) on i Vel, a
’bright’ target of $L-$mag=4.1, observed during approx. 60-min. (Middle) on GQ
Lup, a ’faint’ target of $L-$mag=6.1, observed during approx. 70-min. (Right)
On multiple targets during an entire night of FPC operation.  Figure 9: Time
evolution (color coded) of the QACITS total command measured on i Vel over
1-hr. The curve is typical of differential atmospheric refraction, with a
meridian passage after 15-20 min.

Based on the different results obtained on-sky, we can conjecture that QACITS
would still work up to mag${}_{L}=8$ (although at a reduced pointing
accuracy), and the control loop would eventually break between magnitude 8 and
9. We should stress however that the usefulness of coronagraphic observations
for objects fainter than magnitude $\sim$ 9 is questionable. Indeed, in this
regime, we expect the contrast to be limited by background noise at most
separations, making normal imaging the preferred mode.

###  4.2 Coronagraphic performance

####  4.2.1 Raw contrast

While the vortex coronagraph is expected to perfectly reject the starlight
(with a circular pupil and flat wavefront), four main effects degrade its
rejection in practice [16, 14] : (i) the diffraction of the central
obscuration and the spiders, (ii) the pointing errors, (iii) the adaptive
optics residuals and other sources of residual wavefront errors (e.g., NCPA),
(iv) the intrinsic imperfections of the vortex phase mask. In the following,
we provide typical order of magnitude of the integrated leakage due to those
effects:

  1. 1.

The fractional leakage due to the central obscuration can be approximated to
first order[16] by $(d/D)^{2}$. With the VLT central obscuration $d$ and
telescope diameter $D$, the estimated leakage is $\sim 0.05$. However, this
approximation does not take the spiders into account nor the Lyot stop
blocking a large fraction of this leakage. To have a more realistic estimate,
we perform a Fourier-based simulation of the ERIS vortex coronagraph and
compute a leakage of $\sim 0.016$.

  2. 2.

Next, small pointing errors $\theta$ lead to a leakage given[13] by
$(\pi\theta/(\lambda/D))^{2}/8$. Based on the pointing precision obtained with
QACITS ($\sim 0.01-0.02\lambda/D$), the leakage would be $<5e-4$. However,
this only represents the slow-evolving pointing drifts and we may expect
jitter residuals from atmospheric turbulence and telescope vibration. If we
reasonably assume a fast residual jitter of 10mas, the leakage term would be
$\sim 0.012$.

  3. 3.

Wavefront errors will also produce leakage within the geometric pupil. For a
good AO correction, this can be approximated by $1-S$, with $S$ the Strehl
ratio. With ERIS, we have about 95% Strehl in L band, thus the associated
leakage is approximately 0.05.

  4. 4.

The intrinsic imperfection of the vortex phase mask translates essentially in
a chromatic leakage, i.e. a fraction of the light is not affected by the
vortex phase ramp and thus ‘leak’ inside the geometric pupil. The ERIS
components have been tested on our VODCA bench at ULiège, and perform very
well with rejection ratios above $1000:1$, see Tab. 1. Thus the leakage
associated to the mask themselves is negligible and $<0.001$.

Summing up the various effects, we can expect a total integrated leakage of
about 0.06 - 0.08. In Fig. 10, we show the radial profiles obtained on i Vel
in May 2023, i.e. the off-axis PSF and the on-axis coronagraphic PSF and a
comparison to Fourier-based simulations (under perfect conditions, i.e. using
the actual ERIS pupil and Lyot stops but without any wavefront or pointing
errors). The measured integrated leakage is here $\sim 0.065$ indicating the
good match with our rough error budget (and the good Strehl and low residual
jitter).

Figure 10: Radial intensity profiles with the star centered on the vortex
phase mask (blue) or off-axis (black). Simulated profiles (dashed lines) are
also computed for comparison, assuming perfect wavefront. While the solid blue
line represents the average profile, the light blue one corresponds to the
best profile (best rejection) obtained during the i Vel observation.

Beyond the absolute value of the integrated leakage, the stability of the
leakage and the post-coronagraphic PSF is key to ensure an optimum PSF
subtraction in post-processing. Assuming that the Strehl ratio is stable
within 0.5%, we can derive a pointing stability requirement on par with the AO
residual leakage. The derived stability requirement is $\sim 0.05\lambda/D$ at
L band on the VLT telescope, and can be considered as a reasonable precision
requirement for QACITS.

Other effects such as misalignment (e.g. Lyot stop) or NCPA can also
significantly degrade the performance, if not cautiously tracked down. In the
case of ERIS, as far as we can judge, these effects do not play a significant
role on the raw contrast, but may still significantly affect the sensitivity
limits after Angular Differential Imaging (ADI) processing. Finally, this
integrated leakage error budget does not address where the different leakage
terms end up in the focal plane (e.g. low order distortion of the diffraction
pattern, or speckles) nor their interference effects (which may lead to semi-
static ”pinned” speckles).

####  4.2.2 Post-processed contrast in L’

During commissioning, the FPC on-sky contrast curves have been derived in L
band on two different objects: i Vel (magL = 4.1) and GQ Lup (magL = 6.1). The
details of the observations and the results are presented below.

GQ Lup was observed in a Object-Sky-Object-… sequence with 400 DIT of 0.3s on
Object and 10 DIT of 0.3s on Sky, providing a net exposure time of 52-min on
target (after frame selection). The total exposure time, including sky frame,
was 1h1m and the total execution time was 1h23m. The total ADI sequence
amounts to 58.6 degrees of parallactic angle rotation. The $R$-mag (AO guide
star) of GQ Lup is 10.3. The data reduction used the VIP package[17] and
includes dark subtraction, flat correction, bad pixel removal, recentering of
frames, PCA sky subtraction, absolute recentering, and median-subtraction or
annular PCA for post-processing. The resulting contrast curve is given in
Figure 11. This curve represents the 5-sigma contrast, corrected for small
sample statistics[18]. Prior to computing the contrast curve, the companion GQ
Lup b is removed. The curve accounts for the off-axis transmission of the
vortex, as illustrated in Figure 2. We also produce an ADI post-processed
image, see Fig. 12, providing an exquisite detection of the GQ Lup companion
at about 0.6′′.

Figure 11: 5-sigma post-processed contrast curves obtained with the vortex
focal plane coronagraph and median PSF subtraction. (Blue) GQ Lup with a total
ADI sequence of 70-min and a parallactic angle rotation of 59 degrees.
(Orange) i Vel with a total ADI sequence of 62-min and a parallactic angle
rotation of 39 degrees. With annular PCA, the detection limit goes down by
another magnitude, reaching down to $\Delta\sim 13{\rm mag}$ for i Vel.

i Vel was observed following the same strategy as GQ Lup, with a net exposure
time of 60-min on target. The data processing and post-processing follows the
same principle. The resulting contrast curves are given in Figure 11 . The
performance obtained are similar to GQ Lup up to about 2.5 $\lambda$/D ($\sim
0.25^{\prime\prime}$). At larger separation, the contrast is better with i
Vel. A brighter star means improved AO performance, increased pointing
accuracy with QACITS ($\sim$0.01 $\lambda$/D with i Vel instead of $\sim$0.02
$\lambda$/D for GQ Lup), and a deeper background-limited sensitivity in terms
of contrast, which account for the difference in the contrast curves beyond
2.5 $\lambda$/D.

Figure 12: Post-processed image of GQ Lup using median subtraction, revealing
the companion, GQ Lup b, at about 0.6′′.

####  4.2.3 Comparison with NACO

In an attempt to compare apples with apples, we selected four datasets in the
ESO archive taken with the vortex coronagraph of NACO. The datasets are chosen
to have similar target magnitudes and observing conditions. In addition to the
i Vel ERIS observations with a 57 minutes integration time and 39 degrees
field rotation, we have HD 11171 (2 hours 2 minutes; 108.7 degrees), HD 188228
(1 hours 46 minutes; 63 degrees), HD 71155 (47 minutes, 48 degrees), HD 98058
(1 hours 44 minutes, 88deg). All NACO and ERIS observations were reduced with
the same post-processing pipeline using PynPoint[19]. The pipeline includes
basic calibrations (dark subtraction, flat correction, bad pixel removal,
recentering of frames) as well as a PCA-based[20, 21] PSF subtraction. The
number of PCA components is optimized to reach the deepest contrast for each
separation. Afterwards we compute contrast curves based on a t-test [18] using
applefy[22].

The results are given in Fig. 13. Overall, considering the variety of
integration times and field rotations (which globally favor the NACO
observations), the results indicate a gain of approximately 1 magnitude for
ERIS at all separations, both in the speckle-limited and background-limited
regimes. This comparison provides a first indication of the excellent
performance of the ERIS AO system and the FPC coronagraph, and is promising
for forthcoming science observations.

Figure 13: 5-sigma contrast curves, comparing a set of representative NACO
observations with the observation of i Vel with ERIS. The NACO observations
have different integration times and field rotations, but overall, the ERIS
contrast curve is about 1 mag better than NACO.

##  5 CONCLUSIONS

In this contribution, we have presented the newly commissioned focal plane
coronagraphic mode of ERIS, featuring vortex phase masks. We have reviewed its
design, pointing control with QACITS, template implementation, and on-sky
results in terms of pointing stability, raw coronagraphic performance and
post-processed detection limits.

While the QACITS algorithm is straightforward in theory, its implementation
can be challenging due to its interface between high-contrast imaging,
instrument control, and adaptive optics. This typically requires the
intervention of multiple experts. Access to a mid-infrared calibration source
can facilitate the algorithm’s integration and enable early testing, e.g.
during system AIV, significantly reducing the required on-sky time at
commissioning. This early integration approach is foreseen for the ELT/METIS
instrument.

The implementation of the ERIS FPC mode is now mostly finalized, enabling
robust vortex operations. Its full integration into the ESO environment
provides an easy access for the scientific community. A good pointing
stability is now systematically achieved within 0.02 $\lambda/D$ or better,
and with a rejection bandwidth of about 0.02Hz. In an hour-long observation,
we obtain a 5$\sigma$ background level of $L^{\prime}>17$. However, how close
to the star this performance level can be reached strongly depends on the
residual pointing and wavefront errors. In both speckle and background-limited
regimes, the first ERIS results show a promising improvement of about 1-mag
over previous NACO observations. While, in this proceeding, we have primarily
focused on the L band vortex, the M band mode has just been commissioned and
we have demonstrated similar pointing stability with QACITS. Both vortex modes
are thus ready for scientific exploitation.

###### Acknowledgements.

GOX thanks Valentin Christiaens for discussion on the ERIS data reduction
using the VIP Python package. GOX and OAb acknowledges funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement no. 819155), and from the
Wallonia-Brussels Federation (grant for Concerted Research Actions). This work
has been carried out within the framework of the NCCR PlanetS supported by the
Swiss National Science Foundation under grant 51NF40_205606. JH acknowledges
the financial support from the Swiss National Science Foundation under project
grant number 200020_200399.

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preprint no.

Teukolsky-like equations in
a non-vacuum axisymmetric type D spacetime
Ya Guo1, 2, Hiroaki Nakajima1 and Wenbin Lin${}^{1,\,2,\,*}$
1 School of Mathematics and Physics, University of South China,
Hengyang, 421001, China
2 School of Physical Science and Technology, Southwest Jiaotong University,
Chengdu, 610031, China
∗ Email<EMAIL_ADDRESS>

We study an axisymmetric metric satisfying the Petrov type D property with
some additional ansatze, but without assuming the vacuum condition. We find
that our metric in turn becomes conformal to the Kerr metric deformed by one
function of the radial coordinate. We then study the gravitational-wave
equations on this background metric in the case that the conformal factor is
unity. We find that under an appropriate gauge condition, the wave equations
admit the separation of the variables, and the separated equation for the
radial coordinate gives a natural extension of the Teukolsky equation.

##  1 Introduction

The direct search of the gravitational waves is one of the most important
topic in astrophysics since the observation by LIGO and VIRGO [1]. For the
theoretical computation, the waves from the binary system is mainly studied,
because it is much simpler than the many-body system. In particular, for the
case of the extreme mass-ratio inspiral (EMRI), one can apply the black hole
perturbation theory [3], where the heavier object is regarded as the (black
hole) background, whereas the lighter object is regarded as the perturbation
which appears as the source term of the wave equation. For the background of
the Kerr black hole, the wave equation propagating on it is derived by
Teukolsky [4], and called Teukolsky equation. If the mass ratio of the binary
is not so extreme, one can apply the similar setting by simultaneously
modifying the background and the perturbation by the (symmetric) mass-ratio.
This is called the effective one-boby (EOB) dynamics [5, 6, 7]111Of course,
one can also use the conventional post-Newtonian expansion method [2], and the
result from the EOB dynamics has to be compared with that. .

Since the background of the EOB dynamics is deformed from the black hole
background, it may not satisfy the vacuum condition. Thus we need to study the
wave equation on the non-vacuum background. Since it is however difficult in
general, here we would like to consider some simple cases. In the derivation
of the Teukolsky equation, it plays a crucial role that the Kerr background is
classified as the Petrov type D [8]. Then we will assume that the background
here also belongs to the type D, but not assuming the vacuum condition. Here
we note that in the case of the spherically symmetric background, it always
belongs to the type D [9, 10, 11, 12]. However, for the axisymmetric
background, it does not always belong. We will study a possible form of the
background metric under some ansatze which is introduced to avoid some
technical complexities.

In the previous study on the spherically symmetric background, it has been
found that the gravitational-wave equation in general depends on the gauge [9,
10, 11, 12], unless the background is vacuum, which is unavoidable due to the
gauge-dependence of the source term [12]. In the case of the axisymmetric
background, one also has similar problems for the choice the gauge [13].
Moreover, the separation of the variables is quite nontrivial in the current
case. For instance, a certain gauge taken in [13] does not allow the
separation and needs the expansion in terms of the Kerr-like parameter, which
controls the axial deformation. We here consider the more convenient gauge
such that the wave equation allows the separation of variables, and the
differential equation for the angular part in turn becomes that of the
conventional spin-weighted spheroidal harmonics, as expected.

The remainder of this paper is organized as follows: in section 2, we will
show the background metric, which in turn becomes conformal to the Kerr metric
deformed by one function of the radial coordinate. In section 3, we will
derive the gravitational-wave equation on the background where the conformal
factor is unity. Section 4 is devoted to the summary and discussion. In
appendix A, we will summarize the reduction to the spherically symmetric case.
In appendix B, we will discuss the gauge condition.

##  2 Background metric and tetrads

Here we will consider an axisymmetric stationaly metric satisfying the Petrov
type D condition using the Newman-Penrose formalism [14]. We will also assume
that the metric is asymptotically flat and some of the spin coefficients
vanish, which is to avoid the technical complexities and is for the later use.
However we will not assume the vacuum condition.

We begin with the following form of the metric:

$\displaystyle ds^{2}=A_{0}(dt-E_{1}d\varphi)^{2}-B(d\varphi-
E_{2}dt)^{2}-Cdr^{2}-D_{0}d\theta^{2},$ (2.1)

where $A_{0}$, $B$, $C$, $D_{0}$, $E_{1}$ and $E_{2}$ are the real functions
of $r$ and $\theta$. (2.1) is the most general form of the stationary
axisymmetric metric. We take the corresponding null tetrads as

$\displaystyle l$ $\displaystyle=dt-E_{1}d\varphi-\sqrt{\frac{C}{A_{0}}}dr\ ,$
$\displaystyle n$ $\displaystyle=\frac{A_{0}}{2}\left(dt-
E_{1}d\varphi+\sqrt{\frac{C}{A_{0}}}dr\right),$ $\displaystyle m$
$\displaystyle=\frac{e^{i\xi}}{\sqrt{2}}\left[i\sqrt{B}(d\varphi-
E_{2}dt)+\sqrt{D_{0}}d\theta\right],$ $\displaystyle\bar{m}$
$\displaystyle=\frac{e^{-i\xi}}{\sqrt{2}}\left[-i\sqrt{B}(d\varphi-
E_{2}dt)+\sqrt{D_{0}}d\theta\right].$ (2.2)

Here $\xi$ is a real function of $r$ and $\theta$. The null tetrads satisfy

$\displaystyle ds^{2}=2ln-2m\bar{m}.$ (2.3)

We now consider the type D condition. In order to avoid some technical
complexities, we first require $\kappa=\lambda=\sigma=\nu=0$, and then require
the type D condition $\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0$ itself. In the
vacuum spacetime the former implies the latter, and vice versa, which is
called the Goldberg-Sachs (GS) theorem [15] (for the type D). Our strategy is
inspired from the GS theorem and its non-vacuum extension [16, 17] (see also
[18] for instance). Another reason to require $\kappa=\lambda=\sigma=\nu=0$
first is because we will also use it to derive the wave equation below. The
spin coefficients $\kappa$ and $\sigma$ are computed from (2.2) as

$\displaystyle\kappa$
$\displaystyle=-\frac{e^{i\xi}}{2\sqrt{2D_{0}}A_{0}}\left[\frac{\partial_{\theta}C}{C}-\frac{\partial_{\theta}A_{0}}{A_{0}}+2\frac{E_{2}\partial_{\theta}E_{1}}{1-E_{1}E_{2}}-2i\sqrt{\frac{A_{0}D_{0}}{BC}}\frac{\partial_{r}E_{1}}{1-E_{1}E_{2}}\right],$
(2.4) $\displaystyle\sigma$
$\displaystyle=-\frac{e^{2i\xi}}{4\sqrt{A_{0}C}}\left[\frac{\partial_{r}D_{0}}{D_{0}}-\frac{\partial_{r}B}{B}+2\frac{E_{1}\partial_{r}E_{2}}{1-E_{1}E_{2}}+2i\sqrt{\frac{BC}{A_{0}D_{0}}}\frac{\partial_{\theta}E_{2}}{1-E_{1}E_{2}}\right].$
(2.5)

Then $\kappa=\sigma=0$ implies

$\displaystyle\frac{\partial_{\theta}C}{C}-\frac{\partial_{\theta}A_{0}}{A_{0}}+2\frac{E_{2}\partial_{\theta}E_{1}}{1-E_{1}E_{2}}=0,$
(2.6) $\displaystyle E_{1}=E_{1}(\theta),$ (2.7)
$\displaystyle\frac{\partial_{r}D_{0}}{D_{0}}-\frac{\partial_{r}B}{B}+2\frac{E_{1}\partial_{r}E_{2}}{1-E_{1}E_{2}}=0,$
(2.8) $\displaystyle E_{2}=E_{2}(r),$ (2.9)

where $\lambda=\nu=0$ also gives the same conditions. From (2.6) and (2.9), we
have

$\displaystyle\frac{C}{A_{0}(1-E_{1}E_{2})^{2}}=F(r).$ (2.10)

In a similar way, from (2.7) and (2.8), we have

$\displaystyle\frac{D_{0}}{B(1-E_{1}E_{2})^{2}}=G(\theta).$ (2.11)

By the use of the above conditions and relations, one can find that
$\Psi_{0}=\Psi_{4}=0$ is already satisfied. $\Psi_{1}$ and $\Psi_{3}$ are
computed as

$\displaystyle\Psi_{1}$
$\displaystyle=\frac{-ie^{i\xi}}{8\sqrt{2B}A_{0}(1-E_{1}E_{2})^{3}}(I-iJ),$
(2.12) $\displaystyle\Psi_{3}$
$\displaystyle=\frac{ie^{-i\xi}}{16\sqrt{2B}(1-E_{1}E_{2})^{3}}(I-iJ),$ (2.13)

where $I$ and $J$ are given by

$\displaystyle I$
$\displaystyle=\frac{2}{F}\partial_{r}\left(\frac{B}{A_{0}}E^{\prime}_{2}(r)\right)+\frac{2}{G}\partial_{\theta}\left(\frac{A_{0}}{B}E^{\prime}_{1}(\theta)\right)$
$\displaystyle\qquad{}+(1-E_{1}E_{2})^{2}\frac{B}{A_{0}}E^{\prime}_{2}(r)\partial_{r}\left[\frac{1}{F(1-E_{1}E_{2})^{2}}\right]$
$\displaystyle\qquad{}+(1-E_{1}E_{2})^{2}\frac{A_{0}}{B}E^{\prime}_{1}(\theta)\partial_{\theta}\left[\frac{1}{G(1-E_{1}E_{2})^{2}}\right],$
(2.14) $\displaystyle J$
$\displaystyle=\frac{2(1-E_{1}E_{2})}{\sqrt{FG}}\left[\frac{\partial_{r}\partial_{\theta}B}{B}-\frac{\partial_{r}B\partial_{\theta}B}{B^{2}}-\frac{\partial_{r}\partial_{\theta}A_{0}}{A_{0}}+\frac{\partial_{r}A_{0}\partial_{\theta}A_{0}}{A_{0}^{2}}\right],$
(2.15)

where the prime denotes the ordinary derivative of the functions with respect
to their arguments. $\Psi_{1}=\Psi_{3}=0$ implies $I=J=0$. The condition $J=0$
can be integrated as

$\displaystyle\frac{B(r,\theta)}{A_{0}(r,\theta)}=H(r)M(\theta).$ (2.16)

By substituting (2.10), (2.11) and (2.16), the metric (2.1) becomes

$\displaystyle ds^{2}$ $\displaystyle=A_{0}\Bigl{[}(dt-
E_{1}d\varphi)^{2}-HM(d\varphi-E_{2}dt)^{2}$
$\displaystyle\qquad\qquad{}-F(1-E_{1}E_{2})^{2}dr^{2}-HMG(1-E_{1}E_{2})^{2}d\theta^{2}\Bigr{]}.$
(2.17)

In order to simplify the condition $I=0$, we use the condition such that the
metric (2.1) is asymptotically flat, which gives the behavior of the functions
at large $r$ as

$\displaystyle A_{0}(r,\theta)$ $\displaystyle\sim 1,$ (2.18) $\displaystyle
E_{1}(\theta)-E_{2}(r)H(r)M(\theta)$ $\displaystyle\sim 0,$ (2.19)
$\displaystyle H(r)M(\theta)-E_{1}^{2}(\theta)$ $\displaystyle\sim
r^{2}\sin^{2}\theta,$ (2.20) $\displaystyle F(r)[1-E_{1}(\theta)E_{2}(r)]^{2}$
$\displaystyle\sim 1,$ (2.21) $\displaystyle
H(r)M(\theta)G(\theta)[1-E_{1}(\theta)E_{2}(r)]^{2}$ $\displaystyle\sim
r^{2}.$ (2.22)

From (2.19), the product $E_{2}(r)H(r)$ has to behave as a constant at large
$r$. Thus we define a constant $a$ as

$\displaystyle a=\lim_{r\to\infty}E_{2}(r)H(r).$ (2.23)

We will see later that $a$ plays a similar role of the Kerr parameter. Other
conditions fix the functions of $\theta$ explicitly and behaviors of the
functions of $r$ at large $r$ as

$\displaystyle E_{1}(\theta)$ $\displaystyle=a\sin^{2}\theta,$ $\displaystyle
G(\theta)$ $\displaystyle=\frac{1}{\sin^{2}\theta},$ $\displaystyle M(\theta)$
$\displaystyle=\sin^{2}\theta,$ (2.24) $\displaystyle E_{2}(r)$
$\displaystyle\sim\frac{a}{r^{2}},$ $\displaystyle F(r)$ $\displaystyle\sim
1,$ $\displaystyle H(r)$ $\displaystyle\sim r^{2}.$ (2.25)

From (2.24) and (2.25), $I$ in (2.14) is evaluated as

$\displaystyle I$
$\displaystyle=-4a+8a^{2}E_{2}+P^{\prime}E^{\prime}_{2}+2PE^{\prime\prime}_{2}$
$\displaystyle\qquad{}+a\sin^{2}\theta\left[-4aE_{2}-P^{\prime}E_{2}E^{\prime}_{2}-2PE_{2}E^{\prime\prime}_{2}+2P(E^{\prime}_{2})^{2}\right],$
(2.26)

where $P(r)=H^{2}(r)/F(r)$. Then the condition $I=0$ is split into two
ordinary differential equations with respect to the argument $r$ as

$\displaystyle-4a+8a^{2}E_{2}+P^{\prime}E^{\prime}_{2}+2PE^{\prime\prime}_{2}$
$\displaystyle=0,$ (2.27)
$\displaystyle-4aE_{2}-P^{\prime}E_{2}E^{\prime}_{2}-2PE_{2}E^{\prime\prime}_{2}+2P(E^{\prime}_{2})^{2}$
$\displaystyle=0.$ (2.28)

By eliminating $P^{\prime}$ from two equations, we obtain

$\displaystyle P=\frac{4aE_{2}-4a^{2}E_{2}^{2}}{(E^{\prime}_{2})^{2}}.$ (2.29)

One can confirm that (2.29) satisfies both (2.27) and (2.28). By substituting
(2.24) and (2.29), the metric (2.17) is

$\displaystyle ds^{2}$ $\displaystyle=A_{0}\Biggl{[}(dt-a\sin^{2}\theta
d\varphi)^{2}-H\sin^{2}\theta(d\varphi-E_{2}dt)^{2}$
$\displaystyle\qquad\qquad{}-\frac{H^{2}(E^{\prime}_{2})^{2}}{4aE_{2}-4a^{2}E_{2}^{2}}(1-a\sin^{2}\theta
E_{2})^{2}dr^{2}-H(1-a\sin^{2}\theta E_{2})^{2}d\theta^{2}\Biggr{]}.$ (2.30)

We define the function $R(r)$ as

$\displaystyle R(r)=\sqrt{\frac{a}{E_{2}(r)}-a^{2}},$ (2.31)

and then (2.30) is rewritten as

$\displaystyle ds^{2}$ $\displaystyle=A_{0}\Biggl{[}(dt-a\sin^{2}\theta
d\varphi)^{2}-H\sin^{2}\theta\left(d\varphi-\frac{a}{R^{2}+a^{2}}dt\right)^{2}$
$\displaystyle\qquad\qquad{}-\frac{H^{2}(R^{\prime})^{2}}{(R^{2}+a^{2})^{2}}\left(\frac{\Sigma}{R^{2}+a^{2}}\right)^{2}dr^{2}-H\left(\frac{\Sigma}{R^{2}+a^{2}}\right)^{2}d\theta^{2}\Biggr{]}.$
(2.32)

where $\Sigma$ is defined by

$\displaystyle\Sigma=R(r)^{2}+a^{2}\cos^{2}\theta.$ (2.33)

The existence of the factor $(R^{\prime})^{2}dr^{2}=dR^{2}$ leads us that one
can use $R$ as the radial coordinate, which is equivalent to take the gauge

$\displaystyle R(r)=r.$ (2.34)

In this gauge (2.32) becomes

$\displaystyle ds^{2}$ $\displaystyle=A_{0}\Biggl{[}(dt-a\sin^{2}\theta
d\varphi)^{2}-\frac{H^{2}\Sigma^{2}}{(r^{2}+a^{2})^{4}}dr^{2}$
$\displaystyle\qquad\qquad\quad{}-\frac{H\Sigma^{2}}{(r^{2}+a^{2})^{2}}d\theta^{2}-H\sin^{2}\theta\left(d\varphi-\frac{a}{r^{2}+a^{2}}dt\right)^{2}\Biggr{]}.$
(2.35)

Finally, by defining

$\displaystyle L(r)=\frac{(r^{2}+a^{2})^{2}}{H(r)},\quad
A(r,\theta)=A_{0}(r,\theta)\frac{\Sigma}{L(r)},$ (2.36)

we obtain the metric as

$\displaystyle ds^{2}$
$\displaystyle=A(r,\theta)\Biggl{[}\frac{L(r)}{\Sigma}(dt-a\sin^{2}\theta
d\varphi)^{2}-\frac{\Sigma}{L(r)}dr^{2}$
$\displaystyle\qquad\qquad\quad{}-\Sigma
d\theta^{2}-\frac{(r^{2}+a^{2})^{2}\sin^{2}\theta}{\Sigma}\left(d\varphi-\frac{a}{r^{2}+a^{2}}dt\right)^{2}\Biggr{]},$
(2.37)

One can find that (2.37) is the Kerr metric deformed by the function $L(r)$
and by the conformal factor $A(r,\theta)$. When we restrict these two
functions as

$\displaystyle A(r,\theta)=1,\quad L(r)=r^{2}-2Mr+a^{2},$ (2.38)

the metric (2.37) reduces to the Kerr metric with the mass parameter $M$ and
the Kerr parameter $a$ in the Boyer-Lindquist coordinate, where the choice of
the coordinate is from the gauge condition (2.34). On the other hand, when we
consider the case $A(r,\theta)=A(r)$ and $a=0$, (2.37) reduces to the general
form of the spherically symmetric metric in a particular gauge (see appendix
A).

The null tetrads (2.2) become

$\displaystyle l$ $\displaystyle=dt-a\sin^{2}\theta
d\varphi-\frac{\Sigma}{L}dr\ ,$ $\displaystyle n$
$\displaystyle=\frac{A}{2}\left[\frac{L}{\Sigma}(dt-a\sin^{2}\theta
d\varphi)+dr\right],$ $\displaystyle m$
$\displaystyle=\frac{1}{\varrho}\sqrt{\frac{A}{2}}\left[-i(r^{2}+a^{2})\sin\theta
d\varphi+ia\sin\theta dt-\Sigma d\theta\right],$ $\displaystyle\bar{m}$
$\displaystyle=\frac{1}{\bar{\varrho}}\sqrt{\frac{A}{2}}\left[i(r^{2}+a^{2})\sin\theta
d\varphi-ia\sin\theta dt-\Sigma d\theta\right],$ (2.39)

where $\varrho=r+ia\cos\theta$ and the phase $\xi$ is chosen as

$\displaystyle
e^{-i\xi}=-\frac{\varrho}{\sqrt{\Sigma}}=-\frac{\varrho}{|\varrho|}.$ (2.40)

From the tetrad basis (2.39), the spin coefficients are computed as

$\displaystyle\kappa=\lambda=\sigma=\nu=\epsilon=0,$ (2.41)
$\displaystyle\mu=-\frac{L}{2\bar{\varrho}\Sigma}-\frac{L\partial_{r}A}{4A\Sigma},\quad\tau=-\frac{ia\sin\theta}{\sqrt{2A}\Sigma}-\frac{\partial_{\theta}A}{2\sqrt{2A^{3}}\varrho},\quad\pi=\frac{ia\sin\theta}{\sqrt{2A}\bar{\varrho}^{2}}+\frac{\partial_{\theta}A}{2\sqrt{2A^{3}}\bar{\varrho}},$
(2.42)
$\displaystyle\rho=-\frac{1}{A\bar{\varrho}}-\frac{\partial_{r}A}{2A^{2}},\quad\gamma=\mu+\frac{L^{\prime}}{4\Sigma}+\frac{L\partial_{r}A}{2A\Sigma},\quad\beta=\frac{\cot\theta}{2\sqrt{2A}\varrho},\quad\alpha=\pi-\bar{\beta}-\frac{\partial_{\theta}A}{\sqrt{2A^{3}}\bar{\varrho}}.$
(2.43)

In a similar way, the Weyl scalars are

$\displaystyle\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0,$ (2.44)
$\displaystyle\Psi_{2}=\frac{1}{A\Sigma}\left[-\frac{1}{6}-\frac{a^{2}\sin^{2}\theta}{\bar{\varrho}^{2}}-\frac{ia\cos\theta}{\bar{\varrho}}+\frac{L}{\bar{\varrho}^{2}}-\frac{L^{\prime}}{2\bar{\varrho}}+\frac{L^{\prime\prime}}{12}\right].$
(2.45)

Finally the components of the Ricci tensor are

$\displaystyle\Phi_{00}=\frac{3(\partial_{r}A)^{2}}{4A^{4}}-\frac{\partial_{r}^{2}A}{2A^{3}},$
(2.46)
$\displaystyle\Phi_{01}=\bar{\Phi}_{10}=\frac{\partial_{\theta}A}{2\sqrt{2A^{5}}\varrho^{2}}-\frac{ia\sin\theta\partial_{r}A}{2\sqrt{2A^{5}}\varrho^{2}}+\frac{3\partial_{r}A\partial_{\theta}A}{4\sqrt{2A^{7}}\varrho}-\frac{\partial_{r}\partial_{\theta}A}{2\sqrt{2A^{5}}\varrho},$
(2.47) $\displaystyle\begin{aligned}
\Phi_{11}&=\frac{1}{4A\Sigma}\left[\frac{2r^{2}-2a^{2}+2L-2rL^{\prime}}{\Sigma}+\frac{L^{\prime\prime}}{2}-1-\left(\frac{\cot\theta}{2}+\frac{a^{2}\sin
2\theta}{\Sigma}\right)\frac{\partial_{\theta}A}{A}\right.\\\
&\quad{}+\left.\frac{3}{4}\left(\frac{\partial_{\theta}A}{A}\right)^{2}+\left(\frac{L^{\prime}}{2}-\frac{2rL}{\Sigma}\right)\frac{\partial_{r}A}{A}-\frac{3}{4}L\left(\frac{\partial_{r}A}{A}\right)^{2}-\frac{\partial_{\theta}^{2}A}{2A}+\frac{L\partial_{r}^{2}A}{2A}\right],\end{aligned}$
(2.48)
$\displaystyle\Phi_{02}=\bar{\Phi}_{20}=\frac{3(\partial_{\theta}A)^{2}}{8A^{3}\varrho^{2}}+\frac{\cot\theta\partial_{\theta}A}{4A^{2}\varrho^{2}}-\frac{\partial_{\theta}^{2}A}{4A^{2}\varrho^{2}},$
(2.49)
$\displaystyle\Phi_{22}=\frac{3L^{2}(\partial_{r}A)^{2}}{16A^{2}\Sigma^{2}}-\frac{L^{2}\partial_{r}^{2}A}{8A\Sigma^{2}},$
(2.50)
$\displaystyle\Phi_{12}=\bar{\Phi}_{21}=-\frac{3\bar{\varrho}L\partial_{r}A\partial_{\theta}A}{8\sqrt{2A^{5}}\Sigma^{2}}-\frac{L\partial_{\theta}A}{4\sqrt{2A^{3}}\Sigma^{2}}-\frac{ia\sin\theta
L\partial_{r}A}{4\sqrt{2A^{3}}\Sigma^{2}}+\frac{\bar{\varrho}L\partial_{r}\partial_{\theta}A}{4\sqrt{2A^{3}}\Sigma^{2}},$
(2.51) $\displaystyle\begin{aligned}
\Lambda&=\frac{1}{4A\Sigma}\left[\frac{1}{3}-\frac{L^{\prime\prime}}{6}-\frac{\cot\theta\partial_{\theta}A}{2A}+\frac{1}{4}\left(\frac{\partial_{\theta}A}{A}\right)^{2}\right.\\\
&\qquad\qquad{}-\left.\frac{L^{\prime}\partial_{r}A}{2A}+\frac{L}{4}\left(\frac{\partial_{r}A}{A}\right)^{2}-\frac{\partial_{\theta}^{2}A}{2A}-\frac{L\partial_{r}^{2}A}{2A}\right].\end{aligned}$
(2.52)

Note that in the case of $A(r,\theta)=1$, the spin coefficients, the Weyl
scalars and the components of the Ricci tensor are simplified as

$\displaystyle\kappa=\lambda=\sigma=\nu=\epsilon=0,$ (2.53)
$\displaystyle\rho=-\frac{1}{\bar{\varrho}},\quad\tau=-\frac{ia\sin\theta}{\sqrt{2}\Sigma},\quad\mu=-\frac{L}{2\bar{\varrho}\Sigma},$
(2.54)
$\displaystyle\gamma=\mu+\frac{L^{\prime}}{4\Sigma},\quad\pi=\frac{ia\sin\theta}{\sqrt{2}\bar{\varrho}^{2}},\quad\beta=\frac{\cot\theta}{2\sqrt{2}\varrho},\quad\alpha=\pi-\bar{\beta},$
(2.55) $\displaystyle\Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0,$ (2.56)
$\displaystyle\Psi_{2}=\frac{1}{\Sigma}\left[-\frac{1}{6}-\frac{a^{2}\sin^{2}\theta}{\bar{\varrho}^{2}}-\frac{ia\cos\theta}{\bar{\varrho}}+\frac{L}{\bar{\varrho}^{2}}-\frac{L^{\prime}}{2\bar{\varrho}}+\frac{L^{\prime\prime}}{12}\right],$
(2.57)
$\displaystyle\Phi_{00}=\Phi_{01}=\Phi_{10}=\Phi_{02}=\Phi_{20}=\Phi_{12}=\Phi_{21}=0,$
(2.58)
$\displaystyle\Phi_{11}=\frac{1}{4\Sigma}\left[\frac{2r^{2}-2a^{2}+2L-2rL^{\prime}}{\Sigma}+\frac{L^{\prime\prime}}{2}-1\right],$
(2.59) $\displaystyle\Lambda=\frac{2-L^{\prime\prime}}{24\Sigma}.$ (2.60)

In particular, some of the components of the Ricci tensor vanish as (2.58),
which gives the great simplification when we will study the gravitational-wave
equation on this background in next section. We also note that in the case of
$A(r,\theta)=1$, our background (2.37) coincides the one studied in [13],
which is the special case of the axisymmetric EOB background in [7]. In [7],
the function $L(r)$ in front of $(dt-a\sin^{2}\theta d\varphi)^{2}$ and that
in front of $dr^{2}$ are taken to be different. However in that case, the type
D condition cannot be satesfied.

##  3 Gravitational-wave equation

In prevoius section, we have shown that the metric (2.37) is classified as the
non-vacuum Petrov type D background, which is useful to derive the
gravitational-wave equation for the perturbed Weyl scalars using the Newman-
Penrose formalism, as in the case of the Teukolsky equation [4] on the
Schwarzschild and the Kerr background. For simplicity, hereafter we will focus
on the case $A(r,\theta)=1$, such that among the components of the Ricci
tensor, only $\Phi_{11}$ and $\Lambda$ are nonvanishing as (2.58)–(2.60). The
case with the general $A(r,\theta)$ will be left as the future work. We note
that in the case of the spherically symmetric background, the wave equation
with the general conformal factor can be derived [11, Guo2023-2], where the
different radial coordinate is used. For the comparison of the radial
coordinate, see appendix A.

We begin with the following equations in Newman-Penrose formalism:

$\displaystyle(\delta+4\beta-\tau)\Psi_{4}-(\Delta+4\mu+2\gamma)\Psi_{3}+3\nu\Psi_{2}$
$\displaystyle\qquad=(\bar{\delta}-\bar{\tau}+2\bar{\beta}+2\alpha)\Phi_{22}-(\Delta+2\gamma+2\bar{\mu})\Phi_{21}-2\lambda\Phi_{12}+2\nu\Phi_{11}+\bar{\nu}\Phi_{20},$
(3.1)
$\displaystyle(D+4\epsilon-\rho)\Psi_{4}-(\bar{\delta}+4\pi+2\alpha)\Psi_{3}+3\lambda\Psi_{2}$
$\displaystyle\qquad=(\bar{\delta}-2\bar{\tau}+2\alpha)\Phi_{21}-(\Delta+2\gamma-2\bar{\gamma}+\bar{\mu})\Phi_{20}+\bar{\sigma}\Phi_{22}-2\lambda\Phi_{11}+2\nu\Phi_{10},$
(3.2)
$\displaystyle(\Delta+\mu+\bar{\mu}+3\gamma-\bar{\gamma})\lambda-(\bar{\delta}+\pi-\bar{\tau}+\bar{\beta}+3\alpha)\nu+\Psi_{4}=0.$
(3.3)

We split all the quantities in the above into the background part $(A)$ and
the perturbation part $(B)$, for instance,
$\Psi_{4}=\Psi_{4}^{A}+\Psi_{4}^{B}$, etc. The background part of
eqs.(3.1)–(3.3) are shown to be satisfied, and the part of the first-order
perturbation becomes

$\displaystyle(\delta+4\beta-\tau)^{A}\Psi_{4}^{B}-(\Delta+4\mu+2\gamma)^{A}\Psi_{3}^{B}+3\nu^{B}\Psi_{2}^{A}$
$\displaystyle\qquad=(\bar{\delta}-\bar{\tau}+2\bar{\beta}+2\alpha)^{A}\Phi_{22}^{B}-(\Delta+2\gamma+2\bar{\mu})^{A}\Phi_{21}^{B}+2\nu^{B}\Phi_{11}^{A},$
(3.4)
$\displaystyle(D+4\epsilon-\rho)^{A}\Psi_{4}^{B}-(\bar{\delta}+4\pi+2\alpha)^{A}\Psi_{3}^{B}+3\lambda^{B}\Psi_{2}^{A}$
$\displaystyle\qquad=(\bar{\delta}-2\bar{\tau}+2\alpha)^{A}\Phi_{21}^{B}-(\Delta+2\gamma-2\bar{\gamma}+\bar{\mu})^{A}\Phi_{20}^{B}-2\lambda^{B}\Phi_{11}^{A},$
(3.5)
$\displaystyle(\Delta+\mu+\bar{\mu}+3\gamma-\bar{\gamma})^{A}\lambda^{B}-(\bar{\delta}+\pi-\bar{\tau}+\bar{\beta}+3\alpha)^{A}\nu^{B}+\Psi_{4}^{B}=0,$
(3.6)

where the superscript $A$ $(B)$ on the parentheses denotes that all the
quantities and the operators inside the parentheses are in the background
(perturbation). The quantities in previous section are regarded as the
background ones, and hence they will appear in this section with the
superscript $A$.

We will obtain the wave equation for $\Psi_{4}^{B}$ in a similar way with the
method used to derive the Teukolsky equation [4]. First we note that the
following commutation relation of the differential operators holds not only in
vacuum but also in the current background:

$\displaystyle\left[\Delta+(p+1)\gamma-\bar{\gamma}-q\mu+\bar{\mu}\right]^{A}(\bar{\delta}+p\alpha-q\pi)^{A}$
$\displaystyle\qquad-\left[\bar{\delta}+(p+1)\alpha+\bar{\beta}-\bar{\tau}-q\pi\right]^{A}(\Delta+p\gamma-q\mu)^{A}$
$\displaystyle=\nu^{A}D^{A}-\lambda^{A}\delta^{A}-p\left[(\beta+\tau)\lambda-(\rho+\epsilon)\nu+\Psi_{3}\right]^{A}$
$\displaystyle\qquad+q\left[-D\nu+\delta\lambda+(\bar{\pi}+\tau+3\beta-\bar{\alpha})\lambda-(3\epsilon+\bar{\epsilon}+\rho-\bar{\rho})\nu+2\Psi_{3}\right]^{A}$
$\displaystyle=0,$ (3.7)

where $p$ and $q$ are arbitrary constants and we have just used
$\lambda^{A}=\nu^{A}=\Psi_{3}^{A}=0$222Here $\lambda^{A}=\nu^{A}=0$ is
important. $\kappa^{A}=\sigma^{A}=0$ will also be used for a similar
calculation of the wave equation for $\Psi_{0}^{B}$. . We operate
$(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})^{A}$ to (3.5) and
$(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)^{A}$ to (3.4), and then
subtract one equation from the other. The terms with $\Psi_{3}^{B}$ cancel by
(3.7) with $p=2$, $q=-4$ and the remaining becomes

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)-(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)\right]^{A}\Psi_{4}^{B}$
$\displaystyle\qquad{}+(3\Psi_{2}+2\Phi_{11})^{A}(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})^{A}\lambda^{B}$
$\displaystyle\qquad{}-(3\Psi_{2}-2\Phi_{11})^{A}(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)^{A}\nu^{B}$
$\displaystyle=T_{4}-\lambda^{B}\Delta^{A}(3\Psi_{2}+2\Phi_{11})^{A}+\nu^{B}\bar{\delta}^{A}(3\Psi_{2}-2\Phi_{11})^{A}$
(3.8)

where $T_{4}$ is defined by

$\displaystyle T_{4}$
$\displaystyle=(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})^{A}\left[(\bar{\delta}-2\bar{\tau}+2\alpha)^{A}\Phi_{21}^{B}-(\Delta+2\gamma-2\bar{\gamma}+\bar{\mu})^{A}\Phi_{20}^{B}\right]$
$\displaystyle\quad{}-(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)^{A}\left[(\bar{\delta}-\bar{\tau}+2\bar{\beta}+2\alpha)^{A}\Phi_{22}^{B}-(\Delta+2\gamma+2\bar{\mu})^{A}\Phi_{21}^{B}\right].$
(3.9)

In the right hand side, we use

$\displaystyle\Delta^{A}(3\Psi_{2}-2\Phi_{11})^{A}$
$\displaystyle=-3\mu^{A}(3\Psi_{2}+2\Phi_{11})^{A}+4(\mu+\bar{\mu})^{A}\Phi_{11}^{A},$
(3.10) $\displaystyle\bar{\delta}^{A}(3\Psi_{2}+2\Phi_{11})^{A}$
$\displaystyle=-3\pi^{A}(3\Psi_{2}-2\Phi_{11})^{A}-4(\pi-\bar{\tau})^{A}\Phi_{11}^{A}.$
(3.11)

and then substituting the above into (3.8), we have

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)-(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}\right]^{A}\Psi_{4}^{B}$
$\displaystyle\qquad{}+2\Phi_{11}^{A}(\Delta+3\gamma-\bar{\gamma}+\mu+\bar{\mu})^{A}\lambda^{B}+2\Phi_{11}^{A}(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+\pi)^{A}\nu^{B}$
$\displaystyle=T_{4}-4\lambda^{B}[(\Delta+\mu+\bar{\mu})\Phi_{11}]^{A}-4\nu^{B}[(\bar{\delta}+\pi-\bar{\tau})\Phi_{11}]^{A},$
(3.12)

where we have used (3.6). In the case of vacuum $\Phi_{11}^{A}=0$, (3.12)
gives the decoupled equation for $\Psi_{4}^{B}$, but in the present case,
$\nu^{B}$ and $\lambda^{B}$ are still remaining. Hence by applying appropriate
gauge conditions, one has to eliminate them.

One possible choice is to take the gauge $\Psi_{3}^{B}=0$ [13], under which
both $\lambda^{B}$ and $\nu^{B}$ can be expressed in terms of $\Psi_{4}^{B}$
from (3.4) and (3.5). However the resulting wave equation becomes complicated,
and then we will consider other gauges. Since the relative sign in the second
line in (3.12) is positive, one cannot rewrite the second line in terms of
$\Psi_{4}^{B}$ using (3.6) completely, but instead one can eliminate either
$\nu^{B}$ or $\lambda^{B}$. When $\nu^{B}$ is eliminated, (3.12) becomes

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)\right.$
$\displaystyle\qquad{}-\left.(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}+2\Phi_{11}\right]^{A}\Psi_{4}^{B}$
$\displaystyle=T_{4}-4(\Delta+3\gamma-\bar{\gamma}+2\mu+2\bar{\mu})^{A}(\Phi_{11}^{A}\lambda^{B})-4\nu^{B}[(\bar{\delta}+\pi-\bar{\tau})\Phi_{11}]^{A}.$
(3.13)

Then under the gauge condition

$\displaystyle\Omega_{1}\equiv(\Delta+3\gamma-\bar{\gamma}+2\mu+2\bar{\mu})^{A}(\Phi_{11}^{A}\lambda^{B})+\nu^{B}[(\bar{\delta}+\pi-\bar{\tau})\Phi_{11}]^{A}=0,$
(3.14)

we obtain the decoupled equation for $\Psi_{4}^{B}$ as

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)\right.$
$\displaystyle\qquad{}-\left.(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}+2\Phi_{11}\right]^{A}\Psi_{4}^{B}=T_{4}.$
(3.15)

In the case of spherically symmetric background this gauge is simple, because
the terms with $\nu^{B}$ in the gauge condition (3.14) vanish due to

$\displaystyle(\bar{\delta}+\pi-\bar{\tau})^{A}\Phi_{11}^{A}=0,$ (3.16)

and then (3.14) is satisfied just by setting $\lambda^{B}=0$, which has been
used in the previous studies. In the present case, (3.16) does not hold
generically unless the vacuum case, but instead, one can solve (3.14) in terms
of $\nu^{B}$ as

$\displaystyle\nu^{B}=-\frac{(\Delta+3\gamma-\bar{\gamma}+2\mu+2\bar{\mu})^{A}(\Phi_{11}^{A}\lambda^{B})}{(\bar{\delta}+\pi-\bar{\tau})^{A}\Phi_{11}^{A}}.$
(3.17)

When we eliminate $\lambda^{B}$ from the second line in (3.12), we have

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)\right.$
$\displaystyle\qquad{}-\left.(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}-2\Phi_{11}\right]^{A}\Psi_{4}^{B}$
$\displaystyle=T_{4}-4\lambda^{B}[(\Delta+\mu+\bar{\mu})\Phi_{11}]^{A}-4(\bar{\delta}+3\alpha+\bar{\beta}-2\bar{\tau}+2\pi)^{A}(\Phi_{11}^{A}\nu^{B}).$
(3.18)

Then under the gauge condition

$\displaystyle\Omega_{2}\equiv\lambda^{B}[(\Delta+\mu+\bar{\mu})\Phi_{11}]^{A}+(\bar{\delta}+3\alpha+\bar{\beta}-2\bar{\tau}+2\pi)^{A}(\Phi_{11}^{A}\nu^{B})=0,$
(3.19)

or equivalently333One can find that the denominator of (3.20) is generically
nonvanishing unless the vacuum case.

$\displaystyle\lambda^{B}=-\frac{(\bar{\delta}+3\alpha+\bar{\beta}-2\bar{\tau}+2\pi)^{A}(\Phi_{11}^{A}\nu^{B})}{(\Delta+\mu+\bar{\mu})^{A}\Phi_{11}^{A}},$
(3.20)

we obtain the decoupled equation as

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)\right.$
$\displaystyle\qquad{}-\left.(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}-2\Phi_{11}\right]^{A}\Psi_{4}^{B}=\tilde{T}_{4},$
(3.21)

where we have put the tilde to the source $T_{4}$ in order to emphasize that
we are taking the gauge (3.19) which is different from (3.14), and then
$T_{4}$ in (3.15) and that in (3.21) are different.

One can find that by taking different gauges, the last term in the left hand
side of the wave equations can be changed, namely
$+2\Phi_{11}^{A}\Psi_{4}^{B}$ for (3.15) and $-2\Phi_{11}^{A}\Psi_{4}^{B}$ for
(3.21). Here we will introduce yet another gauge such that the last term is
$-6\Lambda^{A}\Psi_{4}^{B}$, and we refer to as the separable gauge. One can
rewrite (3.12) as

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)\right.$
$\displaystyle\qquad{}-\left.(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}-6\Lambda\right]^{A}\Psi_{4}^{B}$
$\displaystyle=T_{4}-4\lambda^{B}[(\Delta+\mu+\bar{\mu})\Phi_{11}]^{A}-2(\Phi_{11}-3\Lambda)^{A}(\Delta+3\gamma-\bar{\gamma}+\mu+\bar{\mu})^{A}\lambda^{B}$
$\displaystyle\qquad{}-4\nu^{B}[(\bar{\delta}+\pi-\bar{\tau})\Phi_{11}]^{A}-2(\Phi_{11}+3\Lambda)^{A}(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+\pi)^{A}\nu^{B},$
(3.22)

Then the gauge condition for the separable gauge is

$\displaystyle\Omega_{3}$
$\displaystyle\equiv\frac{1}{2}(\Phi_{11}-3\Lambda)^{A}(\Delta+3\gamma-\bar{\gamma}+\mu+\bar{\mu})^{A}\lambda^{B}+\lambda^{B}[(\Delta+\mu+\bar{\mu})\Phi_{11}]^{A}$
$\displaystyle\quad{}+\frac{1}{2}(\Phi_{11}+3\Lambda)^{A}(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+\pi)^{A}\nu^{B}+\nu^{B}[(\bar{\delta}+\pi-\bar{\tau})\Phi_{11}]^{A}=0,$
(3.23)

which can also be rewritten as

$\displaystyle\nu^{B}=-\frac{(\Delta+3\gamma-\bar{\gamma}+2\mu+2\bar{\mu})^{A}(\Phi_{11}^{A}\lambda^{B})}{(\bar{\delta}+\pi-\bar{\tau})^{A}\Phi_{11}^{A}}-\frac{1}{2}\frac{(\Phi_{11}+3\Lambda)^{A}\Psi_{4}^{B}}{(\bar{\delta}+\pi-\bar{\tau})^{A}\Phi_{11}^{A}},$
(3.24)

or

$\displaystyle\lambda^{B}=-\frac{(\bar{\delta}+3\alpha+\bar{\beta}-2\bar{\tau}+2\pi)^{A}(\Phi_{11}^{A}\nu^{B})}{(\Delta+\mu+\bar{\mu})^{A}\Phi_{11}^{A}}+\frac{1}{2}\frac{(\Phi_{11}-3\Lambda)^{A}\Psi_{4}^{B}}{(\Delta+\mu+\bar{\mu})^{A}\Phi_{11}^{A}}.$
(3.25)

The corresponding wave equation is

$\displaystyle\left[(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})(D+4\epsilon-\rho)\right.$
$\displaystyle\qquad{}-\left.(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)(\delta+4\beta-\tau)-3\Psi_{2}-6\Lambda\right]^{A}\Psi_{4}^{B}=\hat{T}_{4},$
(3.26)

where we have put the hat to the source in a similar sense as (3.21). The
advantage of the separable gauge is that the wave equation (3.26) admits the
separation of the variables. We define $\psi_{(-2)}$ and $T_{(-2)}$ as

$\displaystyle\psi_{(-2)}=(\rho^{A})^{-4}\Psi_{4}^{B}=\bar{\varrho}^{4}\Psi_{4}^{B},\qquad
T_{(-2)}=\bar{\varrho}^{4}\hat{T}_{4}.$ (3.27)

Then the wave equation for $\psi_{(-2)}$ is

$\displaystyle\left[\frac{(r^{2}+a^{2})^{2}}{L(r)}-a^{2}\sin^{2}\theta\right]\frac{\partial^{2}\psi_{(-2)}}{\partial
t^{2}}+\left[\frac{2(r^{2}+a^{2})L^{\prime}(r)}{L(r)}-8r-4ia\cos\theta\right]\frac{\partial\psi_{(-2)}}{\partial
t}$ $\displaystyle{}-L^{2}(r)\frac{\partial}{\partial
r}\left(\frac{1}{L(r)}\frac{\partial\psi_{(-2)}}{\partial
r}\right)-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\psi_{(-2)}}{\partial\theta}\right)+\left(\frac{a^{2}}{L(r)}-\frac{1}{\sin^{2}\theta}\right)\frac{\partial^{2}\psi_{(-2)}}{\partial\varphi^{2}}$
$\displaystyle{}+\left[\frac{2a(r^{2}+a^{2})}{L(r)}-2a\right]\frac{\partial^{2}\psi_{(-2)}}{\partial
t\partial\varphi}+\left(\frac{2aL^{\prime}(r)}{L(r)}+\frac{4i\cos\theta}{\sin^{2}\theta}\right)\frac{\partial\psi_{(-2)}}{\partial\varphi}$
$\displaystyle{}+(4\cot^{2}\theta+2)\psi_{(-2)}=2\Sigma T_{(-2)},$ (3.28)

which reduces to the Teukolsky master equation on the Kerr background with the
spin $s=-2$ in the case of $L=r^{2}-2Mr+a^{2}$. For the separation of the
variables, we assume the product form of the solution:

$\displaystyle\psi_{(-2)}=e^{-i\omega t}e^{im\varphi}R(r)S(\theta),$ (3.29)

where $\omega$ is the frequency of the gravitational wave and $m$ is constant.
For homogeneous case $T_{(-2)}=0$, by substituting the above into (3.28), we
obtain the separated equations as

$\displaystyle
L^{2}\frac{d}{dr}\left(\frac{1}{L}\frac{dR}{dr}\right)+\left(\frac{K^{2}+2iKL^{\prime}}{L}-8i\omega
r+2am\omega-a^{2}\omega^{2}-\bm{\lambda}_{(-2)}\right)R=0,$ (3.30)
$\displaystyle\begin{aligned}
&\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{dS}{d\theta}\right)\\\
&{}\qquad+\left(a^{2}\omega^{2}\cos^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}+4a\omega\cos\theta+\frac{4m\cos\theta}{\sin^{2}\theta}-4\cot^{2}\theta-2+\bm{\lambda}_{(-2)}\right)S=0,\end{aligned}$
(3.31)

where $K=(r^{2}+a^{2})\omega-am$ and $\bm{\lambda}_{(-2)}$ is the separation
constant. From (3.31) one can find that $S(\theta)e^{im\varphi}$ coincides
with the $s=-2$ spin-weighted spheroidal harmonics
${}_{-2}S^{a\omega}_{lm}(\theta,\varphi)$, where $l$ and $m$ take the integer
values with $l\geq 2$ and $-l\leq m\leq l$, respectively. For the value of the
separation constant $\bm{\lambda}_{(-2)}$, which becomes the eigenvalue of the
spheroidal harmonics, the exact expression is not known yet. Instead some
approximated expressions are studied. For example, the small-$a\omega$
expansion of $\bm{\lambda}_{(s)}$ for the general spin $s$ has been computed
as [19, 20]

$\displaystyle\bm{\lambda}_{(s)}=(l-s)(l+s+1)-\frac{2s^{2}m}{l(l+1)}a\omega+\left[h(l+1)-h(l)-1\right]a^{2}\omega^{2}+O(a^{3}\omega^{3}),$
(3.32)

where $h(l)$ is given by

$\displaystyle
h(l)=\frac{2(l^{2}-m^{2})(l^{2}-s^{2})^{2}}{(2l-1)l^{3}(2l+1)}.$ (3.33)

The leading term $(l-s)(l+s+1)$ in (3.32) is the eigenvalue of the spin-
weighted spherical harmonics with the spin $s$. For nonhomogeneous case
$T_{(-2)}\neq 0$, we can expand $\psi_{(-2)}$ and $T_{(-2)}$ by
${}_{-2}S^{a\omega}_{lm}(\theta,\varphi)$ as

$\displaystyle\psi_{(-2)}$ $\displaystyle=\int
d\omega\sum_{l,m}R^{(-2)}_{lm\omega}(r)_{-2}S^{a\omega}_{lm}(\theta,\varphi)e^{-i\omega
t},$ (3.34) $\displaystyle-2\Sigma T_{(-2)}$ $\displaystyle=\int
d\omega\sum_{l,m}G^{(-2)}_{lm\omega}(r)_{-2}S^{a\omega}_{lm}(\theta,\varphi)e^{-i\omega
t}.$ (3.35)

Then we obtain the ordinary differential equation for the radial coordinate as

$\displaystyle
L^{2}\frac{d}{dr}\left(\frac{1}{L}\frac{dR^{(-2)}_{lm\omega}}{dr}\right)+\left(\frac{K^{2}+2iKL^{\prime}}{L}-8i\omega
r+2am\omega-a^{2}\omega^{2}-\bm{\lambda}_{(-2)}\right)R^{(-2)}_{lm\omega}=G^{(-2)}_{lm\omega},$
(3.36)

which reduces to the Teukolsky radial equation on the Kerr background with the
spin $s=-2$ in the case of $L=r^{2}-2Mr+a^{2}$, as well as the master
equation.

A similar analysis can be applied for the wave equation for $\Psi_{0}^{B}$,
which in the separable gauge is obtained as

$\displaystyle\left[(D-3\epsilon+\bar{\epsilon}-4\rho-\bar{\rho})(\Delta-4\gamma+\mu)\right.$
$\displaystyle\qquad{}-\left.(\delta+\bar{\pi}-\bar{\alpha}-3\beta-4\tau)(\bar{\delta}+\pi-4\alpha)-3\Psi_{2}-6\Lambda\right]^{A}\Psi_{0}^{B}=T_{0},$
(3.37)

where the source $T_{0}$ is defined as

$\displaystyle T_{0}$
$\displaystyle=(\delta+\bar{\pi}-\bar{\alpha}-3\beta-4\tau)^{A}\left[(D-2\epsilon-2\bar{\rho})^{A}\Phi_{01}^{B}-(\delta+\bar{\pi}-2\bar{\alpha}-2\beta)^{A}\Phi_{00}^{B}\right]$
$\displaystyle\quad{}+(D-3\epsilon+\bar{\epsilon}-4\rho-\bar{\rho})^{A}\left[(\delta+2\bar{\pi}-2\beta)^{A}\Phi_{01}^{B}-(D-2\epsilon+2\bar{\epsilon}-\bar{\rho})^{A}\Phi_{02}^{B}\right].$
(3.38)

The gauge condition is

$\displaystyle\frac{1}{2}(\Phi_{11}-3\Lambda)^{A}(D-3\epsilon+\bar{\epsilon}-\rho-\bar{\rho})^{A}\sigma^{B}+\sigma^{B}[(D-\rho-\bar{\rho})\Phi_{11}]^{A}$
$\displaystyle\quad{}+\frac{1}{2}(\Phi_{11}+3\Lambda)^{A}(\delta+\bar{\pi}-\bar{\alpha}-3\beta-\tau)^{A}\kappa^{B}+\kappa^{B}[(\delta+\bar{\pi}-\tau)\Phi_{11}]^{A}=0,$
(3.39)

Then the explicit form of the wave equation in turn becomes

$\displaystyle\left[\frac{(r^{2}+a^{2})^{2}}{L(r)}-a^{2}\sin^{2}\theta\right]\frac{\partial^{2}\Psi_{0}^{B}}{\partial
t^{2}}+\left[-\frac{2(r^{2}+a^{2})L^{\prime}(r)}{L(r)}+8r+4ia\cos\theta\right]\frac{\partial\Psi_{0}^{B}}{\partial
t}$ $\displaystyle{}-\frac{1}{L^{2}(r)}\frac{\partial}{\partial
r}\left(L^{3}(r)\frac{\partial\Psi_{0}^{B}}{\partial
r}\right)-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Psi_{0}^{B}}{\partial\theta}\right)+\left(\frac{a^{2}}{L(r)}-\frac{1}{\sin^{2}\theta}\right)\frac{\partial^{2}\Psi_{0}^{B}}{\partial\varphi^{2}}$
$\displaystyle{}+\left[\frac{2a(r^{2}+a^{2})}{L(r)}-2a\right]\frac{\partial^{2}\Psi_{0}^{B}}{\partial
t\partial\varphi}-\left(\frac{2aL^{\prime}(r)}{L(r)}+\frac{4i\cos\theta}{\sin^{2}\theta}\right)\frac{\partial\Psi_{0}^{B}}{\partial\varphi}$
$\displaystyle{}+\left[4\cot^{2}\theta+2-2L^{\prime\prime}(r)\right]\Psi_{0}^{B}=2\Sigma
T_{0},$ (3.40)

which reduces to the Teukolsky master equation with $s=2$ in the case of Kerr
background. After the separation of variables, the solution to the angular
part gives the $s=-2$ spin-weighted spheroidal harmonics
${}_{2}S^{a\omega}_{lm}(\theta,\varphi)$. On the other hand, the equation for
the radial part becomes

$\displaystyle\frac{1}{L^{2}}\frac{d}{dr}\left(L^{3}\frac{dR^{(2)}_{lm\omega}}{dr}\right)$
$\displaystyle\qquad{}+\left(\frac{K^{2}-2iKL^{\prime}}{L}+8i\omega
r-4+2L^{\prime\prime}+2am\omega-a^{2}\omega^{2}-\bm{\lambda}_{(2)}\right)R^{(2)}_{lm\omega}=G^{(2)}_{lm\omega},$
(3.41)

where we have used the following expansion

$\displaystyle\Psi_{0}^{B}$ $\displaystyle=\int
d\omega\sum_{l,m}R^{(2)}_{lm\omega}(r)_{2}S^{a\omega}_{lm}(\theta,\varphi)e^{-i\omega
t},$ (3.42) $\displaystyle-2\Sigma T_{0}$ $\displaystyle=\int
d\omega\sum_{l,m}G^{(2)}_{lm\omega}(r)_{2}S^{a\omega}_{lm}(\theta,\varphi)e^{-i\omega
t}.$ (3.43)

We again note that (3.41) reduces to the Teukolsky radial equation with $s=2$
in the case of Kerr background.

##  4 Summary and discussion

In this paper, we have studied a metric satisfying the axial symmetry, the
stationality, the Petrov type D property and the asymptotic flatness with some
additional ansatze, but without assuming the vacuum condition. The metric
(2.37) in turn becomes the Kerr metric deformed by the function $L(r)$ of the
radial coordinate and the conformal factor $A(r,\theta)$. We then have studied
the gravitational-wave equation on the background of this metric in the case
of $A(r,\theta)=1$. For non-vacuum case, the wave equation is usually gauge-
dependent, although the unknown variable $\Psi_{4}^{B}$ is gauge-invariant. We
have found one convenient gauge, refered to as the separable gauge, such that
the wave equation admits the separation of the variables. The solution of the
equation (3.31) for the angular part just becomes the spin-weighted spheroidal
harmonics. The radial part (3.36) of the equation gives the natural extension
of the Teukolsky radial equation.

The metric (2.37) has been obtained using some assumptions, and hence it is
not a classification. In the case of vacuum, the metric which posseses the
type D property is completely classified, called the Kinnersley metric [21].
It would be interestiing to find the non-vacuum extension of the Kinnersley
classification.

We have considered the gravitational-wave equation only for the special case
$A(r,\theta)=1$ of the background, because otherwise all of the components of
the Ricci tensor ($\Phi^{A}$’s and $\Lambda^{A}$) becomes nonzero, and then it
is complicated. We will leave to study this issue as our future work. The wave
equations (3.28) and (3.36) have been obtained under the separable gauge
(3.23). Since the gauge dependence of the wave equation is due to that of the
source term $T_{4}$, in particular $\Phi_{21}^{B}$ as (B.5) below, the gauge
choice seems to be unavoidable. Then we have to consider how to ensure the
gauge condition, which would actually be the problem for any type of the gauge
condition.

The property of the wave equations derived here strongly depends on the
behaviour of the function $L(r)$. In particular, the zeroes of $L(r)$ would
give some of the singularites of the equation (3.36). For example, in the case
of the Kerr background $L(r)=r^{2}-2Mr+a^{2}$, the zeroes of $L(r)$ are just
located at the inner and the outer horizon, which appear as the two regular
singularities of the Teukolsky equation. Since the Teukolsky equation also has
one irregular singularity at the infinity $r\to\infty$, it is mathematically
classified as an example of the confluent Heun equation [22]. Similarly, in
the present case, the behaviour of $L(r)$ is very important. For the large-$r$
behavior, since $L(r)\sim r^{2}$ from the asymptotic flatness, the structure
of the wave equation is similar with that of the Teukolsky equation. However,
for the small $r$, $L(r)$ could have the behaviour different from the Kerr
case. In particular, the number of the zeroes could change, which gives the
quite different behaviour of the wave equation, because the zeroes of $L(r)$
could give the singularities of the differential equation. The small-$r$
behaviour is very important for the scattering problem as well, since the
boundary condition of the wave is determined from that.  It is also important
to calculate the quasi-normal frequency and the fluxes of the gravitational
waves [23, 24, 25, 26].

Finally here we have considered only the gravitational-wave equation. It would
be interesting to study the wave equation on this background with the
different spin, namely the Klein-Gordon equation, the Dirac equation and the
Maxwell equation, as in the case of the spherically symmetric background [12].

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China (Grant No. 11973025).

##  Appendix A Reduction of the background to spherically symmetric case

In the case of $A(r,\theta)=A(r)$ and $a=0$, the metric (2.37) takes the
following form:

$\displaystyle
ds^{2}=A(r)\left[\frac{L(r)}{r^{2}}dt^{2}-\frac{r^{2}}{L(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right].$ (A.1)

On the other hand, the general form of the metric for the spherically
symmetric background is

$\displaystyle
ds^{2}=\mathcal{A}(r)dt^{2}-\mathcal{B}(r)dr^{2}-\mathcal{C}(r)r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}).$ (A.2)

From the comparison of two metrics, we have

$\displaystyle\mathcal{A}(r)=\frac{A(r)L(r)}{r^{2}},\quad\mathcal{B}(r)=\frac{r^{2}A(r)}{L(r)},\quad\mathcal{C}(r)=A(r).$
(A.3)

By eliminating $A(r)$ and $L(r)$ from the above, we obtain

$\displaystyle\mathcal{C}(r)=\sqrt{\mathcal{A}(r)\mathcal{B}(r)},$ (A.4)

which can be regarded as the gauge choice of radial coordinate, and the metric
(2.37) contains the general case of the spherically symmetric background. In
the previous study, we have used the different coordinate such that
$\mathcal{C}(r)$=1. In the case of $A(r)=1$, which gives
$\mathcal{A}(r)\mathcal{B}(r)=1$, these two choices agree.

##  Appendix B Gauge condition

In this paper, we have studied a few types of the gauge condition, which is
imposed in order to obtain the decoupled equation of the gravitational wave.
At the beginning, we have four variables $\Psi_{4}^{B}$, $\Psi_{3}^{B}$,
$\lambda^{B}$ and $\nu^{B}$ in (3.4)–(3.6). Among them $\Psi_{3}^{B}$ is
eliminated by using the commutation relation (3.7), and then the three
variables remain. These variables are not independent and there is one
relation (3.6) between them, which is gauge-covariant. Hence we can have
another one relation as the gauge conditon to obtain the decoupled equation
for one variable. Here we will use the tetrad rotation (the local Lorentz
transformation) as the gauge symmetry, which consists of the following three
kinds [27]:

$\displaystyle l^{\mu}\to l^{\mu},\quad m^{\mu}\to
m^{\mu}+al^{\mu},\quad\bar{m}^{\mu}\to\bar{m}^{\mu}+\bar{a}l^{\mu},\quad
n^{\mu}\to n^{\mu}+\bar{a}m^{\mu}+a\bar{m}^{\mu}+a\bar{a}l^{\mu},$ (B.1)
$\displaystyle n^{\mu}\to n^{\mu},\quad m^{\mu}\to
m^{\mu}+bn^{\mu},\quad\bar{m}^{\mu}\to\bar{m}^{\mu}+\bar{b}n^{\mu},\quad
l^{\mu}\to l^{\mu}+\bar{b}m^{\mu}+b\bar{m}^{\mu}+b\bar{b}n^{\mu},$ (B.2)
$\displaystyle l^{\mu}\to e^{-c}l^{\mu},\quad n^{\mu}\to e^{c}n^{\mu},\quad
m^{\mu}\to e^{i\vartheta}m^{\mu},\quad\bar{m}^{\mu}\to
e^{-i\vartheta}\bar{m}^{\mu},$ (B.3)

where $a$ and $b$ are the complex functions444Only here $a$ is not the Kerr
parameter. We hope the readers may not confuse it.  and $c$ and $\vartheta$
are real functions. Since we do not want to change the background, these
parameter functions has to be regarded as the perturbation quantities, and the
linear order of the transformation is enough. The variables $\lambda^{B}$ and
$\nu^{B}$ are transformed as555$\lambda^{B}$ and $\nu^{B}$ are invariant under
(B.2) and (B.3). A similar thing also holds for (B.5).

$\displaystyle\lambda^{B}\to\lambda^{B}+(\bar{\delta}+2\alpha+\pi)^{A}\bar{a},\quad\nu^{B}\to\nu^{B}+(\Delta+2\gamma+\mu)^{A}\bar{a},$
(B.4)

whereas $\Psi_{4}^{B}$ is invariant, which can also be checked using the
commutation relation (3.7) with $p=2$, $q=-1$.

The source term $T_{4}$ contains the perturbation quantities $\Phi_{20}^{B}$,
$\Phi_{21}^{B}$ and $\Phi_{22}^{B}$, which transforms under (B.1)–(B.3) as

$\displaystyle\Phi_{20}^{B}\to\Phi_{20}^{B},\quad\Phi_{21}^{B}\to\Phi_{21}^{B}+2\Phi_{11}^{A}\bar{a},\quad\Phi_{22}^{B}\to\Phi_{22}^{B}.$
(B.5)

From the above, $T_{4}$ transforms as

$\displaystyle T_{4}$ $\displaystyle\to
T_{4}+4(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)^{A}(\Delta+2\gamma+2\mu+\bar{\mu})^{A}(\Phi_{11}^{A}\bar{a})$
$\displaystyle\qquad\ \
{}-4(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})^{A}[(\bar{\tau}+2\pi)^{A}\Phi_{11}^{A}\bar{a}].$
(B.6)

The gauge dependence of the wave equations is due to the gauge dependence of
$T_{4}$, although $T_{4}$ is written in terms of the energy-momentum tensor of
the matter through the Einstein equation. The above transformation has to be
canceled by that of terms in the right hand side in (3.15), (3.21) and (3.26)
other than $T_{4}$, because the left hand side of each equation is gauge-
invariant. Then the gauge transformation of the gauge conditions (3.14),
(3.19) and (3.23) is

$\displaystyle\Omega_{i}$
$\displaystyle\to\Omega_{i}+(\bar{\delta}+3\alpha+\bar{\beta}-\bar{\tau}+4\pi)^{A}(\Delta+2\gamma+2\mu+\bar{\mu})^{A}(\Phi_{11}^{A}\bar{a})$
$\displaystyle\qquad\ \
{}-(\Delta+3\gamma-\bar{\gamma}+4\mu+\bar{\mu})^{A}[(\bar{\tau}+2\pi)^{A}\Phi_{11}^{A}\bar{a}],$
(B.7)

where $i$ runs from 1 to 3, corresponding to the three gauge conditions. The
gauge parameter $a$ has to be chosen so that the right hand side of (B.7)
vanishes.

## Acknowledgements

The authors thank Prof. Remo Ruffini for useful discussions. This work was
supported in part by the National Natural Science Foundation of China (Grant
No. 11973025).

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