Best-fit light curves for 26 (left) and 10 (right), where the x-axis is shown in observer-frame days minus the time of peak brightness (in MJD). $\it{Left}$: The best-fit light curve for 26 in F115W, F150W, F200W, F277W, F356W, and F444W. The best-fit model, , is a type Ibc model. The thick dashed curve corresponds to the  model at the best-fit redshift of $z$=2.845. The other fainter curves show other type Ibc model fits. $\it{Right}$: The best-fit light curve for 10 for F150W, F200W, F277W, F335M, F356W, and F444W. The best-fit model, , is a type IIP model. The thick dashed lines correspond to the  model at the best-fit redshift of $z$ = 3.61. The other fainter curves show other type II models.

The right panel of Figure <ref> shows the best-fit light curve for 10, which belongs to host JADES-GS+53.16439-27.83877, in F150W, F200W, F277W, F335M, F356W, and F444W. The best-fit model is , which is a type IIP model, and the best-fit model redshift is $z = $ 3.61. 10 was also targeted by NIRSpec with DDT Program 6541, and it is spectroscopically-confirmed to be at z=3.61 based on host galaxy features in the spectrum. See D. Coulter et al. (in prep) for a complete analysis of 10 and its spectrum. As shown in the right panel of Figure <ref>, 10 is post-peak in each epoch of observation, and all of the data points fall either on or within 1 $\sigma$ of the best-fit model curve, making the type IIP classification robust.

Figure <ref> shows 13's Epoch1 SED (left) and light curve (right). Based on the 3 $< z <$ 4 F150W vs. F150W-F444W CMD shown in the bottom left panel of Figure <ref>, 13 is a SLSN candidate at $z_{phot} =$ 3.58 in host galaxy JADES-GS+53.15375-27.82513 (see Section <ref>). Both the SED and light curve display its very blue color, and the light curve is best-fit by the  model, which is a type Ib/c model. The best-fit redshift is 3.437, which broadly agrees with the assigned host redshift of $z_{phot} =$ 3.58. The light curve indicates that 13 is a young blue type Ib/c SN that has not yet reached peak. However, we emphasize that we cannot confidently draw any conclusions regarding 13's classification or phase from the SED, best-fit light curve, or position in the F150W vs. F150W-F444W CMD because they are based on only one epoch of observation. Since we do not know 13's peak absolute magnitude, we cannot confirm that it is a SLSN. It would be a large extrapolation to assume it gets significantly brighter based on just one epoch, so we simply list it as a SLSN candidate and provide the alternate solution that its a young type Ib/c SN. 13 appears to be faintly visible in F200W in Epoch2 and may be visible in F356W and F444W (see Figure <ref> and Section <ref>), but there is still insufficient data coverage of 13 to robustly classify it as a specific SN subtype at a specific phase. There is also a chance that it is some non-SN hot blue object, but that analysis is beyond the scope of this paper.

Epoch1 SED (left) and light curve (right) for 13. $\it{Left}$: 13's Epoch1 SED displays its very blue color, peaking at $m_{F150W} =$ 29.23 $\pm$ 0.09 mag. $\it{Right}$: 13's light curve, with measured PSF photometry shown as circles (upper limits shown as circles with downward arrows), the best-fit model (, type Ib/c at z=3.437) shown as the thick dashed lines, and the other type Ib/c models shown as faint lines in the background. We show the light curves for F150W, F200W, F277W, F335M, F356W, and F444W. The best-fit curve indicates that 13 is a young blue type Ib/c SN that has not yet reached peak. However, this classification is based on just one epoch of observation, so it is not a robust classification.

20, which resides in host galaxy JADES-GS$+$53.14729$-$27.81047, is another interesting source in the sample, as it is very bright in Epoch1 and continues to be visible in Epoch2 and all of the follow-up epochs. Because of its maintained brightness, we measured PSF photometry from the science images rather than the difference images and attempted to perform multi-epoch light curve fitting with this photometry. However, the CC models could not fit the follow-up epoch data because the median duration of the CC models is $\sim$80 rest-frame days and the maximum duration is $\sim$140 rest-frame days. At $z_{phot}$=2.00 (z-rank=3 host redshift), 20 remains bright for at least $\sim$150 rest-frame days (5 months in rest frame; 15 months in the observer frame). We thus do not list  classification results for 20. However, 20 is likely a type II SN because it stayed visible for such a long period of time, and it does not appear to be bright enough to be a SLSN candidate.

33's light curve fitting yields a $z=$6.829$\pm$0.002 type II solution, which is unreliable as it is based on just one epoch of observation and the host redshift is a highly uncertain photometric redshift ($z_{phot}$ = 4.82$\pm$0.49). While 33 is a high-redshift SN candidate (see Section <ref>), we do not claim that it is a $z\sim$6.8 SN.

§.§ Other Variable Sources

There are 9 sources in the JADES Deep Field that show signs of variability but are likely not supernovae. Their positions and the reason for their exclusion from the supernova sample are listed in Table <ref>.

We note that the source at 03:32:38.8660, -27:47:13.465 (42 in Table <ref>) was reported by Hayes et al., 2024 as a $z>6$ variable AGN candidate (object 1052123 in their work). This source fades in Epoch2 but re-brightens in Epoch4, and therefore we removed it from our SN sample. We took a spectrum of this object as part of DDT Program 6541. Despite a low S/N, the spectrum seems not match any supernova, galaxy or AGN template at $z\simeq 5 - 7$. On the contrary, the latest JADES NIRCam SED of 42 can be fit with a late-type star at $z\sim 0$, which could be intrinsically variable. We also note that the significance of other $z>6$ variable sources in Hayes et al., 2024 do not pass our selection criteria.

Any additional analysis of our excluded, non-SN variable sources (e.g., AGN activities) is beyond the scope of this paper.

Other Variable Sources

RA     Dec     Reason for Exclusion from SN Sample

03:32:44.4863     -27:46:42.018     Brightens then fades below initial brightness

03:32:38.8660     -27:47:13.465     Fades then re-brightens; DDT-6541 spectrum does not match SN spectrum

03:32:43.2401     -27:49:14.452     Fades then re-brightens; variability directly over galactic core

03:32:41.8760     -27:44:00.214     Variability directly over galactic core

03:32:38.5752     -27:49:09.357     Fades then re-brightens; variability directly over galactic core

03:32:38.6035     -27:45:00.285     Variability directly over galactic core

03:32:27.4456     -27:48:25.316     Failed SN light curve fitting; highly compact (potentially a star)

03:32:37.5294     -27:47:56.522     Host is obscured MIRI AGN

03:32:43.3201     -27:49:47.317     Host SED fitting indicates AGN is present

§.§ Implications of Supernovae Mis-Classified as z$\sim$16 Galaxy Candidates

The discovery that two $z$ $\sim$ 16 galaxy candidates were actually transient events (or galaxies hosting transient events) has important implications for high-redshift galaxy surveys[There was also an earlier discovery of a probable SN reported with the HST HUDF data that could have been mis-identified as a $z$ $\sim$ 7 “z-band dropout" (the source zD0 reported in Bunker] et al., 2010).. With only one deep imaging epoch, it is often difficult to discern real compact high-redshift galaxy candidates from faint transient events. One of the three $z>$15 galaxy candidates in Hainline et al., 2024 that fall within the JADES Deep Field is a transient interloper (101; see object JADES-GS-53.12692-27.79102 in their Figure 8 for the $z\sim$16 SED fit). In principle, it is possible to discern between high-redshift galaxy candidates and SNe if the SW data are deep enough because the SNe do not drop out as fast with wavelength as Lyman-break galaxies. For example, Hainline et al., 2024 rejected 107 as a $z\sim$16 galaxy candidate due to 4$\sigma$ emission in F090W (see object JADES-GS-53.20055-27.78493 in their Figure 11 for the $z\sim$16 SED fit). However, many surveys do not have adequate depth in the dropout filters to rule out SNe.

The timescale for brightness variation increases as $(1+z)$ in the observer's frame due to the cosmological time dilation, so a sufficiently long time baseline is needed to recognize slowly varying high-redshift transient events, which could be mistaken as galaxies at extremely high redshift based on a single-epoch observation because of their faintness in the SW bands.  In the case of the JADES Transient Survey, the two main imaging epochs are separated by one year, allowing high-redshift transients sufficient time to fade significantly ($\sim$ 4 months at $z$ $\sim$ 2 and $\sim$ 2 months at $z$ $\sim$ 5 in the SN rest-frame).

With faint long-wavelength transient events possibly contaminating the high-redshift galaxy candidate sample, we risk overestimating the galaxy abundance at high-redshift if we base our analysis solely on a single-epoch data. Although the level of such SN contamination is thought to be small ($\sim$ 10% at $z$ $>$ 11 according to Yan et al., 2023), it may become more significant in the sample of the highest-redshift galaxies (e.g., $z$ $>$ 15) as our study seems to indicate.  Since reaching comparable depth in multi-epoch data is very expensive, it is important to explore ways to distinguish high-redshift galaxy candidates and transient events with single-epoch data <cit.>.

In this regard, we also note that although Hainline et al., 2024 present 101 as a candidate at $z >$ 15, they caution that its colors are consistent with a lower redshift solution.

101 is bright source with some morphology seen in the individual images in the filters, and its SED is fairly red from 2 to 5 , which is unexpected for a high-redshift galaxy (Figure 8 in Hainline et al., 2024). These are some key indicators that a $z >$ 15 galaxy candidate may instead be a transient interloper.

The discovery of these $z$ $\sim$ 16 galaxy interlopers also raises the question of how much the observed galaxy SEDs are affected by transient phenomena in general.  At high redshift, SN rates are expected to increase due to enhanced star-forming activities in galaxies and potentially more top-heavy IMFs due to lower metallicities and higher gas temperature, partially heated by the cosmic microwave background. SN emission also will remain for longer periods of time due to time dilation.

Since these two $z$ $\sim$ 16 galaxy interlopers are spatially compact, the possibility remains that what we are seeing may be the light from slowly fading high-redshift SNe from 2023 to 2022.

Important future work involves conducting a systematic search of high-redshift galaxies (e.g., $z$ $\geq$ 8) to determine if JWST has the capability of detecting point-like variability at such high redshift. This will provide information about supernova rates at even higher redshift, and may lend clues for how to discern genuinely high-redshift transient events from those in lower-redshift interlopers.

§ CONCLUSIONS

The JADES Transient Survey is the deepest systematic transient/SN search with JWST/NIRCam targeting the 9-band multi-epoch JADES Deep Field. It covers $\sim$25 arcmin$^2$ and has a depth of $\sim$30.5 magnitude. We discovered the transients by differencing the Epoch1 and Epoch2 images and searching for point-like sources of emission. For a subset of the SNe, we had 1-6 multi-band follow-up NIRCam epochs to build light curves. Here, we summarize the main takeways from the JADES Transient Survey:

* The sample we have presented contains 79 SNe in the JADES Deep Field, 34 of which brightened between Epoch1 and Epoch2 and 45 of which faded between Epoch1 and Epoch2. There are 4 additional marginally-detected SNe that are detailed in the paper but not included in the main statistical sample. This sample contains many of the highest redshift SNe, with 7 SNe at $z\geq$4, 15 SNe at $z\geq$3, and 38 SNe at $z\geq$2. The difference images clearly show emission down to $\sim$30.5 magnitude, making JWST capable of detecting distant SNe as well as SNe in dusty regions close to the galaxy centers. Previous SN surveys with HST probed only to $z$$\sim$2 and $m_{F160W}$$\sim$26.7 [Grogin et al., 2011], so JWST is exploring new redshift and magnitude space for SNe. We adopt the host redshifts as the SN redshifts, $\sim$59% of which are spectroscopic redshifts, $\sim$23% of which are robust photometric redshifts, and $\sim$14% of which are photometric redshifts whose SEDs may suffer from SN contamination. We were unable to determine the redshift for $\sim$4% of the SNe.

* At a depth of $\sim$30.5 AB magnitude, the SN detection rate is $\sim$ 1–2 per arcmin$^2$ per observer-frame year, demonstrating JWST's ability to discover supernovae essentially anywhere it looks.

* We have 1-6 follow-up NIRCam observations for the majority of the JADES-SN-23 supernovae, which allowed us to build light curves and attempt to classify the SNe. We had to rely on single-epoch light curve fitting for the JADES-SN-22 sources. With varying levels of confidence indicated by the best-fit model's $\chi^2$/DOF, we classified 40 of the 45 JADES-SN-22 sources and 30 of the 34 JADES-SN-23 sources. The JADES-SN-22 sample contains 9 Type Ia, 5 Ibc, and 26 Type II SN candidates, and the JADES-SN-23 sample contains 2 Type Ia, 9 Type Ibc, and 19 Type II SN candidates. The best-fit SN redshifts generally agree well with the associated host galaxy redshifts.

* The sample includes a spectroscopically-confirmed type-Ia SN at $z =$ 2.90 (27; see Pierel et al., 2024 for a complete analysis), a spectroscopically-confirmed type Ic-BL SN at $z = $ 2.83 (26; see Siebert et al., 2024 for a complete analysis) and a type IIP SN at $z_{spec} =$ 3.61 (10; see D. Coulter et al. (in prep) for a complete analysis). 27 and 26 are the highest-redshift spectroscopically-confirmed SNe of their respective types, emphasizing the groundbreaking nature of the JADES-SN-23 and JADES-SN-22 samples. Additionally, 10 is the highest-redshift type IIP SN discovered thus far (D. Coulter et al. (in prep).

* Two Epoch1 $z\sim$16 galaxy candidates are actually transients that faded at LW in Epoch2. We treat them as SNe, but there is a possibility that they are some other type of isolated transient event. This result exemplifies how multi-epoch surveys can identify interlopers in high-redshift galaxy surveys, as transients can mimic high-redshift galaxies when they are compact and lack SW emission.

The overwhelming result from this paper is that JWST is a SN discovery machine. JWST opens up an entirely new redshift regime for SN discovery, which allows the possibility of finding more exotic SNe, such as Population III SNe that are expected to exist at high-redshift. It is essential to take advantage of JWST's power to discover SNe and plan additional high-redshift transient surveys to add to our sample of JWST transients and SNe.

CD thanks Ofer Yaron for his extensive support in producing the TNS IDs for each source. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #1180 and 6541.
This research is based (in part) on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
All the HST data used in this paper can be found in MAST: [10.17909/T91019]http://dx.doi.org/10.17909/T91019.
Part of the JWST data used in this paper can be found in MAST: [10.17909/8tdj-8n28]https://dx.doi.org/10.17909/8tdj-8n28 (JADES DR1).
Additionally, this work made use of the lux supercomputer at UC Santa Cruz which is funded by NSF MRI grant AST 1828315, as well as the High Performance Computing (HPC) resources at the University of Arizona which is funded by the Office of Research Discovery and Innovation (ORDI), Chief Information Officer (CIO), and University Information Technology Services (UITS).
AJB acknowledges funding from the “FirstGalaxies” Advanced Grant from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement No. 789056).
PAC, EE, DJE, BDJ, GR, MR, FS, CNAW are supported by JWST/NIRCam contract to the University of Arizona, NAS5-02015.
DJE is also supported as a Simons Investigator.
RM acknowledges support by the Science and Technology Facilities Council (STFC), by the ERC through Advanced Grant 695671 “QUENCH”, and by the UKRI Frontier Research grant RISEandFALL. RM also acknowledges funding from a research professorship from the Royal Society.
BER acknowledges support from the NIRCam Science Team contract to the University of Arizona, NAS5-02015, and JWST Program 3215.
ST acknowledges support by the Royal Society Research Grant G125142.
The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.

JWST, HST

astropy [Astropy Collaboration et al., 2022], EAZY and eazy-py [Brammer et al., 2008], STARDUST2 [Rodney et al., 2014], photutils [Bradley et al., 2024]

§ THE JADES-SN-23 SAMPLE

In Appendix <ref>, we present, (1) multi-band cutout NIRCam images (top), (2) multi-epoch SEDs (lower-left), and (3) multi-band light curves (lower-right) for each of the JADES-SN-23 sources.  At the top of each page, we show the IAU ID (issued by TNS), redshift (spectroscopic $z_{\rm spec}$ or photometric $z_{\rm phot}$), and SN classification based on the light-curve analysis.

The panel of multi-band images (top) displays NIRCam images in three filters (from left to right in the order of increasing wavelengths).  For each filter, we show Epoch1 (middle), Epoch2 (top), and Epoch2-Epoch1 images (bottom). We also show three-color images for each epoch and the difference image (the right column).

The SED plot (lower-left) shows the PSF photometry measured as described in Section <ref> in nJy. For sources covered by multiple epochs, we color code the multi-epoch SEDs by observer-frame days since the first observation. The SED associated with the first observation is always shown in lime green.

The light-curve plot (lower-right) is presented only when, (i) the fitting converged to a solution, and (ii) the $\chi^2$/DOF $\leq$ 50 for the best-fit model. See Section <ref> for more details.
The Y axis on the left shows the measured apparent AB magnitude while the Y axis on the right shows the corresponding absolute AB magnitude calculated with the best-fit model redshift (see Table <ref>).
Observer-frame days minus the mean Modified Julian Date (MJD) of peak brightness are shown on the bottom X axis while the top X axis shows the corresponding rest-frame days. The circles are the measured photometry (or upper limits when shown with a downward arrow), the thick dashed line shows the best-fit model with $\pm 1\sigma$ error bars as the shaded region, and the faint lines show the other models of the same type (i.e., type II, type Ibc, or type Ia).

§ THE JADES-SN-22 SAMPLE

In Appendix <ref>, we present the same information for each of the JADES-SN-22 sources. The bottom row of the multi-band images show Epoch1-Epoch2 difference images. Note that 20's SED has multiple epochs shown because it remained visible in all of the follow-up observations. We measured its photometry from the science images rather than the difference images to avoid underestimating its brightness. For the light curves, the best-fit model redshifts that were used to convert apparent to absolute AB magnitude are listed in Table <ref>.

[Astier et al., 2006]
Astier, P., Guy, J., Regnault, N., et al. 2006, , 447, 31

[Astropy Collaboration et al., 2022]
Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, , 935, 167

[Bradley et al., 2024]
Bradley, L., Sipőcz, B., Robitaille, T., et al. 2024, astropy/photutils: 1.12.0, 1.12.0,  Zenodo

[Brammer et al., 2008]
Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, , 686, 1503

[Brout et al., 2022]
Brout, D., Scolnic, D., Popovic, B., et al. 2022, , 938, 110

[Bunker et al., 2010]
Bunker, A. J., Wilkins, S., Ellis, R. S., et al. 2010, , 409, 855

[Bunker et al., 2023]
Bunker, A. J., Cameron, A. J., Curtis-Lake, E., et al. 2023, arXiv e-prints, arXiv:2306.02467

[Conroy et al., 2009]
Conroy, C., Gunn, J. E., & White, M. 2009, , 699, 486

[Cooke et al., 2012]
Cooke, J., Sullivan, M., Gal-Yam, A., et al. 2012, , 491, 228

[Cooper et al., 2012]
Cooper, M. C., Yan, R., Dickinson, M., et al. 2012, , 425, 2116

[Curtin et al., 2019]
Curtin, C., Cooke, J., Moriya, T. J., et al. 2019, , 241, 17

[D'Andrea et al., 2010]
D'Andrea, C. B., Sako, M., Dilday, B., et al. 2010, , 708, 661

[DeCoursey et al., 2023]
DeCoursey, C., Egami, E., Rieke, M., et al. 2023a, Transient Name Server AstroNote, 16, 1

[DeCoursey et al., 2023]
—. 2023b, Transient Name Server AstroNote, 164, 1

[D'Eugenio et al., 2024]
D'Eugenio, F., Cameron, A. J., Scholtz, J., et al. 2024, arXiv e-prints, arXiv:2404.06531

[Drout et al., 2011]
Drout, M. R., Soderberg, A. M., Gal-Yam, A., et al. 2011, , 741, 97

[Eisenstein et al., 2023]
Eisenstein, D. J., Willott, C., Alberts, S., et al. 2023, arXiv e-prints, arXiv:2306.02465

[Frieman et al., 2008]
Frieman, J. A., Bassett, B., Becker, A., et al. 2008, , 135, 338

[Frye et al., 2024]
Frye, B. L., Pascale, M., Pierel, J., et al. 2024, , 961, 171

[Garilli et al., 2021]
Garilli, B., McLure, R., Pentericci, L., et al. 2021, , 647, A150

[Giavalisco et al., 2004]
Giavalisco, M., Ferguson, H. C., Koekemoer, A. M., et al. 2004, , 600, L93

[Golubchik et al., 2023]
Golubchik, M., Zitrin, A., Pierel, J., et al. 2023, , 522, 4718

[Graur et al., 2014]
Graur, O., Rodney, S. A., Maoz, D., et al. 2014, , 783, 28

[Grogin et al., 2011]
Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, , 197, 35

[Gupta et al., 2016]
Gupta, R. R., Kuhlmann, S., Kovacs, E., et al. 2016, , 152, 154

[Hainline et al., 2024]
Hainline, K. N., Johnson, B. D., Robertson, B., et al. 2024, , 964, 71

[Hamuy et al., 2006]
Hamuy, M., Folatelli, G., Morrell, N. I., et al. 2006, , 118, 2

[Hayes et al., 2024]
Hayes, M. J., Tan, J. C., Ellis, R. S., et al. 2024, arXiv e-prints, arXiv:2403.16138

[Hsiao et al., 2007]
Hsiao, E. Y., Conley, A., Howell, D. A., et al. 2007, , 663, 1187

[Inami et al., 2017]
Inami, H., Bacon, R., Brinchmann, J., et al. 2017, , 608, A2

[Ji et al., 2023]
Ji, Z., Williams, C. C., Tacchella, S., et al. 2023, arXiv e-prints, arXiv:2305.18518

[Kasen et al., 2011]
Kasen, D., Woosley, S. E., & Heger, A. 2011, , 734, 102

[Kessler et al., 2009]
Kessler, R., Bernstein, J. P., Cinabro, D., et al. 2009, , 121, 1028

[Kool et al., 2018]
Kool, E. C., Ryder, S., Kankare, E., et al. 2018, , 473, 5641

[Le Fèvre et al., 2013]
Le Fèvre, O., Cassata, P., Cucciati, O., et al. 2013, , 559, A14

[Le Fèvre et al., 2015]
Le Fèvre, O., Tasca, L. A. M., Cassata, P., et al. 2015, , 576, A79

[Lyu et al., 2022]
Lyu, J., Alberts, S., Rieke, G. H., & Rujopakarn, W. 2022, , 941, 191

[Lyu et al., 2024]
Lyu, J., Alberts, S., Rieke, G. H., et al. 2024, , 966, 229

[Mignoli et al., 2005]
Mignoli, M., Cimatti, A., Zamorani, G., et al. 2005, , 437, 883

[Momcheva et al., 2016]
Momcheva, I. G., Brammer, G. B., van Dokkum, P. G., et al. 2016, , 225, 27

[Moriya et al., 2022]
Moriya, T. J., Quimby, R. M., & Robertson, B. E. 2022, , 925, 211

[Moriya et al., 2019]
Moriya, T. J., Tanaka, M., Yasuda, N., et al. 2019, , 241, 16

[Morrell, 2012]
Morrell, N. I. 2012, in Death of Massive Stars: Supernovae and Gamma-Ray Bursts, ed. P. Roming, N. Kawai, & E. Pian, Vol. 279, 361–362

[Morris et al., 2015]
Morris, A. M., Kocevski, D. D., Trump, J. R., et al. 2015, , 149, 178

[Oesch et al., 2023]
Oesch, P. A., Brammer, G., Naidu, R. P., et al. 2023, , 525, 2864

[Oke & Gunn, 1983]
Oke, J. B., & Gunn, J. E. 1983, , 266, 713

[Pan et al., 2017]
Pan, Y. C., Foley, R. J., Smith, M., et al. 2017, , 470, 4241

[Pierel et al., 2018]
Pierel, J. D. R., Rodney, S., Avelino, A., et al. 2018, , 130, 114504

[Pierel et al., 2022]
Pierel, J. D. R., Jones, D. O., Kenworthy, W. D., et al. 2022, , 939, 11

[Pierel et al., 2024]
Pierel, J. D. R., Newman, A. B., Dhawan, S., et al. 2024a, , 967, L37

[Pierel et al., 2024]
Pierel, J. D. R., Frye, B. L., Pascale, M., et al. 2024b, , 967, 50

[Pierel et al., 2024]
Pierel, J. D. R., Engesser, M., Coulter, D. A., et al. 2024c, arXiv e-prints, arXiv:2406.05089.

[Postman et al., 2012]
Postman, M., Coe, D., Benítez, N., et al. 2012, , 199, 25

[Rieke et al., 2023]
Rieke, M. J., Robertson, B., Tacchella, S., et al. 2023, , 269, 16

[Rodney et al., 2014]
Rodney, S. A., Riess, A. G., Strolger, L.-G., et al. 2014, , 148, 13

[Sako et al., 2008]
Sako, M., Bassett, B., Becker, A., et al. 2008, , 135, 348

[Siebert et al., 2024]
Siebert, M. R., Decoursey, C., Coulter, D. A., et al. 2024, arXiv e-prints, arXiv:2406.05076.

[Skilling, 2004]
Skilling, J. 2004, in American Institute of Physics Conference Series, Vol. 735, Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 24th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, ed. R. Fischer, R. Preuss, & U. V. Toussaint (AIP), 395–405

[Smith et al., 2018]
Smith, M., Sullivan, M., Nichol, R. C., et al. 2018, , 854, 37

[Stetson, 1987]
Stetson, P. B. 1987, , 99, 191

[Stritzinger et al., 2009]
Stritzinger, M., Mazzali, P., Phillips, M. M., et al. 2009, , 696, 713

[Strolger et al., 2015]
Strolger, L.-G., Dahlen, T., Rodney, S. A., et al. 2015, , 813, 93

[Taddia et al., 2018]
Taddia, F., Stritzinger, M. D., Bersten, M., et al. 2018, , 609, A136

[Urrutia et al., 2019]
Urrutia, T., Wisotzki, L., Kerutt, J., et al. 2019, , 624, A141

[Vanzella et al., 2008]
Vanzella, E., Cristiani, S., Dickinson, M., et al. 2008, , 478, 83

[Xue et al., 2011]
Xue, Y. Q., Luo, B., Brandt, W. N., et al. 2011, , 195, 10

[Yan et al., 2023]
Yan, H., Wang, L., Ma, Z., & Hu, L. 2023a, , 947, L1

[Yan et al., 2023]
Yan, H., Ma, Z., Sun, B., et al. 2023b, , 269, 43


# Motion of a rotating black hole in a homogeneous scalar field

Valeri P. Frolov<EMAIL_ADDRESS>Theoretical Physics Institute,
Department of Physics, University of Alberta,
Edmonton, Alberta, T6G 2E1, Canada

###### Abstract

In the present paper, we consider a rotating black hole moving in a
homogeneous massless scalar field. We assume that the field is weak and
neglect its backreaction, so that the metric at far distance from the black
hole is practically flat. In this domain one can introduce two reference
frames, $K$ and $\tilde{K}$. The frame $\tilde{K}$ is associated with the
homogeneous scalar field, in which its constant gradient has only time
component. The other frame, $K$, is the frame in which the black hole is at
rest. To describe the Kerr metric of the black hole we use its Kerr-Schild
form $g_{\mu\nu}=\eta_{\mu\nu}+\Phi l_{\mu}l_{\mu}$, where $\eta_{\mu\nu}$ is
the (asymptotic) flat metric in $K$ frame. We find an explicit solution of the
scalar field equation which is regular at the horizon and properly reproduce
the asymptotic form of the scalar field at the infinity. Using this solution
we calculate the fluxes of the energy, momentum and the angular momentum of
the scalar field into the black hole. This allows us to derive the equation of
motion of the rotating black hole. We discuss main general properties of
solutions of these equations and obtain explicit solutions for special type of
the motion of the black hole.

Alberta Thy 9-23

##  I Introduction

Scalar field plays an immense role in the modern physics and cosmology. In the
high energy physics, a scalar Higgs field is used to provide the particles
their mass as a result of the spontaneous symmetry breaking. In cosmology, the
inflation can be driven by the potential part of the scalar field
(”inflaton”). The scalar field and Higgs mechanism are important parts of the
models describing possible symmetry breaking and phase transitions in the
cosmology. It is believed that in the early Universe there might be several of
such phase transitions which played an important role in its evolution. During
these transitions formation of primordial black holes and cosmic strings might
become possible. More recently, another mechanism of symmetry breaking known
as the ghost condensation was proposed Arkani-Hamed  _et al._ (2004a, b). The
corresponding phase of the ghost condensate is formed due to the special form
of the kinetic part of the scalar field action. In such a model the ground
state is the scalar field with nonvanishing vector of its gradient. The
presence of such a field breaks the Lorentz invariance.

The ghost condensation model belongs to a wide class of scalar field models
which are invariant with respect to the scalar field shift
$\Psi\to\Psi+\mbox{const}$. At the lowest order in derivatives the Poincaré
invariant Lagrangian for such a theory takes the form

$L=L(X),\hskip 14.22636ptX=\eta^{\mu\nu}\Psi_{,\mu}\Psi_{,\nu}\,.$ (1)

Such a model can also be used as the low-energy effective field theory for
zero- and finite-temperature relativistic superfluids Son (2002); Nicolis
(2011). For the superfluid state with finite charge density and vanishing
spatial current the corresponding solution is $\Psi=\mu t$, where $\mu$ is the
chemical potential Son (2002). Similar solutions with a constant spacelike
gradient of the scalar field were considered in application to the cosmology
in the framework of a so-called solid inflation model Endlich  _et al._
(2013). Let us also mention an interesting approach in which the scalar field
is used to describe a phenomenon emergence of time and dynamics in originally
Euclidean spacetime (see e.g. Moffat (1993); Babichev  _et al._ (2008);
Mukohyama and Uzan (2013); Mukohyama (2013)).

A natural and interesting question is how a black hole interacts with a scalar
field in different models and under different conditions. Let us note that
black holes cannot have their own scalar field. Discussion of the no-hair
theorem for the scalar field and further references can be found in Bekenstein
(1995). At the same time a black hole can exist in the presence of an external
scalar field. Accretion of the ghost condensate by black holes in the
expanding universe was discussed in Frolov and Kofman (2003); Frolov (2004);
Mukohyama (2005); Barranco  _et al._ (2011). Primordial black holes and Higgs
field vacuum decay were considered in Gregory  _et al._ (2014); Chadburn and
Gregory (2014); Gregory  _et al._ (2018); Gregory (2023). Scalar field
accretion by black holes was also discussed in de Cesare and Oliveri (2022).

In this paper we consider a motion of a rotating black hole in an external
homogeneous massless minimally coupled scalar field. This model allows a
rather complete analysis. In the absence of the black hole, the scalar field
equation in the flat spacetime

$\Box\Psi=0\,,$ (2)

has a simple solution

$\Psi=\Psi_{0}t\,.$ (3)

Such a field has constant gradient and is homogeneous in space. This form of
the solution is valid in a specially chosen inertial reference frame, and, in
this sense, it breaks the Lorentz invariance. This choice of the solution is
motivated by the ghost condensate model. We consider a rotating black hole
moving in such a scalar field. In the presence of the black hole the scalar
field $\Psi$ is distorted. We assume that the scalar field is weak and its
backreaction on the metric can be neglected.

Our first goal is to find a solution for the scalar field which is regular at
the horizon of the moving black hole and has the asymptotic form (3) at far
distance from it. We shall demonstrate that this problem allows an exact
solution. To find this solution we proceed as follows.

A remarkable property of the Kerr metric is that it can be written in the
Kerr-Schild form Kerr and Schild (2009)

$g_{\mu\nu}=\eta_{\mu\nu}+\Phi l_{\mu}l_{\nu}\,,$ (4)

where $\eta_{\mu\nu}$ is a flat metric, $\Phi$ is a scalar field, and
${\boldsymbol{l}}$ is a tangent vector to a shear-free geodesic null
congruence. It has been shown that these solutions of the Einstein equations
can be obtained by complex coordinate transformations from the Schwarzschild
metric Newman and Janis (1965); Newman (1973). In particular, the potential
$\Phi$ for the Kerr metric can be obtained as a solution of the Laplace
equation in flat coordinates $(X,Y,Z)$

$\triangle\Phi=4\pi j\,,$ (5)

with a point-like source $j$ located at the complex coordinate $Z+ia$, where
$a$ is the rotation parameter of the Kerr black hole Israel (1970); Kaiser
(2003). A comprehensive review of the Kerr-Schild metrics and complex space
approaches can be found in Adamo and Newman (2014). More recently, the Kerr-
Newman representation of the spacetime geometry attracted a lot of attention
in the so-called double copy formalism. This formalism is based on the
following result: Einstein equations for the metrics which allow the Kerr-
Schild representation can be reduced to the linear equations for Maxwell and
scalar fields. At the moment there exist dozens of publications on this
subject. Related references can be found e.g. in the following review articles
White (2018); Bern  _et al._ (2019, 2022).

One can interpret this result as follows. The form (4) of the metric allows
one to treat the metric ${\boldsymbol{\eta}}$ as the metric of the background
flat spacetime, while the term $\Phi l_{\mu}l_{\nu}$ describes its
”perturbation” due to the black hole located at the origin of the background
space. Denote by $M$ the mass of the black hole. The gravitational field of
the black hole is strong in its vicinity. For an observer located at the
distance $L\gg M$ from the black hole this field is weak. One can say that
such an observer ”lives” in the space with the background metric
${\boldsymbol{\eta}}$. Such an observer can describe the black hole as a small
compact object and use for the description of its motion a ”point particle”
approximation.

If the scalar field is present and has the form (3) the interaction of this
field results in its accretion by the black hole. In this paper we consider a
black hole moving in such a homogenous scalar field. In this case their exist
two natural reference frames. One of them, which we denote by $\tilde{K}$, is
the frame in which at far distance from the black hole the scalar field has
the form (3). The other frame, moving with respect to $\tilde{K}$ with the
velocity $\vec{V}$, is the frame in which the black hole is at rest. We denote
it by $K$. In this frame the form of the solution for the scalar field differs
from (3) and it can be obtained by making the correspondent Lorentz
transformation. We shall use both frames. Namely, we use the Kerr-Schild form
of the Kerr metric associated with $K$ frame to solve the scalar field
equations in the presence of the black hole. Using this result we calculate
the force acting on the black hole due to its motion in the scalar field and
obtain the equation of motion of the black hole in $\tilde{K}$ frame.

The paper is organized as follows. In section II we collect useful formulas
connected with the Kerr-Schild form of the Kerr metric. Section III describes
a solution for the scalar field in the presence of the moving rotating black
hole. Fluxes of the scalar field in $K$ frame are obtained in section IV.
Section V contains calculation of the 4D force acting on the black hole in the
frame $\tilde{K}$. It also discusses the equations of motion of the black
hole, general properties of their solutions, and special cases. Useful
information concerning complex null tetrads is collected in the appendix A.
The appendix B contains details of the calculations of the fluxes of the
energy and angular momentum of the scalar field through the horizon of the
black hole. In the paper we use units in which $G=c=1$ and sign conventions of
the book Misner  _et al._ (1973).

##  II Kerr metric and its Kerr-Schild form

###  II.1 The Kerr metric

The Kerr metric, describing a vacuum stationary rotating black hole, written
in the Boyer-Lindquist coordinates is

$\begin{split}ds^{2}=&-\left(1-\frac{2Mr}{\Sigma}\right)dt^{2}-\frac{4Mar\sin^{2}\theta}{\Sigma}dtd\varphi\\\
+&\left(r^{2}+a^{2}+\frac{2Ma^{2}r}{\Sigma}\sin^{2}\theta\right)\sin^{2}\theta
d\varphi^{2}\\\ +&\frac{\Sigma}{\Delta}dr^{2}+\Sigma d\theta^{2}\,,\\\
&\Sigma=r^{2}+a^{2}\cos^{2}\theta,\hskip
14.22636pt\Delta=r^{2}-2Mr+a^{2}\,.\end{split}$ (6)

Here $M$ is the black hole mass, and $a$ is its rotation parameter. This
metric has two commuting Killing vectors ${\boldsymbol{\xi}}=\partial_{t}$ and
${\boldsymbol{\zeta}}=\partial_{\phi}$. Let us denote

$r_{\pm}=M\pm b,\hskip 14.22636ptb=\sqrt{M^{2}-a^{2}}\,.$ (7)

Equation $r=r_{+}$, where $\Delta=0$, describes the event horizon. The surface
area of the horizon is

$\mathcal{A}=4\pi(r_{+}^{2}+a^{2})=8\pi Mr_{+}\,.$ (8)

Coordinates $(t,r,\theta,\varphi)$ are singular at this surface. To describe
both the exterior and interior of a rotating black hole one can use so called
Kerr incoming coordinates $(v,r,\theta,\tilde{\varphi})$ which are regular at
the future event horizon Kerr (1963)

$\displaystyle dv=dt+dr_{*},\hskip
14.22636ptdr_{*}=(r^{2}+a^{2})\dfrac{dr}{\Delta}\,,$ (9) $\displaystyle
d\tilde{\varphi}=d\varphi+a\dfrac{dr}{\Delta}\,.$

In these coordinates the metric (6) takes the form

$\displaystyle
ds^{2}=-\dfrac{\Delta}{\Sigma}\left(dv-\dfrac{1}{a}\Delta_{y}^{(0)}\,d\tilde{\varphi}\right)^{2}+\dfrac{\Delta_{y}^{(0)}}{\Sigma}\left(dv-\dfrac{1}{a}\Delta_{r}^{(0)}\,d\tilde{\varphi}\right)^{2}$
$\displaystyle+\dfrac{\Sigma}{\Delta_{y}^{(0)}}dy^{2}+2dr\,\left(dv-\dfrac{1}{a}\Delta_{y}^{(0)}\,d\tilde{\varphi}\right)\,.$

Similarly, one can introduce Kerr outgoing coordinates
$(u,r,\theta,\tilde{\varphi})$

$dv=dt-dr_{*},\hskip
14.22636ptd\tilde{\varphi}=d\varphi-a\dfrac{dr}{\Delta}\,,$ (10)

which are regular at the past horizon and cover the white hole domain.

###  II.2 Useful coordinates

For $M=0$ the Riemann curvature of the Kerr metric vanishes and the metric (6)
becomes flat. We write it the form111Let us note that we use notations $T$ and
$\phi$ for the time and angle variables in the flat spacetime. These
coordinates are used in the Kerr-Schild form of the Kerr metric while the
standard Bouer-Lindquist coordinates $t$ and $\varphi$ are related to $T$ and
$\phi$ by means of relations (LABEL:KST).

$\begin{split}dS^{2}&=-d{T}^{2}+dh^{2}\,,\\\
dh^{2}&=\frac{\Sigma}{r^{2}+a^{2}}dr^{2}+\Sigma
d\theta^{2}+(r^{2}+a^{2})\sin^{2}\theta d\phi^{2}\,.\end{split}$ (11)

The coordinates $(r,\theta,\phi)$ are oblate spheroidal coordinates taking the
following values $r\geq 0$, $\theta\in[0,\pi]$, $\phi\in[0,2\pi]$. These
coordinates are related to the Cartesian coordinates as follows

$\begin{split}X&=\sqrt{r^{2}+a^{2}}\sin\theta\cos\phi\,,\\\
Y&=\sqrt{r^{2}+a^{2}}\sin\theta\sin\phi\,,\\\ Z&=r\cos\theta\,.\end{split}$
(12)

In these coordinates the flat metric $dS^{2}$ takes the standard form

$dS^{2}=\eta_{\mu\nu}dX^{2}\,dX^{\nu}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2}\,.$ (13)

Figure 1:  Coordinate lines of the oblate spheroidal coordinates $(r,\theta)$
in the plane $Y=0$

For $r>0$ the surfaces $r=$const are oblate ellipsoids. Figure 1 shows the
coordinate lines of the oblate spheroidal coordinates $(r,\theta)$ in the
plane $Y=0$ ($\phi=0$). For $r=0$ and $\theta\in[0,\pi]$, $\phi\in[0,2\pi]$
one has a disc of radius $a$ located in the $Z=0$ plane. The coordinate
$\theta$ is discontinuous on the disc. For $(0,\pi/2)$ the coordinate $\theta$
covers the upper part of the disc, while for $(\pi/2,\pi)$, it covers the
lower part of it. The boundary of this disc is a ring of radius $a$. Equations
$\theta=0$ and $\theta=\pi$ describe the axis of symmetry $X=Y=0$. For
$\theta=0$ and $Z=r$ it is positive, while for $\theta=\pi$ $Z=-r$ it is
negative.

In what follows we shall also use another coordinate, $y$, related to the
angle $\theta$ as follows

$y=a\cos\theta.$ (14)

The flat metric $dS^{2}$ in the spheroidal coordinates $(T,r,y,\phi)$ is

$\begin{split}dS^{2}&=-dT^{2}+\Sigma\left(\dfrac{dr^{2}}{\Delta^{0}_{r}}+\dfrac{dy^{2}}{\Delta^{0}_{y}}\right)+\dfrac{\Delta^{0}_{r}\Delta^{0}_{y}}{a^{2}}d\phi^{2}\,,\\\
\Sigma&=r^{2}+y^{2},\hskip 5.69046pt\Delta^{0}_{r}=r^{2}+a^{2},\hskip
5.69046pt\Delta^{0}_{y}=a^{2}-y^{2}\,.\end{split}$ (15)

In these coordinates the Cartesian coordinates take the form

$\begin{split}X&=\dfrac{1}{a}\sqrt{\Delta_{r}^{(0)}\Delta_{y}^{(0)}}\cos\phi\,,\\\
Y&=\dfrac{1}{a}\sqrt{\Delta_{r}^{(0)}\Delta_{y}^{(0)}}\sin\phi\,,\\\
Z&=\dfrac{1}{a}ry\,.\end{split}$ (16)

We denote

$e_{(X)\mu}=X_{,\mu},\hskip 14.22636pte_{(Y)\mu}=Y_{,\mu},\hskip
14.22636pte_{(Z)\mu}=Z_{,\mu}\,.$ (17)

We also denote by ${\boldsymbol{e}}_{i}$, $i=1,2,3$ a set of 4D unit vectors
$e_{i}^{\mu}$ along $X$, $Y$ and $Z$ axes.

###  II.3 Kerr metric in the Kerr-Schild form

Let us consider the following 1-form

$l_{\mu}dx^{\mu}=-dT-\frac{\Sigma}{\Delta^{0}_{r}}dr+\dfrac{\Delta^{0}_{y}}{a}d\phi\,.$
(18)

We define a metric

$d{s}^{2}=dS^{2}+\Phi(l_{\mu}dx^{\mu})^{2}\,,$ (19)

where $\Phi=\Phi(r,y)$ is some function. The metric coefficients of the
metrics $ds^{2}$ and $dS^{2}$ are related as follows

$g_{\mu\nu}=\eta_{\mu\nu}+\Phi l_{\mu}l_{\nu}\,.$ (20)

The following statements are valid for each of the metrics $d{s}^{2}$ and
$dS^{2}$. In other words, these statements are valid for an arbitrary function
$\Phi$, including $\Phi=0$. The vector field ${\boldsymbol{l}}$ has the
following properties

  * •

The contravariant components of the vector ${\boldsymbol{l}}$ in
$(T,r,\theta,\phi)$ coordinates are

$l^{\mu}=\left(1,-1,0,\dfrac{a}{r^{2}+a^{2}}\right)\,.$ (21)

  * •

${\boldsymbol{l}}$ is a null vector ${\boldsymbol{l}}^{2}=l_{\mu}l^{\mu}=0$;

  * •

Vectors ${\boldsymbol{l}}$ are tangent vectors to incoming null geodesics in
the affine parametrization, $l^{\nu}l^{\mu}_{\ ;\nu}=0$.

  * •

$l^{\mu}_{\ ;\mu}=-\dfrac{2r}{\Sigma}$;

  * •

$l_{(\mu;\nu)}l^{(\mu;\nu)}-\frac{1}{2}(l^{\mu}_{\ ;\mu})^{2}=0$ .

The last property implies that the congruence of null vectors
${\boldsymbol{l}}$ is shear-free (for more details see e.g. Sommers (1976);
Frolov (1979)). Such a null geodesic congruence is related to the light cones
with apex on the world-line in the complex space. The twist is a measure of
how far the complex world-line is from the real slice Newman (2004).

It is easy to check that for a special choice of the function $\Phi$

$\Phi_{0}=\frac{2Mr}{\Sigma}\,,$ (22)

the metric $d{s}^{2}$ given by (19) is Ricci flat, and in fact, it coincides
with the Kerr metric. In order to prove this it is sufficient to make the
following coordinate transformation

$\begin{split}&T=t+t_{0}(r),\hskip 14.22636pt\phi=\varphi+\varphi_{0}(r)\,,\\\
&t_{0}(r)=\int\frac{2Mr}{\Delta}dr\\\
&=\dfrac{M}{\sqrt{M^{2}-a^{2}}}\left[r_{+}\ln(r-r_{+})-r_{-}\ln(r-r_{-})\right]\,,\\\
&\varphi_{0}(r)=\int\frac{2Mar}{(r^{2}+a^{2})\Delta}dr\\\
&=\dfrac{a}{2\sqrt{M^{2}-a^{2}}}\ln\left(\dfrac{r-r+}{r-r_{-}}\right)-\arctan(r/a)+\dfrac{1}{2}\pi\,.\end{split}$
(23)

Here $\Delta$ is defined in (6). These coordinates $(t,r,\theta,\varphi)$ are
chosen so that the non-diagonal components $g_{rt}$ and $g_{r\varphi}$ of the
metric $d{s}^{2}$ vanish. One can check that the metric $d{s}^{2}$ written in
the $(t,r,\theta,\varphi)$ coincides with the Kerr metric $dS^{2}$, provided
one identifies the coordinates $t$ and $\varphi$ in $d{s}^{2}$ with the
standard Boyer-Lindquist coordinates $t$ and $\varphi$ in the metric (6). The
integration constant in the expression for $\varphi_{0}$ is chosen so that
this quantity vanishes when $r\to\infty$. Hence in this limit the angle
variables $\varphi$ and $\phi$ coincide.

It is easy to check that the coordinates $T$ and $\phi$ are related to the
Kerr incoming coordinates $v$ and $\tilde{\varphi}$ as follows

$T=v-r,\hskip 14.22636pt\phi=\tilde{\varphi}-\arctan(r/a)\,.$ (24)

Coordinates $(T,r,y,\phi)$ are regular at the future event horizon and cover
the exterior region of the black hole as well as a part of its interior.
Similarly, by a simple change of the sign of the coefficient of $dr$ term in
the expression (18) one can obtain outgoing null vector and use it to
construct the Kerr-Schild metric which is regular at the past horizon.

##  III Scalar field

###  III.1 Flat spacetime

Let us consider a minimally coupled massless scalar field $\Psi$ which obeys
the equation

$\Box\Psi=0\,$ (25)

Its stress-energy tensor is

$T_{\mu\nu}=\Psi_{,\mu}\Psi_{,\nu}-\dfrac{1}{2}g_{\mu\nu}\Psi_{,\alpha}\Psi^{,\alpha}\,.$
(26)

Let us consider first the flat spacetime. We choose some inertial frame in it.
We denote it by $\tilde{K}$ and denote by
$\tilde{X}^{\mu}=(\tilde{T},{X},Y,\tilde{Z})$ the Cartesian coordinates
associated with this frame. In this frame there exists a simple solution of
the (25)

$\Psi=\Psi_{0}\tilde{T}\,,$ (27)

describing a homogeneous scalar field. The stress-energy tensor for this
solution is diagonal and has the following non-vanishing components

$T_{\tilde{T}\tilde{T}}=T_{\tilde{X}\tilde{X}}=T_{\tilde{Y}\tilde{Y}}=T_{\tilde{Z}\tilde{Z}}=\dfrac{1}{2}\Psi_{0}^{2}\,.$
(28)

The gradient of the field $\Psi$ has only time component. In this sense this
solution breaks the Lorentz invariance and singles out an inertial frame in
which the field $\Psi$ does not depend on spatial coordinates. As we already
mentioned such solutions with a constant gradient of the field play an
important role the shift-invariant scalar field theories (see e.g. Arkani-
Hamed  _et al._ (2004a, b); Nicolis (2011); Endlich  _et al._ (2013). )

Figure 2: Two inertial frames, $K$ and $\tilde{K}$, are schematically shown at
this figure. The Cartesian coordinates in these frames are $(T,X,Y,Z)$ and
$(\tilde{T},\tilde{X},\tilde{Y},\tilde{Z})$, respectively. The frame
$\tilde{K}$ is the ”rest frame” of the homogeneous scalar field, in which it
has the form $\Psi=\Psi_{0}\tilde{T}$. The frame $K$ is the rest frame of the
rotating black hole. The origin of this frame $O$ is at the position of the
black hole. The frame $K$ moves with respect to $\tilde{K}$ with the velocity
$\vec{V}$. The coordinate axes in both frames are parallel. $Z$ axis in $K$
frame coincides with the direction of the spin $\vec{J}$ of the black hole.
The velocity $\vec{V}$ in the frame $K$ can be decomposed as follows
$\vec{V}=\vec{V}_{\parallel}+\vec{V}_{\perp}$, where $\vec{V}_{\parallel}$ is
parallel to the spin of the black hole, and $\vec{V}_{\perp}$ is orthogonal to
it.

Consider a rotating black hole moving with respect to $\tilde{K}$ with a
constant velocity $\vec{V}$ (see figure 2). We denote by $K$ a reference frame
associated with the black hole and denote by $X^{\mu}=(T,X,Y,Z)$ the Cartesian
coordinates associated with this frame. We choose the coordinate axes in the
both frames to be in the same directions. We also choose $Z$ axis to be
parallel to the spin of the black hole.The vector $\vec{V}$ has the following
components

$\begin{split}&\vec{V}=(V_{X},V_{Y},V_{Z})\,,\\\
&V^{2}=(\vec{V})^{2}=V_{X}^{2}+V_{Y}^{2}+V_{Z}^{2}\,.\end{split}$ (29)

Denote by $\vec{R}=(X,Y,Z)$ a 3D vector connecting the origin $\tilde{O}$ of
the frame $\tilde{K}$ with the origin $O$ of the moving frame $K$. Then the
Lorentz transformation implies

$\begin{split}\tilde{T}&=\gamma\left(T+(\vec{V},\vec{R})\right)\,,\\\
\gamma&=1/\sqrt{1-V^{2}}\,.\end{split}$ (30)

The solution (27) written in the frame $K$ comoving with the black hole takes
the form

$\begin{split}\Psi=&\bar{\Psi}_{0}(T+V_{X}X+V_{Y}Y+V_{Z}Z)\,,\\\
\bar{\Psi}_{0}=&\dfrac{\Psi_{0}}{\sqrt{1-V^{2}}}\,.\end{split}$ (31)

###  III.2 Scalar field solution in the presence of a moving rotating black
hole

In the previous subsection we ignore the gravitational field of the black
hole. To obtain a solution for the scalar field in the presence of a moving
rotating black hole we proceed as follows. We use the Kerr-Schild form of the
metric (19) associated with $K$ frame, in which the black hole is at rest. In
this form the Einstein equations are linearized. One can identify the metric
$dS^{2}$ with the flat background geometry in $K$ frame, while the $\Phi
l_{\mu}l_{\nu}$ describes its ”perturbation” due to the presence of the black
hole. We are looking for a solution of the scalar field equation (25) which is
regular at the horizon of the black hole and at far distance has the
asymptotic form (31).

The required solution satisfying the imposed boundary conditions in the
coordinates $(T,r,y,\phi)$ is

$\begin{split}&\Psi=\bar{\Psi}_{0}\left(T+T_{0}(r)+V_{X}\mathcal{V}_{X}+V_{Y}\mathcal{V}_{Y}+V_{Z}\mathcal{V}_{Z}\right)\,,\\\
&T_{0}(r)=-2M\ln(r-r_{-})\,,\\\
&\mathcal{V}_{X}=\dfrac{1}{a}\sqrt{\dfrac{\Delta_{y}^{(0)}}{\Delta_{r}^{(0)}}}\left((\Delta_{r}^{(0)}-Mr)\cos\phi-
Ma\sin\phi\right)\,,\\\
&\mathcal{V}_{Y}=\dfrac{1}{a}\sqrt{\dfrac{\Delta_{y}^{(0)}}{\Delta_{r}^{(0)}}}\left((\Delta_{r}^{(0)}-Mr)\sin\phi+Ma\cos\phi\right)\,,\\\
&\mathcal{V}_{Z}=\dfrac{y}{a}(r-M)\,.\end{split}$ (32)

To check the validity of the boundary conditions at the infinity it is
sufficient to use following asymptotics of the functions $\mathcal{V}$

$\begin{split}&\mathcal{V}_{X}\approx r\sin\theta\cos\phi=X\,,\\\
&\mathcal{V}_{Y}\approx r\sin\theta\sin\phi=Y\,,\\\ &\mathcal{V}_{Z}\approx
r\cos\theta=Z\,.\end{split}$ (33)

Since the coordinates $(T,r,y,\phi)$ are regular on the future horizon, and
the components of its gradient $\Phi_{;\mu}$ are regular functions of these
coordinates, one can conclude that the presented solution (32) does satisfy
the required condition of the regularity at the horizon.

The same solution written in the Boyer-Lindquist coordinates
$(t,r,\theta,\varphi)$ is

$\begin{split}&\Psi=\bar{\Psi}_{0}\big{(}t+\tilde{t}_{0}+V_{X}\mathcal{U}_{X}+V_{Y}\mathcal{U}_{Y}+V_{Z}\mathcal{U}_{Z}\big{)}\,,\\\
&\mathcal{U}_{X}=\dfrac{\sin\theta}{\sqrt{\Delta_{r}^{(0)}}}\left[(\Delta_{r}^{(0)}-Mr)\cos\psi-
Ma\sin\psi\right]\,,\\\
&\mathcal{U}_{Y}=\dfrac{\sin\theta}{\sqrt{\Delta_{r}^{(0)}}}\left[(\Delta_{r}^{(0)}-Mr)\sin\psi+Ma\cos\psi\right]\,,\\\
&\mathcal{U}_{Z}=V(r-M)\cos\theta\,,\\\
&\tilde{t}_{0}=t_{0}(r)+T_{0}(r)=\dfrac{M}{\sqrt{M^{2}-a^{2}}}\big{[}r_{+}\ln(r-r_{+})\\\
&\quad-r_{-}\ln(r-r_{-})\big{]}-2M\ln(r-r_{-})\,,\\\
&\varphi_{0}(r)=\dfrac{a}{2\sqrt{M^{2}-a^{2}}}\ln\big{(}\dfrac{r-r_{+}}{r-r_{-}}\big{)}-\arctan(r/a)\,,\\\
&\psi=\varphi+\varphi_{0}(r)\,.\end{split}$ (34)

##  IV Fluxes

###  IV.1 Energy and angular momentum fluxes through $r=$const surface

Let us calculate the fluxes of the energy, angular momentum, and the momentum
through a 2D surface of constant radius $r=r_{0}$ surrounding the black hole.
For these calculations it is convenient to use the Boyer-Lindquist coordinates
$(t,r,\theta,\varphi)$. Denote by $\Sigma_{0}$ a 3D timelike surface
describing the ”evolution” of $S$ for the time interval $(t_{-},t_{+})$.

Denote by ${\boldsymbol{q}}$ a 3D metric on $\Sigma$ induced by its embedding
in the 4D space. Then

$q\equiv\sqrt{-\mbox{det}({\boldsymbol{q}})}=\sqrt{\Delta\Sigma}\sin\theta\,.$
(35)

A unit vector ${\boldsymbol{n}}$ orthogonal to $\Sigma_{0}$ and inward
directed is

$n^{\mu}=-\sqrt{\Delta/\Sigma}\delta^{\mu}_{r}\,.$ (36)

The vector of the volume element of the surface $\Sigma$ is

$d\sigma^{\mu}=n^{\mu}dt\,d\theta\,d\varphi=-\delta_{r}^{\mu}\Delta\sin\theta\,dt\,d\theta\,d\varphi\,.$
(37)

We denote the fluxes of the energy and angular momentum through 3D surface
$\Sigma_{0}$ into its interior per a unit time $t$ by $\mathcal{E}$ and
$\mathcal{J}$, respectively. Then one has

$\begin{split}\mathcal{E}=&-\dfrac{1}{t_{+}-t_{-}}\int_{\Sigma}\xi^{\mu}T_{\mu\nu}d\sigma^{\mu}=\Delta\int_{S}T_{tr}d\omega\,,\\\
\mathcal{J}=&\dfrac{1}{t_{+}-t_{-}}\int_{\Sigma}\zeta^{\mu}T_{\mu\nu}d\sigma^{\mu}=-\Delta\int_{S}T_{r\varphi}d\omega\,,\\\
&d\omega=\sin\theta d\theta\,d\varphi\,,\\\ \end{split}$ (38)

The signs in these expressions are chosen so that these quantities describe
the flux into the surface $S$ from its exterior.

Since $g_{tr}$ and $g_{r\varphi}$ components of the Kerr metric in the Boyer-
Lindquist coordinates vanish one has

$T_{tr}=\Psi_{t}\Psi_{r},\hskip
14.22636ptT_{t\varphi}=\Psi_{r}\Psi_{\varphi}\,.$ (39)

Calculating these expressions and taking integrals in (38) one obtains

$\begin{split}&\mathcal{E}=8\pi\bar{\Psi}_{0}^{2}Mr_{+}\,,\\\
&\mathcal{J}=-\dfrac{4}{3}\pi\bar{\Psi}_{0}^{2}aM^{2}(V_{X}^{2}+V_{Y}^{2})\,.\end{split}$
(40)

Let us emphasize that these quantities do not depent on the radius $r=r_{0}$
of the surface $S$ which was used for their calculation. (For explanation of
this property and more details, see appendix B.)

###  IV.2 Momentum fluxes

We use the vectors $e_{i}^{\mu}$, defined in (17) to define the following
objects222 Let us note that the norm of the vectors $e_{i}^{\mu}$, calculated
in the metric $g_{\mu\nu}$, at far distance slightly differs from one.
However, it is possible to check that using the normalized versions of these
vectors in the expressions given below does not change the results when the
limit $r=r_{0}\to\infty$ is taken.

$\begin{split}\mathcal{P}_{X}=&\dfrac{1}{t_{+}-t_{-}}\int_{\Sigma}T_{\mu\nu}e_{(X)}^{\mu}d\sigma^{\nu}=-\Delta\int_{S}T_{Xr}d\omega\,,\\\
\mathcal{P}_{Y}=&\dfrac{1}{t_{+}-t_{-}}\int_{\Sigma}T_{\mu\nu}e_{(Y)}^{\mu}d\sigma^{\nu}=-\Delta\int_{S}T_{Yr}d\omega\,,\\\
\mathcal{P}_{Z}=&\dfrac{1}{t_{+}-t_{-}}\int_{\Sigma}T_{\mu\nu}e_{(Z)}^{\mu}d\sigma^{\nu}=-\Delta\int_{S}T_{Zr}d\omega\,,\\\
T_{Xr}=&T_{\mu r}e_{(X)}^{\mu},\hskip 5.69046ptT_{Yr}=T_{\mu
r}e_{(Y)}^{\mu},\hskip 5.69046ptT_{Zr}=T_{\mu r}e_{(Z)}^{\mu}\,.\end{split}$
(41)

Calculating these integrals and taking the limit $r_{0}\to\infty$ one obtains
the following results

$\begin{split}&\mathcal{P}_{X}=-8\pi\bar{\Psi}_{0}^{2}\left[Mr_{+}V_{X}+\dfrac{2}{3}aMV_{Y}\right]\,,\\\
&\mathcal{P}_{Y}=-8\pi\bar{\Psi}_{0}^{2}\left[Mr_{+}V_{Y}-\dfrac{2}{3}aMV_{X}\right]\,,\\\
&\mathcal{P}_{Z}=-8\pi\bar{\Psi}_{0}^{2}Mr_{+}V_{Z}\,.\end{split}$ (42)

The obtained results can be presented in the following 3D vector form. Let us
denote

$\begin{split}&\beta=8\pi{\Psi}_{0}^{2},\hskip
14.22636pt\vec{V}=(V_{X},V_{Y},V_{Z})\,,\\\
&\vec{\mathcal{P}}=(\mathcal{P}_{X},\mathcal{P}_{Y},\mathcal{P}_{Z}),\hskip
5.69046pt\vec{J}=M\vec{a},\hskip
5.69046pt\vec{\mathcal{J}}=\dfrac{\vec{a}}{a}\mathcal{J}\,.\end{split}$ (43)

Here $\vec{a}$ is a 3D vector with the norm $a$ and directed along $Z-$axis.
Consider an observer which is located at very far distance from the black and
assume that the black hole is initially at rest in his/her frame. As a result
of the interaction with ”moving” scalar field, the black hole absorbs energy
and momentum. To describe its further evolution as a result of this effect, a
far-distant observer can neglect the black hole size and approximate it by the
massive point with spin which has energy and momentum. In this interpretation
the calculated quantities $\mathcal{E}$ and $\vec{\mathcal{P}}$ are nothing
but the components of the 4D force ${\boldsymbol{F}}$ acting on such a massive
point.

In the asymptotic flat coordinates $(T,X,Y,Z)$ the components of this 4D force
are

$\begin{split}&F^{\mu}=(\mathcal{E},\mathcal{P}_{X},\mathcal{P}_{Y},\mathcal{P}_{Z})\,,\\\
&F^{T}=\dfrac{\beta Mr_{+}}{1-V^{2}}\,,\\\
&\vec{F}=-\dfrac{\beta}{1-V^{2}}\left(Mr_{+}\vec{V}+\dfrac{2}{3}\vec{J}\times\vec{V}\right)\,.\end{split}$
(44)

In addition to these expressions for the 4D force acting on the black hole,
there exists one more equation which demonstrates that the spin of the black
changes when the black hole has a non-vanishing transverse component of the
velocity $V_{\perp}$

$\begin{split}&\vec{\mathcal{J}}=-\dfrac{1}{6}\beta
MV_{\perp}^{2}\vec{J}\,,\\\
&V_{\perp}^{2}=\vec{V}^{2}-\dfrac{1}{J^{2}}(\vec{J}\cdot\vec{V})^{2}\,.\end{split}$
(45)

##  V Motion of a rotating black hole through the scalar field

###  V.1 Friction force in $\widetilde{K}$ frame

The 4D force $F^{\mu}$ is calculated in the frame where the black hole is
initially at rest. Under the action of this force the black hole has a non-
vanishing acceleration. Since the velocity of the black hole interacting with
the scalar fields changes in time, the frame in which the black hole is at
rest is not inertial. For this reason it is more convenient to write the
equations of motion in the frame $\tilde{K}$ associated with the scalar field.
Let us denote by $f^{\mu}\equiv\tilde{F}^{\mu}$ the components of the 4D force
in $\tilde{K}$ frame. The corresponding Lorentz transformation implies

$\begin{split}&f^{0}=\gamma(F^{T}+\vec{V}\cdot\vec{F})\,,\\\
&\vec{f}=\vec{F}+\gamma
F^{T}\vec{V}+(\gamma-1)\dfrac{\vec{V}\cdot\vec{F}}{V^{2}}\vec{V}\,.\end{split}$
(46)

Here $\gamma=1/\sqrt{1-V^{2}}$. Using (44) one obtains

$\begin{split}&f^{0}=\dfrac{\beta Mr_{+}}{\sqrt{1-V^{2}}}\,,\\\
&\vec{f}=-\dfrac{2}{3}\dfrac{\beta}{\sqrt{1-V^{2}}}\
[\vec{J}\times\vec{V}]\,.\end{split}$ (47)

Let us notice that the following two useful relations are valid

$\vec{V}\cdot\vec{f}=0,\hskip 14.22636pt\vec{J}\cdot\vec{f}=0\,.$ (48)

###  V.2 Equations of motion

We denote by ${\boldsymbol{P}}$ the 4D momentum of the black hole in
$\tilde{K}$ frame. In Cartesian coordinates it has the following form

$P^{\mu}=(E,\vec{P}),\hskip 14.22636pt\vec{P}=(P_{X},P_{Y},P_{Z})\,.$ (49)

The energy $E$ and momentum $\vec{P}$ of the black hole with mass $M$ and
velocity $\vec{V}$ are

$E=\dfrac{M}{\sqrt{1-V^{2}}},\hskip
14.22636pt\vec{P}=\dfrac{M\vec{V}}{\sqrt{1-V^{2}}}\,.$ (50)

Such a black hole is similar to a massive relativistic particle, with two
important differences: (i) The mass of the black hole changes as a result of
the absorption of the scalar field, and (ii) the black hole has spin
$\vec{J}$, which also changes with time.

The equations of motion are

$\dfrac{dE}{d{\tau}}=f^{0},\hskip
14.22636pt\dfrac{d\vec{P}}{d{\tau}}=\vec{f}\,,$ (51)

where $f^{0}$ and $\vec{f}$ are given in (LABEL:fff), and $\tau$ is the proper
time along the worldline of the black hole. There exists an additional
equation for the spin evolution

$\dfrac{dJ}{d{\tau}}=-\mathcal{J}\,,$ (52)

where $\mathcal{J}$ is given by (LABEL:FLUXES).

Since vector $\vec{P}$ is collinear with $\vec{V}$, one has

$\dfrac{1}{2}\dfrac{d\vec{P}^{2}}{d\tau}=\vec{P}\cdot\dfrac{d\vec{P}}{d{\tau}}=\dfrac{M}{\sqrt{1-V^{2}}}\,\vec{V}\cdot\vec{f}=0\,.$
(53)

Hence $\vec{P}^{2}=P_{0}^{2}=$const.

Let us write the momentum vector in the form

$\vec{P}=\vec{P}_{\parallel}+\vec{P}_{\perp}\,,$ (54)

where $\vec{P}_{\parallel}$ is parallel to the spin vector $\vec{J}$ and
$\vec{P}_{\perp}$ is perpendicular to it. Then one has

$\vec{P}_{\parallel}\cdot\dfrac{d\vec{P}_{\parallel}}{d{\tau}}=0\,.$ (55)

This means that the norms of the both vectors
$P_{\parallel}=\sqrt{\vec{P}_{\parallel}^{2}}$ and
$P_{\perp}=\sqrt{\vec{P}_{\perp}^{2}}$ are conserved quantities.

Using relations

$\begin{split}&E=\sqrt{M^{2}+P_{0}^{2}}\,,\\\
&\dfrac{1}{\sqrt{1-V^{2}}}=\dfrac{1}{M}\sqrt{M^{2}+P_{0}^{2}}\,,\end{split}$
(56)

one can rewrite the first equation in (51) as follows

$\begin{split}&\dfrac{dM}{d\tau}=\beta\dfrac{r_{+}}{M}(M^{2}+P_{0}^{2})\,,\\\
&r_{+}=M+\sqrt{M^{2}-J^{2}/M^{2}}\,.\end{split}$ (57)

Using relation

$V_{\perp}^{2}=\dfrac{P_{\perp}^{2}}{M^{2}+P_{0}^{2}}$ (58)

one can write (52) in the form

$\dfrac{dJ}{d\tau}=-\dfrac{1}{6}\beta\dfrac{MP_{\perp}^{2}}{M^{2}+P_{0}^{2}}J\,.$
(59)

The second equation in (51) implies

$\dfrac{d\vec{P}_{\perp}}{d\tau}=-\dfrac{2}{3}\beta\dfrac{1}{M}[\vec{J}\times\vec{P}_{\perp}]\,.$
(60)

Let us denote

$P_{X}+iP_{Y}={P}_{\perp}\exp(i\alpha)\,.$ (61)

Then the equation (60) implies

$\dfrac{d\alpha}{d\tau}=-\dfrac{2}{3}\dfrac{\beta
J}{\sqrt{M^{2}+P_{0}^{2}}}\,.$ (62)

The obtained equations (57) and (59) allow one to find the time dependence of
the black hole mass $M$ and spin $J$ for a given initial values $M_{0}$ and
$J_{0}$ of these parameters. The equation (62) determines the time evolution
of the transverse momentum $\vec{P}_{\perp}$. In order to find a unique
solution one needs also specify the conserved parameters $P_{0}$ and
$P_{\perp}$.

Let us make a following remark. In the above consideration we assumed that a
distant observer is located at far distance $L$ from the black hole, $L>>M$.
This allows one to describe a black hole as a ”point-like particle” which has
mass $M$ and spin $J$. We also assume that these parameters change slowly in
time, so that

$\dot{M}/M\ll 1/M,\hskip 14.22636pt\dot{J}/J\ll 1/M\,.$ (63)

In this adiabatic approximation one can describe the black hole metric by the
Kerr solution with a slowly changing parameters $M(t)$ and $J(t)$. Note that
such an adiabatic approximation is broken in the vicinity of the moment of
time, at which the mass of the black hole formally becomes infinite.

###  V.3 General properties of solutions

####  V.3.1 Mass evolution

Let us discuss general properties of the solutions of the equations of motion
for the rotating black hole moving in the homogeneous scalar field. First of
all let us notice that equations (57) and (59) show that the black hole mass
$M$ is a monotonically increasing function of time $\tau$, while its spin $J$
monotonically decreases in time.

One can obtain a more detailed information about the time dependence of these
parameters by using the following trick. Let us denote $r_{+}=bM$, where
$1\leq b\leq 2$. The quantity $b$ takes the value 1 for the extremely rotating
black hole when $J=M^{2}$, and $b=2$ for a nonrotating black hole. Solving
(57) for $b$=const one gets

$\arctan(M_{b}/P_{0})-\arctan(M_{0}/P_{0})=b\beta P_{0}\tau\,,$ (64)

where $M_{0}$ is the mass at the initial moment of time $\tau=0$. If $M(\tau)$
is the exact solution of the equation (57) then for the same initial mass
$M_{0}$ one has

$M_{1}(\tau)\leq M(\tau)\leq M_{2}(\tau)\,.$ (65)

Denote

$\tau_{0}=\dfrac{1}{\beta
P_{0}}\left(\dfrac{\pi}{2}--\arctan(M_{0}/P_{0})\right)\,.$ (66)

Then the black hole mass $M$ becomes infinite at the finite proper time
$\tau=\tau_{fin}$

$\dfrac{1}{2}\tau_{0}\leq\tau_{fin}\leq\tau_{0}\,.$ (67)

####  V.3.2 Spin evolution

Let us consider the spin of the black hole as a function of its mass,
$J=J(M)$, Then using (57) and (59) one obtains

$\dfrac{1}{J}\dfrac{dJ}{dM}=-\dfrac{1}{6}\dfrac{P_{\perp}^{2}M^{2}}{r_{+}(M^{2}+P_{0}^{2})^{2}}\,.$
(68)

Substituting $r_{+}=bM$ and solving the obtained equation with fixed value of
$b$ one finds

$J_{b}(M)=J_{0}\exp\big{[}-\dfrac{1}{6b}\dfrac{P_{\perp}^{2}(M^{2}-M_{0}^{2})}{(M^{2}+P_{0}^{2})(M_{0}^{2}+P_{0}^{2})}\big{]}\,.$
(69)

The exact solution $J(M)$ of (68) satisfies the inequalities

$J_{1}(M)\leq J(M)\leq J_{2}(M)\,.$ (70)

For $M\to\infty$

$J_{b,fin}=J_{0}\exp\big{[}-\dfrac{P_{\perp}^{2}}{6b(M_{0}^{2}+P_{0}^{2})}\big{]}\,.$
(71)

Hence at the moment when the mass of the black hole becomes infinitely large,
its spin remains finite $J=J_{fin}$

$J_{1,fin}\leq J_{fin}\leq J_{2,fin}\,.$ (72)

##  VI Special cases

###  VI.1 Black hole at rest in $\widetilde{K}$ frame

If the black hole is at rest then $P_{0}=0$ and the spin of the black hole
remains constant. The equation for the black hole mass takes the form

$\dfrac{dM}{d\tau}=\beta
Mr_{+}=\beta\big{(}M^{2}+\sqrt{M^{4}-J^{2}}\big{)}\,.$ (73)

If the spin $J$ does not vanish one can denote

$\mu=M/\sqrt{J}\,,$ (74)

and write a solution of the equation (73) in the form

$\sqrt{J}\beta\tau=\hat{\tau},\hskip
14.22636pt\hat{\tau}=\int\dfrac{d\mu}{\mu^{2}+\sqrt{\mu^{4}-1}}\,.$ (75)

Taking the integral one obtains

$\hat{\tau}=\dfrac{1}{3}\big{[}\mu^{3}-1-\mu\sqrt{\mu^{4}-1}+2(F(i\mu,i)-F(i,i))\big{]}\,.$
(76)

Here $F(a,b)$ is the incomplete elliptic integral of the first kind, and the
integration constant in this expression is chosen so that $\hat{\tau}=0$ for
$\mu=1$. For $\mu\to\infty$ the parameter $\hat{\tau}$ has a limit $0.54068$.
The plot of $\hat{\tau}$ as a function of $\mu$ is shown at figure 3.

Figure 3:  $\hat{\tau}$ as a function of $\mu$. Dashed line shows the limiting
value of $\hat{\tau}$ for $\mu\to\infty$, which is equal to 0.54068.

If $J=0$ one obtains the following solution of the equation (73) for the mass

$M=\dfrac{M_{0}}{1-2\beta M_{0}\tau}\,.$ (77)

###  VI.2 Motion of a nonrotating black hole

When the spin of the black hole vanishes, one can always chose the orientation
of the axes so that $P_{\perp}=0$ and $P_{\parallel}=P_{0}$. One also has
$r_{+}=2M$. The equation (57) simplifies and takes the form

$\dfrac{dM}{d\tau}=2\beta(M^{2}+P_{0}^{2})\,.$ (78)

It can be easily integrated with the following result

$2\beta P_{0}\tau=\arctan(M/P_{0})\,.$ (79)

This expression correctly reproduces the result obtained in Frolov  _et al._
(2023).

###  VI.3 Motion in the spin direction

For the motion in the spin direction one has $P_{\perp}=0$ and, as a result,
the spin of the black hole remains constant. If its initial value does not
vanish, we denote

$\mu=M/\sqrt{J},\hskip 14.22636ptp=P_{0}/\sqrt{J}\,.$ (80)

Then one has

$\beta\sqrt{J}\tau=\int\dfrac{\mu^{2}d\mu}{(\mu^{2}+p^{2})(\mu^{2}+\sqrt{\mu^{4}-1})}\,.$
(81)

The integral in the right-hand side of this relation can be expressed in terms
of the incomplete elliptic integrals.

###  VI.4 Transverse to the spin motion

For this case $P_{\parallel}=0$ and $P_{0}=P_{\perp}$. We denote

$M=P_{\perp}m,\hskip 14.22636ptJ=P_{\perp}j,\hskip 14.22636pt\hat{\tau}=\beta
P_{\perp}\tau\,.$ (82)

Then the equations for the mass and spin evolution take the form

$\begin{split}&\dfrac{dm}{d\hat{\tau}}=\dfrac{\hat{r}_{+}(m^{2}+1)}{m}\,,\\\
&\dfrac{dj}{d\hat{\tau}}=-\dfrac{1}{6}\dfrac{mj}{m^{2}+1}\,,\\\
&\hat{r}_{+}=m+\sqrt{m^{2}-j^{2}/m^{2}}\,.\end{split}$ (83)

##  VII Discussion

In this paper we discussed a motion of a rotating black hole in the
homogeneous massless scalar field. For this purpose we used the Kerr-Schild
form of the Kerr metric. We introduced two frames. One which we denoted by
$\tilde{K}$ is the frame in which the asymptotic scalar field does not depend
on the spatial coordinates. The other frame, which we denoted by $K$, is a
frame moving with respect to $\tilde{K}$ with a constant velocity $V$ and in
which the black hole is at rest. We first solved the scalar field equation in
$K$ frame and found a solutions satisfying the condition of the regularity at
the black hole horizon and proper behavior at the infinity. After this we
calculated the fluxes of the energy, momentum and angular momentum through a
surface surrounding the black hole. This allowed us to find the force acting
on the black hole and to obtain the equation of its motion in $\tilde{K}$
frame.

The main results of the analysis of the solutions of these equations are the
following:

  * •

For a general type of motion the components of the momentum of the black hole
both in the direction of its spin $P_{\parallel}$ and in the transverse plane
$P_{\perp}$ are conserved.

  * •

The black hole mass $M$ monotonically grows and formally becomes infinitely
large in a finite interval of time $\tau_{fin}$, which depends both on the
strength of the scalar field and initial value of the mass, spin and velocity
of the black hole.

  * •

The spin $J$ of the black hole monotonically decrease, but does not vanish at
$\tau=\tau_{fin}$. However, since the mass of the black hole becomes
infinitely large, the rotation parameter $s=J/M$ vanishes in this limit, and
the Kerr metric reduces to its Schwarzschild limit.

Let us note that these results are obtained in the adiabatic approximation in
which the mass $M$ and the spin $J$ of the black hole change slowly in time.
At time close to $\tau_{fin}$, where $\dot{M}$ is not small, this
approximation is broken.

In this paper we considered a simple model and assumed that the scalar field
obeys a linear equation $\Box{\Phi}=0$. This approximation can be violated in
application to some interesting cases, e.g. for the scalar field of the ghost
condensate, where the effects of the non-linearity might become important. In
application to the cosmology, the scalar field may depend on the general
expansion of the universe, which would affect the imposed boundary conditions
at the infinity. For the black hole in the expanding universe the adopted form
(4) of the metric should also be modified. It is interesting to study how
these modification of the model would affect the interaction of the black hole
with a scalar field and its motion.

##  Appendix A Complex null tetrads

In this appendix we collect useful formulas and expressions for the complex
null tetrads in the Kerr-Schild geometry. Let us denote

$\begin{split}m^{\mu}&=\sqrt{\dfrac{\Delta_{y}^{(0)}}{2\Sigma}}\left(1,0,i,\dfrac{a}{\Delta_{y}^{(0)}}\right)\,,\\\
\bar{m}^{\mu}&=\sqrt{\dfrac{\Delta_{y}^{(0)}}{2\Sigma}}\left(1,0,-i,\dfrac{a}{\Delta_{y}^{(0)}}\right)\,.\end{split}$
(84)

These vectors satisfy the following relations

$\begin{split}&g_{\mu\nu}m^{\mu}m^{\nu}=\eta_{\mu\nu}m^{\mu}m^{\nu}=0\,,\\\
&g_{\mu\nu}\bar{m}^{\mu}\bar{m}^{\nu}=\eta_{\mu\nu}\bar{m}^{\mu}\bar{m}^{\nu}=0\,,\\\
&g_{\mu\nu}m^{\mu}\bar{m}^{\nu}=\eta_{\mu\nu}m^{\mu}\bar{m}^{\nu}=1\,.\end{split}$
(85)

The complex null vectors ${\boldsymbol{m}}$ and $\bar{{\boldsymbol{m}}}$ are
orthogonal to ${\boldsymbol{l}}$ both in ${\boldsymbol{g}}$ and
${\boldsymbol{\eta}}$ metrics.

The forth vector of the null tetrad has a slightly different form for
${\boldsymbol{g}}$ and ${\boldsymbol{\eta}}$ metrics. We denote

$K^{\mu}=\dfrac{\Delta_{r}^{(0)}}{2\Sigma}\left(1,1,0,\dfrac{a}{r^{2}+a^{2}}\right)\,.$
(86)

This vector is null in the metric ${\boldsymbol{\eta}}$ and normalized so that

$\eta_{\mu\nu}l^{\mu}K^{\nu}=-1\,.$ (87)

A similar vector ${\boldsymbol{k}}$ for ${\boldsymbol{g}}$ metric is

$k^{\mu}=K^{\mu}+\dfrac{1}{2}\Phi l^{\mu}\,.$ (88)

It satisfies the relations

$\begin{split}&g_{\mu\nu}k^{\mu}k^{\nu}=0,\hskip
14.22636ptg_{\mu\nu}l^{\mu}k^{\nu}=-1\,,\\\
&g_{\mu\nu}m^{\mu}k^{\nu}=g_{\mu\nu}\bar{m}^{\mu}k^{\nu}=0\,.\end{split}$ (89)

The complex null tetrad in the metric ${\boldsymbol{g}}$ regular at the future
horizon of the Kerr black hole is

${\boldsymbol{z}}^{a}=({\boldsymbol{l}},{\boldsymbol{k}},{\boldsymbol{m}},\bar{{\boldsymbol{m}}})\,.$
(90)

The index $a$ enumerating the basis vectors takes the values $0,1,2,3$. By the
construction the vectors of the basis ${\boldsymbol{z}}^{a}$ are regular the
future event horizon.

##  Appendix B Energy and angular momentum fluxes through the horizon

Let $\xi^{\mu}$ be a Killing vector and $T_{\mu\nu}$ be a conserved stress-
energy tensor, $T^{\mu\nu}_{\ \ ;\nu}=0$. Then the following vector

$\mathcal{P}^{\mu}=T^{\mu\nu}\xi_{\nu}\,.$ (91)

is conserved

$\mathcal{P}^{\mu}_{\ ;\mu}=0\,.$ (92)

Figure 4:  Illustration to the Stockes’ theorem.

Let us consider a rotating black hole and use the coordinates $(T,r,y,\phi)$.
We denote by $\Sigma_{\pm}$ two 3D slices determined by the equations
$T=T_{\pm}$ and restricted from one side by the horizon $r=r_{+}$ and from the
other side by a surface $r=r_{0}>r_{+}$. We denote by $\Sigma_{H}$ a part of
the horizon surface between $T=T_{-}$ and $T_{+}$. Similarly, we denote be
$\Sigma_{0}$ a 3D surface $r=r_{0}$ between $T=T_{-}$ and $T_{+}$ (see figure
4).

Let $\mathcal{V}$ be a four volume restricted by $\Sigma_{\pm}$, $\Sigma_{H}$
and $\Sigma_{0}$. Using the Stockes’ theorem one can write

$\begin{split}0=&\int_{\mathcal{V}}\sqrt{-g}d^{4}x\mathcal{P}^{\mu}_{\
;\mu}=\\\
=&\big{[}\int_{\Sigma_{+}}-\int_{\Sigma_{-}}\big{]}\mathcal{P}^{\mu}d\sigma_{\mu}+\big{[}\int_{\Sigma_{0}}-\int_{\Sigma_{H}}\big{]}\mathcal{P}^{\mu}d\sigma_{\mu}\,.\end{split}$
(93)

The surface elements $d\sigma^{\mu}$ are chosen so that for $\Sigma_{\pm}$ and
at the horizon $\Sigma_{H}$ they are both future directed, while at
$\Sigma_{0}$ it is directed into this surface’s interior.

For the problem under the consideration the gradient of the scalar field
$\Psi$ does not depend on time, and, since the metric is also time
independent, the stress-energy tensor has the same property. As a result the
expression in the first square bracket in (93) vanishes and one has

$\int_{\Sigma_{0}}\mathcal{P}^{\mu}d\sigma_{\mu}=\int_{\Sigma_{H}}\mathcal{P}^{\mu}d\sigma_{\mu}\,.$
(94)

This relation shows that

  * •

The flux of ${\boldsymbol{\mathcal{P}}}$ inside $\Sigma_{0}$ during the time
interval $T_{+}-T_{-}$ is equal to the flux through the horizon for the same
interval of time $T$.

  * •

The flux of ${\boldsymbol{\mathcal{P}}}$ inside $\Sigma_{0}$ in fact does not
depend in the choice of the radius $r_{0}$.

Let us emphasize that at the surface of constant radius $r=r_{0}$ the
coordinates $T$, $v$, and $t$ differs only by constant values. For this
reason, one has

$T_{+}-T_{-}=v_{+}-v_{-}=t_{+}-t_{-}\,.$ (95)

These remarks can be used to confirm the results (LABEL:EEE) for the energy
and angular momentum fluxes through $\Sigma_{0}$.

Let us calculate the energy and angular momentum fluxes through the horizon of
the black hole. Denote by

$\eta^{\mu}=\xi^{\mu}+\Omega\zeta^{\mu},\hskip
14.22636pt\Omega=\dfrac{a}{2Mr_{+}}\,.$ (96)

Then for the horizon surface $\Sigma_{H}$ one has

$d\sigma^{\mu}=-\eta^{\mu}\dfrac{2Mr_{+}}{a}dy\,d\phi\,dT\,.$ (97)

The energy and angular momentum fluxes through the horizon are

$\begin{split}\mathcal{E}=&-\dfrac{1}{T_{+}-T_{-}}\dfrac{2Mr_{+}}{a}\int_{\Sigma_{H}}\xi^{\mu}\eta^{\nu}T_{\mu\nu}dy\,d\phi\,dT\\\
=&\dfrac{2Mr_{+}}{a}\int_{-a}^{a}(T_{TT}+\Omega T_{T\phi})dy\,d\phi\,,\\\
\mathcal{J}=&\dfrac{1}{T_{+}-T_{-}}\dfrac{2Mr_{+}}{a}\int_{\Sigma_{H}}\zeta^{\mu}\eta^{\nu}T_{\mu\nu}dy\,d\phi\,dT\\\
=&-\dfrac{2Mr_{+}}{a}\int_{-a}^{a}(T_{T\phi}+\Omega
T_{\phi\phi})dy\,d\phi\,.\end{split}$ (98)

Calculations give

$\begin{split}&T_{TT}+\Omega T_{T\phi}\\\ &\stackrel{{\scriptstyle
H}}{{=}}\bar{\Psi}^{2}_{0}\big{[}1-\dfrac{(r_{-}\cos\phi+a\sin\phi)\sqrt{a^{2}-y^{2}}}{2\sqrt{2Mr_{+}}a}V_{X}\\\
&-\dfrac{(r_{-}\sin\phi-a\cos\phi)\sqrt{a^{2}-y^{2}}}{2\sqrt{2Mr_{+}}a}V_{Y}\big{]}\,.\end{split}$
(99)

After integration of this expression over the angle $\phi$ the terms which
depend on the velocity components $V_{X}$ and $V_{Y}$ vanish. The further
integration over $y$ gives

$\mathcal{E}\stackrel{{\scriptstyle H}}{{=}}8\pi\bar{\Psi}^{2}_{0}Mr_{+}\,.$
(100)

This expression correctly reproduces the result (LABEL:FLUXES) as it should
be.

To calculate the angular momentum flux through the horizon one need first to
find the value of $T_{T\phi}+\Omega T_{\phi\phi}$ on the horizon. The
corresponding expression is rather long and we do not reproduce it here.
Instead of this, we give the expression which is obtained after integration of
this object over the angle $\phi$

$\int_{0}^{2\pi}(T_{T\phi}+\Omega T_{\phi\phi})d\phi=-\bar{\Psi}^{2}\dfrac{\pi
M}{2ar_{+}}(V_{X}^{2}+V_{Y}^{2})(a^{2}-y^{2})\,.$ (101)

Using (98) and performing the integration over $y$ one obtains

$\mathcal{J}\stackrel{{\scriptstyle
H}}{{=}}-\dfrac{4}{3}\pi\bar{\Psi}_{0}^{2}aM^{2}(V_{X}^{2}+V_{Y}^{2})\,.$
(102)

This result correctly reproduces the expression obtained earlier in
(LABEL:FLUXES). The calculations presented in this appendix provide an
additional check of the results presented in section IV.

###### Acknowledgements.

The author would like to thank Shinji Mukohyama and Andrei Frolov for
stimulating discussions. This work was supported by the Natural Sciences and
Engineering Research Council of Canada. The author is also grateful to the
Killam Trust for its financial support.

## References

  * Arkani-Hamed  _et al._ (2004a) Nima Arkani-Hamed, Hsin-Chia Cheng, Markus A. Luty, and Shinji Mukohyama, “Ghost condensation and a consistent infrared modification of gravity,” JHEP 05, 074 (2004a), arXiv:hep-th/0312099 .
  * Arkani-Hamed  _et al._ (2004b) Nima Arkani-Hamed, Paolo Creminelli, Shinji Mukohyama, and Matias Zaldarriaga, “Ghost inflation,” JCAP 04, 001 (2004b), arXiv:hep-th/0312100 .
  * Son (2002) D. T. Son, “Low-energy quantum effective action for relativistic superfluids,” (2002), arXiv:hep-ph/0204199 .
  * Nicolis (2011) Alberto Nicolis, “Low-energy effective field theory for finite-temperature relativistic superfluids,” (2011), arXiv:1108.2513 [hep-th] .
  * Endlich  _et al._ (2013) Solomon Endlich, Alberto Nicolis, and Junpu Wang, “Solid Inflation,” JCAP 10, 011 (2013), arXiv:1210.0569 [hep-th] .
  * Moffat (1993) J. W. Moffat, “Quantum gravity, the origin of time and time’s arrow,” Found. Phys. 23, 411–437 (1993), arXiv:gr-qc/9209001 .
  * Babichev  _et al._ (2008) Eugeny Babichev, Viatcheslav Mukhanov, and Alexander Vikman, “k-Essence, superluminal propagation, causality and emergent geometry,” JHEP 02, 101 (2008), arXiv:0708.0561 [hep-th] .
  * Mukohyama and Uzan (2013) Shinji Mukohyama and Jean-Philippe Uzan, “From configuration to dynamics: Emergence of Lorentz signature in classical field theory,” Phys. Rev. D 87, 065020 (2013), arXiv:1301.1361 [hep-th] .
  * Mukohyama (2013) Shinji Mukohyama, “Emergence of time in power-counting renormalizable Riemannian theory of gravity,” Phys. Rev. D 87, 085030 (2013), arXiv:1303.1409 [hep-th] .
  * Bekenstein (1995) J. D. Bekenstein, “Novel “no-scalar-hair” theorem for black holes,” Phys. Rev. D 51, R6608 (1995).
  * Frolov and Kofman (2003) Andrei V. Frolov and Lev Kofman, “Inflation and de Sitter thermodynamics,” JCAP 05, 009 (2003), arXiv:hep-th/0212327 .
  * Frolov (2004) Andrei V. Frolov, “Accretion of ghost condensate by black holes,” Phys. Rev. D 70, 061501 (2004), arXiv:hep-th/0404216 .
  * Mukohyama (2005) Shinji Mukohyama, “Black holes in the ghost condensate,” Phys. Rev. D 71, 104019 (2005), arXiv:hep-th/0502189 .
  * Barranco  _et al._ (2011) Juan Barranco, Argelia Bernal, Juan Carlos Degollado, Alberto Diez-Tejedor, Miguel Megevand, Miguel Alcubierre, Dario Nunez, and Olivier Sarbach, “Are black holes a serious threat to scalar field dark matter models?” Phys. Rev. D 84, 083008 (2011), arXiv:1108.0931 [gr-qc] .
  * Gregory  _et al._ (2014) Ruth Gregory, Ian G. Moss, and Benjamin Withers, “Black holes as bubble nucleation sites,” JHEP 03, 081 (2014), arXiv:1401.0017 [hep-th] .
  * Chadburn and Gregory (2014) Sarah Chadburn and Ruth Gregory, “Time dependent black holes and scalar hair,” Class. Quant. Grav. 31, 195006 (2014), arXiv:1304.6287 [gr-qc] .
  * Gregory  _et al._ (2018) Ruth Gregory, David Kastor, and Jennie Traschen, “Evolving Black Holes in Inflation,” Class. Quant. Grav. 35, 155008 (2018), arXiv:1804.03462 [hep-th] .
  * Gregory (2023) Ruth Gregory, “Primordial Black Holes and Higgs Vacuum Decay,” Lect. Notes Phys. 1022, 289–311 (2023).
  * de Cesare and Oliveri (2022) Marco de Cesare and Roberto Oliveri, “Evolving black hole with scalar field accretion,” Phys. Rev. D 106, 044033 (2022), arXiv:2205.01712 [gr-qc] .
  * Kerr and Schild (2009) R. P. Kerr and A. Schild, “Republication of: A new class of vacuum solutions of the Einstein field equations,” General Relativity and Gravitation 41, 2485–2499 (2009).
  * Newman and Janis (1965) E. T. Newman and A. I. Janis, “Note on the Kerr spinning particle metric,” J. Math. Phys. 6, 915–917 (1965).
  * Newman (1973) E. T. Newman, “Complex coordinate transformations and the Schwarzschild-Kerr metrics,” J. Math. Phys. 14, 774 (1973).
  * Israel (1970) Werner Israel, “Source of the kerr metric,” Phys. Rev. D 2, 641–646 (1970).
  * Kaiser (2003) Gerald Kaiser, “Physical wavelets and their sources: real physics in complex spacetime,” Journal of Physics A: Mathematical and General 36, R291–R338 (2003).
  * Adamo and Newman (2014) Tim Adamo and E. T. Newman, “The Kerr-Newman metric: A Review,” Scholarpedia 9, 31791 (2014), arXiv:1410.6626 [gr-qc] .
  * White (2018) C. D. White, “The double copy: gravity from gluons,” Contemporary Physics 59, 109–125 (2018).
  * Bern  _et al._ (2019) Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, and Radu Roiban, “The duality between color and kinematics and its applications,” (2019), arXiv:1909.01358 [hep-th] .
  * Bern  _et al._ (2022) Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, and Radu Roiban, “The sagex review on scattering amplitudes, chapter 2: An invitation to color-kinematics duality and the double copy,” (2022), arXiv:2203.13013 [hep-th] .
  * Misner  _et al._ (1973) C. W. Misner, K. S. Thorne, and J. A. Wheeler,  _Gravitation_ (W. H. Freeman, San Francisco, 1973).
  * Kerr (1963) Roy P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,” Phys. Rev. Lett. 11, 237–238 (1963).
  * Sommers (1976) P Sommers, “Properties of shear-free congruences of null geodesics,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 349, 309–318 (1976).
  * Frolov (1979) V. P. Frolov, “The newman-penrose method in the theory of general relativity,” in  _Problems in the General Theory of Relativity and Theory of Group Representations_, edited by N. G. Basov (Springer US, Boston, MA, 1979) pp. 73–185.
  * Newman (2004) Ezra T Newman, “Maxwell fields and shear-free null geodesic congruences,” Classical and Quantum Gravity 21, 3197–3221 (2004).
  * Frolov  _et al._ (2023) Andrei V Frolov, Valeri Frolov, and Shinji Mukohyama, “Paper under preparation,” (2023).