# Formation of a long filament through the connection of two filament segments
observed by CHASE

H. T. Li School of Astronomy and Space Science, Nanjing University, Nanjing
210023, People′s Republic of China<EMAIL_ADDRESS>Key Laboratory of
Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210023, People′s Republic of China  X. Cheng School of Astronomy and
Space Science, Nanjing University, Nanjing 210023, People′s Republic of China;
<EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and Astrophysics
(Nanjing University), Ministry of Education, Nanjing 210023, People′s Republic
of China  Y. W. Ni School of Astronomy and Space Science, Nanjing University,
Nanjing 210023, People′s Republic of China<EMAIL_ADDRESS>Key
Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry
of Education, Nanjing 210023, People′s Republic of China  C. Li School of
Astronomy and Space Science, Nanjing University, Nanjing 210023, People′s
Republic of China<EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and
Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023,
People′s Republic of China  S. H. Rao School of Astronomy and Space Science,
Nanjing University, Nanjing 210023, People′s Republic of China;
<EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and Astrophysics
(Nanjing University), Ministry of Education, Nanjing 210023, People′s Republic
of China  J. H. Guo School of Astronomy and Space Science, Nanjing University,
Nanjing 210023, People′s Republic of China<EMAIL_ADDRESS>Key
Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry
of Education, Nanjing 210023, People′s Republic of China  Centre for
Mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven,
Celestijnenlaan 200B,B-3001 Leuven, Belgium  M. D. Ding School of Astronomy
and Space Science, Nanjing University, Nanjing 210023, People′s Republic of
China<EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and
Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023,
People′s Republic of China  P. F. Chen School of Astronomy and Space Science,
Nanjing University, Nanjing 210023, People′s Republic of China;
<EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and Astrophysics
(Nanjing University), Ministry of Education, Nanjing 210023, People′s Republic
of China

###### Abstract

We present imaging and spectroscopic diagnostics of a long filament during its
formation with the observations from the Chinese H$\alpha$ Solar Explorer and
Solar Dynamics Observatory. The seed filament first appeared at about 05:00 UT
on 2022 September 13. Afterwards, it grew gradually and connected to another
filament segment nearby, building up a long filament at about 20:00 UT on the
same day. The CHASE H$\alpha$ spectra show an obvious centroid absorption with
mild broadening at the main spine of the long filament, which is interpreted
as the evidence of filament material accumulation. More interestingly, near
the footpoints of the filament, persistent redshifts have been detected in the
H$\alpha$ spectra during the filament formation, indicating continuous
drainage of filament materials. Furthermore, through inspecting the extreme
ultraviolet images and magnetograms, it is found that EUV jets and
brightenings appeared repeatedly at the junction of the two filament segments,
where opposite magnetic polarities converged and canceled to each other
continuously. These results suggest the occurrence of intermittent magnetic
reconnection that not only connects magnetic structures of the two filament
segments but also supplies cold materials for the filament channel likely by
the condensation of injected hot plasma, even though a part of cold materials
fall down to the filament footpoints at the same time.

Solar filaments (1495) — Solar magnetic reconnection (1504)

##  1 Introduction

Solar filaments are cold and dense plasma suspended in the hot and tenuous
corona above the polarity inversion lines (PILs) of magnetic field at the
photosphere (Tandberg-Hanssen, 1974; Martin, 1998; Chen et al., 2020). They
often appear as dark structures on the solar disk and are associated with an
absorption in the H$\alpha$ line. When appearing above the solar limb, they
are also called prominences with brighter emissions than the background
(Mackay et al., 2010).

Studies about solar filaments can be roughly divided into two categories. The
first category is focusing on the nature and formation mechanisms of the
filament magnetic structure, which is also named filament channel (Schmieder
et al., 2014; Patsourakos et al., 2020). At present, it is widely accepted
that the magnetic structure of filaments could be either sheared arcades
(Antiochos et al., 1994) or twisted magnetic flux ropes (Aulanier et al.,
1998; Cheng et al., 2014, 2015), as both of them contain concave magnetic dips
that are able to support filament materials against the gravity.

In the past decades, more attention was paid to the formation of filament
channels consisting of a twisted flux rope, possibly because about 89% of
filaments are supported by flux ropes Ouyang et al. (2017). The twisted
filament channel can be gradually formed through photospheric rotational
motions that continuously inject twists into the channel (Török & Kliem, 2003;
Yan et al., 2015, 2016). Besides, magnetic reconnection is argued to be the
more important mechanism because it can directly build up a coherent flux rope
by reconnecting sheared arcades (van Ballegooijen & Martens, 1989; Moore et
al., 2001; Török et al., 2011). This was well supported by observations where
a long filament (hot channel) was formed by the reconnection of two filament
segments (sheared arcades) (Schmieder et al., 2004; Joshi et al., 2014; Cheng
et al., 2015; Zhu et al., 2015; Chen et al., 2016). In addition, erupting
filaments were also observed occasionally to interact with a nearby filament,
giving rise to a distinct filament with a new footpoint (Jiang et al., 2014;
Joshi et al., 2016; Yang et al., 2017; Fang et al., 2023)

The second category of studies is on the fine structures of filaments with
particular interest in the origin of the filament materials (Zhou et al.,
2020; Wang et al., 2022). One possibility is that the cold materials within a
filament come from the emergence of a magnetic flux rope containing cold
plasma in the lower atmosphere (e.g., Manchester et al., 2004; Okamoto et al.,
2008; Cheung & Isobe, 2014; Zhao et al., 2017). However, it should be pointed
out that the entire emergence of a twisted flux rope from below the
photosphere to the corona is difficult (Manchester et al., 2004). Besides flux
rope emergence, the cold materials in the chromosphere can be directly
transported to the filament channel by surges/jets in a manner of direct
injection (Wang, 1999; Wang et al., 2018). When the injected plasmas are hot,
they need to experience a condensation process, i.e., so-called evaporation-
condensation model (Antiochos et al., 1999; Xia et al., 2012; Xia & Keppens,
2016). Given that the main difference between the injection and evaporation-
condensation models is the temperature of injected plasma, Huang et al. (2021)
performed MHD simulations and unified the two models in the same framework.
They proposed that if heating takes place at the lower chromosphere, the
enhanced pressure will push the cold plasma in the upper chromosphere to the
corona through the direct injection way. In contrast, if heating is in the
upper chromosphere, the chromospheric plasma will be first heated and
evaporated into the corona and then undergoes a condensation process to cold
filaments.

In this paper, we perform a detailed investigation of the formation of a long
filament through the interaction of two filament segments with a combination
of imaging and spectroscopic observations for the first time. The interesting
finding is the coexistence of material accumulation along the filament spine
and material drainage toward the filament footpoints, indicating a highly
dynamic circulation of materials during the filament formation. In the
following, the instruments and the observations are introduced in Section 2,
the main results are shown in Section 3, which is followed by a summary and
discussions in Section 4.

##  2 Instruments and Observations

The data we use are mainly from the H$\alpha$ Imaging Spectrograph (HIS; Liu
et al. 2022) on board the Chinese H$\alpha$ Solar Explorer (CHASE; Li et al.
2019, 2022), which was designed to acquire the spectroscopic information of
the solar lower atmosphere. It contains two observational
modes111https://ssdc.nju.edu.cn/: raster scanning mode (RSM) and continuum
imaging mode (CIM). The former provides spectra at two wavebands of H$\alpha$
($6562.8\pm 3.1$ Å) and Fe I ($6569.2\pm 1.4$ Å) with a spectral resolution of
0.024 Å per pixel. The reconstructed monochromatic images of the full solar
disk have a spatial resolution of $\sim$ 0.52′′ per pixel and time cadence of
$\sim$ 60 s at all wavelengths. The CIM yields photospheric images at 6689 Å
with a full width at half maximum of 13.4 Å.

We also use the data from the Atmospheric Imaging Assembly (AIA; Lemen et al.
2012) on board Solar Dynamics Observatory (SDO; Pesnell et al. 2012), which
provides 7 extreme ultraviolet (EUV) images of the full solar disk with a
pixel size of 0.6′′ and time cadence of 12 s. Simultaneously, the Helioseismic
and Magnetic Imager (HMI; Schou et al., 2012; Scherrer et al., 2012) on board
SDO provides the line-of-sight (LOS) magnetograms with a pixel size of $\sim$
0.6′′ and time cadence of $\sim$ 45 s, which are used for investigating the
magnetic property during the filament formation. Note that, the alignment
between the CHASE and SDO images is done by comparing their white-light
images, and all images have been reprojected to the same reference time
(11:48:10 UT) for a better comparison.

On 2022 September 13, the CHASE observed the Sun for 15 orbits, each of which
lasted approximately 25 minutes. Figure 1a shows the H$\alpha$ line center
image of the full Sun observed by CHASE at 21:10:20 UT. The target filament is
located near the disk center as indicated by the white box. Figure 1b displays
the average H$\alpha$ spectra of the entire target region and filament, which
are derived through integrating all pixels within the white box and the
filament, respectively (as shown in Figure 1c). Following the method in Qiu et
al. (2022), the H$\alpha$ line center for all spectra is calibrated assuming
that the average spectrum of nearby quiescent regions (as shown in Figure 1d)
has a zero Doppler shift. Note that, we take advantage of the waveband of
6561–6565 Å (as shown by the shaded area) for calculating the line centroid,
the purpose of which is to avoid the contamination by the Si I line (6560.58
Å; Hong et al. 2022).

##  3 Results

###  3.1 Filament Formation and Mass accumulation

The formation and evolution process of the long filament was completely
captured by CHASE, as shown in Figures 2a–2d. From a time sequence of
H$\alpha$ line center images, it is found that the seed filament (F1) first
appeared as a slightly elongated dark structure at 02:00 UT (pointed out by
the blue arrows), lying over the PIL of the active region 13099. At about
05:00 UT, the second seed filament appeared above the PIL of the adjacent
active region 13096, as represented by the yellow arrows. Afterwards, with
time elapsing, F1 extended toward the northeast and F2 toward the northwest.
At 18:00 UT, the right endpoint of F1 merged with the left endpoint of F2,
forming an elongated $\Omega$-shaped long filament F3, as indicated by the red
arrow. More details can be found in the attached animation of Figure 2.

During the formation of F3, an obvious feature is that more and more filament
materials were accumulated in the filament channel, which is indicated by the
fact that F3 became darker and darker. To quantify the accumulation of
filament materials, we plot the evolution of the H$\alpha$ spectrum of a
selected point on the filament spine in Figure 2e, where the point is marked
as a plus sign in Figure 2d. It is found that the intensity at the central
part of the H$\alpha$ line gradually decreased with the formation of F3, in
particular after T0 (11:48 UT). At the same time, the line width increased
slightly. According to the cloud model (Beckers, 1964), the intensity of a
filament can be solved by the radiative transfer equation as:
$I=I_{0}\text{exp}(-\tau)+S[1-\text{exp}(-\tau)]$, where $I_{0}$ is the
intensity of the background quiescent region, $S$ and $\tau$ denote the source
function and optical depth of the cloud, respectively. Assuming the source
function is constant and the line profile is only subject to Doppler
broadening, the optical depth $\tau$ can be expressed as a Gaussian profile:
$\tau=\tau_{0}\text{exp}[-(\lambda-\lambda_{0}-\lambda_{0}v/c)^{2}/W^{2}]$,
where $\tau_{0}$ is optical depth at the line center, $v$ and $W$ are the
Doppler velocity and width, respectively. Applying this method to the
H$\alpha$ profiles for the filament spine, we derive the optical depth
$\tau_{0}$, which is plotted in Figure 2f. Moreover, we estimate the area of
the filament, the boundary of which is determined by a threshold, i.e., 1.1
times the minimal intensity of the filament region, through the trial-and-
error method. Obviously, the filament area increased almost synchronously with
the optical depth $\tau_{0}$. These results clearly confirm that the filament
materials were also accumulated gradually in the filament channel during its
formation.

###  3.2 Persistent Drainage of Filament Plasma

A very interesting feature is that H$\alpha$ redshifts appeared at the
footpoints of F1, F2, and F3 for most of the time during the filament
formation. In the H$\alpha$ red-wing images (Figure 3a–3d and attached
animation), one can see that absorption prominently appeared at the two
footpoints of F1 (i.e., F1L and F1R in Figure 3a). It was also observed at the
footpoints of F2 (i.e., F2L and F2R in Figure 3b) when it appeared. In order
to show the redshift more clearly, we select a curved slice along the filament
as indicated in Figure 2d. The corresponding distance-spectrum diagram is
displayed in Figure 3i, in which the H$\alpha$ spectra at the four footpoints
are significantly shifted toward the red wing. It is found that the two red-
shifted footpoints in the middle even merged together after $\sim$19:20 UT, as
indicated by F3M in Figure 3c–3d and Figure 3i–3j.

We also apply the cloud model to all H$\alpha$ spectra at the entire filament
region. The resulting Dopplergram is shown in Figure 4a, which clearly shows
the appearance of downflows at the two footpoints (F3L and F3R) of F3. Figures
4b–4c show two representative H$\alpha$ profiles. The fitting results give the
corresponding red-shift velocities being more than 10 km s-1. These results
show that, during the formation of F3, the filament materials also drained
down continuously to the footpoints, which may partly counteract the
accumulation of materials within the filament channel as revealed in Section
3.1.

Besides, it is interesting to see that the H$\alpha$ profiles also present
redshifts with a velocity of around 10 km s-1 at the merging site F3M between
F1 and F2. This even persisted for a long time after F3 has been well formed
visually. Comparing with F1R and F2L, the redshifts at F3m are stronger as
shown in Figures 3c–3d. It is interpreted that the reconnection may take place
between two dark threads of F1 and F2, thus enhancing the drainage of filament
materials. In addition, during the formation of F3, the drainage at the
footpoints of the fluxes of F1 and F2 that have not reconnected yet likely
also has a contribution to the redshifts at F3M. By contrast, in the H$\alpha$
blue-wing images, the absorption feature was also detectable but primarily on
the spines of F1, F2, and F3 (Figures 3e–3h). This may be an indication of the
filament materials moving out of the corresponding magnetic dips before
falling down.

###  3.3 Plasma Injection by EUV Jets

The formation of the magnetic configuration of F3 and its material circulation
are argued to be closely related to continuous outflow jets occurring near the
joint region of filaments F1 and F2. These jets can be seen most obviously at
the AIA 304 Å passband, indicating that they have a temperature similar to the
transition region and are thus hotter than the filament but cooler than the
corona. To see it more clearly, we take a curved slice AB (Figure 5a) along
the spine of F3 to create a time-distance diagram of AIA 304 Å images (Figure
5b). It clearly shows that, at the joint region of F1 and F2, the emission was
remarkably enhanced. Moreover, from the bright joint region, some prominent
transition-region jets were quickly ejected outward with a velocity of 60–170
km s-1. Then, they moved along the filament spine either toward the east or
the west (Figure 5b). In addition, some weak upward jets were also observed.
Figures 5c and 5d show the zoom-in of boxes 1 and 2 in Figure 5b,
respectively. It is found that the weak jets, resembling spicules, were
continuously injected from their bright base to the filament but with
relatively small velocities ranging from 10 to 85 km s-1. It is argued that
these ejected hot plasma likely provide a source of the cold materials within
the filament channel through the condensation process as suggested by Xia et
al. (2012) and Xia & Keppens (2016). However, it seems that not all erupted
hot plasmas were supplied to the filament because part of them were observed
to return to the base of the jets (see Figure 5c).

The merging of the magnetic structures of F1 and F2 and the transition-region
jets could be driven by flux cancellation. In the sub-panels of Figure 5a, we
plot the HMI line-of-sight magnetograms at three instants, which show an
obvious flux cancellation occurring at the joint region of F1 and F2. We
further integrate the positive and negative magnetic fluxes within the region
of interest and plot their temporal evolutions and find that both of them
decrease constantly during the entire formation process of F3. Moreover, the
time variation of the average AIA 304 Å intensity within the same field-of-
view is over-plotted for a comparison (Figure 5e). The big spikes roughly
correspond to the decreases of the magnetic fluxes, in particular of the
positive one. It is suggested that magnetic reconnection should play a crucial
role not only in reconnecting magnetic structures of F1 and F2 to form that of
F3 but also in injecting the transition-region plasma to the channel of F3,
which then condense to cold materials of the filament.

##  4 Summary and Discussions

In this paper, we present a detailed study of the formation of a long filament
via merging two filament segments with the imaging and spectroscopic
observations from CHASE and SDO. The H$\alpha$ line at the filament spine went
through an obviously enhanced absorption and moderate broadening, suggesting
the accumulation of cold materials. Meanwhile, the H$\alpha$ line at the
filament foopoints presented prominent redshifts for most of the time with a
downward Doppler velocity more than 10 km s-1, indicating the drainage of
filament materials. Moreover, abundant transition-region plasmas were
continuously injected into the filament channel through intermittent jets. The
observations document that the filament formation process involves a highly
dynamic circulation of plasma that includes injection, accumulation, and
drainage of filament materials simultaneously.

The circulation of hot and cold materials is believed to be an important
process for the filament formation as well reproduced in previous observations
and simulations (e.g., Liu et al., 2012; Luna et al., 2012; Xia & Keppens,
2016; Huang et al., 2021). For the current event, the heating and injection of
the chromospheric materials into the filament are supposed to be through
magnetic reconnection. As indicated by the flux cancellation occurring near
the joint region of the two filament segments, the reconnection is argued to
take place between the adjacent legs of F1 and F2 flux ropes. As illustrated
in Figure 6, the right leg of F1 and the left leg of F2 approach each other,
as driven by the converging motions, and reconnect over the polarity inversion
line, forming a longer filament channel F3, with the submergence of the
reconnected short loop manifesting as the flux cancellation (Cheng et al.,
2015).

Besides connecting the magnetic structures of F1 and F2, the reconnection also
heats the chromospheric plasma, which is then injected to the newly formed
filament channel F3, as evidenced by the continuous AIA 304 Å outflow jets.
Once thermal instability occurs, the hot plasma within the filament channel
starts to condense and gradually form the cold filament materials. It is
worthy of mentioning that, after the formation of F3, we can not exclude the
other possibility that the local hot coronal plasma directly condenses into
the filament materials. As the longer magnetic field lines are created, on the
one hand, the condensation will become easier as suggested by the simulations
of Kaneko & Yokoyama (2017), on the other hand, the dips become deeper to
allow more cold materials to be accumulated. In case the newly formed lines
without cold materials, the condensation will produce new cold materials; in
case these lines already with cold materials, the condensation will continue
to accumulate the filament materials. At the same time, the filament materials
also drained out of their magnetic dips with the descending motions, thus
giving rise to the redshifts at the footpoints of F3.

Our observations suggest that magnetic reconnection not only forms magnetic
configuration of the filament channel but also supplies filament materials
mainly through an injection-condensation process, which is comparable with the
conclusion of Yang et al. (2021). However, the hot plasma in their work is
evaporated into the filament channel during the flare, differing from the
direct injection by a series of outflow jets as found in the current study.
Moreover, the filament formation in Yang et al. (2021) only lasted for tens of
minutes, which is much shorter than the duration of the event we study here
(about 20 hours). Such a significant difference in lifetime could be due to
the fact that magnetic reconnection in Yang et al. (2021) is highly efficient
and involves abundant magnetic flux in a short time, while it is relatively
moderate and of small-scale in the current event and thus a series of
analogous individual jets are needed (e.g., Cheng et al., 2023). In addition,
we also observe the drainage of the filament materials, which also has a role
in prolonging the timescale of the filament formation.

The magnetic dips are the key for the condensation of hot plasma within the
filament channel. In previous observations, magnetic dips have been observed
explicitly to be formed in the high corona through the reconnection between
large-scale open and closed coronal loops (Li et al., 2018, 2021; Chen et al.,
2022). In contrast, in the current event, the reconnection probably occurs in
the lower atmosphere because it seems that the observed jets are composed of
plasma from the transition zone.

We appreciate the referee for his/her comments and suggestions. We thank the
CHASE, SDO/AIA, and SDO/HMI teams for providing the high-quality data. CHASE
mission is supported by China National Space Administration. The project was
supported by the National Key R&D Program of China under grants 2021YFA1600504
and by NSFC under grants 12127901 and 12333009.

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Figure 1: (a) CHASE H$\alpha$ line center image of the full disk showing the
location of the target filament pointed out by the white box. The two black
boxes show the nearby quiescent regions. (b) The average H$\alpha$ profiles of
the entire active region (red) and filament (grey) as indicated by the cyan
curve in in panel (c). (d) The H$\alpha$ profile of the quiescent regions that
is used for calibration. The shadow region indicates the waveband where the
H$\alpha$ line centre is calculated.  Figure 2: (a)–(d) H$\alpha$ line center
images showing the formation and mass accumulation of the long filament. The
blue, yellow and red arrows represent the filament F1, F2 and F3,
respectively. The contours in cyan indicate the filament region. The white and
black contours in panel (a) represent the positive and negative magnetic
polarities of $\pm$50G. The blue/yellow circles in pannel (c) mark the
footpoints of F1/F2. The plus in panel (d) indicates the position to make the
spectrum-time diagram in panel (e). The dashed line in red indicates a slit AB
along the filament channel to make the spectrum-distance diagram in Figure 3.
(e) H$\alpha$ spectrum-time diagram for the filament spine with the horizontal
dashed line showing the line center. (f) Temporal evolutions of the filament
area and optical depth. The vertical dashed blue lines in panels (e)–(f)
represent the time gaps between the two adjacent orbits, which are hidden
here. The start and end times of each orbit are pointed out at above and below
of the timeline, respectively. The animation that starts at 00:17 UT and ends
at 22:51 UT is available online to show the formation of the long filament F3
with a duration of 28 s.  Figure 3: (a)–(d) H$\alpha$ red-wing images showing
the drainage of the filament materials at the footpoints. The blue, yellow and
red circles represent the footpoints of F1, F2 and F3, respectively. (e)–(h)
H$\alpha$ blue-wing images at the same time as panels (a)–(d). (i)–(j)
H$\alpha$ spectrum-distance diagrams for the slit AB showing the Doppler
redshifts at the footpoints. The dashed lines indicate the H$\alpha$ line
center. The animation that starts at 00:17 UT and ends at 22:51 is available
online to show the evolution of the redshifts at all footpoints with a
duration of 28 s.  Figure 4: (a) Dopplergram of the filament F3. The grey and
black contours represent the positive and negative polarities of $\pm$50G.
(b)–(d) H$\alpha$ profiles (black) at points C, D, E in panel (a). The gray
lines display the H$\alpha$ spectrum of the quiescent region. The dashed lines
in red show the fitted results.  Figure 5: (a) AIA 304 Å image overlaid by the
contours ($\pm$50 G) of the HMI line-of-sight magnetogram. The inserted panels
are the zoom-in of the magnetograms for the region of interest with the
contours in blue (green) representing the positive (negative) polarities of 50
G (–50 G). (b) Distance-time diagram of the AIA 304 Å images along the slit AB
in panel (a). The dashed lines in white show the trajectories of the ejected
jets. (c)–(d) The zoom-in of the time-distance diagram for the white boxes 1
and 2 in panel (b). The dashed red/blue lines indicate the trajectories of the
upward/downward plasma. (e) Temporal evolutions of the integrated fluxes for
the positive (purple) and negative (green) polarities, as well as the
integrated AIA 304 Å intensity (red) as shown in the black box in panel (a).
Figure 6: Schematic drawing of the formation of the long filament F3. The
lines with different colors indicate magnetic field lines, which indicate
magnetic flux rope structures of F1, F2 and F3. The blue arrow indicates the
upward EUV jets caused by the reconnection between two threads of F1 and F2.
The black threads represent the cold filament materials. The arrows in red
represent downward-moving filament materials shown as H$\alpha$ redshifts.


#  A new class of regular black hole solutions with quasi-localized sources of
matter in $(2+1)$ dimensions

R. V. Maluf<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC),
Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  C. R. Muniz
<EMAIL_ADDRESS>Universidade Estadual do Ceará (UECE), Faculdade de
Educação, Ciências e Letras de Iguatu, Av. Dário Rabelo s/n, Iguatu-CE,
63.500-00 - Brazil.  A. C. L. Santos<EMAIL_ADDRESS>Universidade
Federal do Ceará (UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  Milko Estrada
<EMAIL_ADDRESS>Facultad de Ingeniería, Ciencia y Tecnología,
Universidad Bernardo O’Higgins, Santiago, Chile.

(September 3, 2024)

###### Abstract

This paper investigates a new class of regular black hole solutions in (2 +
1)-dimensions by introducing a generalization of the quasi-localized matter
model proposed by Estrada and Tello-Ortiz. Initially, we try to physically
interpret the matter source encoded in the energy-momentum tensor as
originating from nonlinear electrodynamics. We show, however, that the
required conditions for the quasi-locality of the energy density are
incompatible with the expected behavior of nonlinear electrodynamics, which
must tend to Maxwell’s theory on the asymptotic limit. Despite this, we
propose a generalization for the quasi-localized energy density that
encompasses the existing models in the literature and allows us to obtain a
class of regular black hole solutions exhibiting remarkable features on the
event horizons and their thermodynamic properties. Furthermore, since the
usual version of the first law of thermodynamics, due to the presence of the
matter fields, leads to incorrect values of entropy and thermodynamics volume
for regular black holes, we propose a new version of the first law for regular
black holes.

regular black hole, nonlinear electrodynamics, $(2+1)$-dimensional spacetime

##  I Introduction

Some solutions of gravitational field equations in general relativity (GR)
have divergences (singularities), as is the case with the usual black hole
solutions. These singularities could be related to a limitation in a classical
theory and would be corrected in a quantum version Bambi:2013ufa . In 1968,
yet inside a classical description, James Bardeen BARDEEN  found the first
regular black hole solution in (3 + 1)-dimensions. In other words, a solution
without divergences in the metric and their curvature invariants. Afterward,
Ayón-Beato and García Ayon-Beato:2000mjt  proposed the interesting
interpretation that the metric chosen by Bardeen is equivalent to taking
nonlinear electrodynamics as the source of the energy-momentum tensor. Since
then, these solutions for regular black holes have been widely explored in
literature Balart:2014cga ; Burinskii:2002pz ; Bronnikov:2000vy ; Ma:2015gpa ;
Toshmatov:2017zpr , and recent works include quantum corrections Sharif:2010pj
; Maluf:2018ksj , thermodynamics analysis Akbar , cosmological constant, and
quintessence Rodrigues:2022qdp ; Celio . For details about the issue involving
nonlinear electrodynamics as a matter source, one can refer to the review
article Sorokin:2021tge .

There are also regular solutions in lower-dimensional spacetime Cataldo:2000ns
; He:2017ujy , and their studies have attracted attention for the special
properties which they exhibit. In particular, (2 + 1)-dimensional gravity
models work as a simplest theoretical laboratory than its higher-dimensional
counterpart, bringing new insights to fundamental questions of classical and
quantum gravity Carlip:1995qv . In (2+1)-dimensions, the best-known black hole
solution was discovered in the early 1990s by Bañados, Teitelboim, and Zanelli
(BTZ) Banados:1992wn ; Banados:1992gq . It is known that in (2+1)-dimensions,
general relativity does not have a Newtonian limit in the weak field regime,
and the gravitational field has no dynamical degrees of freedom Gott:1982qg ;
Barrow1986 . In this case, the curvature is determined by the Ricci tensor,
and in the absence of matter, we have $R_{\mu\nu}=0$, so we cannot have a
black hole solution Padmanabhan:2010zzb . To avoid this problem, a negative
cosmological constant was introduced into the theory allowing black hole
solutions to existing in a spacetime asymptotically anti-de Sitter (AdS).
Since the BTZ solution was found, several other (2+1)-dimensional exact black
hole solutions have been obtained in different scenarios Garcia-Diaz:2017cpv .

Regular black hole solutions have been proposed in (2 + 1)-dimensional
spacetime Bueno:2021krl , considering a profile of mass density that obeys the
quasi-locality of the corresponding mass-energy Estrada:2020tbz ;
Hendi:2022opt . In this context, Estrada and Tello-Ortiz Estrada:2020tbz
proposed a mass density that satisfies the requirements of a position-
dependent mass function that is quasi-localized, i.e., the mass function must
reach its finite maximum value at infinity. Inspired by this proposal, we seek
to interpret the origin of the matter source as coming from nonlinear
electrodynamics. As a result, we show that the quasi-locality requirements and
Maxwell’s theory limit for nonlinear electrodynamics are incompatible. Despite
this result, we propose a generalization for the mass density function that
gives rise to a new class of regular black hole solutions.

On the other hand, the standard version of the first law of thermodynamics
must be modified for regular black holes due to the presence of matter fields
in the energy-momentum tensor Ma:2014qma . In fact, the usual first law,
namely, $dm=TdS-PdV$, leads to entropy and thermodynamics volume values that
do not coincide with the Bekenstein-Hawking area law Bekenstein:1973ur .
Aiming to fix this problem, Ma and Zhao included an additional factor in the
first law, which corresponds to an integration of the radial coordinate up to
infinity Ma:2014qma . In Ref. Estrada:2020tbz , a local definition of energy
at the horizon was proposed, imposing constraints on the evolution along the
space parameters. Thus, including matter fields requires redefining the energy
term to get the correct values for temperature, entropy, and volume, so the
first law of thermodynamics for regular black holes is still an open question.
In this context, we propose a new version of the first law for regular black
holes based on the inclusion of a potential $dX$, which leads to a local
definition of internal energy at the horizon, $dU=dm+dX$. So, we study the
thermodynamic properties of these new solutions, calculating Hawking
temperature, entropy, Gibbs potential, and heat capacity.

This paper is organized as follows: In section II, we determine the mapping
among perfect fluids densities and nonlinear electrodynamics. We proceed with
the thermodynamics analysis in section III. Finally, in section IV, we present
our remarks and conclusions.

##  II (2 + 1)-Regular black hole with nonlinear electrodynamics

The (2 + 1)-Einstein gravity action coupled to nonlinear electrodynamics can
be defined as

$S=\int d^{3}x\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi}+L(F)\right],$ (1)

where $g$ is the determinant of the metric tensor, $\Lambda=-1/\ell^{2}$ is
the cosmological constant written in terms of the anti-de Sitter (AdS) radius,
$\ell$, and $L(F)$ is the Lagrangian of the nonlinear electrodynamics with
$F=F^{\mu\nu}F_{\mu\nu}$ Cataldo:2000ns ; He:2017ujy .

The Einstein field equations are obtained by varying the action (1) with
respect to the metric tensor, yielding,

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu},$ (2)

with the energy momentum tensor given by

$T_{\mu\nu}\equiv-\frac{\displaystyle
2}{\displaystyle\sqrt{-g}}\frac{\displaystyle\delta(\sqrt{-g}L_{m})}{\displaystyle\delta
g^{\mu\nu}}=g_{\mu\nu}L(F)-4L_{,F}F_{\mu\alpha}F_{\nu}^{\leavevmode\nobreak\
\alpha},$ (3)

where $L_{,F}$ represents the derivative of $L(F)$ with respect to $F$. The
electromagnetic field equations are determined by varying with respect to the
potential $A_{\mu}$, remembering that the field-strength tensor $F_{\mu\nu}$
is defined as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, we
obtain

$\nabla_{\mu}(L_{,F}F^{\mu\nu})=0.$ (4)

To find exact solutions to the above field equations, let us assume a static,
circularly symmetric spacetime whose metric ansatz is given by

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\phi^{2},$ (5)

where $f(r)$ is an arbitrary function of the radial coordinate $r$.
Furthermore, it was demonstrated in Ref. Cataldo:2000ns  that the magnetic
field vanishes and only the electric field plays the role of source of the
gravitational field in a geometry defined by the metric ansatz (5). Therefore,
we can assume that the electromagnetic field tensor is expressed in the
simplified form

$F_{\mu\nu}=E(r)(\delta^{t}_{\mu}\delta^{r}_{\nu}-\delta^{t}_{\nu}\delta^{r}_{\mu}),$
(6)

and correspondingly $F=-2E^{2}$. For the line element (5) together with the
electromagnetic tensor (6), it can be seen that the relevant components of the
equations of motion (2) and (4) result in

$\displaystyle\frac{f^{\prime}(r)}{2r}+\Lambda$ $\displaystyle=$
$\displaystyle 8\pi\left[L(F)+4E^{2}(r)L_{,F}\right],$ (7) $\displaystyle
E(r)L_{,F}$ $\displaystyle=$ $\displaystyle-\frac{q}{r},$ (8)

where $q$ is an integration constant related the electric charge. Note that
for the particular case of Maxwell theory, namely, $L(F)=F$, the electric
field in (8) corresponds to the field of a point charge in (2 + 1)-dimensions
$E(r)\sim 1/r$. It is convenient to express the derivative $L_{,F}$ as a
function of $r$, and this is accomplished from Eq. (6), leading to

$E(r)L_{,F}=-\frac{L^{\prime}(r)}{4E^{\prime}(r)},$ (9)

which compared with (8), results in

$L^{\prime}(r)=\frac{4q}{r}E^{\prime}(r),$ (10)

where the prime (′) stands for the total derivative with respect to the radial
coordinate $r$.

Thus, we can rewrite Einstein equation (7) equivalently as

$\frac{f^{\prime}(r)}{2}+\Lambda r=8\pi\left[rL(r)-{4q}E(r)\right].$ (11)

In principle, we can integrate the above equation for all values of $r$. For
the metric ansatz that we assume here, it can be verified that this condition
is guaranteed if $f^{\prime}(r)$, $rL(r)$, and $E(r)$ are free of
singularities everywhere. Furthermore, to obtain regular black hole solutions,
one must ensure that the curvature invariants, i.e., the Ricci scalar $R$ and
Kretschmann scalar $K$ (built from the Riemann tensor) are singularity free.
For this latter, we need the function $f(r)$ and its first and second
derivatives to be free of singularities everywhere Cai:2006pq .

The previous requirements impose constraints on the geometric side of Einstein
equations, encoded in the general form of the metric function $f(r)$. The
simplest regular black hole solution in (2 + 1)-dimensions is just the
celebrated BTZ solution, with $f(r)=-m+r^{2}/l^{2}$. If the presence of matter
fields is taken into account, then the metric function may include additional
terms, and a standard ansatz to represent the metric function of a general
regular black hole in (2 + 1)-dimensions is

$f(r)=-m+\frac{r^{2}}{l^{2}}+k(r),$ (12)

where $k(r)$ is a well-behaved function (continuous and differentiable) which
because Eq. (11) must obey the equation

$\frac{k^{\prime}(r)}{2r}=8\pi\left[L(r)-\frac{4q}{r}E(r)\right].$ (13)

So, one can construct several regular black holes through a convenient choice
of the Lagrangian $L(F)$ or the electric field $E(r)$ He:2017ujy .

In addition to the geometric aspects discussed above, one must consider some
conditions on the matter-energy content. To ensure a more physically
acceptable origin for the solutions found, we still need to impose
requirements on the matter fields, which in our case is represented by the
Lagrangian of the nonlinear electrodynamics $L(F)$. It is customary to impose
two requirements or conditions on $L(F)$: (1) the correspondence to the
Maxwell theory in the weak field limit, namely $L(F)\rightarrow F$, for large
$r$ and (2) the weak energy condition should be fulfilled, in our case
requires $L(F)+4E^{2}L_{,F}\leqslant 0$. The conditions described above are
satisfied by many regular black hole solutions described in the literature.

As was pointed out before, one of our goals is to verify if it is possible to
associate the matter source of the regular black holes solutions obtained by
Estrada and Tello-Ortiz Estrada:2020tbz  and recently generalized by in Ref.
Hendi:2022opt  to a nonlinear electromagnetic origin. For this purpose, we
define an effective energy-momentum tensor $T^{\mu(eff)}_{\
\nu}=\mbox{diag}(-\rho,p_{r},p_{\theta})$ as follows,

$T^{(eff)}_{\mu\nu}\equiv
T_{\mu\nu}=g_{\mu\nu}L(F)-4L_{,F}F_{\mu\alpha}F_{\nu}^{\leavevmode\nobreak\
\alpha}.$ (14)

In this way, we can carry out the following identification:

$\rho(r)=-\left[L(r)-\frac{4q}{r}E(r)\right],$ (15)

and with the help of Eq. (10), we can express the Lagrangian $L(r)$ and the
electric field $E(r)$ in terms of the energy density $\rho(r)$ as

$L(r)=-\left[\rho(r)+r\rho^{\prime}(r)\right],$ (16)

and

$E(r)=-\frac{r^{2}}{4q}\rho^{\prime}(r).$ (17)

According to Estrada and Tello-Ortiz Estrada:2020tbz , an energy density
compatible with the regular black hole solution in (2 + 1)-dimensions must
meet the following requirements 111These requirements were enunciated
initially in work Eur.Phys.J.C 79 (2019) 3, 259 , e-Print: 1901.08724 [gr-qc].
(i) The energy density must be positive and continuously differentiable to
avoid singularities; (ii) The energy density must have a single finite maximum
at the origin; (iii) the energy density must be a decreasing radial function
and must vanish at infinity. Moreover, for the case where the Lagrangian
$L(r)$ and the electric field $E(r)$ are written in terms of the energy
density $\rho(r)$, it is necessary to include the following additional
requirement (iv) The energy density must be such that $r$ multiplied by the
Lagrangian $rL(r)$ and the electric field $E(r)$ are free of singularities
everywhere, as was mentioned after the equation (11). In our case, for our
election given for equations (16) and (17), it is direct to check that this
latter is fulfilled due to condition (i).

Thus, they proposed the following functional form for the energy density:

$\rho(r)=\frac{mb^{2}}{\pi(b^{2}+r^{2})^{2}},$ (18)

so that $\displaystyle\lim_{r\rightarrow\infty}\rho(r)=0,$ and $\rho(0)=m/\pi
b^{2}$. Here, we are employing a slightly different definition, with $b$ being
a positive constant with units of length $[b]=\ell$. The quasi-locality of the
mass-energy Aoki:2020prb ; Sorge:2020pdj  is assured, since
$\int_{0}^{\infty}\rho(r)2\pi rdr=m$ is finite .

Thus, the Lagrangian $L(r)$ and the electric field $E(r)$ associated with this
energy density are given by:

$L(r)=-\frac{mb^{2}(b^{2}-3r^{2})}{\pi(b^{2}+r^{2})^{3}},$ (19)

and

$E(r)=\frac{mb^{2}r^{3}}{\pi q(b^{2}+r^{2})^{3}}.$ (20)

We can easily see that the electric field does not tend to the expected
asymptotic Maxwell-like behavior, $E(r)\sim 1/r$, in (2 + 1)-dimensions.

The functional form for the energy density profile given in Eq. (18) can be
generalized to arbitrary powers in $r$ and still preserve the quasi-locality
properties (i)-(iv). Thus, we propose the following expression for the energy
density:

$\rho(r)=\frac{Ab^{n}}{(r^{2}+b^{2})^{(n+3)/2}},$ (21)

with $n\geq 0$. Notice that, for $n=1$, we retrieve the form of Estrada and
Tello-Ortiz energy density profile. The quasi-locality of mass-energy is
guaranteed since we have

$\int_{0}^{\infty}\rho(r)2\pi rdr=m\Rightarrow A=\frac{(n+1)mb}{2\pi}.$ (22)

The electric field associated to this energy density, according to Eq. (17),
is given by

$E(r)=\frac{(n+1)(n+3)mb^{n+1}r^{3}}{8\pi
q\left(b^{2}+r^{2}\right)^{(5+n)/2}}.$ (23)

Once more, we can observe that the field does not present an asymptotic
Maxwell-like behavior, since in this limit $E(r)\sim r^{-2-n}$.

It is worth pointing out that another generalization for regular (2 +
1)-dimensional black holes was recently discussed in Ref. Hendi:2022opt .
However, our proposal contains it as a subset of sources for the possible
regular black hole geometries.

At this point, we must raise the issue of the incompatibility between the
mass-energy quasi-locality condition and the asymptotic Maxwell limit for the
electric field, at least in (2 + 1)-dimensions. To one get the field behavior
at the infinity as $E(r)\sim r^{-1}$, the required $\rho(r)$ must be
represented by a series expansion, according to Eq. (17), in the form

$\rho(r)=a_{0}r^{-2}+\sum_{n=0}^{\infty}b_{n}r^{n},$ (24)

where the sum converges in all domain. Since the mass is given by

$m=\int_{0}^{\infty}2\pi r\rho(r)dr,$ (25)

we have that this quantity diverges. Therefore, we conclude that any energy
density profile obeying the weak energy condition and the quasi-local energy
requirement cannot be associated with an electrodynamics source since the
Maxwell electrodynamic limit is not recovered for large $r$.

From the results obtained above, we can study the horizons associated with the
regular black hole solution obtained from the quasi-localized energy density
profile proposed by us. Thus, given that, by Eqs. (13) and (15), $k(r)$ is
proportional to $\widetilde{m}(r)=\int 2\pi r\rho(r)dr$ plus an integration
constant, the form of the metric coefficient should be, according to (12) and
(21)

$f(r)=1-8m+r^{2}/\ell^{2}+8\widetilde{m}(r)=1-8m+\frac{r^{2}}{\ell^{2}}+\frac{8mb^{n+1}}{(r^{2}+b^{2})^{(n+1)/2}},$
(26)

with the referred constant being $1-7m$, where we have chosen to get an anti-
de Sitter space nearby the origin for the regular black holes and, at
infinity, $\widetilde{m}(r)\to 0$, as expected from the quasi-locality of
mass-energy. Since our solution is asymptotically similar to the BTZ metric,
namely, $f(r)\approx 1-8m+r^{2}/\ell^{2}$, the curvature is entirely
determined by $\Lambda<0$, and the spacetime is asymptotically AdS rather than
asymptotically flat.

Figure 1: Metric coefficient for the regular black holes given from Eq. (26),
as a function of the radial coordinate, for some powers $n$, with $m=2.5$,
$b=2.2$, and $\ell=0.7$ in Planck units.

Considering the Descartes rule, for $8m\leq 1$, there is no real roots for
$f(r)=0$ and the black holes do not possess any horizons. For $8m>1$, these
objects can have up to two horizons, as it is depicted in Fig. 1. Notice that
the radius of the outer (inner) horizon, $r_{h}$, is greater (smaller) the
greater the power, $n$.

On the regularity of the obtained solution, we consider the curvature
invariants, namely, the Ricci $R$ and Kretschmann $K$ scalars calculated below
for the metric (26):

$\displaystyle R$ $\displaystyle=$
$\displaystyle-\frac{6}{\ell^{2}}+\frac{8mb^{n+1}(n+1)(3b^{2}-nr^{2})}{(r^{2}+b^{2})^{\frac{n+5}{2}}},$
(27) $\displaystyle K$ $\displaystyle=$ $\displaystyle
8\left(\frac{1}{\ell^{2}}-\frac{4mb^{n+1}(n+1)}{(r^{2}+b^{2})^{\frac{n+3}{2}}}\right)^{2}$
(28)
$\displaystyle+\frac{4}{\ell^{4}(r^{2}+b^{2})^{n+5}}\left[(r^{2}+b^{2})^{\frac{n+5}{2}}+4(n+1)\ell^{2}mb^{n+1}\left((n+2)r^{2}-b^{2}\right)\right]^{2}.$

As we can see, Ricci and Kretschmann do not diverge at the origin $r=0$ or any
other value of $r$ since we assume $b>0$. Thus, our solution (26) represents a
genuine class of singularity-free spacetime.

##  III Thermodynamics of the three dimensional regular black holes

We start with the study of Hawking temperature expressed by
$T_{H}=f^{\prime}(r_{h})/4\pi$. The form of the metric function $f(r)$ (26)
allows us to determine this quantity analytically in terms of the horizon
radius, and it is given by

$T_{H}=\frac{r_{h}}{2\pi\ell^{2}}\Bigg{\\{}1+\frac{(n+1)b^{n+1}\left(\ell^{2}+r_{h}^{2}\right)}{2\left(b^{2}+r_{h}^{2}\right)\left[b^{n+1}-\left(b^{2}+r_{h}^{2}\right)^{\frac{n+1}{2}}\right]}\Bigg{\\}}.$
(29)

For $b=0$, we retrieve the Hawking temperature of BTZ black hole.

We depict Eq. (29) in the left panel of Fig. 2 for some black hole solutions,
depending on $n$. Notice that the temperature is greater, the greater this
power. From this figure, we can also infer that for the regular black holes
under consideration, a critical horizon radius for which the temperature
vanishes exists. That last statement means that when the black hole reaches
this radius, it stops radiating, leaving a remnant behind. Note that the
greater the power in $n$, the smaller the critical radius.

Figure 2: Hawking temperature (left panel) and heat capacity (right panel) of
the regular black holes under consideration, as functions of the event horizon
radius for some powers, $n$, with $b=2.2$ and $\ell=0.7$ in Planck units.

For the regular black hole solution in which $n=1$, we can exactly calculate
this critical radius, which is given by

$r_{h}^{c}=\sqrt{b\ell},$ (30)

and for any $n$, when $b=\ell$ we get

$r_{h}^{c}=b\sqrt{4^{-\frac{1}{n+1}}(n+3)^{\frac{2}{n+1}}-1}.$ (31)

###  III.1 A new propose for the structure of the first law of thermodynamics
for regular black holes

As mentioned in the introduction, the standard version of the first law,
namely $dm=TdS-PdV$, leads to entropy and thermodynamics volume values that do
not coincide with the usual definitions (for example, the entropy does not
follow the area law in GR). So, to address this problem, we propose the
following structure of the first law of thermodynamics for regular black holes

$dU=dm+dX=T_{H}dS-PdV,$ (32)

where $dX$ is a quantity that modifies the internal energy regarding the BTZ
black hole, since we are working with a different class of regular three-
dimensional black holes.

It is worth mentioning that the equation (32) is locally defined at the
horizon, $r_{h}$. Thus, the term $dU$ can be understood as a local definition
of the variation of the internal energy at the horizon.

Following Ref. Kothawala:2007em , we identify the radial pressure of the
energy-momentum tensor as the thermodynamics pressure, which in our case is
given by $P(r)=-\rho(r)=-m(n+1)b^{(n+1)}/[2\pi(r^{2}+b^{2})^{(n+3)/2}]$, with
$V=\pi r_{h}^{2}$. If we express $m$ as a function of $r_{h}$ from Eq. (26),
and considering $S=\pi r_{h}/2$ (i.e., following area law Frodden:2012nu ), we
can analytically find the quantity $X$, after integration of the Eq. (32).
Hence,

$X=\frac{b^{n+1}\left(\ell^{2}+r_{h}^{2}\right)}{8\ell^{2}\left[b^{n+1}-\left(b^{2}+r_{h}^{2}\right)^{\frac{n+1}{2}}\right]},$
(33)

which corrects the internal energy of the regular black hole under
consideration. For $n=1$, it assumes the simplest form

$X=-\frac{b^{2}\left(\ell^{2}+r_{h}^{2}\right)}{8\ell^{2}r_{h}^{2}}.$ (34)

On adding $m$ and $X$ in order to obtain the internal energy, we find the
unexpected simple quantity

$U=\frac{1}{8}\left(1+\frac{r_{h}^{2}}{\ell^{2}}\right),$ (35)

for all $n$. Notice that the differential of this quantity is equal to the
static BTZ black hole up to a multiplicative factor. Therefore, the internal
energy is not identified with the quasi-localized mass of the regular black
holes under inspection. This is because, as was above mentioned, the term $dU$
corresponds to a local definition of the variation of the internal energy at
the horizon.

Identifying the cosmological constant as another component of the
thermodynamics pressure, i.e., $P_{\Lambda}=-\dfrac{\Lambda}{8\pi}$
Estrada:2020tbz , from equation (35) we get to

$U=P_{\Lambda}V+\frac{1}{8}\Rightarrow dU=P_{\Lambda}dV=dW_{\Lambda}$ (36)

where $dW_{\Lambda}$ can be understood as the work done by the thermodynamics
pressure associated with the cosmological constant. It is worth mentioning
that we are considering only variations along the horizon radius. In a future
work could be considered variations along the cosmological constant, i.e., in
the extended phase space. Thus, we can say that considering the variation of
the internal energy as the variation of the work done by the thermodynamics
pressure associated with the cosmological constant in the first law of
thermodynamics for regular black holes, the correct values of entropy and
thermodynamics volume are obtained.

###  III.2 Stability analysis

Since variations in volume and entropy are both functions of the horizon
radius, $r_{h}$ in the parameter space, they are not mutually independent.
Then, it is not possible to use the conventional definition of heat capacity
given by the equation $(dQ/dT)_{v\sim r_{+}=\mbox{fixed}}$ with $dQ=TdS$. This
is only possible in the extended phase space, which could be studied in a
future work. However, following Estrada:2019qsu , it is possible to define the
heat capacity in the form

$C=T_{H}\frac{dS}{dT_{H}}=T_{H}\frac{dS/dr_{h}}{dT_{H}/dr_{h}},$ (37)

in order to analyze the evolution of the solution.

For $n=1$, we find

$C=\frac{\pi
r\left(b^{2}+r^{2}\right)\left(r^{4}-b^{2}L^{2}\right)}{2\left[b^{4}L^{2}+3b^{2}r^{2}\left(L^{2}+r^{2}\right)+r^{6}\right]}.$
(38)

In the right panel of Fig. 2, we depict $C$ for some powers of $n$. Notice
that the system presents thermodynamical local stability in the intervals
where $r_{h}\geq r_{h}^{c}$, namely, until where the Hawking temperature
vanishes, since there $C\geq 0$.

Completing our thermodynamic analysis, we will calculate the Gibbs free
energy, $G$, via $G=U-T_{H}S+PV$, and investigate the possible phase
transitions. Thus, we get from Eqs. (29), (35), and $S=\pi r_{h}/2$, that the
Gibbs free energy is given by, for $n=1$

$G=\frac{2b^{2}\ell^{2}+\ell^{2}r_{h}^{2}-r^{4}}{8\ell^{2}\left(b^{2}+r_{h}^{2}\right)}.$
(39)

From Fig. 3, we can verify that the regular black holes realize phase
transitions of zero order (i.e., continuous) at a critical horizon radius,
$r^{pt}_{h}$, where the Gibbs free energy vanishes.

Such a quantity goes thus from a globally thermodynamic stable region ($G<0$)
to an unstable one ($G>0$) as it evaporates before reaching the critical
horizon $r_{h}^{c}$ for which the remnant arises.

For $n=1$, we can exactly determinate the point of this phase transition:

$r^{pt}_{h}=\sqrt{\frac{1}{2}\ell\left(\sqrt{8b^{2}+\ell^{2}}+\ell\right)},$
(40)

which is always larger than $r_{h}^{c}=\sqrt{b\ell}$. We can also notice that
the greater the free energy, the smaller is the power of $n$.

Figure 3: Gibbs free energy of the regular black holes, as a function of the
event horizon radius, for some powers of $n$, with $b=2.2$, and $\ell=0.7$ in
Planck units.

##  IV Conclusion

We have obtained a new class of regular black hole solutions in (2 +
1)-dimensions whose source is a quasi-localized mass-energy distribution.
These solutions depend on a characteristic distance, $b$, and an exponent
parameter, $n$, with no horizons if the mass is smaller or equal to $m=1/8$ or
with up to two horizons if $m>1/8$. The inner (outer) horizon is smaller
(greater), with the greater power of $n$. The associated energy density
satisfies the requirements described in Ref. Estrada:2020tbz , namely, a
positive profile, a single finite maximum at the origin, going to zero at
infinity, and integrable in this interval, in what is known as quasi-locality
of the mass-energy. Furthermore, it is satisfied a fourth requirement that we
have included to write the electric field and the Lagrangian $L(F)$ as
functions of the energy density. We show that, in general, such a source of
matter cannot be associated with nonlinear electrodynamics since it does not
lead to the Maxwell theory limit in the weak field limit for large $r$. In
addition, we have shown that the referred conditions of quasi-locality to the
energy density are incompatible with the expected electromagnetic asymptotic
limit for the electric field, namely, $E\sim 1/r$. Our view is that the quasi-
localized energy conditions are additional constraints on the matter sector.
Similar to the fact that in (2 + 1)-dimensional spacetimes, general relativity
does not reduce to Newtonian theory in the weak field limit. Ultimately, we
say that our energy-matter source cannot come from nonlinear electrodynamics.

In the sequel, we studied the Hawking temperature for the regular black holes
from the surface gravity on the event horizons. We found that such a
temperature is greater, the greater the power, $n$, and that there exists a
critical event horizon position, $r_{h}^{c}$, for which this temperature
vanishes and the evaporation process halts, giving rise to a classical remnant
mass.

Since the usual version of the first law of thermodynamics, $dm=TdS-PdV$, due
to the presence of the matter fields, leads to incorrect values of entropy and
thermodynamics volume for regular black holes, we have proposed a new version
of the first law for regular black holes. So, we have included a potential
$dX$ in the first law, which leads to a local definition of internal energy at
the horizon, $dU=dm+dX$. We found the internal energy associated with the BTZ
black hole, up to a constant, for all the found regular solutions.
Furthermore, by identifying the cosmological constant as another component of
the thermodynamics pressure, we could interpret the internal energy as the
variation of the work done by the thermodynamics pressure associated with the
cosmological constant $dU=dW_{\Lambda}$. Remarkably, our form proposed for the
first law allows us to obtain the correct values of entropy and thermodynamics
volume. Hence, we found the heat capacity. It is positive for the horizon
radius in which $r_{h}\geq r_{h}^{c}$, denoting thermodynamical local
stability.

Finally, we determined the Gibbs free energy of the regular black holes under
consideration, finding that a zero-order phase transition occurs at a critical
event horizon larger than $r_{h}^{c}$, which separates a region globally
stable from an unstable one, according to Fig. 3.

————————————————————————-

## Acknowledgments

The authors thank the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), Grant no 88887.621648/2021-00, and the Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), Grants no 311732/2021-6 (RVM)
and 308268/2021-6 (CRM), for financial support.

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