# Key-Value Transformer

Ali Borji
Quintic AI
San Francisco, CA
<EMAIL_ADDRESS>
###### Abstract

Transformers have emerged as the prevailing standard solution for various AI
tasks, including computer vision and natural language processing. The widely
adopted Query, Key, and Value formulation (QKV) has played a significant role
in this. Nevertheless, no research has examined the essentiality of these
three components for transformer performance. Therefore, we conducted an
evaluation of the key-value formulation (KV), which generates symmetric
attention maps, along with an asymmetric version that incorporates a 2D
positional encoding into the attention matrix. Remarkably, this transformer
requires fewer parameters and computation than the original one. Through
experiments encompassing three task types—synthetics (such as reversing or
sorting a list), vision (mnist or cifar classification), and NLP (character
generation and translation)—we discovered that the KV transformer occasionally
outperforms the QKV transformer. However, it also exhibits instances of
underperformance compared to QKV, making it challenging to draw a definitive
conclusion. Nonetheless, we consider the reported results to be encouraging
and anticipate that they may pave the way for more efficient transformers in
the future111Code is available at https://github.com/aliborji/kv-transformer.

##  1 Introduction

Transformers (Vaswani et al.  [4]) have gained significant attention in recent
times due to their effectiveness in various domains such as language, vision,
and reinforcement learning. The research community has witnessed a remarkable
surge in the number of proposed Transformer model variants. These "X-former"
models, including Reformer, Linformer, Performer, and Longformer, to name a
few, aim to enhance the original Transformer architecture with advancements in
computational and memory efficiency. For a detailed and comprehensive review
of this topic, please refer to the review on efficient Transformer models by
Tay et al.  [3].

To our current understanding, the majority of X-formers have primarily focused
on the Query, Key, and Value (QKV) formulation. However, it remains unclear
whether these three components are essential or if solely having the key-value
aspect is sufficient. Furthermore, the extent to which the performance may be
compromised by removing the query component is not well-established. It is
still uncertain whether the QKV formulation holds inherent theoretical
significance or if it is primarily a matter of parameters. In simpler terms,
is the inclusion of queries, keys, and values crucial to the success of these
models? In this study, our aim is to investigate this problem.

##  2 Key-Value Transformer

Transformer blocks are known for incorporating several key components,
including a multi-head self-attention mechanism, a position-wise feed-forward
network, layer normalization modules, and residual connections. Here, we
assume the reader has a basic understanding of the overall transformer
architecture and specifically direct our focus towards the multi-head
attention, which serves as a central component of the transformer’s design.

The self-attention mechanism, also known as intra-attention, is a crucial and
distinguishing feature of Transformer models. Its purpose is to establish
relationships between different positions within a sequence, enabling the
computation of a representation for that very sequence. This mechanism has
proven to be highly valuable in various tasks such as machine translation,
abstractive summarization, and image description generation.

The fundamental concept underlying self attention mechanism is for each
element within the sequence to learn the ability to gather information from
other tokens present in the same sequence. The operation for a single head can
be defined as follows:

$A_{h}=\text{Softmax}(\alpha Q_{h}K_{h}^{T})V_{h},$ (1)

where X is a matrix in $\mathbb{R}^{N\times d}$, $\alpha$ is a scaling factor
typically set to $1/\sqrt{d}$, $Q_{h}=XW_{q}$, $K_{h}=XW_{k}$, and
$V_{h}=XW_{v}$ are linear transformations applied on the temporal dimension of
the input sequence. $W_{q}$, $W_{k}$, $W_{v}$ $\in\mathbb{R}^{d\times d/H}$
are the weight matrices (parameters) responsible for the query, key, and value
projections, respectively. These matrices project the input X into an output
tensor of $d$ dimensions. $H$ represents the number of heads. Softmax is
applied row-wise.

The heads $A_{1}\cdots A_{H}$ are computed in parallel, and their outputs are
concatenated and passed through a dense layer. The attention matrix $A=QK^{T}$
is primarily responsible for learning alignment scores between tokens in the
sequence. Notably, this formulation involves taking the dot product between
each element/token in the query (Q) and the corresponding element/token in the
key (K).

Figure 1: Key-value transformer. The attention with positional encoding is
denoted as KV+Pos. In scenarios that require cross-attention, such as
translation, we continue to employ the QKV attention mechanism. Removing the Q
component is different from weight sharing between Q and K. In our
formulation, we completely eliminate Q’s parameters and computations.

In our formulation, we simply replace Q with K, resulting in:

$A_{h}=\text{Softmax}(\alpha K_{h}K_{h}^{T})V_{h},$ (2)

As depicted in Figure 1, our approach computes the attention matrix in a
manner that maintains symmetry. To introduce asymmetry, a 2D positional
encoding of dimension $m$ is added to the $n\times n$ attention matrix222added
to the broadcast attention matrix.. The resulting tensor, with dimensions
$n\times n\times m$, is then projected back into an $n\times n$ matrix using a
linear layer consisting of $m$ neurons. In our experiments, we refer to these
two variations as KV and KV+Pos attention, respectively.

| Computational Complexity | # Parameters
---|---|---
QKV | $2nd^{2}$ | $2d^{2}$
KV+Pos | $nd^{2}+n^{2}m$ | $d^{2}+m$
KV | $nd^{2}$ | $d^{2}$
Table 1: Comparison of the attention mechanisms in terms of computational
complexity and number of parameters. $d$ is the embedding dimensionality, $n$
is the sequence length, and $m$ is the dimensionality of the 2D positional
encoding layer (refered to as pos dim or positional dimension across the
paper).

Table 1 demonstrates the computational complexity of the two models. For the
KV (symmetric) attention, the number of parameters and FLOPS is half of that
of the QKV attention. On the other hand, the KV+Pos attention incurs higher
parameter and FLOPS counts, which are dependent on the parameter $m$. It is
worth noting that $m$ can be set to a much lower value than $d$. However, the
drawback of the KV+Pos attention is that its computational complexity is
dependent on $n^{2}$, which can be computationally expensive.

The choice of $m$, whether the addition of positional encoding is beneficial,
and whether a symmetric KV attention is sufficient, all depend on the specific
problem being addressed. Notice that our formulation provides a trade-off
between model complexity and model performance, which becomes particularly
important during inference time.

Below is the implementation in PyTorch:

att = torch.matmul(k, k.transpose(-2, -1)) att /= math.sqrt(self.head_dim)

if add_pos: pos = pos_embeddings_2d[:, :T, :T, :] att = att.unsqueeze(-1) +
pos.unsqueeze(0) att = self.map_pos(att).squeeze(-1)

if apply_mask: att = att.masked_fill(self.tril[:T, :T] == 0, float(’-inf’))
att = att.softmax(dim=-1)

It is important to mention that certain tasks, such as translation, may
necessitate the use of cross attention. In such cases, we retain the QKV
attention mechanism when needed but replace the self-attention with KV
attention. Self-attention refers to the scenario where the keys and values are
derived from the same source as the queries. On the other hand, in cross-
attention, the queries are still generated from the input sequence, but the
keys and values are obtained from an external source, such as an encoder
module.

##  3 Experiments and Results

We conduct empirical experiments on 13 tasks, exploring various variations of
transformers. All models are trained from scratch, except the set anomaly
detection. Our objective is not to achieve state-of-the-art performance, but
rather to compare the attention mechanisms employed.

Some of the tasks are sequence-to-sequence where the input and the output is a
sequence, not necessarily of the same length. Example tasks in this domain is
translation. Here a Transformer encoder is used for interpreting the input
sequence, and a decoder is used for generating the output in an autoregressive
manner. Some other tasks include image classification, list reversal, etc. In
these tasks only an encoder is used to map the input to a set of labels.

###  3.1 Synthetic tasks

We focus on five specific tasks outlined below. The input list, which has a
predetermined length, consists of numbers ranging from 0 to 9, inclusive of
both 0 and 9.

Reverse. In this task, a list of numbers is subjected to a reversal operation.
For instance, the input list [4, 3, 9, 8, 1] would be transformed into [1, 8,
9, 3, 4].

Sort. The objective of this task is to arrange the input list in ascending
order. For example, [4, 3, 9, 8, 1] would be transformed into [1, 3, 4, 8, 9].

Swap. In this scenario, the first half of the list is exchanged with the
second half. For instance, the list [4, 3, 9, 8, 1, 7] would be transformed
into [8, 1, 7, 4, 3, 9].

Sub. In this case, each element of the list is subtracted from 9. For example,
the array [4, 3, 9, 8, 1] would be transformed into [5, 6, 0, 1, 8].

Copy. In this task, the objective is to retain the input list as is. For
example, [4, 3, 9, 8, 1] remains unchanged as [4, 3, 9, 8, 1].

Figure 2: Loss over time for the synthetics tasks.

In training, we feed the input sequence into the Transformer encoder to
generate predictions for each token in the input. We utilize the standard
cross entropy loss for this purpose. Each number is encoded as a one-hot
vector. We apply a gradient clip value of 5 and set the dimensions pos dim to
10 (_i.e._ m). Additionally, we employ the Adam optimizer along with the
CosineWarmupScheduler, using a warm-up period of 5.

We conduct experiments with different configurations of the transformer models
by varying the embedding dimension (32, 64, 256), the number of layers (2, 4),
the number of heads (2, 4), a learning rate of 1e-3, and the input sequence
length (16, 64, 128). Each configuration is run three times for two epochs,
and the results are then averaged across the variants.

Figure 2 illustrates the loss curves for both training and validation sets
over time. The QKV attention mechanism exhibits faster convergence compared to
the KV attention mechanism. However, all attention mechanisms demonstrate good
performance on synthetic tasks, as indicated by the accuracies presented in
Table 2. Notably, the KV+Pos model outperforms other models in this scenario,
while the KV transformer model performs slightly better than the QKV
transformer model.

Table 2: The performance of attention mechanisms on synthetic tasks. Standard deviations across various architecture hyperparameters (such as model architecture, learning rate, sequence length, etc.) are indicated within parentheses. Multiple runs are conducted for each setting, and the results are averaged across these settings. | reverse | sort | swap | sub | copy | avg.
---|---|---|---|---|---|---
QKV | 0.698 (0.38) | 0.971 (0.033) | 1.0 (0.0) | 0.588 (0.42) | 1.0 (0.0) | 0.851
KV+Pos | 0.718 (0.35) | 0.963 (0.038) | 1.0 (0.0) | 0.671 (0.37) | 1.0 (0.0) | 0.87
KV | 0.705 (0.37) | 0.967 (0.035) | 1.0 (0.0) | 0.597 (0.41) | 1.0 (0.0) | 0.854

Figure 3 displays sample attention maps. It is worth noting that the attention
maps of the KV transformer exhibit symmetry around the line $y=x$. Notable
patterns can be observed within the attention maps. For instance, in the
reversing task, the QKV model has learned to attend to the token located at
the flipped index of itself. However, it also allocates some attention to
values near the flipped index. This behavior arises because the model does not
require precise, strict attention to solve this problem but rather benefits
from an approximate, noisy attention map.

Figure 3: Attention maps over synthetic tasks. Rows from top to bottom:
Reverse, Sort, Swap, Sub, and Copy. Columns from left to right: QKV, KV, and
KV+Pos.

###  3.2 Vision tasks

We consider classification over MNIST, FashionMNIST, CIFAR-10, and CIFAR-100
datasets, as well as finding an anomaly image in a set of images.

Classification. We explore various settings for the patch size (4, 7),
learning rate (1e-3, 1e-4), embedding dimension (64, 256, 512), number of
layers (2, 4), and number of heads (2, 4). For each configuration, we run two
experiments, with each experiment lasting k epochs. The value of k differs
depending on the dataset: 20 epochs for MNIST and FashionMNIST, 40 epochs for
CIFAR-10, and 50 epochs for CIFAR-100. We employ the cross entropy loss
function and utilize the Adam optimizer with the MultiStepLR scheduler for
optimization. In the case of 2D positional encoding, we set pos dim to 50.

The loss curves are depicted in Figure 4, while the corresponding accuracies
are presented in Table 3. It is evident that the KV+Pos attention mechanism
achieves superior performance compared to the other variants, with the KV
attention mechanism following closely behind, and the QKV attention mechanism
performing slightly lower in terms of accuracy.

Set Anomaly Detection. The CIFAR-100 dataset was utilized, consisting of 600
images distributed across 100 classes, each with a resolution of 32x32 pixels.
Our objective is to present the model with a set of 9 images from one class
and 1 image from a different class. The task is to identify the image that
belongs to the different class among the set. To extract high-level, low-
dimensional features from the images, we employ a pre-trained ResNet34 model
[2] trained on the ImageNet dataset [1]. To monitor the training progress and
determine when to stop, a validation set is created. In this scenario, we
divide the training set into 90% for training purposes and 10% for validation,
ensuring a balanced distribution across classes.

In our approach, we define an epoch as a sequence in which each image within
the dataset is considered as an “anomaly" exactly once. Therefore, the length
of the dataset is determined by the total number of images it contains.

When constructing the training set, we follow a two-step process. First, we
randomly sample a class that is different from the class of the image at the
corresponding index, denoted as “idx." Then, in the second step, we sample N-1
images from the newly selected class. This process ensures that the resulting
set consists of 10 images, with 9 images belonging to a single category and 1
image serving as the anomaly.

Here, we have a classification of the whole set. For the prediction to be
permutation-equivariant, we will output one logit for each image. Over these
logits, we apply a softmax and train the anomaly image to have the highest
score/probability. This is slightly different than a standard classification
layer as the softmax is applied over images, not over output classes in the
classical sense. However, if we swap two images in their position, we
effectively swap their position in the output softmax. Hence, the prediction
is equivariant with respect to the input.

Our transformer model consists of four layers, each equipped with four
attention heads. The hidden dimensionality of the model is set to 256, and we
incorporate a dropout rate of 0.1 throughout the model to facilitate
regularization.

To control the model’s learning rate, we utilize the CosineWarmupScheduler. We
configure the warm-up parameter (set to 100) to gradually initiate the model
training process.

In our experiments, we vary the embedding dimension, selecting from the
options of 256 and 512. Additionally, we explore different depths and numbers
of heads, choosing values of 2 and 4. We set the learning rate to 5e-4 for all
configurations. Each setting is executed twice for a total of 20 epochs, and
the results are subsequently averaged to obtain reliable performance
measurements.

Figure 4: Loss over time for the vision tasks. Table 3: The performance of attention mechanisms on vision tasks. | mnist | f-mnist | cifar10 | cifar100 | anomaly | avg.
---|---|---|---|---|---|---
QKV | 0.981 (0.002) | 0.887 (0.004) | 0.663 (0.019) | 0.363 (0.016) | 0.943 (0.003) | 0.767
KV+Pos | 0.982 (0.002) | 0.884 (0.006) | 0.662 (0.017) | 0.366 (0.012) | 0.965 (0.012) | 0.772
KV | 0.981 (0.002) | 0.885 (0.005) | 0.666 (0.021) | 0.369 (0.010) | 0.927 (0.01) | 0.766

As indicated in Table 3, the KV+Pos transformer exhibits comparable
performance to the QKV transformer across the MNIST, FashionMNIST, and
CIFAR-10 datasets. However, when it comes to the CIFAR-100 dataset and anomaly
tasks, the KV+Pos transformer surpasses the QKV transformer in terms of
accuracy, resulting in an overall higher average performance. The KV
transformer, while slightly behind these two variants, still performs at a
reasonably competitive level.

###  3.3 NLP tasks

Character generation. The objective in this task is to generate text by
predicting the next character333For more detailed information about the model
used for this task, please refer to
https://www.youtube.com/watch?v=kCc8FmEb1nY&t=6064s&ab_channel=AndrejKarpathy..
We utilize the tiny Shakespeare
dataset444https://raw.githubusercontent.com/karpathy/char-
rnn/master/data/tinyshakespeare/input.txt, which consists of 1,115,394
characters.

For the training process, we allocate 90% of the data for training purposes,
while the remaining portion is used for validation. During each training step,
we track the loss across the entire validation set. Several hyperparameters
are considered, including the context size (8, 32, 64), maximum iteration set
to 1000, learning rate of 5e-4, embedding dimension chosen from 64 and 192,
number of heads selected from 1 and 4, number of layers chosen from 1, 2, 3,
and 6, dropout rate set to 0.2, and positional dimension set to 20. The
optimization process involves using the Adam optimizer along with cross
entropy loss. In total, four models are trained. The results are illustrated
in Fig. 5. It is evident that the KV+Pos and KV transformers perform
similarly, while the QKV transformer exhibits faster convergence.

Figure 5: Loss over time for the character generation task.

Number generation. In this task, our objective is to generate the next token
in a dataset that comprises written numbers. The dataset includes consecutive
numbers from 1 to 9999, spelled out as words. The task involves predicting the
subsequent token given the preceding $l$ tokens. For instance, if the sequence
length is $l=3$, we would have examples like:

([’thirteen’, ’.’, ’fourteen’], ’.’)
([’.’, ’fifteen’, ’.’], ’sixteen’)
$\cdots$

The vocabulary size for this dataset is 30, and there are a total of 63,095
tokens.

To train transformers, we vary the sequence length $l$ from 16, 32, 64, to
128. Additionally, we adjust the learning rate in the range of 0.001 to
0.0001. Other hyperparameters include a positional dimension of 10, embedding
dimension of 64, four layers, and eight heads. Each variation is trained five
times for 15 epochs, utilizing the cross entropy loss function.

The results are depicted in Figure 6. It can be observed that the KV+Pos
variant performs closely to the QKV variant, while the KV variant exhibits
significantly lower performance.

Figure 6: Epoch-wise loss in the number generation task.

Translation. The objective of this task is to train a transformer model from
scratch to perform sentence translation between two languages. Specifically,
we utilize the Multi30k dataset to train a German to English translation model
as well as an English to German translation model.

In contrast to using a one-hot target distribution, we adopt a different
approach. We set the probability of the target word to a predetermined
“confidence value" (typically 0.9) and allocate the remaining “smoothing
value" mass (usually 0.1) evenly across the rest of the vocabulary. This
technique, known as label smoothing, aims to provide a more robust training
signal.

To optimize the model, we employ the KL divergence loss and the Adam
optimizer. The learning rate follows a linear ramp-up for a specified number
of warm-up steps (usually 4000) and then decays according to the inverse
square root law based on the current training step number. During the
translation process, we utilize a greedy decoding strategy, starting with a
designated start token.

For the model configuration, we set the positional dimension to 10 and apply a
dropout rate of 0.1. We explore different variations by varying the number of
layers (1 or 2), the number of heads (1 or 4), and the embedding dimension
(64, 128, or 256). Each variant is trained twice for 20 epochs.

The results for both German to English and English to German translation are
depicted in Figure 7. Interestingly, incorporating 2D positional encoding
negatively affects the model’s performance. However, the KV transformer
demonstrates competitive performance compared to the QKV transformer in this
task.

Figure 7: Loss and Bleu per epoch in the translation task. Bleu is computed
over the test set. Top: German to English, Bottom: English to German.

##  4 Conclusion

While transformers have brought significant advancements to various domains in
AI, our comprehension of self-attention remains somewhat limited. In light of
this, we have posed a fundamental question regarding the indispensability of
the QKV formulation and endeavored to address it through empirical means. To
achieve this, we have put forth two alternative models known as KV and KV+Pos.
It is important to note that removing the Q component is distinct from weight
sharing between Q and K. In our proposed formulation, we have completely
eliminated the parameters and computations associated with the Q component.

We conducted an evaluation comparing the performance of KV attention, both
with and without 2D positional encoding, to the commonly used QKV attention
across 13 different tasks. Our findings indicate that in certain cases, such
as the synthetic and vision datasets, KV attention outperforms QKV attention.
It is worth noting that there is a trade-off involved: KV attention (without
positional encoding) achieves lower accuracy but requires fewer parameters
compared to QKV attention.

Overall, our results highlight the potential benefits of exploring symmetric
attention mechanisms. However, it remains unclear how and when symmetry can
enhance or hinder model accuracy. To foster further research in this area, we
have made the code we used to train and evaluate our models available at:
https://github.com/aliborji/kv-transformer.

## References

  * [1] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
  * [2] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
  * [3] Yi Tay, Mostafa Dehghani, Dara Bahri, and Donald Metzler.  Efficient transformers: A survey.  ACM Computing Surveys, 55(6):1–28, 2022.
  * [4] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  Advances in neural information processing systems, 30, 2017.


# Black Holes with Scalar Hair in Three Dimensions

Thanasis Karakasis<EMAIL_ADDRESS>Physics Department,
National Technical University of Athens, 15780 Zografou Campus, Athens, Greece
George Koutsoumbas<EMAIL_ADDRESS>Physics Department, National
Technical University of Athens, 15780 Zografou Campus, Athens, Greece
Eleftherios Papantonopoulos<EMAIL_ADDRESS>Physics Department,
National Technical University of Athens, 15780 Zografou Campus, Athens, Greece

###### Abstract

Three-dimensional static and spinning black hole solutions of the Einstein-
Klein-Gordon system are obtained for a particular scalar field configuration.
At large distances, and for small scalar field, the solutions reduce to the
BTZ black hole. The scalar field dresses the black hole with secondary scalar
hair, since the scalar charge is related to the conserved black hole mass. A
self interacting potential is included, containing a mass term that is above
the Breitenlohner-Freedman bound in three dimensions. Independence of the
scalar potential from the conserved black hole charges, imposes fixed mass and
angular momentum to scalar charge ratios. The thermodynamic properties as well
as the energy conditions of the black hole are analysed.

###### Contents

  1. I Introduction
  2. II Static Black Hole Solutions
    1. II.1 Energy Conditions
    2. II.2 Thermodynamics
  3. III Rotating Black Hole Solutions
  4. IV Conclusions
  5. V Acknowledgements

##  I Introduction

Gravity in three dimensions has been studied extensively and it is interesting
in many ways. One of the main advantages of working in three dimensions is
that space-time has simple enough topology and it can be characterized
completely and explicitly, giving important insights into black hole physics
and the structure of quantum gravity and it also can provide information on
how the dimensionality of space-time affects the resulting physics. Einstein’s
theory of gravitation, General Relativity (GR) in four or higher dimensions
and in the absence of matter, admits Schwarzchild black holes, while in three
dimensions the Weyl tensor, which describes the distortion of the shape of a
body in the presence of the gravitational force, vanishes by definition while
the Ricci tensor, describing how this force changes the volume of the body,
vanishes in the absence of matter. Therefore, since $Riemann=Weyl+Ricci$ we
can only have flat space-time. Hence the theory can only admit a conical
singularity as a solution of the field equations [1]. As a result, a negative
cosmological constant was included to make the Ricci tensor non-zero and the
BTZ black hole has been found [2, 3] by Bañados, Teitelboim, and Zanelli. The
BTZ black hole possesses two black hole horizons, it can arise from collapsing
matter [4], and it has well posed thermodynamic properties: it quantizes, for
example, the entropy is given by a Bekenstein-Hawking area law, where the area
is replaced by the circumference of the black hole. The $fundamental$ length
scale of the three-dimensional black hole is the AdS radius, since the mass
term turns out to be a dimensionless constant.

After the discovery of the BTZ black hole, scalar fields minimally and non-
minimally coupled to gravity were introduced as matter fields. In [5, 6]
three-dimensional black holes with a conformally coupled scalar field, being
regular everywhere, were discussed. The entropy in these cases is modified due
to the non-minimal coupling between matter and gravity. After these first
results other hairy black holes in three dimensions were discussed [7, 8, 9,
10, 11, 12]. In [13] three-dimensional gravity with negative cosmological
constant in the presence of a scalar field and an Abelian gauge field was
introduced. Both fields are conformally coupled to gravity, the scalar field
through a non-minimal coupling with the curvature and the gauge field by means
of a Lagrangian given by a power of the Maxwell field. A sixth-power self-
interaction potential, which does not spoil conformal invariance is also
included in the action, resulting to a very simple relation between the scalar
curvature and the cosmological constant. The cases where the entropy is
positive were pointed out, the authors arguing that the appearance of non-
physical negative entropy in the Jordan frame results in a naked singularity
in the Einstein frame.

$(2+1)$-dimensional charged black holes with scalar hair where derived, where
the scalar potential is not fixed ad hoc, but derived from Einstein’s
equations in [14] and [15]. In [16] exact three-dimensional black holes with a
non-minimal scalar field were discussed, the coupling parameter being left
arbitrary. In [18, 17] the motion and trajectories of photons and the quasi-
normal modes of three-dimensional rotating Hořava-AdS black hole were
calculated. In [19, 20, 21], static black holes in three-dimensional dilaton
gravity and modifications of the BTZ black hole by a dilaton scalar were
investigated, the dilaton case attracting more attention over the years [22,
23, 24, 25, 26]. Three-dimensional black holes in modified theories of gravity
have also been studied. $f(R)$ gravity black holes are found in [27, 28, 29,
30], while BTZ-like black holes in the Einstein Gauss Bonnet theory have been
recently obtained [31, 32, 33, 34, 35], while scalar fields in three-
dimensional modified gravity have also been investigated in the Horndeski
theory [36] also in a general scalar tensor theory as well [37]. Finally,
scalar fields have been added to three-dimensional $f(R)$ gravity and the
resulting black holes were discussed [38, 39].

As we already discussed, a scale has to be introduced with the presence of a
cosmological constant, to generate the BTZ black hole in $(2+1)$-dimensions.
In this work we will follow another approach. We will introduce a scalar
field, parametrizing a matter distribution, which back-reacts on a
$(2+1)$-dimensional metric. The scalar field is characterized by a parameter
called scalar charge. When the scalar field back - reacts to the space-time
metric, this scalar charge appears in the metric introducing an effective
cosmological constant. Then a hairy black hole is generated which at small
distances because matter is strongly back - reacting to the metric a new BTZ-
like black hole is generated, while at large distances because the back -
reaction is weak, the BTZ black hole is found. We discuss the thermodynamic
properties and the energy conditions of the hairy BTZ-like black hole solution
as well as the rotating case.

This work is organized as follows. In section II we derive static solutions of
the Einstein-Klein-Gordon system and discuss the geometric properties of the
new black hole solutions, as well as the energy conditions and the
thermodynamic properties of these solutions. In section III we derive novel
rotating solutions that reduce to the ones obtained in section II, once the
angular momentum of the black hole vanishes and discuss their thermodynamics.
We discuss the resultant scalar potentials, which from a field theory point-
of-view have to be independent of the black hole parameters, a fact implying
that the black holes produced have a fixed mass to angular momentum ratio.
Finally, in IV we point out the prospects of our work.

##  II Static Black Hole Solutions

We consider a simple model of a scalar field minimally coupled to gravity in
three dimensions in the action

$S=\frac{1}{8\pi}\int
d^{3}x\sqrt{-g}\left\\{\frac{R}{2}-\frac{1}{2}\partial^{\alpha}\phi\partial_{\alpha}\phi-V(\phi)\right\\}~{},$
(1)

which consists of the Ricci scalar and a self interacting scalar field
minimally coupled to gravity. By variation of this action we obtain the field
equations

$\displaystyle G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=T_{\mu\nu}~{},$ (2)
$\displaystyle\Box\phi-\frac{\partial V}{\partial\phi}=0~{},$ (3)
$\displaystyle
T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\partial^{\alpha}\phi\partial_{\alpha}\phi-
g_{\mu\nu}V(\phi)~{}.$ (4)

To solve the field equations we assume that the scalar field has a coulomb-
like form as in the four dimensional theory [40], with the solution first
appearing in [41]. We find that the scalar field dresses the black hole with
secondary hair, with the scalar charge appearing in the conserved black hole
mass. The Null Energy Condition is violated inside the event horizon, a
feature also appearing in the four-dimensional sibling [40], and we point out
that this is a global feature in any space-time having a vanishing
$1/g_{rr}(r_{+})=0$ at the event horizon of the black hole, where only radial
dependence on the scalar field is assumed. At large distances, the solution
reduces to the BTZ black hole, while the scalar potential admits an even power
law expansion, with the mass term being above the Breitenlohner-Freedman
bound.

We consider a $(2+1)$-dimensional metric ansatz of the form

$ds^{2}=-h(r)dt^{2}+\frac{1}{b(r)}dr^{2}+r^{2}d\theta^{2}~{},$ (5)

where $h(r),b(r)$ are the two unknown metric functions to be found solving the
field equations. Since the metric only depends on $r,$ we will consider that
$\phi$ is only $r$-dependent, hence $\phi=\phi(r),V=V(r)$. The
$tt,rr,\theta\theta$ components of the Einstein field equation and the Klein-
Gordon equation read

$\displaystyle b^{\prime}(r)+rb(r)\phi^{\prime}(r)^{2}+2rV(r)=0~{},$ (6)
$\displaystyle
b(r)\left(\frac{h^{\prime}(r)}{rh(r)}-\phi^{\prime}(r)^{2}\right)+2V(r)=0~{},$
(7)
$\displaystyle-b(r)h^{\prime}(r)^{2}+2h(r)^{2}\left(b(r)\phi^{\prime}(r)^{2}+2V(r)\right)+h(r)\left(b^{\prime}(r)h^{\prime}(r)+2b(r)h^{\prime\prime}(r)\right)=0~{},$
(8)
$\displaystyle\frac{1}{2}b^{\prime}(r)\phi^{\prime}(r)+b(r)\left(\left(\frac{h^{\prime}(r)}{2h(r)}+\frac{1}{r}\right)\phi^{\prime}(r)+\phi^{\prime\prime}(r)\right)-\frac{V^{\prime}(r)}{\phi^{\prime}(r)}=0~{}.$
(9)

Using the Bianchi identity one can prove that (9) can be obtained from the
Einstein equations. Hence, we have a system of with three independent
equations in four unknown functions. As a result one of the unknowns has to be
fixed $ad~{}hoc$. However, if one fixes the scalar potential from the begining
in (1), then the system can be, at least in principle, integrated, since one
will have three unknown functions and three equations. Hence, we fix the form
of the scalar field as

$\phi(r)=\frac{A}{r}~{},$ (10)

where $A$ is a scalar length scale that controls the behaviour of the scalar
field, which we will call scalar charge. Note that the scalar charge has been
defined as the term controlling the $\mathcal{O}(r^{-1})$ term of the scalar
field at infinity in four dimensions and we will use the same terminology in
our work. We have checked that using $\phi(r)=A/r^{n}$ with $n>0,$ exact
results can be obtained for different $n,$ however for $n=1$ we can find
simple exact solutions.

There is a pole in the scalar field function for $r\to 0$, however this will
not be a problem because of the presence of a curvature singularity at $r=0$
as we will discuss, thence the solution will be valid for $r>0$. Then, we find
that

$\displaystyle
b(r)=\frac{c_{1}r^{2}}{A^{2}c_{3}^{2}}e^{\frac{A^{2}}{2r^{2}}}\left(A^{2}c_{2}e^{\frac{A^{2}}{2r^{2}}}+c_{3}\right)~{},$
(11) $\displaystyle
h(r)=r^{2}\left(\frac{c_{3}}{A^{2}}e^{-\frac{A^{2}}{2r^{2}}}+c_{2}\right)~{},$
(12) $\displaystyle
V(r)=\frac{c_{1}c_{2}}{2c_{3}^{2}r^{2}}e^{\frac{A^{2}}{r^{2}}}\left(A^{2}-2r^{2}\right)-\frac{c_{1}}{A^{2}c_{3}}e^{\frac{A^{2}}{2r^{2}}}~{},$
(13)

where $c_{1},c_{2},c_{3}$ are constants of integration to be determined from
the boundary conditions. Note here that at large distances, the solution
asymptotes to

$\displaystyle
b(r\to\infty)\sim\frac{c_{1}r^{2}}{c_{3}^{2}}\left(\frac{c_{3}}{A^{2}}+c_{2}\right)+\frac{c_{1}\left(2A^{2}c_{2}+c_{3}\right)}{2c_{3}^{2}}+\frac{A^{2}c_{1}\left(4A^{2}c_{2}+c_{3}\right)}{8c_{3}^{2}r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{},$
(14) $\displaystyle
h(r\to\infty)\sim+\frac{A^{2}c_{3}}{8r^{2}}+\frac{c_{3}r^{2}}{A^{2}}+c_{2}r^{2}-\frac{c_{3}}{2}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{},$
(15) $\displaystyle
V(r\to\infty)\sim-\frac{c_{1}}{c_{3}^{2}}\left(\frac{c_{3}}{A^{2}}+c_{2}\right)-\frac{A^{2}c_{1}c_{2}+c_{1}c_{3}}{2c_{3}^{2}r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{}.$
(16)

From the above relations we can see that we have some $\mathcal{O}(r^{2})$
terms that survive at large distances which we can identify as cosmological
constant terms, that are generated by the vacuum of the scalar field theory
and depend on $A$. For
$c_{1}=A^{4}(c_{2}+\Lambda)^{2},c_{3}=-A^{2}(c_{2}+\Lambda)$, where $\Lambda$
is an effective cosmological constant, we rewrite the asymptotic expressions
of the metric functions at large distances

$\displaystyle b(r\to\infty)\sim-\Lambda
r^{2}+\frac{1}{2}A^{2}(c_{2}-\Lambda)+\frac{A^{4}(3c_{2}-\Lambda)}{8r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{},$
(17) $\displaystyle h(r\to\infty)\sim-\Lambda
r^{2}+\frac{1}{2}A^{2}(c_{2}+\Lambda)-\frac{A^{4}(c_{2}+\Lambda)}{8r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{}.$
(18)

To compute the mass of the black hole, we will use the quasi-local method
[20]. The quasi-local energy at a finite distance $r_{0}$ is defined as

$E(r_{0})=2\left(\sqrt{b_{0}(r_{0})}-\sqrt{b(r_{0})}\right)~{},$ (19)

where $b_{0}$ determines the zero of the energy (which we take to be the pure
AdS space-time $b_{0}(r)=-\Lambda r^{2}$) and the quasi-local mass at $r_{0}$
can be obtained as

$m(r_{0})=\sqrt{h(r_{0})}E(r_{0})~{}.$ (20)

Now, the ADM mass of the black hole can be read off by taking the limit at
$r_{0}\to\infty$

$\mathcal{M}=-\frac{1}{2}A^{2}\left(c_{2}-\Lambda\right)~{}.$ (21)

Setting $c_{2}=\cfrac{A^{2}\Lambda-2\mathcal{M}}{A^{2}}$ we can rewrite the
solution in terms of the black hole mass $\mathcal{M}$, the scalar charge $A$
and the effective cosmological constant $\Lambda$, which are the parameters of
our solution

$\displaystyle
h(r)=r^{2}\left(\frac{A^{2}\Lambda-2\mathcal{M}}{A^{2}}-e^{-\frac{A^{2}}{2r^{2}}}\left(\frac{A^{2}\Lambda-2\mathcal{M}}{A^{2}}+\Lambda\right)\right),$
(22) $\displaystyle
b(r)=\frac{2r^{2}e^{\frac{A^{2}}{2r^{2}}}\left(\mathcal{M}-A^{2}\Lambda\right)-r^{2}e^{\frac{A^{2}}{r^{2}}}\left(2\mathcal{M}-A^{2}\Lambda\right)}{A^{2}}~{},$
(23) $\displaystyle
V(r)=\frac{e^{\frac{A^{2}}{r^{2}}}\left(A^{2}-2r^{2}\right)\left(A^{2}\Lambda-2\mathcal{M}\right)}{2A^{2}r^{2}}+\frac{2e^{\frac{A^{2}}{2r^{2}}}\left(A^{2}\Lambda-\mathcal{M}\right)}{A^{2}},$
(24) $\displaystyle
V(\phi)=\frac{2e^{\frac{\phi^{2}}{2}}\left(A^{2}\Lambda-\mathcal{M}\right)}{A^{2}}+\frac{e^{\phi^{2}}\left(\phi^{2}-2\right)\left(A^{2}\Lambda-2\mathcal{M}\right)}{2A^{2}}~{}.$
(25)

Their asymptotic expressions now read

$\displaystyle h(r\to\infty)\sim-\Lambda
r^{2}+\left(A^{2}\Lambda-\mathcal{M}\right)-\frac{A^{2}\left(A^{2}\Lambda-\mathcal{M}\right)}{4r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{},$
(26) $\displaystyle b(r\to\infty)\sim-\Lambda
r^{2}-\mathcal{M}+\frac{A^{4}\Lambda-3A^{2}\mathcal{M}}{4r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{},$
(27) $\displaystyle
V(r\to\infty)\sim\Lambda+\frac{A^{2}\Lambda}{2r^{2}}+\frac{A^{2}\left(A^{2}\Lambda-\mathcal{M}\right)}{4r^{4}}+\frac{3A^{6}\Lambda-5A^{4}\mathcal{M}}{24r^{6}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{8}\right)~{},$
(28) $\displaystyle V(\phi\to
0)\sim\Lambda+\frac{\Lambda\phi^{2}}{2}+\phi^{4}\left(\frac{\Lambda}{4}-\frac{\mathcal{M}}{4A^{2}}\right)+\phi^{6}\left(\frac{\Lambda}{8}-\frac{5\mathcal{M}}{3A^{2}}\right)+\mathcal{O}\left(\phi^{8}\right)~{}.$
(29)

We can see that at large distances, the solution resembles the BTZ black hole,
while corrections in the structure of space-time appear as
$\mathcal{O}\left(\left(\cfrac{1}{r}\right)^{s}\right)$ terms, where $s\geq 2$
which are supported by the existence of the scalar field. Moreover, the scalar
field dresses the black hole with secondary scalar hair, since the conserved
mass is given by the scalar charge in addition to an integration constant. The
potential has a mass term given by $m^{2}=V^{\prime\prime}(\phi=0)=\Lambda,$
which is above the Breitenlohner-Freedman bound in three dimensions [42, 43]
and for small $\phi$ (large $r$), the potential admits an even power series
expansion. It is also invariant under the substitution $\phi\to-\phi$. The
potential (25) contains both the mass and the scalar charge of the black hole
space-time. However this should not happen. As a result we have to find a way
to render the potential independent of the black hole mass and scalar charge.
By inspection we can see that we can define the conserved mass to scalar
charge ratio $q$ as

$q=\frac{\mathcal{M}}{A^{2}}~{},$ (30)

and now the potential will be

$\displaystyle
V(\phi)=e^{\frac{\phi^{2}}{2}}(2\Lambda-2q)+e^{\phi^{2}}\left(\frac{\Lambda\phi^{2}}{2}-\Lambda-q\phi^{2}+2q\right)~{},$
(31) $\displaystyle V(\phi\to
0)\sim\Lambda+\frac{\Lambda\phi^{2}}{2}+\frac{\Lambda-q}{4}\phi^{4}+\frac{3\Lambda-5q}{24}\phi^{6}+\mathcal{O}\left(\phi^{8}\right)$
(32)

where now $q$ is the parameter of our theory. Now the potential is general
enough and the theory (1) can yield black holes with different masses and
scalar charges. The mass of the resulting compact object might reduce through
Hawking evaporation for example, but a varying mass implies a varying scalar
charge, so that their ratio $q$ is constant. Therefore, from a field theory
point-of-view we can argue that the scalar charge $A$ is kind of a
thermodynamic variable, since it has to vary when $\mathcal{M}$ is changing.
In Fig. 1 we plot the scalar potential as a function of $r$ and $\phi$, where
we can see that the potential is always negative in order to support the hairy
structure and to violate the no-hair theorem. Moreover from the plot of
$V(\phi)$ it is clear that the theory contains a global maximum located at
$V_{\text{max}}(\phi)=\Lambda$. We have also checked that the on-shell action
is constant at large distances. The $1/g_{rr}$ component has two roots given
by

$r_{\pm}=\pm
A\left(\ln\left(\frac{4\left(A^{2}\Lambda-\mathcal{M}\right)^{2}}{\left(A^{2}\Lambda-2\mathcal{M}\right)^{2}}\right)\right)^{-1/2}~{}.$
(33)

We work on the solution in AdS space-time, the horizon being $r_{+}$. From now
on, we will set $\Lambda=-\ell^{-2}$, where $\ell$ denotes the AdS radius. In
addition, note that there always exist a horizon when $\mathcal{M}$ and $A$
are positive and the scalar field does not imply any bound for the existence
of a horizon. In the limit of small scalar charge $A$, we obtain, at zero
order, the BTZ black hole

$\displaystyle h(r)\sim
r^{2}\left(\frac{1}{\ell^{2}}-\frac{\mathcal{M}}{r^{2}}\right)+A^{2}r^{2}\left(\frac{\mathcal{M}}{4r^{4}}-\frac{1}{r^{2}\ell^{2}}\right)+\mathcal{O}\left(A^{4}\right)~{},$
(34) $\displaystyle b(r)\sim
r^{2}\left(\frac{1}{\ell^{2}}-\frac{\mathcal{M}}{r^{2}}\right)-\frac{3A^{2}\mathcal{M}}{4r^{2}}+\mathcal{O}\left(A^{4}\right)~{},$
(35) $\displaystyle
V(r)\sim-\frac{1}{\ell^{2}}-\frac{A^{2}\left(2r^{2}+\ell^{2}\mathcal{M}\right)}{4\left(r^{4}\ell^{2}\right)}-\frac{A^{4}\left(3r^{2}+20\ell^{2}\mathcal{M}\right)}{12\left(r^{6}\ell^{2}\right)}+\mathcal{O}\left(A^{6}\right)~{}.$
(36)

This is related to the asymptotic nature of the scalar field. In our case, the
scalar field decays fast enough ($\mathcal{O}(r^{-1})$) and hence its impact
on the conserved mass is mild, in the sense that the mass is not completely
determined by the scalar hair parameter $A$. As a result, sending $\phi$ to
zero we can obtain the massive BTZ black hole. We note that this is not the
case when $\phi$ falls like $\mathcal{O}(r^{-1/2})$ [5, 6, 7, 9, 12, 13, 14]
where the mass is given explicitly in terms of the scalar hair parameter. In
our case, the mass (21) depends on the scalar hair $A$ but also on an
independent integration constant $c_{2}$. Therefore, the integration constant
can take any particular value and we can always have a massive black hole
solution when $A$ approaches zero and the scalar field vanishes. Recently
another solution appeared [44], where the scalar field falls faster at
infinity ($\mathcal{O}(r^{-1})$ as in our case) and the no-hair limit in this
case is also well-defined and the BTZ black hole is obtained in the limit of
vanishing scalar field.

Figure 1: The potential $V(r)$ and $V(\phi)$ for $\mathcal{M}=-\Lambda=1$,
while changing the scalar charge $A$.

Figure 2: Left: $h(r)$ versus $r$ for $\mathcal{M}=\ell=1$, while changing the
scalar charge $A$. Right: $r_{+}$ as a function of $A$, while changing the
mass of the black hole for $\ell=1$.

In Fig. 2 we plot $h(r)$ as a function of $r$ for different scalar charges and
$r_{+}$ as a function of the scalar charge $A$ and we can see that as $A$
grows, the bigger the event horizon radius becomes. We should also note that
the horizon $r_{+}$ is also a root of $h(r)$ even though $h(r)\neq b(r)$ (in
fact $h(r)/b(r)=e^{-\frac{A^{2}}{r^{2}}}$), therefore, the black hole has the
same causal structure as the static BTZ black hole, where inside the horizon
we still have one time and two position coordinates. There exists a
singularity at the origin as can be seen by calculating the Kretschmann
scalar, which is plotted in Fig. 3. Its expression is complicated, but by
checking the limits we can see that it is divergent at the origin, while
regular for any other $r>0$ and at large distances is related to the
cosmological constant

$R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}(r\to\infty)\sim\frac{12}{\ell^{4}}+\frac{8A^{2}}{r^{2}\ell^{4}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)~{}.$
(37)

Figure 3: The Kretschmann scalar
$R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}(r)$ for
$\mathcal{M}=-\Lambda=1/\ell^{2}=1$ while changing $A$.

###  II.1 Energy Conditions

In this subsection we will discuss the energy conditions of the obtained
space-time. For this reason we rewrite the Einstein field equation as

$G_{\mu}^{~{}\nu}=T_{\mu}^{~{}\nu}~{}.$ (38)

In this frame of reference, we can identify
$T_{t}^{~{}t}=-\rho,~{}T_{r}^{~{}r}=p_{r},~{}T_{\theta}^{~{}\theta}=p_{t}=-\rho$
being the energy density, the radial pressure and the transverse pressure
respectively. The energy conditions are obtained from these expressions. The
weak energy condition (WEC) states that given a time-like vector field
$t^{a}$, the quantity $T_{ab}t^{a}t^{b}$ is positive, $T_{ab}t^{a}t^{b}\geq
0\to\rho>0$. The null energy condition (NEC) states that $T_{ab}l^{a}l^{b}\geq
0\to\rho+p_{r}>0$, where $l^{a}l_{a}=0$, so that the geometry will have a
focussing effect on null geodesics. The explicit expressions read

$\displaystyle\rho=\mathcal{T}+V=b(r)\phi^{\prime
2}/2+V(r)=\frac{e^{\frac{A^{2}}{2r^{2}}}\left(A^{2}-2r^{2}\right)\left(A^{2}+\ell^{2}\mathcal{M}\right)-e^{\frac{A^{2}}{r^{2}}}(A-r)(A+r)\left(A^{2}+2\ell^{2}\mathcal{M}\right)}{A^{2}r^{2}\ell^{2}}~{},$
(39) $\displaystyle p_{r}=\mathcal{T}-V=b(r)\phi^{\prime
2}/2-V(r)=\frac{e^{\frac{A^{2}}{2r^{2}}}\left(A^{2}+2r^{2}\right)\left(A^{2}+\ell^{2}\mathcal{M}\right)-r^{2}e^{\frac{A^{2}}{r^{2}}}\left(A^{2}+2\ell^{2}\mathcal{M}\right)}{A^{2}r^{2}\ell^{2}}~{},$
(40)
$\displaystyle\rho+p_{r}=2\mathcal{T}=b(r)\phi^{\prime}(r)^{2}=\frac{2e^{\frac{A^{2}}{2r^{2}}}\left(A^{2}+\ell^{2}\mathcal{M}\right)-e^{\frac{A^{2}}{r^{2}}}\left(A^{2}+2\ell^{2}\mathcal{M}\right)}{r^{2}\ell^{2}}~{},$
(41)

where $\mathcal{T}=b(r)\phi^{\prime 2}/2$ is the kinetic energy of the scalar
field. The energy density $\rho$ is negative inside and on the black hole
horizon. Inside the horizon, $b(r)$ is negative, $\phi^{\prime 2}$ is always
positive and the potential is negative everywhere as we can see from Fig. 1,
resulting in negative energy density inside the black hole. On the horizon, we
have $b(r_{+})=0$, hence the contribution from the kinetic energy of the
scalar field vanishes and the potential makes the energy density negative.
Outside of the black hole horizon $b(r)$ is positive, but the effect of the
potential energy is stronger than the kinetic energy, resulting in negative
energy density everywhere, as we can see in Fig. 4. At large distances, the
kinetic energy asymptotes as

$\mathcal{T}(r\to\infty)\sim\frac{A^{2}}{2r^{2}\ell^{2}}-\frac{A^{2}\mathcal{M}}{2r^{4}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{6}\right)~{},$
(42)

while the expression for the potential energy in the limit $r\to+\infty$ is
given in (28). The leading order term in the kinetic energy is positive and
since all constants $A,\ell,\mathcal{M}$ are finite constants, the kinetic
energy is positive for large $r$. We can see in (28) that the potential will
cancel this positive contribution of the kinetic energy, therefore the sum
$\rho=\mathcal{T}+V$ will be always negative. It is known that too negative a
potential might threaten the WEC of a regular scalar field. In our case the
potential is the quantity that violates WEC. For the NEC, we can see that at
the event horizon $r_{+}$, we have $\rho+p_{r_{+}}=0$ due to the fact that
$b(r_{+})=0$. Outside and on the horizon the NEC is satisfied, while inside
the event horizon the NEC is violated. This is a common feature of black hole
space-times that arise from an action that consists of the Ricci scalar of
Einstein’s gravity and a simple non-minimally self interacting scalar field in
arbitrary dimensions that satisfies $g^{rr}(r_{+})=0,\phi=\phi(r)$, and not a
peculiar case of our model. This behaviour is indeed present in the four-
dimensional case [40, 45]. In Fig. 4 we plot the energy density (WEC) and the
sum of the energy density and radial pressure (NEC) of our black hole in order
to illustrate the discussion above. In FIG. 4 we also plot the radial
pressure. We can see that the radial pressure is negative for some region
inside the black hole horizon, while at the horizon and outside of the
horizon, the radial pressure is positive.

Figure 4: The energy density $\rho$, the radial pressure $p_{r}$ and the
radial pressure $\rho+p_{r}$ for $\mathcal{M}=\ell=1$ while changing the
scalar charge $A$.

###  II.2 Thermodynamics

In this subsection we will discuss the thermodynamics of the black hole
solution. We begin with the temperature. To derive the black hole temperature,
at first we perform a Wick rotation and move to imaginary time $t\to i\tau$
where $\tau$ will now be periodic, the period of which we have to find in
order to specify the temperature. We will now ignore the angular part of the
space-time metric and thence we are left with

$ds^{2}=h(r)d\tau^{2}+\frac{1}{b(r)}dr^{2}~{}.$ (43)

We now expand the metric functions near the horizon

$\displaystyle h(r\to
r_{+})=h(r_{+})+h^{\prime}(r_{+})(r-r_{+})+...=h^{\prime}(r_{+})(r-r_{+})~{},$
(44) $\displaystyle b(r\to
r_{+})=b(r_{+})+b^{\prime}(r_{+})(r-r_{+})+...=b^{\prime}(r_{+})(r-r_{+})~{},$
(45)

and the reduced space-time element reads

$ds^{2}=h^{\prime}(r_{+})(r-r_{+})d\tau^{2}+\frac{1}{b^{\prime}(r_{+})(r-r_{+})}dr^{2}~{}.$
(46)

Even if it is not clear at this point, the above line element describes a cone
in Euclidean space and has a conical singularity at the tip $r\to 0$, unless
we fix the period of $\tau$ in a particular way. Therefore, we will now
compare this line element with the line element of two-dimensional flat space-
time in polar coordinates that reads

$dS^{2}=dR^{2}+R^{2}d\Theta^{2}~{},$ (47)

where $\Theta$ is periodic of period $T_{\Theta}=2\pi$ and we will treat
$\tau$ as an angular coordinate, in order for the space-time to be truly
Euclidean. By setting $ds^{2}=dS^{2}$ we can relate the two radial coordinates

$dR^{2}=\frac{1}{b^{\prime}(r_{+})(r-r_{+})}dr^{2}~{},$ (48)

which by integration will yield the relation

$R=2\sqrt{\frac{r-r_{+}}{b^{\prime}(r_{+})}}~{},$ (49)

and now we are left with the angular coordinates

$h^{\prime}(r_{+})(r-r_{+})d\tau^{2}=R^{2}d\Theta^{2}~{},$ (50)

which again by integration yields

$\Theta=\frac{\sqrt{h^{\prime}(r_{+})b^{\prime}(r_{+})}}{2}\tau\to\frac{\Theta}{\tau}=\frac{\sqrt{h^{\prime}(r_{+})b^{\prime}(r_{+})}}{2}~{}.$
(51)

$\Theta$ is periodic with $T_{\Theta}=2\pi$ and by denoting $\beta$ the period
of $\tau$ we have

$\beta=\frac{4\pi}{\sqrt{h^{\prime}(r_{+})b^{\prime}(r_{+})}}\to\mathfrak{T}\equiv\frac{1}{\beta}=\frac{\sqrt{h^{\prime}(r_{+})b^{\prime}(r_{+})}}{4\pi}~{},$
(52)

which is the temperature of our black hole space-time. Substituting the
functions we find

$\mathfrak{T}(r_{+})=\frac{\left(A^{2}+2r_{+}^{2}\right)\left(A^{2}+\ell^{2}\mathcal{M}\right)-r_{+}^{2}e^{\frac{A^{2}}{2r_{+}^{2}}}\left(A^{2}+2\ell^{2}\mathcal{M}\right)}{2\pi
A^{2}r_{+}\ell^{2}}~{},$ (53)

and substituting the horizon radius we can express the temperature as a
function of the black hole mass

$\mathfrak{T}({\mathcal{M}})=\frac{\left(A^{2}+\ell^{2}\mathcal{M}\right)\sqrt{\ln\left(A^{2}+\ell^{2}\mathcal{M}\right)-\ln\left(A^{2}+2\ell^{2}\mathcal{M}\right)+\ln(2)}}{\sqrt{2}\pi
A\ell^{2}}~{}.$ (54)

The temperature is always real and positive. In Fig. 5 we plot the temperature
of the black hole.

Figure 5: The temperature of the black hole for $\ell=1$ as a function of
$\mathcal{M}$, while changing the scalar charge.

We can see that the temperature increases as the mass of the black hole is
growing. Moreover, the temperature is non-zero and finite for zero mass, which
happens because a horizon exists even for the massless case, given by
$r_{+}(\mathcal{M}=0)=A/(2\ln 2)$. For small $A$, it becomes

$\mathfrak{T}(r_{+})\sim\frac{r_{+}}{2\pi\ell^{2}}+\frac{A^{2}\left(2r_{+}^{2}-\ell^{2}\mathcal{M}\right)}{8\pi
r_{+}^{3}\ell^{2}}+\mathcal{O}\left(A^{4}\right)~{},$ (55)

while the BTZ temperature corresponds to
$\mathfrak{T}_{\text{BTZ}}=r_{+}/(2\pi\ell^{2})=\sqrt{\mathcal{M}}/2\pi\ell$.
As a result, the hairy black hole possesses a larger temperature at the event
horizon. Using the Wald formula [46, 47], we can calculate the entropy of the
black hole as

$\mathcal{S}=-2\pi\int
d\theta\sqrt{r_{+}^{2}}\left(\frac{\partial\mathcal{L}}{\partial
R_{\alpha\beta\gamma\delta}}\right)\Bigg{|}_{r=r_{+}}\hat{\varepsilon}_{\alpha\beta}\hat{\varepsilon}_{\gamma\delta}~{},$
(56)

where $\hat{\varepsilon}_{\alpha\beta}$ is the bi-normal to the horizon
surface [48], $\mathcal{L}$ is the Lagrangian of the theory
$\mathcal{L}=(8\pi)^{-1}\left(R/2-\partial^{\alpha}\phi\partial_{\alpha}\phi/2-V(\phi)\right)$,
and

$\frac{\partial\mathcal{L}}{\partial
R_{\alpha\beta\gamma\delta}}\Big{|}_{r=r_{+}}=\frac{1}{2}\left(g^{\alpha\gamma}g^{\beta\delta}-g^{\beta\gamma}g^{\alpha\delta}\right)~{},$
(57)

hence the entropy will be given by the Bekenstein-Hawking area law

$\mathcal{S}=\frac{\mathcal{A}}{4}=\frac{\pi r_{+}}{2}~{},$ (58)

where $\mathcal{A}=2\pi r_{+}$ is the circumference of the three-dimensional
black hole. The hairy black holes possess a larger event horizon radius in
comparison to the BTZ black hole which has an event horizon at
$r_{+}=\ell\sqrt{\mathcal{M}}$

$r_{+}\sim\ell\sqrt{\mathcal{M}}+\frac{3A^{2}}{8\ell\sqrt{\mathcal{M}}}+\mathcal{O}\left(A^{3}\right)~{},$
(59)

hence they are thermodynamically preferred over the BTZ black hole, having
higher entropy, when the scalar charge is small. The black hole space-time is
thermally stable in the canonical ensemble, which can be seen by evaluating
the heat capacity

$C(r_{+})=T\frac{dS}{dT}\Big{|}_{r=r_{+}}=\frac{\pi
A}{2\sqrt{2}\sqrt{\ln\left(A^{2}+\ell^{2}\mathcal{M}\right)-\ln\left(A^{2}+2\ell^{2}\mathcal{M}\right)+\ln(2)}}~{},$
(60)

which is always positive. For negligible scalar hair we obtain the heat
capacity of the BTZ black hole $C=\pi r_{+}/2=\pi\ell\sqrt{\mathcal{M}}/2$.

In general, one cannot impose that the internal energy of the black hole
coincides with the mass of the black hole, as the matter fields may contribute
to the internal energy. As a result the first law cannot simply be
$d\mathcal{M}=TdS$ but $dE=TdS$, where $E$ is the internal energy. We can re-
express the horizon of the black hole as

$r_{+}=\frac{A}{\sqrt{\ln\left(\frac{4\left(q\ell^{2}+1\right)^{2}}{\left(2q\ell^{2}+1\right)^{2}}\right)}}~{},$
(61)

where we have substituted the mass parameter with $q$ via (30). It is
therefore clear, that the only free parameter of the system that one is
allowed to vary is $A$. Now, we have that

$TdS=T\frac{\partial S}{\partial r_{+}}dr_{+}=T\frac{\partial S}{\partial
r_{+}}\frac{\partial r_{+}}{\partial A}dA=T\frac{\partial S}{\partial
A}dA=\frac{A\left(q\ell^{2}+1\right)\sqrt{\ln\left(2q\ell^{2}+2\right)-\ln\left(2q\ell^{2}+1\right)}}{2\sqrt{2}\ell^{2}\sqrt{\ln\left(\frac{4\left(q\ell^{2}+1\right)^{2}}{\left(2q\ell^{2}+1\right)^{2}}\right)}}~{}.$
(62)

Integrating the above expression we get:

$E(A)=\frac{A^{2}\left(q\ell^{2}+1\right)\sqrt{\ln\left(2q\ell^{2}+2\right)-\ln\left(2q\ell^{2}+1\right)}}{4\sqrt{2}\ell^{2}\sqrt{\ln\left(\frac{4\left(q\ell^{2}+1\right)^{2}}{\left(2q\ell^{2}+1\right)^{2}}\right)}}~{},$
(63)

which is the internal energy of the black hole. It is clear that when the
effect of the scalar field theory is negligible we have that

$E(q\to
0)\sim\frac{A^{2}q}{8}+\frac{A^{2}}{8\ell^{2}}=\frac{\mathcal{M}_{\text{BTZ}}}{8}+\frac{A^{2}}{8\ell^{2}}~{},$
(64)

which is a nice way to express that the fields contribute to the total energy
of the black hole. We note that the conserved black hole mass and the
geometric mass in the BTZ black hole differ by a factor of $1/8$.

##  III Rotating Black Hole Solutions

To discuss rotating solutions, we impose the metric ansatz

$ds^{2}=-h(r)dt^{2}+\frac{1}{b(r)}dr^{2}+r^{2}\left(d\theta+u(r)dt\right)^{2}~{},$
(65)

where we have introduced the angular shift function $u(r)$. Inserting this
ansatz in the field equations (2),(3) we obtain the following solution

$\displaystyle
u(r)=\frac{J}{A^{2}}e^{-\frac{A^{2}}{2r^{2}}}-\frac{J}{A^{2}}~{},$ (66)
$\displaystyle
b(r)=\frac{r^{2}e^{\frac{A^{2}}{2r^{2}}}\left(-\frac{2J^{2}}{A^{2}}+\frac{2A^{2}}{\ell^{2}}+2\mathcal{M}\right)}{A^{2}}+\frac{r^{2}e^{\frac{A^{2}}{r^{2}}}\left(-A^{2}\left(-\frac{2J^{2}}{A^{2}}+\frac{2A^{2}}{\ell^{2}}+2\mathcal{M}\right)+\frac{A^{4}}{\ell^{2}}-J^{2}\right)}{A^{4}}+\frac{J^{2}r^{2}}{A^{4}}~{},$
(67) $\displaystyle h(r)=e^{-A^{2}/r^{2}}b(r)~{},$ (68) $\displaystyle
V(r)=-\frac{8r^{4}e^{\frac{A^{2}}{2r^{2}}}\left(A^{2}\ell^{2}\mathcal{M}+A^{4}-J^{2}\ell^{2}\right)+2r^{2}e^{\frac{A^{2}}{r^{2}}}\left(A^{2}-2r^{2}\right)\left(2A^{2}\ell^{2}\mathcal{M}+A^{4}-J^{2}\ell^{2}\right)+J^{2}\ell^{2}\left(2A^{2}r^{2}+A^{4}+4r^{4}\right)}{4A^{4}r^{4}\ell^{2}}~{},$
(69) $\displaystyle
V(\phi)=-\frac{8e^{\frac{\phi^{2}}{2}}\left(A^{2}\ell^{2}\mathcal{M}+A^{4}-J^{2}\ell^{2}\right)+2e^{\phi^{2}}\left(\phi^{2}-2\right)\left(2A^{2}\ell^{2}\mathcal{M}+A^{4}-J^{2}\ell^{2}\right)+J^{2}\ell^{2}\left(\phi^{4}+2\phi^{2}+4\right)}{4A^{4}\ell^{2}}~{},$
(70)

while the scalar field is the same as (10) and $J$ is the angular momentum of
the black hole. We used the quasi-local method to derive the angular momentum
and the conserved black hole mass [20]

$\displaystyle
J=\lim_{r_{0}\to\infty}\frac{\sqrt{b(r_{0})}u^{\prime}(r_{0})r_{0}^{3}}{\sqrt{h(r_{0})}}~{},$
(71)
$\displaystyle\mathcal{M}=\lim_{r_{0}\to\infty}\left(\sqrt{h(r_{0})}E(r_{0})-Ju(r_{0})\right)~{}.$
(72)

The asymptotic expressions at large distances yield

$\displaystyle
u(r\to\infty)\sim-\frac{J}{2r^{2}}+\frac{A^{2}J}{8r^{4}}-\frac{A^{4}J}{48r^{6}}+\frac{A^{6}J}{384r^{8}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{10}\right)~{},$
(73) $\displaystyle
h(r\to\infty)\sim\frac{r^{2}}{\ell^{2}}+\left(-\frac{A^{2}}{\ell^{2}}-\mathcal{M}\right)+\frac{\frac{A^{4}}{\ell^{2}}+A^{2}\mathcal{M}+J^{2}}{4r^{2}}-\frac{3A^{2}J^{2}\ell^{2}+A^{4}\ell^{2}\mathcal{M}+A^{6}}{24r^{4}\ell^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{6}\right)~{},$
(74) $\displaystyle
b(r\to\infty)\sim\frac{r^{2}}{\ell^{2}}-\mathcal{M}+\frac{-\frac{A^{4}}{\ell^{2}}-3A^{2}\mathcal{M}+J^{2}}{4r^{2}}+\frac{A^{2}\left(-\frac{3A^{4}}{\ell^{2}}-7A^{2}\mathcal{M}+3J^{2}\right)}{24r^{4}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{6}\right)~{},$
(75) $\displaystyle
V(r\to\infty)\sim-\frac{1}{\ell^{2}}-\frac{A^{2}}{2r^{2}\ell^{2}}-\frac{A^{2}\ell^{2}\mathcal{M}+A^{4}}{4r^{4}\ell^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{6}\right)~{},$
(76) $\displaystyle V(\phi\to
0)\sim-\frac{1}{\ell^{2}}-\frac{\phi^{2}}{2\ell^{2}}+\phi^{4}\left(-\frac{\mathcal{M}}{4A^{2}}-\frac{1}{4\ell^{2}}\right)+\frac{1}{24}\phi^{6}\left(\frac{3J^{2}}{A^{4}}-\frac{5\mathcal{M}}{A^{2}}-\frac{3}{\ell^{2}}\right)+\mathcal{O}\left(\phi^{8}\right)~{}.$
(77)

We can see that at large distances we obtain a solution similar to the
rotating BTZ black hole, with changes in the structure of space-time being
related to $A$, and in the small hair case, we also obtain the rotating BTZ
black hole. The horizon is given by $b(r)=0$ [49, 34], which is also a root of
$h(r)$. $b(r)$ has two roots, which can be computed analytically but these
expressions are lengthy, so we will not given them here. The existence of
horizons provides bounds for the angular momentum of the black hole.
Therefore, black holes can only exist when

$A>0~{}\&~{}\ell>0~{}\&~{}\mathcal{M}>0~{}\&~{}\left(J^{2}\leq\frac{A^{4}}{\ell^{2}}+2A^{2}\mathcal{M}\text{~{}or~{}}\frac{A^{4}}{\ell^{2}}+2A^{2}\mathcal{M}<J^{2}\leq\frac{\left(A^{2}+\ell^{2}\mathcal{M}\right)^{2}}{\ell^{2}}\right)~{}.$
(78)

In the cases where the inequalities are saturated
($J^{2}=A^{4}/\ell^{2}+2A^{2}\mathcal{M}$ or
$J^{2}=\left(A^{2}+\ell^{2}\mathcal{M}\right)^{2}/\ell^{2}$) in the above
expressions we have black holes with a single event horizon, while in any
other case the black holes develop two horizons, an inner and an event
horizon. The rotating solution admits a region of space-time where the Killing
field $\partial/\partial t$ is space - like, which transforms into the
condition

$g_{tt}>0\to-h(r)+r^{2}u(r)^{2}>0~{}.$ (79)

It is clear that at the event horizon, the previous condition holds, and also
for some region outside of the horizon [34]. In Fig. 6 we plot the scalar
potential where it is obvious that between the inner and event horizon,
negative potential wells are developed, while as we increase the mass
parameter, the wells become deeper. In addition, it is clear form both
figures, that in the rotating case, the vaccum of the field theory represents
a local maximum, and not a global one. One should also note the fact that the
potential energy is bounded from below, which will play a crucial role in the
stability of the system. Moreover, in Fig. 7 we plot the metric function
$b(r)$ and $g_{tt}$ in order to study the geometry.

Figure 6: $V(r)$ having set $\ell=A=1$, $J=2$, while varying the mass
parameter.

Figure 7: $b(r)$ and $g_{tt}(r)$ having set $\ell=A=1$, $J=2$, while varying
the mass parameter.

From Fig. 7 (left) we can see that the black hole develops two horizons, while
for the degenerate case
$J^{2}=\left(A^{2}+\ell^{2}\mathcal{M}\right)^{2}/\ell^{2}$, the horizons
coincide. The $g_{tt}$ component is positive at the horizon and for some
region after the horizon, denoting the presence of an ergo-region, as in the
BTZ black hole [2, 3, 4]. The scalar potential depends on the conserved black
hole charges. We can eliminate these charges by introducing a new constant
$\chi$, such as

$\chi=\frac{J^{2}}{A^{4}}~{},$ (80)

besides $q=\mathcal{M}/A^{2}$. Now the potential will read

$\displaystyle
V(\phi)=e^{\phi^{2}}\left(\phi^{2}\left(-q+\frac{\chi}{2}-\frac{1}{2\ell^{2}}\right)+2q-\chi+\frac{1}{\ell^{2}}\right)+e^{\frac{\phi^{2}}{2}}\left(-2q+2\chi-\frac{2}{\ell^{2}}\right)-\chi-\frac{\chi\phi^{4}}{4}-\frac{\chi\phi^{2}}{2}~{},$
(81) $\displaystyle V(\phi\to
0)\sim-\frac{1}{\ell^{2}}-\frac{\phi^{2}}{2\ell^{2}}+\frac{\phi^{4}\left(-q\ell^{2}-1\right)}{4\ell^{2}}+\frac{\phi^{6}\left(-5q\ell^{2}+3\chi\ell^{2}-3\right)}{24\ell^{2}}+\mathcal{O}\left(\phi^{8}\right)~{}.$
(82)

Consequently, our theory will give black hole space-times with a fixed angular
momentum to conserved mass ratio given by

$\frac{\chi}{q^{2}}=\frac{J^{2}}{\mathcal{M}^{2}}~{}.$ (83)

Both the conserved mass and the angular momentum are allowed to vary, in a
consistent way, so the ratio $J^{2}/\mathcal{M}^{2}$ will always be constant.
To compute the Hawking temperature we will use the concept of surface gravity,
which is defined by

$\kappa=\sqrt{-\frac{1}{2}\nabla_{\mu}X_{\nu}\nabla^{\mu}X^{\nu}}~{},$ (84)

where $X^{\mu}$ is a Killing vector field of our space-time defined as
$X^{\mu}=(1,0,\Omega)$, where $\Omega$ is the angular velocity at the horizon
of the black hole, defined as

$\Omega=-\frac{g_{t\theta}}{g_{\theta\theta}}\bigg{|}_{r=r+}~{}.$ (85)

Evaluating the surface gravity we find

$\kappa=\frac{1}{2}\sqrt{-\frac{b(r)\left(h(r)\left(ru^{\prime}(r)+2(u(r)+\Omega)\right)^{2}-\left(h^{\prime}(r)-r^{2}(u(r)+\Omega)u^{\prime}(r)\right)^{2}\right)}{h(r)}}\Bigg{|}_{r=r+}~{}.$
(86)

Now, we can find the temperature as

$\mathfrak{T}(r_{+})=\frac{\kappa}{2\pi}=\frac{1}{4\pi}\sqrt{b(r)\left(\frac{h^{\prime}(r)^{2}}{h(r)}-r^{2}u^{\prime}(r)^{2}\right)}\Bigg{|}_{r=r+}=\frac{A^{4}e^{\frac{A^{2}}{r_{+}^{2}}}-J^{2}\ell^{2}\left(e^{\frac{A^{2}}{2r_{+}^{2}}}-1\right){}^{2}}{4\pi
A^{2}r_{+}\ell^{2}\left(e^{\frac{A^{2}}{r_{+}^{2}}}-e^{\frac{A^{2}}{2r_{+}^{2}}}\right)}~{},$
(87)

which of course reduces to the temperature of the BTZ black hole, when $A$
approaches $0$. The entropy is given by the same formula as in the non-
rotating case (58). The heat capacity is found to be

$C(r_{+})=\pi
r_{+}^{3}\left(2\left(A^{2}\left(J\ell\left(\frac{1}{J\ell-e^{\frac{A^{2}}{2r_{+}^{2}}}\left(A^{2}+J\ell\right)}+\frac{1}{e^{\frac{A^{2}}{2r_{+}^{2}}}\left(A^{2}-J\ell\right)+J\ell}\right)-3\right)+r_{+}^{2}\right)\right)^{-1}~{},$
(88)

which for small $A$ reduces to the heat capacity of the rotating BTZ black
hole

$C(r_{+},A\to 0)=\frac{1}{6}\pi
r_{+}\left(\frac{16r_{+}^{4}}{3J^{2}\ell^{2}+4r_{+}^{4}}-1\right)>0~{}.$ (89)

Due to the complexity of (78) in order for the existence of horizons, we will
perform a numerical analysis for the thermodynamics of this solution and plot
the temperature, the entropy, and the heat capacity as functions of the black
hole mass $\mathcal{M}$ for appropriate values of $A,J,\ell$. In Fig. 8 we fix
$A=0.5,\ell=1,J=2$ and plot the corresponding thermodynamic quantities for the
allowed values of mass.

Figure 8: The temperature $\mathfrak{T}$, the entropy $S$ and the heat
capacity $C$ as functions of the black hole mass

It is clear all thermodynamic quantities are positive and growing with the
increase of mass, in accordance with the BTZ black hole case. Moreover, these
black holes are thermally stable in the canonical ensemble and do not develop
any phase transition. The system under consideration is complicated and
therefore we cannot present simple calculations to show that the internal
energy of the black hole do not coincide, however we do not expect a different
behaviour from the non-rotating case.

##  IV Conclusions

In this work we consider a three-dimensional Gravity Theory with a self-
interacting scalar field minimally coupled to Gravity. We do not specify the
form of the scalar potential and to solve the field equations we choose a
specific form of the scalar field which is characterized by a scalar charge.
Solving the field equations we find a solution in which the scalar field back
- reacts to the metric and a hairy black hole is generated. The presence of
the scalar charge is introducing a scale in the theory and in the scalar
potential an effective cosmological constant appears. Motivated by field
theory, we introduce a parameter as the ratio of the black hole mass to the
scalar charge and this parameter appears in the potential. In this way the
potential is independent of the mass of the black hole and the scalar charge.
The introduction of this parameter allows the black hole mass to vary, along
with the scalar charge in a particular way, so that their ratio is kept
constant.

We find that the form of the hairy black hole depends on the way matter
parametrized by the scalar field back - reacts to the background gravitational
metric. At large distances because matter is weekly back - reacting to
Gravity, the BTZ black hole is generated with the scalar charge playing the
role of the cosmological constant which is necessary for the existence of the
BTZ black hole. At small distances a novel hairy BTZ-like black hole is
produced, because matter is strongly back - reacting to the gravitational
metric. We discussed the thermodynamics of the solutions, where we found that
the hairy black hole possesses a larger event horizon than the BTZ case and
hence has a higher temperature and entropy at the event horizon. We also
discuss the first law of the thermodynamics which for the hairy BTZ-like black
hole has to be modified.

Finally we discuss rotating solutions introducing angular momentum through an
angular shift function. We find a similar behaviour as in the non-rotating
case, at large distances we obtain a solution similar to the rotating BTZ
black hole, with changes in the structure of space-time being related to the
scalar charge, while in the small scalar charge case, we obtain the rotating
BTZ black hole. At small distances the rotating BTZ black hole is modified but
because of the complexity we did not give an explicit form of the rotating
hairy BTZ-like black hole at small distances. Performing a numerical analysis
for the thermodynamics of this solution and studying the temperature, the
entropy, and the heat capacity as functions of the black hole mass, the scalar
charge and the angular momentum we found a similar behaviour for the non-
rotating case of the hairy BTZ-like black hole.

##  V Acknowledgements

We thank Athanasios Bakopoulos, Christos Charmousis, Alex Kehagias, Nick E.
Mavromatos and Theodoros Nakas for useful discussions. The research project
was supported by the Hellenic Foundation for Research and Innovation
(H.F.R.I.) under the “3rd Call for H.F.R.I. Research Projects to support Post-
Doctoral Researchers” (Project Number: 7212). E.P and G.K. are also supported
by the research project of the National Technical University of Athens (NTUA)
65232600-ACT-MTG, entitled: Alleviating Cosmological Tensions Through Modified
Theories of Gravity.

## References

  * [1] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional Einstein Gravity: Dynamics of Flat Space,” Annals Phys. 152 (1984), 220
  * [2] M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69 (1992), 1849-1851 [arXiv:hep-th/9204099 [hep-th]].
  * [3] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D 48 (1993), 1506-1525 [erratum: Phys. Rev. D 88 (2013), 069902] [arXiv:gr-qc/9302012 [gr-qc]].
  * [4] S. Carlip, “The (2+1)-Dimensional black hole,” Class. Quant. Grav. 12 (1995), 2853-2880 [arXiv:gr-qc/9506079 [gr-qc]].
  * [5] C. Martínez and J. Zanelli, “Conformally dressed black hole in (2+1)-dimensions,” Phys. Rev. D 54, 3830 (1996) [gr-qc/9604021].
  * [6] M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, “Black holes and asymptotics of 2+1 gravity coupled to a scalar field,” Phys. Rev. D 65, 104007 (2002) [hep-th/0201170].
  * [7] F. Correa, C. Martínez and R. Troncoso, “Scalar solitons and the microscopic entropy of hairy black holes in three dimensions,” JHEP 1101, 034 (2011). [arXiv:1010.1259 [hep-th]].
  * [8] F. Correa, C. Martínez and R. Troncoso, “Hairy Black Hole Entropy and the Role of Solitons in Three Dimensions,” JHEP 1202, 136 (2012). [arXiv:1112.6198 [hep-th]].
  * [9] F. Correa, A. Faúndez and C. Martínez, “Rotating hairy black hole and its microscopic entropy in three space-time dimensions,” Phys. Rev. D 87, 027502 (2013) [arXiv:1211.4878 [hep-th]].
  * [10] M. Natsuume and T. Okamura, “Entropy for asymptotically AdS(3) black holes,” Phys. Rev. D 62, 064027 (2000) [hep-th/9911062].
  * [11] J. Aparício, D. Grumiller, E. Lopez, I. Papadimitriou and S. Stricker, “Bootstrapping gravity solutions,” JHEP 1305, 128 (2013) [arXiv:1212.3609 [hep-th]].
  * [12] W. Xu, L. Zhao and D. -C. Zou, “Three - dimensional rotating hairy black holes, asymptotics and thermodynamics,” arXiv:1406.7153 [gr-qc].
  * [13] M. Cardenas, O. Fuentealba and C. Martínez, “Three-dimensional black holes with conformally coupled scalar and gauge fields,” Phys. Rev. D 90, no.12, 124072 (2014) [arXiv:1408.1401 [hep-th]].
  * [14] W. Xu and D. C. Zou, “$(2+1)$ -Dimensional charged black holes with scalar hair in Einstein–Power–Maxwell Theory,” Gen. Rel. Grav. 49, no.6, 73 (2017) [arXiv:1408.1998 [hep-th]].
  * [15] W. Xu and L. Zhao, “Charged black hole with a scalar hair in (2+1) dimensions,” Phys. Rev. D 87 (2013) no.12, 124008 [arXiv:1305.5446 [gr-qc]].
  * [16] Z. Y. Tang, Y. C. Ong, B. Wang and E. Papantonopoulos, “General black hole solutions in ( 2+1 )-dimensions with a scalar field non - minimally coupled to gravity,” Phys. Rev. D 100, no.2, 024003 (2019) [arXiv:1901.07310 [gr-qc]].
  * [17] P. A. González, M. Olivares, E. Papantonopoulos and Y. Vásquez, “Motion and trajectories of photons in a three-dimensional rotating Hořava-AdS black hole,” Phys. Rev. D 101, no.4, 044018 (2020) [arXiv:1912.00946 [gr-qc]].
  * [18] R. Bécar, P. A. González, E. Papantonopoulos and Y. Vásquez, “Quasinormal modes of three-dimensional rotating Ho$\check{r}$ava AdS black hole and the approach to thermal equilibrium,” Eur. Phys. J. C 80, no.7, 600 (2020) [arXiv:1906.06654 [gr-qc]].
  * [19] K. C. K. Chan and R. B. Mann, ‘̀Static charged black holes in (2+1)-dimensional dilaton gravity,” Phys. Rev. D 50, 6385 (1994) [erratum: Phys. Rev. D 52, 2600 (1995)] [arXiv:gr-qc/9404040 [gr-qc]].
  * [20] K. C. K. Chan and R. B. Mann, “Spinning black holes in (2+1)-dimensional string and dilaton gravity,” Phys. Lett. B 371 (1996), 199-205 [arXiv:gr-qc/9510069 [gr-qc]].
  * [21] K. C. K. Chan, “Modifications of the BTZ black hole by a dilaton / scalar,” Phys. Rev. D 55, 3564-3574 (1997) [arXiv:gr-qc/9603038 [gr-qc]].
  * [22] R. Yamazaki and D. Ida, “Black holes in three-dimensional Einstein-Born-Infeld dilaton theory,” Phys. Rev. D 64, 024009 (2001) [arXiv:gr-qc/0105092 [gr-qc]].
  * [23] S. Hossein Hendi, B. Eslam Panah, S. Panahiyan and M. Hassaine, “BTZ dilatonic black holes coupled to Maxwell and Born-Infeld electrodynamics,” Phys. Rev. D 98, no.8, 084006 (2018) [arXiv:1712.04328 [physics.gen-ph]].
  * [24] M. Dehghani, “Thermodynamics of novel charged dilatonic BTZ black holes,” Phys. Lett. B 773, 105-111 (2017) doi:10.1016/j.physletb.2017.08.003
  * [25] M. Dehghani, “Nonlinearly charged three-dimensional black holes in the Einstein-dilaton gravity theory,” Eur. Phys. J. Plus 133, no.11, 474 (2018); M. B. Tataryn and M. M. Stetsko, “Three-dimensional slowly rotating black hole in Einstein-power-Maxwell theory,” Int. J. Mod. Phys. D 29, no.16, 2050111 (2020);
S. H. Hendi, B. Eslam Panah, S. Panahiyan and A. Sheykhi, “Dilatonic BTZ black
holes with power-law field,” Phys. Lett. B 767, 214-225 (2017)
[arXiv:1703.03403 [gr-qc]];
T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “Rotating
(2+1)-dimensional black holes in Einstein-Maxwell-dilaton theory,” Phys. Rev.
D 107 (2023) no.2, 024043 [arXiv:2210.15704 [gr-qc]].

  * [26] M. Dehghani, “Thermodynamics of (2+1)-dimensional black holes in Einstein-Maxwell-dilaton gravity,” Phys. Rev. D 96, no.4, 044014 (2017)
  * [27] G. G. L. Nashed and A. Sheykhi, “New black hole solutions in three-dimensional $\mathit{f(R)}$ gravity,” [arXiv:2110.02071 [physics.gen-ph]].
  * [28] Z. Y. Tang, B. Wang and E. Papantonopoulos, “Exact charged black hole solutions in $D$-dimensions in $f(R)$ gravity,” Eur. Phys. J. C 81 (2021) no.4, 346 [arXiv:1911.06988 [gr-qc]].
  * [29] S. H. Hendi, “(2+1)-Dimensional Solutions in $F(R)$ Gravity,” Int. J. Theor. Phys. 53 (2014) no.12, 4170-4181 [arXiv:1410.7527 [gr-qc]].
  * [30] S. H. Hendi, B. Eslam Panah and R. Saffari, “Exact solutions of three-dimensional black holes: Einstein gravity versus $F(R)$ gravity,” Int. J. Mod. Phys. D 23 (2014) no.11, 1450088 [arXiv:1408.5570 [hep-th]].
  * [31] R. A. Konoplya and A. Zhidenko, “BTZ black holes with higher curvature corrections in the 3D Einstein-Lovelock gravity,” Phys. Rev. D 102 (2020) no.6, 064004 [arXiv:2003.12171 [gr-qc]].
  * [32] L. Ma and H. Lu, “Vacua and Exact Solutions in Lower-$D$ Limits of EGB,” Eur. Phys. J. C 80 (2020) no.12, 1209 [arXiv:2004.14738 [gr-qc]].
  * [33] R. A. Hennigar, D. Kubiznak, R. B. Mann and C. Pollack, “Lower-dimensional Gauss–Bonnet gravity and BTZ black holes,” Phys. Lett. B 808 (2020), 135657 [arXiv:2004.12995 [gr-qc]].
  * [34] R. A. Hennigar, D. Kubiznak and R. B. Mann, “Rotating Gauss-Bonnet BTZ Black Holes,” Class. Quant. Grav. 38 (2021) no.3, 03LT01 [arXiv:2005.13732 [gr-qc]].
  * [35] K. Jusufi, M. Jamil and A. Sheykhi, “Regular 3D Gauss-Bonnet black holes with finite electrodynamics,” [arXiv:2302.10799 [physics.gen-ph]].
  * [36] M. Bravo-Gaete and M. Hassaine, “Thermodynamics of a BTZ black hole solution with an Horndeski source,” Phys. Rev. D 90 (2014) no.2, 024008 [arXiv:1405.4935 [hep-th]].
  * [37] O. Baake, M. F. Bravo Gaete and M. Hassaine, “Spinning black holes for generalized scalar tensor theories in three dimensions,” Phys. Rev. D 102 (2020) no.2, 024088 [arXiv:2005.10869 [hep-th]].
  * [38] T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “Black holes of (2+1)-dimensional $f(R)$ gravity coupled to a scalar field,” Phys. Rev. D 103 (2021) no.6, 064063 [arXiv:2101.06410 [gr-qc]].
  * [39] T. Karakasis, E. Papantonopoulos, Z. Y. Tang and B. Wang, “(2+1)-dimensional black holes in f(R,$\phi$) gravity,” Phys. Rev. D 105 (2022) no.4, 044038 [arXiv:2201.00035 [gr-qc]].
  * [40] A. Bakopoulos and T. Nakas, “Analytic and asymptotically flat hairy (ultra-compact) black-hole solutions and their axial perturbations,” JHEP 04 (2022), 096 [arXiv:2107.05656 [gr-qc]].
  * [41] C. A. R. Herdeiro and E. Radu, “Asymptotically flat black holes with scalar hair: a review,” Int. J. Mod. Phys. D 24 (2015) no.09, 1542014 [arXiv:1504.08209 [gr-qc]].
  * [42] P. Breitenlohner and D. Z. Freedman, “Stability in Gauged Extended Supergravity,” Annals Phys. 144 (1982), 249
  * [43] L. Mezincescu and P. K. Townsend, “Stability at a Local Maximum in Higher Dimensional Anti-de Sitter Space and Applications to Supergravity,” Annals Phys. 160 (1985), 406
  * [44] C. Desa, W. Ccuiro and D. Choque, “Exact hairy black holes asymptotically $AdS_{2+1}$,” [arXiv:2210.06421 [hep-th]].
  * [45] A. Bakopoulos and T. Nakas, “Novel exact ultra-compact and ultra-sparse hairy black holes emanating from regular and phantom scalar fields,” [arXiv:2303.09116 [gr-qc]].
  * [46] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48, no.8, R3427-R3431 (1993) [arXiv:gr-qc/9307038 [gr-qc]].
  * [47] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50, 846-864 (1994) [arXiv:gr-qc/9403028 [gr-qc]].
  * [48] S. Dutta and R. Gopakumar, “On Euclidean and Noetherian entropies in AdS space,” Phys. Rev. D 74, 044007 (2006) [arXiv:hep-th/0604070 [hep-th]].
  * [49] G. Clement, “Spinning charged BTZ black holes and selfdual particle - like solutions,” Phys. Lett. B 367 (1996), 70-74 [arXiv:gr-qc/9510025 [gr-qc]].