# LSTM-CNN Network for Audio Signature Analysis in Noisy Environments

1st Praveen Damacharla  Research Scientist
KineticAI Inc.
The Woodlands, TX, USA
<EMAIL_ADDRESS>2nd Hamid Rajabalipanah  Electrical Engineering
Iran University of Science and Technology
Tehran, Iran
3rd Mohammad Hosein Fakheri  Electrical Engineering
Iran University of Science and Technology
Tehran, Iran

###### Abstract

There are multiple applications to automatically count people and specify
their gender at work, exhibitions, malls, sales, and industrial usage.
Although current speech detection methods are supposed to operate well, in
most situations, in addition to genders, the number of current speakers is
unknown and the classification methods are not suitable due to many possible
classes. In this study, we focus on a long-short-term memory convolutional
neural network (LSTM-CNN) to extract time and / or frequency-dependent
features of the sound data to estimate the number / gender of simultaneous
active speakers at each frame in noisy environments. Considering the maximum
number of speakers as 10, we have utilized 19000 audio samples with diverse
combinations of males, females, and background noise in public cities,
industrial situations, malls, exhibitions, workplaces, and nature for learning
purposes. This proof of concept shows promising performance with
training/validation MSE values of about 0.019/0.017 in detecting count and
gender.

###### Index Terms:

Sound processing, artificial intelligence, deep learning

##  I Introduction

Presence detection is used for many applications. It can be used to
automatically control the indoor climate, blinds or to inform staff on usage
of certain rooms or office. Information on how many people are in a certain
place is also useful for exhibition and product introduction to see which
product attracts people and which gender[1, 2]. It is also useful for the
visually or hearing impaired by giving them a better understanding. Many
methods are used for people counting such as Vision systems (cameras) in
addition to WiFi and Bluetooth tracking are some of the most frequently used
options. However, cameras are not placed everywhere, and WiFi tracking is
unable to narrow a position down to a specific floor or room [3]. An
alternative or additional solution which is considered in the following is to
count people and specifying gender using sound. Using a variety of devices
that are already present such as smartphones and economic recorders, sound can
be recorded at minimal costs and difficulty.

Figure 1: The illustration of the overall architecture for the proposed
multiple speaker counting method.

Figure 2: The proposed hybrid CNN-LSTM-FC architecture for estimating the
gender/number of the speakers in noisy environments

Figure 3: The MSE values corresponding to (a) FC, (b) CNN-FC, (c) LSTM-FC, and
(d) CNN-LSTM-FC networks

Simultaneous conversations have been studied for several years yet still have
many errors in many speech technology applications such as Speaker
Identification and Automatic Speech Recognition [4, 5, 6, 7]. Many papers
developed systems to recognize multiple speakers though the majority of them
assume that the number of concurrent sources is known and also neglect gender
and ambient noise [8, 9], which is not a realistic assumption. Therefore, a
semi-real-time, simple, yet effective speaker counts and gender estimation
method is needed in noisy environment to become beneficial in real-world
applications. Methods based on clustering also often require a priori
information, e.g., the maximum number of concurrent sources in the processed
sequence.

Artificial intelligence (AI) has made its entrance in the real-world
applications in the last decades and is gaining considerable interest from the
industrial community. Machine learning, Neural network and deep learning have
become a popular data analysis tool across scientific society. Using machine
learning with audio data has proven to be particularly useful in domains such
as automatic speech recognition, [10] language processing, [11] and music
classification [12] and event detection for urban and sport environments.

Estimation of speaker-count usually needs two fundamental steps: (i)
extracting the audio features related to the number of concurrent speakers,
and (ii) classifying the number of speakers in the feature space. Recently,
some attempts have been made to apply deep learning to audio source counting.
In [13], the convolutional neural network is used to classify the audio
signals into 3 classes: 1, 2, and 3-or-more sources. Two main solutions may be
found in the literature for this purpose, namely, regression and
classification. Nevertheless, the clustering method is no longer efficient
when both gender and number of active speakers are requested in a crowded
environment. Although several studies have investigated speaker count
estimation [4, 14, 15, 16], they have not identified the gender of speakers.
While, both gender/number data of customers may bring valuable information in
various real-life industrial applications.

This paper will focus on a regression-based deep learning approach to
determine number and genders of people that are concurrently talking via the
Long-Short-Term Memory Convolutional Neural Network (LSTM-CNN). We directly
estimate a speaker count instead of counting them after identification. We
have created 19000 samples of 5-s simultaneous speech with different
combinations of male and female sounds. The contribution of this paper are
summarized as follows:

  * •

Beside estimating the count, the proposed algorithm is also able to detect the
gender of speakers.

  * •

Unlike to previous proposals, the basis of our algorithm relies on regression
model, which performs efficiently for gender detection in higher number of
speakers.

  * •

The proposed method retains its performance in noisy environments.

##  II Data Sets

Many previous studies [17, 18], have employed simulated overlapping speech for
speaker counting with only 5-10% of overlap speech. In this research, a big
data set of noisy realistic conversation scenarios may happen in day life and
industrial applications has been created.

###  II-A Speech

In order to train and test our network, a dataset containing concurrent
speaking of different speakers with different gender, age, and accents. 3682
males and 2311 females have contributed to construct the dataset as speakers.
For each sample, speaking sound of n males and m females (n+m$\leq$10) are
randomly merged. The dataset includes 19000 audio samples of concurrent talks
each of which has 5-s duration and a sampling rate of 16000 frames per second
where, silence in the beginning and the end of audios has been removed.

###  II-B Background Noise

We include 50 non-speech (environmental sounds) examples in our training data,
allowing network to better understand the possible background noises. The
background sounds have been randomly selected among possibale real-life scenes
such as industrial companies, work environments, exhibitions, public cities,
malls, nature, farms, stores and etc., and then added to the original files as
background noise (SNR$\approx$10 dB).

##  III System Design

Since the raw temporal waveforms are dense, it is not computationally
efficient to use them, directly, as the input of the network model. Firstly,
each of 19000 samples have been divided into 32000-frame overlapping samples
with a 8000-frame shift. Through windowing the short-term signals and then
applying the discrete Fourier transform, the Mel-frequency cepstral
coefficients (MFCCs) of the raw data has been extracted, allowing us to use a
compact representation of the spectrum as the input of the network model.

Fig. 1 displays the end-to-end procedure for gender/count detection based on
audio signals. First, the speech voices of multiple persons ($\leq$ 10) are
randomly selected among the available dataset and then mixed. A randomly
chosen environmental noise (SNR=$10$ dB) is added to the mixed signal. After
normalizing the resultant signals, a time-windowing process is applied, making
$q$-sample shorter signal. For each short-time window, the corresponding MFCCs
are extracted to feed the regression model. The train-test splitting procedure
is performed so as to allow the network see diverse combinations of men and
women voices. Finally, the proposed regression network is learned by using the
created dataset with two outputs, i.e., number of men and women in speakers.

Our proposed network architecture is depicted in Fig. 2, where the MFCCs are
considered as the input data. The network is designed so as to extract both
spatial and temporal features of the audio signals. The 2D convolutional
network (2D-CNN) is responsible for extracting the spatial features and the
temporal attributes are achieved by the bi-directional Long short-term memory
(BLSTM) layer. Each CNN layer applies the linear operation by sliding the
kernal of specified size on the output of the previous layer. The 2D-CNN
overcomes the 1D-CNN for our application since it reduces the number of
weights within a convolution layer through the parameter sharing concept.

We use 7 CNN channels of kernal size 5$\times$5 with 256 filters in the
network. Each convolution layer is followed by a Max-pooling layer with a
dimension of 2$\times$2 to minimize the computational cost. Padding is applied
to keep the same dimensions after the convolutions. The Leaky Rectified Linear
Unit (LeakyReLU, $\alpha=0.1$) function is used as the activation layer to
increase the non-linearity degree and avoid the conventional dying ReLU
problem. We should note that the audio signal have temporal correlation which
should be investigated. Since the recurrent neural network (RNN) is unable to
keep the information for the long sequences and suffers from small gradient
updates, especially in the earlier layers, we use bidirectional Long short
term memory (BLSTM, 128 units). Due to using both past and future information
to update weights, the BLSTM network is more robust compared to the LSTM one.
The feature maps extracted by CNN layers are then reshaped and then fed into 3
BLSTM layers. We will show that integrating the CNN with LSTM can
significantly extract the spatial and temporal attributes of the raw data and
therefore improve the overall accuracy of the speaker counting system.

To enhance the accuracy of regression, we employed a fully connected (FC)
model with two hidden layers of 512 and 64 sizes. The dropout methods have
been established to reduce the effect of overfitting and to enhance the
capability of the detection model in unseen data. Also, the layers are
initialized using Xavier Glorot’s initialization. The final stage includes two
nodes to reveal the number of men and women in the speakers. Evidently, the
sum of these values determines the total number of speakers.

##  IV Results

We took care of using diverse combinations of man-woman speakers and
environmental noises in the train, validation and test sets. The overall
duration of the generated mixture dataset is 18 hours for training, and 5
hours for validation and test. The normalization pre-processing is applied to
map the value of the features and also, the output values between 0 and 1. The
train and test sizes are selected as 0.7 and 0.3, respectively.

TABLE I: The MSE values resulted by different network models. Model | FC | FC-CNN | FC-LSTM | FC-CNN-LSTM
---|---|---|---|---
MSE (Training) | 0.06 | 0.0497 | 0.0438 | 0.019
MSE (Validation) | 0.0589 | 0.0485 | 0.0429 | 0.017

To evaluate the performance of our regression problem, MSE performance
indicator is employed to compare the exact value of the outputs with the
observed ones. We investiage the MSE valuse for four different networks: 1)
FC, 2) CNN-FC, 3) LSTM-FC, and 4) CNN-LSTM-FC to show that the proposed
architecture has the minimum MSE value among all other combinations. The
evaluation results are plotted in Figs. 3a-d, respectively. As expected, the
FC model (four hidden layers of 1024, 512, 256, 64 nodes) has the highest MSE
value of about 0.06 (Fig. 3a). Although the MSE values corresponding to CNN-FC
(Fig. 3b) and LSTM-FC (Fig. 3c) are smaller, but they are not still convincing
in our application. The simultaneous use of CNN and LSTM networks leads to the
best training/validation MSE values of about 0.019/0.017 (Fig. 3d). The reason
can be attributed to the fact that both spatial and temporal attributes of the
short-frame audio signals are elaborately included in the learning process.
The results are summarized in Table I which proves the effectiveness of the
proposed hybrid model in the well-known speaker counting problem, even in the
presence of the environmental noise.

TABLE II: The MSE values of the proposed model corresponding to different Kernel sizes. Kernel Size | 3$\times$3 | 5$\times$5 | 7$\times$7
---|---|---|---
MSE (Training) | 0.0313 | 0.019 | 0.028
MSE (Validation) | 0.0325 | 0.017 | 0.025
TABLE III: The MSE values of the proposed model corresponding to different CNN channel numbers. CNN Channel Number | 3 | 5 | 7
---|---|---|---
MSE (Training) | 0.0511 | 0.0311 | 0.019
MSE (Validation) | 0.0482 | 0.0282 | 0.017
TABLE IV: The MSE values of the proposed model corresponding to different CNN filter numbers. CNN Filter Number | 64 | 128 | 256
---|---|---|---
MSE (Training) | 0.039 | 0.0335 | 0.019
MSE (Validation) | 0.037 | 0.0316 | 0.017

Figure 4:  Illustration of the effects of pre-windowing of the input audio
signal on the MSE values of the proposed model. a) window size=q=32000
samples, shift=16000 samples, and b) window size=q=16000 samples, shift=8000
samples, and c) window size=q= 8000 samples, shift= 4000 samples.

We should note that all contributing parameters of our proposed model have
been through a comprehensive parametric study. For instance, we investigate
the effect of change in the number of CNN filters, CNN channels and also, the
kernel size. The corresponding MSE results are tabulated in Tables II, III,
IV, respectively. Based on the results, we should remark the following points.
As the CNN layers are responsible for extracting the complex features, the
greater number of filters allowing the model to learn deeper features,
yielding higher accuracy in gender/number estimation of speakers. As Table I
shows, the model with 256 filters has the best MSE value. Using a greater
number of filters may lead better response but at the expense of higher
computing costs. Similar behaviors can be found in results of Table II, where
considering the time-consumption, 7 layers are found high enough to attain
acceptable MSE values. Finally, Table III reveals the optimum value of the CNN
kernel size as which results in the training and validation MSE values of
0.019 and 0.017, respectively.

As mentioned before, the mixed audio signals have been divided to q-sample
short-time windows feeding the network model. Figs. 4a-c depict the MSE values
of the training system for different overlapped windows with q=8000, 16000,
32000 samples and q/2 shift corresponding to 0.5s, 1s, and 2s time duration
(0.25 s, 0.5s, and 1s time shift), respectively. Evidently, a higher time span
allows the model to deal with more information and learn better. The MSE
values corresponding to 2-sec signals reach 0.019 (training) and 0.017
(validation), respectively. Therefore, the proposed model requires the audio
input of at least 2-sec length to estimate well.

Additionally, we report the performance of our CNN-LSTM-FC architecture for
different random sets of train-test data to show the robustness of the model.
The network is trained by three different sets and the corresponding result
are displayed in (Fig. 5). As can be seen, the MSE values of all three sets
are close to 0.017, indicating that the proposed model is robust against the
random splitting of the train-test data.

Figure 5: The MSE values of the proposed CNN-LSTM-FC architecture upon
variation of train-test data splitting

##  V Conclusions

In this study, we proposed a hybrid CNN-LSTM-FC deep learning model for
simultaneous detection of gender/number of speakers in multi-speaker noisy
environments. The CNN and LSTM layers were used to extract the spatial and
temporal attributes of the speech, respectively. The dataset includes 19000
audio samples of concurrent talks each of which has 5-s duration and a
sampling rate of 16000 frames per second where, silence in the beginning and
the end of audios has been removed. A comprehensive parametric study was
carried out to find the best set of contributing parameters in the training
model. The proposed network has been trained and our results showed the MSE
value of about 0.017 in estimating the number of men and women between the
speakers. The MSE value has been reported for different network architectures
to prove that the effectiveness of the proposed model overcomes the other
possible solutions.

## References

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Generalized Vaidya spacetime: horizons, conformal symmetries, surface gravity
and diagonalization

Vitalii Vertogradov1, Dmitriy Kudryavcev2

1 Physics department, Herzen state Pedagogical University of Russia,

48 Moika Emb., Saint Petersburg 191186, Russia

SPb branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg 196140, Russia

<EMAIL_ADDRESS>
2 Physics department, Herzen state Pedagogical University of Russia,

48 Moika Emb., Saint Petersburg 191186, Russia

<EMAIL_ADDRESS>
Abstract: In this paper the different properties of generalized Vaidya
spacetime are considered. We located and compared different types of horizons.
We show that the apparent horizon can contain the event horizon. The locations
of all types of horizons are compared with ones in usual Vaidya spacetime.
Also, we consider the conformal Killing vector and transform the metric to
conformally-static coordinates. We introduce a new constant of motion along
null and timelike geodesics, which is generated by homothetic Killing vector.
The conformally-static coordinates allow to diagonalize the generalized Vaidya
spacetime. The surface gravity has been calculated for the dust and stiff
fluid cases.

Key words: Generalized Vaidya spacetime, Apparent horizon, Conformal Killing
vector, Surface gravity, Diagonalization, Event horizon.

## Introduction

Nowadays, black hole plays an important role in modern theoretical physics and
astrophysics. In the real world, astronomical bodies gain mass when they
absorb radiation and they lose mass when they emit radiation, which means that
the space-time around them is time-dependent. Vaidya solution [1] is one of
the exact dynamical solution of the Einstein equations. It can be regarded as
a dynamical generalization of the static Schwarzschild solution. The Vaidya
spacetime is widely used in many astrophysical applications with strong
gravitational fields. In general relativity, this spacetime assumed added
importance with the completion of the junction conditions at the surface of
the star by Santos [2]. The pressure at the surface is non-zero, and the star
dissipates energy in the form of heat flux. This made it possible to study
dissipation and physical features associated with gravitational collapse, as
shown by Herrera et al. [3, 4, 5]. The question about the dynamical shadow
formation in Vaidya spacetime is discussed in [6, 7]. The horizon structure,
entropy of this solution is investigated for an empty background in [8, 9],
for Vaidya surrounded by cosmological fields in [10, 11, 12]. Some recent
studies of the temperature properties inside the radiating star have been done
in [13, 14, 15]. The Vaidya spacetime can be extended to include both null
dust and null string fluids leading to the generalized Vaidya spacetime [16].
The detail investigation of the properties of the generalized Vaidya spacetime
can be found in [17, 18, 19]. The generalized Vaidya spacetime has been used
to investigate the gravitational collapse [20, 21, 22, 23, 24, 25, 26]. The
conformal symmetries and embedding and other properties are discussed in [27,
28, 29, 30].

The generalized Vaidya spacetime has the off-diagonal term which can lead to
the negative energy for a particle. The absence of such particles has been
proven in [31]. The forces in Vaidya spacetime are discussed in [32, 33].
Recently, a new generalization of Vaidya spacetime has been found by
gravitational decoupling method [34] which can describe the Vaidya black hole
distorted by dark matter.

In the present work we investigate different horizons locations. Two main
approaches exist for locating the horizon of a black hole:

  1. 1.

The causal approach, familiar, especially in the form of the global event
horizon.

  2. 2.

the quasi-local approach, typically based on marginally outer trapped
surfaces.

Defined horizons in this way coinide in globally static spacetimes such as the
Schwarzschild solution, but differ often in dynamical spacetimes. The key
difference is that the event horizon is always a null surface. The apparent
horizon, on the other hand, might be not only null, but also spacelike and
timelike one. A natural physical phenomenon to associate with the boundary of
a black hole is Hawking radiation [35]. However, it is a hard task to define
the surface gravity in dynamical spacetime [9]. Here we consider some of the
definitions and calculate the surface gravity in generalized Vaidya spacetime
case.

One more important question is the symmetries related to the conformal
symmetry of the spacetime. For the certain choice of the mass function, the
Generalized Vaidya metric possesses the homothetic Killing vector. This extra
symmetry can help to define constant of motion related to the angular momentum
and energy. We transform the metric to conformally-static coordinates and show
the simple method how one can diagonalize the generalized Vaidya spacetime,
which can play an important role for an interpretation of physical results
obtained in this spacetime.

This paper is organized as follows. In sec. II we define the apparent and
event horizons and calculate their approximate location for different mass
functions. Also the null surface with constant area change is discussed. The
difference in horizons locations between Vaidya and generalized Vaidya cases
are discussed. In sec. III We define the homothetic Killing vector and
transform the metric to the conformally-static coordinates. Also, we define
the energy and angular momentum of a particle which goes along the geodesic.
In sec. IV we discuss the different method of dynamical surface gravity
definition in the generalized Vaidya spacetime. The obtained results are
compared to the Vaidya spacetime case. Sec. V has dealt with the
diagonalization of generalized Vaidya spacetime. Sec. VI is the conclusion.

The system of units $c=G=1$and signature $-\,,+\,,+\,,+$ will be used
throughout the paper.

## Generalized Vaidya horizons

The generalized Vaidya spacetime in Eddington-Finkelstein coordinates has the
following form [16]:

$\begin{split}ds^{2}=-\left(1-\frac{2M(v,r)}{r}\right)dv^{2}+2dvdr+r^{2}d\omega^{2}\,,\\\
d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\,.\end{split}$ (2.1)

Here $M(v,r)$ is the mass function which depends upon the advanced Eddington
time $v$ and space areal coordinate $r$ i.e. $r$ is a coordinate such that the
surfaces of spherical isometry have area $A=4\pi r^{2}$.

The energy-momentum tensor of (2.1) represents the combination of the null
dust (type-I) and the null fluid (or null strings, type-II) [36]:

$\begin{split}T^{(m)}_{ik}=\mu n_{i}n_{k}\,,\\\
T^{(n)}_{ik}=(\rho+P)(n_{i}l_{k}+n_{k}l_{i})+Pg_{ik}\,,\\\
\mu=\frac{2\dot{M}}{r^{2}}\,,\\\ \rho=\frac{2M^{\prime}}{r^{2}}\,,\\\
P=-\frac{M^{\prime\prime}}{r}\,,\\\ n_{i}=\delta^{0}_{i}\,,\\\
l_{i}=\frac{1}{2}\left(1-\frac{2M}{r}\right)\delta^{0}_{i}-\delta^{1}_{i}\,,\\\
n_{i}n^{i}=l_{i}l^{i}=0\,,\\\ l_{i}n^{i}=-1\,.\end{split}$ (2.2)

here $P$ \- pressure, $\rho$ \- density, $\mu$ \- the energy density of the
null dust. And $l,n$ \- two null vectors. Here $T^{(M)}_{ik}$ is the energy-
momentum tensor of null dust and $T^{(N)}_{ik}$ \- null fluid.

To calculate a future outer trapping horizon (FOTH), one needs to know the
radial outgoing $\Theta_{l}$ and ingoing $\Theta_{n}$ null expansions. The
apparent horizon exists if at some radius $r_{ah}$ the following conditions
are held:

$\Theta_{l}(r_{ah})=0\,,\Theta_{n}(r_{ah})<0\,.$ (2.3)

In generalized Vaidya spacetime (2.1), the expansions are given by:

$\begin{split}\Theta_{l}=\frac{1}{r}\left(1-\frac{2M(v,r)}{r}\right)\,,\\\
\Theta_{n}=-\frac{2}{r}\,.\end{split}$ (2.4)

From (2.4) one can see that if we satisfy the conditions (2.3), then we obtain
the following apparent horizon equation:

$r_{ah}=2M(v,r_{ah})\,.$ (2.5)

If we consider the equation of the state $P=\alpha\rho$ for the type-II matter
field, then, by virtue of the Einstein equations, the mass function is given
by [23, 24, 16]:

$\begin{split}M(v,r)=C(v)+D(v)r^{1-2\alpha}\,,\\\
\alpha\in[-1\,,1]\,,\alpha\neq\frac{1}{2}\,.\end{split}$ (2.6)

Where $C(v)$ and $D(v)$ are arbitrary functions of the time $v$. If
$D(v)\equiv 0$ then one has the usual Vaidya solution and (2.5) gives the
well-known result $r=2M(v)=2C(v)$. In the dust case $\alpha=0$ (2.5) gives
[22]:

$r_{ah}=\frac{2C(v)}{1-2D(v)}\,.$ (2.7)

In this case the energy conditions [37] demands $C(v)\geq 0\,,D(v)\geq 0$.
Hence, one can conclude if $D(v)\neq 0$ then the location of the apparent
horizon in this case doesn’t coincide with the apparent horizon in Vaidya
spacetime. Moreover, $r_{ah}^{generalized_{V}aidya}>r_{ah}^{Vaidya}$. The
apparent horizon area in the dust case is given by:

$A^{dust}=\frac{16\pi C^{2}(v)}{(1-2D(v)^{2}}\,.$ (2.8)

Cases when $D(v)\geq\frac{1}{2}$ is considered in [22].

If the type-II represents the stiff fluid [38] $\alpha=1$, then we have two
apparent horizons:

$r^{\pm}_{ah}=C(v)\pm\sqrt{C^{2}(v)+2D(v)}\,.$ (2.9)

Here, the energy conditions demand $C(v)\geq 0\,,D(v)\leq 0$ [39]. We are
interested in outer horizon $r^{+}_{ah}$. In comparison with the dust case
$r^{generalized_{V}aidya}_{ah}<r^{Vaidya}_{ah}$. As the result, the area of
the outer apparent horizon is:

$A^{stifffluid}=4\pi(C(v)+\sqrt{C^{2}(v)+2D(v)})^{2}\,.$ (2.10)

We can conclude that the location of the apparent horizon depends upon the
function $D(v)$. The sign of this function depends upon the energy conditions.
The week energy condition demands the positivity of the energy density $\rho$.
Substituting (2.6) into (2.2), one obtains the following energy density
expression:

$\rho=2\left(1-2\alpha\right)\frac{D(v)}{r^{2+2\alpha}}\geq 0\,.$ (2.11)

From (2.11) one can see that we have two cases:

$\begin{split}\alpha<\frac{1}{2}\rightarrow D(v)\geq 0\longrightarrow
r^{generalized_{V}aidya}_{ah}>r^{Vaidya}_{ah}\,,\\\
\alpha>\frac{1}{2}\rightarrow D(v)\leq 0\longrightarrow
r^{generalized_{V}aidya}_{ah}<r^{Vaidya}_{ah}\,.\end{split}$ (2.12)

It is a hard task to find the location of the event horizon in dynamical
spacetimes. Here, we will consider to what extent strictly null horizons can
be defined quasi-locally.

The event horizon is defined as connected components of the past causal
boundary of future null infinity and is generated by null geodesics that fail
to reach infinity. The event horizon is always a null surface since it is a
causal boundary. In generalized Vaidya spacetime (2.1) the coordinate $v$ is
constant on ingoing radial null geodesics. Any outgoing radial null geodesic
must satisfy:

$\frac{dr}{d\lambda}=\frac{1}{2}\left(1-\frac{2M}{r}\right)\frac{dv}{d\lambda}\,,$
(2.13)

for some affine parameter $\lambda$.

We consider the metric (2.1)for accreting matter. In this case, one can also
find the approximate location of the event horizon by imposing the condition:

$\frac{d^{2}v}{dr^{2}}=0\,.$ (2.14)

Taking the second derivative (2.13):

$\frac{d^{2}r}{d\lambda^{2}}=\frac{1}{2}\left(1-\frac{2M}{r}\right)\frac{d^{2}v}{d\lambda^{2}}+\left(\frac{M}{r^{2}}-\frac{M^{\prime}}{r}\right)\frac{dv}{d\lambda}\frac{dr}{d\lambda}-\frac{\dot{M}}{r}\left(\frac{dv}{d\lambda}\right)^{2}\,,$
(2.15)

and imposing the condition (2.14), one finds:

$\left(\frac{M}{r^{2}}-\frac{M^{\prime}}{r}\right)\frac{dv}{d\lambda}\frac{dr}{d\lambda}-\frac{\dot{M}}{r}\left(\frac{dv}{d\lambda}\right)^{2}=0\,.$
(2.16)

And using (2.13), one obtains:

$\left(\frac{1}{2}\left(1-\frac{2M}{r}\right)\left(\frac{M}{r^{2}}-\frac{M^{\prime}}{r}\right)-\frac{\dot{M}}{r}\right)\left(\frac{dv}{d\lambda}\right)^{2}=0\,.$
(2.17)

And approximate location of the event horizon in the generalized Vaidya
spacetime (2.1) is given by:

$(2\dot{M}+M^{\prime})r^{2}-M(1+2M^{\prime})r+2M^{2}=0\,.$ (2.18)

The solutions of (2.18) allow us to define the location of the event horizon
quasi-locally. The equation (2.18), in general, has a lot of solutions but
only the outermost horizon is of immediate interest and comments will be
restricted mostly to that case.

We can’t find the general solution of (2.18) because the mass function also
depends upon $r$ coordinate. Here, we again consider two cases of the dust and
stiff fluid. Let’s suppose again that the mass function is in form (2.6). we
assume that functions $C(v)$ and $D(v)$ in the dust case has the particular
form:

$\begin{split}C(v)=\nu v\,,\nu>0\,,D(v)=\mu=const.\,,\mu>0\,.\end{split}$
(2.19)

In Vaidya spacetime, one has the following restriction $\nu\leq\frac{1}{16}$
[8]. Also, substituting (2.19) into (2.7), one obtains the restriction
$\mu<\frac{1}{2}$. By using (2.19) in (2.18), one has the equation for
defining the approximate location of the event horizon:

$2r^{2}-(1-2\mu)vr+2\nu v^{2}=0\,,$ (2.20)

which gives us the radius $r_{eh}$ of the event horizon:

$r_{eh}=\frac{v}{4}\left(1-2\mu+\sqrt{(1-2\mu)^{2}-16\nu}\right)\,.$ (2.21)

We have dropped the minus sign because the inner event horizon is hidden for
distant observer.

To get the approximate location of the event horizon in Vaidya spacetime, one
needs to put $\mu=0$ [8]. Again, like in the apparent horizon case above, the
event horizon in usual Vaidya is bigger than the event horizon radius in
generalized Vaidya spacetime in the dust case ($r^{gv}_{eh}<r^{v}_{eh}$). Also
one should note, that the apparent horizon is inside the event horizon
($r_{ah}<r_{eh}$). However, if either $\nu>\frac{1}{16}$ or
$(1-2\mu)^{2}<16\lambda$, then the event horizon is absent.

Although, we have stated that only outer horizon is of immediate interest, one
should realize that if we put $v=const.$ then the radius of the outer horizon
tends to infinity and the inner horizon

$r^{-}_{eh}\approx\frac{2\nu v_{c}}{1-2\mu}+O(\nu^{2})\,,$ (2.22)

in the limit $\mu\rightarrow 0$ becomes the Schwarzschild event horizon
($v_{c}$ is some positive constant and $M=\nu v_{c}$).

If we consider the stiff fluid, i.e. $\alpha=1$ then we should define the
functions $C(v)$ and $D(v)$ in the following way:

$\begin{split}C(v)=\nu v\,,\nu>0\,,\\\ D(v)=\mu v^{2}\,,\mu<0\,.\end{split}$
(2.23)

Now if we put (2.23) into (2.17), then we obtain:

$\left(1-\frac{2\nu v}{r}-\frac{2\mu v^{2}}{r^{2}}\right)\left(\frac{\nu
v}{r}+\frac{2\mu v^{2}}{r^{2}}\right)-2\nu-\frac{4\mu v}{r}=0\,.$ (2.24)

One of the solution of (2.24) is

$r_{eh}=-\frac{2\mu v}{\nu}\,.$ (2.25)

The apparent horizon for the choice of functions $C(v)$ and $D(v)$ (2.23) is

$r_{ah}^{\pm}=\left(\nu\pm\sqrt{\nu^{2}+2\mu}\right)v\,.$ (2.26)

For the existence both and the apparent horizon (2.26) and the event horizon
(2.25), the following condition $-\frac{\nu^{2}}{2}\leq\mu\leq 0$ must be
held. If we satisfy this condition then the event horizon is hidden inside the
apparent one. If this condition is violated, then one has only the event
horizon.

The equation (2.24) can be rewritten as:

$\left(1+\frac{2\mu v}{\nu
r}\right)\left[2\mu\left(\frac{v}{r}\right)^{3}+2\nu\left(\frac{v}{r}\right)^{2}-\frac{v}{r}-2\right]=0\,.$
(2.27)

The cubic equation in square brackets has at least one real root. The simplest
solution (2.27) has if we put $\nu=2\mu$, then the apparent horizon is absent
due to (2.26) but the event horizon approximate location is:

$r_{eh}=\nu v\,,$ (2.28)

which is the particular case ($2\mu=\nu^{2}$) in (2.25).

One should realize that the location of the event horizon (2.25) is defined
quasi-locally. However, if the condition $\nu^{2}=-2\mu$ is held and we
consider the case $v=v_{rn}=const.$, then (2.25) coincides with the event
horizon of the extremal Reissner–Nordström black hole.

We have considered two models. The dust case confirms the state that the
apparent horizon is hidden inside the event one [40]. The stiff fluid model,
on the other hand, contradicts to this statement because we have shown that
the event horizon is inside the apparent one.

To conclude this chapter, let’s find the approximate location of the null
surfaces which parameter rate of the area change is constant. The change in
the surface area $A=4\pi r^{2}$ along outgoing null geodesic congruence is
given by:

$\frac{dA}{d\lambda}=4\pi r\left(1-\frac{2M}{r}\right)\frac{dv}{d\lambda}$
(2.29)

We remind that for any parameterization $\lambda$, the following condition
$\frac{dr}{d\lambda}=\frac{1}{2}\left(1-\frac{2M}{r}\right)\frac{dv}{d\lambda}$
is always held.

Now, we consider the second derivative of (2.29):

$\frac{d^{2}A}{d\lambda^{2}}=4\pi\left[\left(\frac{1}{2}\left(1-\frac{2M}{r}\right)(1-2M^{\prime})-2\dot{M}\right)\left(\frac{dv}{d\lambda}\right)^{2}+r\left(1-\frac{2M}{r}\right)\frac{d^{2}v}{d\lambda^{2}}\right]$
(2.30)

the location of null surfaces with constant area change
$\frac{d^{2}A}{d\lambda^{2}}=\frac{d^{2}v}{d\lambda^{2}}=0$ is defined by the
equation:

$(r-2M)(1-2M^{\prime})-4\dot{M}r=0\,.$ (2.31)

Here, we consider only the dust case. For the dust case, we use the mass
function definition (2.19). In this case the equation (2.31) gives:

$r_{ac}=\frac{2\nu v(1-2\mu)}{1-2\mu-4\nu}\,.$ (2.32)

From this equation we obtain a new restriction $1-2\mu-4\nu>0$. It is easy to
show that if we can define the outer event horizon, then the null surface of
constant area change is hidden inside it. However, if we compare the location
of the apparent horizon $r_{ah}$ and the constant area change surface $r_{ac}$
then they can quasi-locally coincide or contain each other. All three options
depend upon parameters $\nu$ and $\mu$.

  1. 1.

Two horizons quasi-locally coincide. In this case $\mu=\mu^{\pm}$. Where

$\mu^{\pm}=\frac{1}{4}\left(1\pm\sqrt{1-16\nu}\right)\,.$ (2.33)

  2. 2.

The apparent horizon is hidden inside of the constant area change null
surface. In this case $0\leq\mu<\mu^{-}<\frac{1}{2}$ and
$\mu^{+}<\mu<\frac{1}{2}$.

  3. 3.

The null surface with constant area change is inside the apparent horizon. In
this case $0\leq\mu^{-}<\mu<\mu^{+}<\frac{1}{2}$.

## Conformal symmetry

In this section we consider the existence only homothetic Killing vectors
which allows us to define an extra constant of motion.

Generalized Vaidya spacetime, in the general case, admits only one constant of
motion associated with spherical-symmetry, i.e. angular momentum $L$, which is
given by:

$L=r^{2}\sin^{2}\theta\frac{d\varphi}{d\lambda}\,,$ (3.34)

and there are not any other symmetries (in general case) to reduce geodesic
equations to the first order differential ones.

However, we can seek for an additional symmetries related to homotetic Killing
vector. If a spacetime admits conformal symmetry, then there exists a
conformal Killing vector field in the spacetime. If the metric is Lie dragged
along this vector field the causal structure of the spacetime remains
invariant.

Any spacetime is said to possess a conformal Killing vector $K^{i}$ if it
solves the following conformal Killing equation:

$K_{i;k}+K_{k;i}=c(x^{l})g_{ik}\,.$ (3.35)

Where $c(x^{l})$ some function which, in general, depends upon all
coordinates. If $c(x^{l})=const.$ then $K^{i}$ is Homothetic Killing vector
and if $c(x^{l})\equiv 0$ then $K^{i}$ is just a Killing vector. The general
conformal Killing vector has been considered in [27]. Usual Vaidya spacetime
possesses only Homothetic Killing vector [41]. Here, we restrict the
consideration by Homothetic Killing vector in order to obtain the metrec (2.1)
in conformally static coordinates in which the Homothetic Killing vector
becomes $\frac{d}{dt}$ which allows us to define a new constant of motion
along null geodesics related to the particle energy.

Let’s consider the following vector field:

$\mathbf{K}=v\frac{d}{dv}+r\frac{d}{dr}\,.$ (3.36)

The vector $K^{i}$ (3.36) is the Homothetic Killing vector if it satisfies the
equation (3.35) with $c(x^{l})\equiv c=const.$ Substituting (3.36) into
(3.35), we obtain that $K^{i}$ is the Homothetic Killing vector if the
following differential equation is held:

$\frac{\dot{M}v}{r}-\frac{M}{r}+M^{\prime}=0$ (3.37)

The solution of this equation is:

$M(r,v)=\mu r^{\xi}v^{1-\xi}+\nu v\,.$ (3.38)

Here $\mu\,,\nu$ and $\xi$ are some constants. The solution (3.38) is the mass
function (2.6) if:

$\begin{split}C(v)=\nu v\,,\nu>0\,,\\\ D(v)=\mu v^{2\alpha}\,,\\\
\xi=1-2\alpha\,.\end{split}$ (3.39)

Where the sign of $\mu$ depends upon the energy condition and $+$ if
$\alpha<\frac{1}{2}$ and $-$ if $\alpha>\frac{1}{2}$ and $\alpha$ is the
constant from the equation of the state $P=\alpha\rho$.

Let’s define the location of the conformal Killing horizon. This horizon is
located at $r=r_{ckh}$ and indicates that the vector $K^{i}$ becomes null:

$k^{i}k_{i}=-\left(1-\frac{2M}{r}\right)v^{2}+2rv=0$ (3.40)

Thus, to obtain the location of the conformal Killing horizon, one needs to
solve the following equation:

$2r^{2}-vr+2Mv=0$ (3.41)

Substituting (3.38) into (3.41), one obtains:

$2r^{2}-rv+2\mu r^{\xi}v^{2-\xi}+2\nu v^{2}=0$ (3.42)

In the dust case ($\alpha=0\rightarrow\xi=1$), we get:

$r_{ckh}=\frac{v}{4}\left(1-2\mu+\sqrt{(1-2\mu)^{2}-16\nu}\right)\,.$ (3.43)

One should note that the conformal Killing horizon location (3.43) coincides
with the event horizon location (2.21). However, by comparing the event
horizon (2.18) and conformal Killing horizon (3.42) equations in general case
with the mass function in the form (3.38), one can easily see that for
different values of $\alpha$ locations of these horizons don’t coincide.

Now, to obtain the generalized Vaidya spacetime in conformally-static
coordinates, one should do the following coordinate transformation [6]:

$\begin{split}v=r_{0}e^{\frac{t}{r_{0}}}\,,\\\
r=Re^{\frac{t}{r_{0}}}\,,\end{split}$ (3.44)

and substituting it into (2.1), we have:

$ds^{2}=e^{\frac{2t}{r_{0}}}\left[-\left(1-\frac{2\nu r_{0}}{R}+\frac{\mu
r_{0}^{2\alpha}}{R^{2\alpha}}-2\frac{R}{r_{0}}\right)dt^{2}+2dtdR+R^{2}d\Omega^{2}\right]\,.$
(3.45)

In these coordinates the homothetic Killing vector $K^{i}$(3.36), becomes:

$v\frac{\partial}{\partial v}+r\frac{\partial}{\partial
r}=\frac{\partial}{\partial t}\,.$ (3.46)

This homotetic Killing vector is timelike if

$1-\frac{2\nu r_{0}}{R}+\frac{\mu
r_{0}^{2\alpha}}{R^{2\alpha}}-2\frac{R}{r_{0}}>0\,,$ (3.47)

and the location of the conformal Killing horizon is:

$1-\frac{2\nu r_{0}}{R_{ckh}}+\frac{\mu
r_{0}^{2\alpha}}{R^{2\alpha}_{ckh}}-2\frac{R_{ckh}}{r_{0}}=0\,.$ (3.48)

From the fact that $\frac{\partial}{\partial t}$ is the homothetic Killing
vector, one obtains that the energy $E$:

$E=e^{\frac{2t}{r_{0}}}\left[\left(1-\frac{2\nu r_{0}}{R}+\frac{\mu
r_{0}^{2\alpha}}{R^{2\alpha}}-2\frac{R}{r_{0}}\right)\frac{dt}{d\lambda}-\frac{dr}{d\lambda}\right]\,,$
(3.49)

is the conserved quantity along a null geodesic. The angular momentum now has
the following form:

$L=e^{\frac{t}{r_{0}}}R^{2}\sin^{2}\theta\frac{d\varphi}{d\lambda}\,.$ (3.50)

From the fact that $K^{i}$ is homothetic Killing vector, one has an additional
constant of motion $\varepsilon$ along any type of geodesics. Let $u^{i}$ is
four velocities along the geodesic then:

$\varepsilon=K_{i}u^{i}-\lambda cg_{ik}u^{i}u^{k}\,.$ (3.51)

This constant of motion depends upon the affine parameter $\lambda$. $c$ is
the conformal factor (3.35). By using (3.46), (3.49) and the fact that $c=1$,
one has for timelike geodesics

$-\varepsilon=E-\lambda\,.$ (3.52)

So, conformal symmetry of the generalized Vaidya spacetime allows us to get a
new constant of motion related to the fact that the homothetic Killing vector
is timelike at the region where the condition (3.47) is held.

## The surface gravity

In black hole thermodynamics the surface gravity of a black hole plays a role
analogous to temperature. However, In a fully dynamical situation, the surface
gravity will probably not be directly analogous to a temperature of any
thermal spectrum. The surface gravity is likely to play a key role in the
emission of Hawking radiation, even in non-equilibrium processes. The surface
gravity is usually defined on the Killing horizon. It works well in stationary
case, but it breaks down in dynamical situation where there is no the Killing
horizon. The key question where one should define the surface gravity in the
case, black hole either emits Hawking radiation or accreting matter. Here, we
give some well-know definitions and compare them with usual Vaidya spacetime.

The first definition which we will cover is the one given by Fodor et al [42].
Let’s consider an affinely-parameterized ingoing null geodesic $n^{i}$ which
asymptotic behaviour such that $t_{i}n^{i}=-1$ where $t^{i}=(1\,,0\,,0\,,0)$
is an asymptotic Killing vector. This definition works only if the spacetime
admits an asymptotically flat spatial infinity. Then for outgoing geodesic
$l^{i}$ one demands that the condition $l^{i}n_{i}=-1$ is held everywhere in
the spacetime. Then the surface gravity is given by:

$\varkappa_{F}=-n^{i}l^{k}l_{k;i}\,.$ (4.53)

For $n^{i}$ and $L^{i}$ which we have used to calculate the expansion
$\Theta$, one has:

$\varkappa_{F}=\frac{1}{4M}\left(1-2M^{\prime}\right)\,.$ (4.54)

From energy condition, we have $M^{\prime}>0$ which leads to the fact that the
surface gravity in generalized Vaidya is less than the one in usual Vaidya
solution by this definition. Also, if we consider the static limit $M=M(r)$
then the Hawking temperature associated with this surface gravity is less than
in Schwarzschild case.

Another definition is associated with Kodama vector $K^{i}$ [43] and has been
proposed by Hayward [44]. The Kodama vector has the property that the
combination $K_{i}T^{ik}$ is divergence free in spherical symmetry. At spatial
infinity, it reduces $K^{i}K_{i}=-1$. The surface gravity for the apparent
horizon is defined by:

$\frac{1}{2}g^{ik}K^{j}\left(K_{i;j}-K_{j;i}\right)=\varkappa_{K}K^{k}\,.$
(4.55)

In generalized Vaidya solution, the Kodama vector has the form
$K^{i}=(1\,,0\,,0\,,0)$ and we have:

$\varkappa_{K}=\frac{1}{4M}\left(1-2M^{\prime}\right)\,.$ (4.56)

This surface gravity, in generalized Vaidya solution, coincides with the
previous definition given by Fodor et all (4.54).

In spherically-symmetric spacetime, one can use the Misner-Sharp mass to
define the surface gravity [45]. We know, that the apparent horizon in
generalized Vaidya spacetime is given by:

$r_{ah}=2M(v,r_{ah})\,.$ (4.57)

By Differentiating this equation with respect to any parameter $\lambda$
labeling spherically symmetric foliations of the horizon, gives:

$\frac{dr_{ah}}{d\lambda}=\frac{2M}{dv}\frac{dv}{d\lambda}+\frac{dM}{dr_{ah}}\frac{dr_{ah}}{d\lambda}\,.$
(4.58)

We take $\lambda=v$ and use the fact that $A=4\pi r_{ah}^{2}$, we obtain:

$\frac{dM}{dv}=\frac{1}{8\pi}\left(1-2M^{\prime}\right)\frac{1}{2r_{ah}}\frac{dA}{dv}\,.$
(4.59)

By applying the first law of black hole dynamics
$dM=\frac{\varkappa}{8\pi}dA$, one gets:

$\varkappa=\frac{1}{4M}\left(1-2M^{\prime}\right)\,.$ (4.60)

Which again coincides with (4.54).

One should realize that we are interested only in the case $2M^{\prime}<1$. If
we have $2M^{\prime}=1$ then the surface gravity vanishes and in this case
$r_{ah}$ is so-called putative horizon [49].

Let’s calculate the surface gravity for the cases, dust and stiff fluid. In
the dust case, by applying the mass definition (3.38), one obtains:

$\begin{split}M(v,r_{ah})=\nu v+\mu r_{ah}\,,\\\ r_{ah}=\frac{2\nu
v}{1-2\mu}\,,\\\ \varkappa=\frac{1-2\mu}{4\nu v}\,.\end{split}$ (4.61)

In the stiff fluid case:

$\begin{split}M(v,r_{ah})=\nu v+\mu v^{2}r^{-1}\,,\mu<0\,,\\\
r_{ah}=\left(nu+\sqrt{\nu^{2}+2\mu}\right)v\,,\\\
\varkappa=\frac{\sqrt{\nu^{2}+2\mu}}{2\nu v+2\nu v\sqrt{\nu^{2}+2\mu}+2\mu
v}\,.\end{split}$ (4.62)

## The diagonalization of the generalized Vaidya spacetime

As we have pointed out in the introduction, the Vaidya spacetime (2.1) has a
large number of astrophysical and theoretical applications. However, the
interpretation of physical results obtained in this metric is complicated
because this metric is written in off-diagonal coordinates. The problem is
that the null coordinate $v$ is not directly measurable physical quantity.
Transition to more physical diagonal coordinates involves analytic
difficulties, and the explicit form of the corresponding coordinate
transformation is generally unknown [46]. The diagonal form of Vaidya
spacetime for linear muss function has been obtained in [47]. Later, the
diagonal form for some simple models has been obtained for the generalized
Vaidya spacetime [48]. Here we show the most general coordinate transformation
to the diagonal form in the case if the metric (2.1) possesses the homothetic
Killing vector.

Let’s consider the metric in the conformally-static coordinates (3.45). Now we
introduce new coordinate $T$ by the relation:

$t=f(T,R)\,.$ (5.63)

Substituting (5.63) into (3.45) one obtains:

$\begin{split}ds^{2}=e^{\frac{f(R,T)}{r_{0}}}\left[-\delta\left(\frac{\partial
f(R,T)}{\partial T}\right)^{2}dT^{2}+2\frac{\partial f(R,T)}{\partial
T}\left(1-\delta\frac{\partial f(R,T)}{\partial R}\right)dTdR+\right.\\\
\left.+\frac{\partial f(R,T)}{\partial R}\left(2-\delta\frac{\partial
f(R,T)}{\partial R}\right)dR^{2}+R^{2}d\Omega^{2}\right]\,.\end{split}$ (5.64)

Here

$\delta=1-\frac{2\nu r_{0}}{R}+\frac{\mu
r_{0}^{2\alpha}}{R^{2\alpha}}-2\frac{R}{r_{0}}\,.$ (5.65)

We demand that $\frac{\partial f}{\partial T}\neq 0$. From (5.64) one can
easily see that this metric is in the diagonal form if $f(R,T)$ satisfies the
following differential equation:

$1-f^{\prime}\delta=0\,.$ (5.66)

Further, in the paper dash and dot mean the particular derivative with respect
to $R$ and $T$ respectively.

From (5.66) one obtains the solution in the form:

$f(R,T)=\int\frac{dR}{\delta}+E(T)\,.$ (5.67)

Where $E(T)=T$ to ensure that $\dot{f}\neq 0$. Substituting this $f$ into
(5.64) one obtains the generalized Vaidya spacetime in the diagonal form:

$ds^{2}=e^{\frac{2T+2\int\delta^{-1}dR}{r_{0}}}\left[-\delta
dT^{2}+\delta^{-1}dR^{2}+R^{2}d\Omega^{2}\right]\,.$ (5.68)

One should note that this spacetime describes a black hole and if we put
$0\neq E(T)\neq 1$ then $\dot{E}^{2}$ might best be interpreted as a sort of
anomalous redshift that describes how far the total gravitational redshift
deviates from that implied by the shape function $M_{s}$ [49]:

$M_{s}(T,R)=e^{\frac{t+\int\delta^{-1}dR}{r_{0}}}\left(\nu r_{0}+\mu
r_{0}^{2\alpha}R^{1-2\alpha}\right)\,.$ (5.69)

## Conclusion

In this work we have considered the horizon structure of the generalized
Vaidya spacetime. If the type-II matter field satisfies the equation of the
state $P=\alpha\rho$ then the presence of this type of matter field affects
the Vaidya solution in the following way:

  * •

The location of the generalized Vaidya apparent horizon is less than the
location of the apparent horizon of usual Vaidya if $\alpha<\frac{1}{2}$.

  * •

and vice versa if $\alpha>\frac{1}{2}$.

In general, the apparent horizon is hidden inside the event horizon. However,
we have shown that there are some models when the event horizon is inside the
apparent horizon or it might absent but the apparent horizon can still exist.

The conformal symmetries of generalized spacetime have been considered in
[27], here we use the existence of the homothetic Killing vector to transform
the generalized Vaidya spacetime to the conformally-static coordinates. Also,
these coordinates allow to transform the metric to the diagonal form. This
diagonalization includes the obtained results in [47, 48]. We have calculated
the location of the conformal Killing horizon and show that it coincides with
the event horizon in the dust case but can differ in general.

There are several ways how one can calculate the Hawking temperature. We have
considered several models and shown that the Hawking temperature is the
largest when only type-I of matter field presence i.e. in usual Vaidya case.
If one adds null strings, then the temperature is decreasing.

acknowledgments: The author says thanks to grant NUM. 22-22-00112 RSF for
financial support. The work was performed within the SAO RAS state assignment
in the part ”Conducting Fundamental Science Research”.

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