2Tashkent Institute of Economics and Pedagogy
Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
4. Samatov B.T., Umaraliyeva N.T., Uralova S.I. Differential games with the Langenhop type constrains on controls, Lobachevskii Journal of Mathematics (2021), 42, b. 2942-2951.
5. Samatov B.T, Uralova S.I. Differential games with inertial players under the Langenhop type constrains on controls, Lobachevskii Journal of Mathematics, (2023),44(10), b.4363-4373.
6. Samatov B.T, Uralova S.I. A Linear Pursuit Problem with Langenhop type constraints , Bulletin of the Institute of Mathematics, (2023), 4(4), b.66-72.
7. Samatov B.T, Uralova S.I. Pursuit-evasion linear differential games with exponential integral constraints ,Scientific Bulletin of NamSU. (2022) 8, b.9-13.
FREQUENCY SHIFT OF PHOTON RADIATED FROM ACCRETION DISC OF ACOUSTIC BLACK HOLE
D. D. Urinboeva
Institute of Fundamental and Applied Research, "THAME" NRU. [email protected]. Uzbekistan.
1. Introduction. The black hole (neutron star, white dwarf) is highly energetic object as due to its strong gravitational attraction, a matter falling onto it loses its angular momentum in an accretion disc and emits radiation Accretion of the matter onto both stellar-mass black holes in galactic X-ray binaries (XRBs) and supermassive black holes in the Active Galactic Nuclei (AGNs) usually proceeds through an accretion disc, which may be truncated at the ISCO because of the general relativistic effect of black hole. Accretion process includes many physical processes such as, gravity, hydrodynamics, viscosity, radiation and magnetic fields.
2. Acoustic black hole spacetime. In this section we briefly present the spacetime metric of the acoustic black hole adopted in this paper for calculations. For details related to obtaining this solution, please see [1]. The acoustic black hole solution is generated from the Gross-Pitaevskii theory that yields the action [2-5]
(1)
Klein-Gordon equation, is given by:
ç + m2ç — 2 —1\ç>Ç ç = 0.
P
(2)
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Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current
_g ^aLge^^Pr^
by
(3)
(4)
We solve this equation for the curved, spherically symmetric background given
ds2 = gttdt2 + grrdr2 + geedO2 + g(pvdq)1,
by choosing the scalar field as
p = ¿v (x w p(x )•
In the fixed spacetime, one can assume the background solution of the scalar field as (p, v0).. Then, we introduce the fluctuations (p, v) • around solution as
P = Po + P v = v0 + vi (5)
After some algebraic coordinate transformations, this effective metric is obtained (for details — see Ref. [1]), as
ds2 =y[3c
-f (r )dt2
dr2
f (r )
r dQ2
(6)
where d Q = dd2 + sin2 ddp2 is a metric of the 2-sphere and the metric function f (r) has the form
f
f (r) =
2M
1 -
v r
1S
2M
f
2M
1 -
vr
(7)
with M is total mass of the black hole and % is the tuning parameter related to the radial component of the background fluid 4-velocity, ^ 2M%/ p. Therefore, the tuning parameter is always nonnegative, %> 0. In the case of % = 0, the spacetime reduces to the one of the Schwarzschild black hole. In order to see the acoustic horizon we produce Fig. 1 and see the effect of the tuning parameter of the spacetime to the formation and location of the horizons. One can see from this figure that for the values of the tuning parameter % e [0,4) the single acoustic horizon and its location at r = 2M
do not change, but when the tuning parameter reaches % = 4, the additional new degenerate acoustic horizon appears at r = am.Beyond, i.e. for the values
%e( 0,4), % the degenerate acoustic horizon separates into two, the
2
s
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Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
first and second of which are located at 2M <r < 4M and 2M < r respectively. If one compares this spacetime with the Kerr one, all the properties are different, including the effect of the tuning parameter increases the location of horizon, unlike the case in the Kerr spacetime that the rotation of the central black hole decreases the gravitational attraction of the black hole [2-5] and consequently, radius of the horizon becomes smaller than the one of the Schwarzschild black hole r < 2M.
g = •
f - f
i ±Jf V 2f
(8)
5 06
Q
£ 0.5
It is well known that the inner edge of the accretion disc is restricted by the ISCO. Therefore, in Fig. 2 we present the ratio of frequencies (8) for the case that the photon is emitted by the matter moving at the ISCO of the acoustic black hole and it depends strongly on the observer-accretion disc orientation, as this frequency in the case that the disc is viewed edge-on, it is significantly less than the one in the case that the disc is viewed face-on. With increasing the value of tuning parameter of the central black hole, this frequency ratio decreases. The maximum value of the ratio corresponds to the Schwarzschild black hole case (£ = 0),
■J2 42
with the values — and — for the disc viewed edge-on and face on case 3 2
o 3 bi ÏÎ 2
I
— disc viewed edge —on
— disc viewed face -on
___
0.0 0.5 1.0 1.5 2.0 2.5 3.0 4
Fig. 2. The g-factor (top panel) and redshift factor (bottom panel) of photon around acoustic black when the accretion disc is viewed edge-on and face-on.
Fig. 3. The g-factor around acoustic black when the accretion disc is viewed edge-on (left) and face-on (right).
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Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
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edge-on and face on cases, respectively. If the matter emits photon at other orbits of the accretion disc (r > rISCO) the energy of the photon emitted at the accretion
disc tends to the value of the one of the detected by observer at infinity — see Fig. 3.
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