International Journal of Astrophysics and Space Sci ence [624273]

International Journal of Astrophysics and Space Sci ence
2013; 1(4): 45-51
Published online October 30, 2013 (http://www.scien cepublishinggroup.com/j/ijass)
doi: 10.11648/j.ijass.20130104.15

Frequency of Hawking radiation of black holes
Dipo Mahto 1, Brajesh Kumar Jha 2, Krishna Murari Singh 1, Kamala Parhi 3
1Dept. of Physics, Marwari College, T.M.B.U. Bhagalp ur-812007, India
2Deptartment of Physics, L.N.M.U. Darbhanga, India
3Dept. of Mathematics, Marwari College, T.M.B.U. Bhag alpur-812007, India
Email address:
[anonimizat](D. Mahto), brajeshjha111@gmai l.com(B. K. Jha), [anonimizat](K. M. Singh),
[anonimizat] (K . Parhi)
To cite this article:
Dipo Mahto, Brajesh Kumar Jha, Krishna Murari Singh , Kamala Parhi. Frequency of Hawking Radiation of Bl ack Holes. International
Journal of Astrophysics and Space Science. Vol. 1, No. 4, 2013, pp. 45-51. doi: 10.11648/j.ija ss.20130104.15

Abstract: In the present research work, we calculate the freq uencies of Hawking radiations emitted from differen t test
black holes existing in X-ray binaries (XRBs) and a ctive galactic nuclei (AGN) by utilizing the propos ed formula for the
frequency of Hawking radiation 33 8.037 10 kg f Hz M×= and show that these frequencies of Hawking radiati ons may be the
components of electromagnetic spectrum and gravitat ional waves. We also extend this work to convert th e frequency of
Hawking radiation in terms of the mass of the sun ( M⊙) and then of Chandrasekhar limit ( ch M), which is the largest unit of
mass.
Keywords: Electromagnetic Spectrum, Hawking Radiation, XRBs a nd AGN

1. Introduction
A black hole is a solution of Einstein’s gravitatio nal field
equations in the absence of matter that describes t he space
time around a gravitationally collapsed star. Its g ravitational
pull is so strong that even light cannot escape fro m it
(Dabholkar 2005, Mahto et al. 2012&2013). The space -time
having a black hole in it, first, has a singularity , and second,
has a horizon preventing an external observer from seeing it.
The singularity in GR is radically different from f ield theory
singularities because it is a property not of some field but of
the space-time itself. The topology of space-time i s changed
when it acquires a black hole (Soloviev, 2005).
In classical theory black holes can only absorb and not
emit particles. However, it is shown that quantum
mechanical effects cause black holes to create and emit
particles as if they were hot bodies with temperatu re
610 2M h
k M κκπ−  ≈    ⊙, where κ is surface gravity of
black holes and k is the Boltzmann constant. This t hermal
emission leads to show a decrease in the mass of th e black
hole and to its eventual disappearance: any primord ial black
hole of mass less than about 10 15 gram would have
evaporated by now (Hawking, 1975).
Hawking’s work followed his visit to Moscow, where
Soviet scientists Yakov Zeldovich and Alexander Starobinsky showed him that according to the quantu m
mechanical uncertainty principle, rotating black ho les
should create and emit particles (Hawking, 1988).
Before Hawking, in 1969, Leonard Parker obtained an
expression for the average particle density as a fu nction of
time, and showed that particles creation is in pair s. The
canonical creation and annihilation operators corre sponding
to physical particles during the expansion of the u niverse are
specified and argued that in the present predominan tly
dust-filled universe, only massless particles of ze ro spin
might possibly be produced in significant amounts b y the
present expansion and also showed that massless par ticles of
arbitrary non-zero spin, such as photons or gravito ns, are not
created by the expansion regardless of its form (Pa rker,
1969).
Hawking radiation arises for any test field on any
Lorentzian geometry containing an event horizon reg ardless
of whether or not the Lorentzian geometry satisfies the
dynamical Einstein equations of general relativity (Matt.,
1997).
The outgoing wave packet is composed of wave vector s
around K+s (and has positive Killing frequency) and arises
from a pair of packets composed of wave vectors aro und K+
and K– respectively which have positive and negativ e
free-fall frequency respectively (Corely and Jacobs on 1998).
Steven Corely and Ted Jacobson found that there are two
qualitatively different types of particle productio n in this

46 Dipo Mahto et al. : Frequency of Hawking radiation of black holes

model: a thermal Hawking flux generated by “mode
conversion” at the black hole horizon, and a non-th ermal
spectrum generated via scattering off the backgroun d into
negative free-fall frequency modes. This second pro cess has
nothing to do with black holes and does not occur f or the
ordinary wave equation because such modes do not
propagate outside the horizon with positive Killing
frequency (Corely and Jacobson 1996).
Compact binaries (two neutron stars, two black hole s, one
BH and one NS, binary stars, rotating neutron stars , neutron
star instabilities, super nova, super massive black holes and
stochastic background) are the sources emitting
gravitational waves (Daniel Sigg, 1998). Gravitatio nal wave
detection effects focus on four frequency bands (Cu rt and
Thorne, 2002) (i) The extremely low frequency band (10 -15
to 10 -18 Hz).
(ii) The very low frequency band (10 -7 to 10 -9 Hz).
(iii) The low frequency band (10 -4 to 1Hz).
(iv) The high frequency band (1 to 10 4 Hz).
In the present research paper, we have calculated t he
frequency of Hawking radiation of different types o f the test
black holes existing in X-ray binaries (XRBs) and a ctive
galactic nuclei (AGN) emitted by black holes and we have
also tried to show that frequencies of Hawking radi ations
emitted by black holes may be the components of
electromagnetic spectrum. This work is further exte nded to
convert the frequency of Hawking radiation in terms of mass
of the sun and then in Chandrasekhar limit (ch M ).
2. Theoretical Discussion
Classically, the black holes are perfect absorbers and do
not emit anything; their temperature is absolute ze ro.
However, in quantum theory black holes emit Hawking
radiation with a perfect thermal spectrum. This all ows a
consistent interpretation of the laws of black hole mechanics
as physically corresponding to the ordinary laws of
thermodynamics (Wald 2001).
In quantum physics, the empty space is not empty at all. In
it there are always particles flashing into existen ce and
disappear again. They always come in pairs; one par ticle and
one anti-particle, like an electron and a positron, or a photon
and another photon with opposite spin and impulse. These
particles are called virtual particles. They only e xist for a
very short time given by (Hawking radiation,
htt://library.thinkquest.org/c007571/English/printc ore.htm,
2011)
1
8tfπΔ = (1)
where fis the frequency of radiation. If one virtual parti cle
falls into the black hole and the other escapes, it escapes as
Hawking radiation from black hole. This radiation s hows the
same allocation as black body radiation. This assum es that
black holes do have a temperature too.
Once, we accept that black holes can radiate, then it is not
hard to estimate the wavelength of the radiation th at they emit. The only length-scale in the problem is the s ize of the
horizon. A photon with a wavelength ( λ) equal to the radius
of the black hole has energy equal to
E h ν= (2)
The energy of a pair of virtual photons will be giv en by
the following equation
2E h ν= (3)
The energy of a photon of Hawking radiation is give n by
the following equation (Hawking radiation,
htt://library.thinkquest.org/c007571/English/printc ore.htm,
2011)
3
16 hc EGM π= (4)
From equation (2) and (4), we have
3
16 c
GM νπ= (5)
All the terms like gravitational constant (G), Plan ck
constant (h) and velocity of light(c) on the right hand side of
the equation (5) are constant except mass (M) of th e black
hole. These constants have vital role discussed as:
The three fundamental constants of nature – the spe ed of
light (c), Planck’s constant (h) and Newton’s gravi tational
constant (G) are present in the eq n (5). Planck’s constant (h)
governs the law of quantum world. The speed of ligh t (c) is
the cornerstone of the special theory of relativity . The fact
that light is an electromagnetic wave travelling at the speed
of light (c) is very important consequences of Maxw ell’s
equations for electromagnetic field. In general rel ativity,
Newton’s gravitational constant G has an entirely n ew
meaning. For Newton, G is the constant of proportio nality
that appears in inverse square law of gravitation, while for
Einstein; G is a constant that determines the degre e to which
a given distribution of matter warps space and time
(Dabholkar, 2005).
The equation (5) can be written as
1
Mν∝ (6)
If λ be the wavelength of Hawking radiation emitted
from the black hole, then we have
Mλ∝ (7)
The relation (6) shows that the frequency of Hawkin g
radiation emitted by the black holes is inversely p roportional
to the mass of the black hole, whereas from relatio n (7) it is
clear that the wavelength of Hawking radiation emit ted by
the black hole is directly proportional to the mass of the
black holes. This means that the heavier black hole will emit
the Hawking radiation of lower frequency or longer

International Journal of Astrophysics and Space Sc ience 2013; 1(4): 45-51 47

wavelength and vice-versa.
Putting the values of h, c, G and π in equation (5), we
have
33 8.037 10 kg Hz Mν×= (8)
Or 33 30
30 8.037 10 1.99 10
1.99 10 kg Hz Mν× × × =× × (9)
The mass of the sun ( M⊙) is given by (Narayan, 2005 &
Mahto et al. 2011)
30 1.99 10 M kg = × ⊙ (10)
Putting the value of eq n (10) in eq n (9), we have
34.0386 10 MHz Mν× × =⊙ (11)
34.0386 10 MHz Mν  = ×     ⊙ (12)
4.0386 MKHz Mν  =    ⊙ (13)
1.4 4.0386
1.4 MKHz Mν  =    ⊙ (14)
2.8847 Mch KHz Mν  =    (15)
where 1.4 Mch M =⊙ is called Chandrasekhar limit
which is itself the largest unit of mass used for t he
measurement of astronomical stellar bodies.
3. Data in Support of Mass of Black
Holes in XRBs and AGN
There are two categories of black holes classified on the
basis of their masses clearly very distinct from ea ch other,
with very different masses M ~ 5 – 20 M ʘ for stellar – mass
black holes in X-ray binaries and M ~ 10 6 – 10 9.5 Mʘ for
super massive black holes in Galactic nuclei. Masse s in the
range 6 9.5 10 3 10 M to M ×⊙ ⊙ have been estimated by this
means in about 20 galaxies (Narayan 2005). The most viable scenario for modeling of active gal actic
nuclei includes a super massive black hole with the mass
6 9 10 10 M−⊙ accreting the galaxian matter from its
vicinity (Madejski, 2003).
At the distance of the Virgo cluster, 15 Mpc, the s phere of
influence of a 73 10 M×⊙∼ super-massive black holes
would shrink to a projected radius of 0''.07 , not only well
beyond the reach of any ground based telescope, bey ond
even HST capabilities (Ferrarese & Ford, 2005).
Assuming an isotropic, spherically symmetric system ,
Sargent et al. detected a central dark mass
95 10 M×⊙∼ within the inner 110pc of M87 (Sargent et al.,
1978).
Assuming the disk is Keplerian, Greenhill and Gwinn
estimated the mass enclosed within 0.65pc to be
71.5 10 M×⊙∼ (Ferrarese, & Ford, 2005).
In NGC 4041, acquiescent Shc spiral, Marconi et al.
(2003) remark that the systematic blue shift of the disk
relative to systemic velocity might be evidence tha t the disk
is kinematically decoupled. They conclude that only an
upper limit, of 72 10 M×⊙, can be put on the central mass.
Cappellari et al. (2002) conclude that non-gravitat ional
motions might indeed be present in the case of IC 1 459, for
which the ionized gas shows no indication of rotati on in the
inner 1'' 1459 IC is the only galaxy for which a super
massive black hole mass estimate exists based both on gas
and stellar kinematics. Three-integral models appli ed to the
stellar kinematics produce, 9(2.6 1.1) 10 M M •= ± × ⊙,
while the gas kinematics produces estimates between a
8 9 10 10 few and M ×⊙,depending on the assumptions
made regarding nature of the gas velocity
dispersion(Ferrarese & Ford, 2005).
Masses of “central dark object” have been estimated in
about forty cases, using stellar dynamics, emission lines of
orbiting gas and, most accurately, using water mase rs. They
range from 6 9 2 10 3 10 M to M × × ⊙ ⊙ ∼ ∼ and, in many
cases, the compactness is sufficient to rule out st ar clusters
with confidence (Blandford, 1999).
Most detected super massive black holes are in the
8 9 10 10 M M •≤ ≤ ⊙range, there are no detections below
10 6 M⊙ (the “building block” range) or above 10 10 M⊙
(the brightest quasar range), and even the
6 7 10 10 M M •≤ ≤ ⊙ range is very poorly sampled
(Ferrarese & Ford, 2005).

48 Dipo Mahto et al. : Frequency of Hawking radiation of black holes

4. Table
4.1. Table 1
Wavelength and frequency of Hawking radiation in XR Bs.
Sl. No Mass of BHs (M)
30 1.99 10 M kg = × ⊙ Frequency of Hawking radiation (Hz) Wavelength of Hawking radiation ( λ in
metre).
1 5M ʘ 8.037×10 2 3.714×10 5
2 6 M ʘ 6.731 x10 2 4.456 x10 5
3 7 M ʘ 5.769 x10 2 5.119 x10 5
4 8 M ʘ 5.048 x10 2 5.942 x10 5
5 9 M ʘ 4.487 x10 2 6.685 x10 5
6 10 M ʘ 4.038 x10 2 7.428 x10 5
7 11 M ʘ 3.671 x10 2 8.170 x10 5
8 12 M ʘ 3.365 x10 2 8.913 x10 5
9 13 M ʘ 3.106 x10 2 9.656 x10 5
10 14 M ʘ 2.884 x10 2 10.399 x10 5
11 15 M ʘ 2.692 x10 2 11.142 x10 5
12 16 M ʘ 2.524 x10 2 11.885 x10 5
13 17 M ʘ 2.375 x10 2 12.627 x10 5
14 18 M ʘ 2.243 x10 2 13.370 x10 5
15 19 M ʘ 2.125 x10 2 14.113 x10 5
16 20 M ʘ 2.019 x10 2 14.852 x10 5
4.2. Table 2
Wavelength and frequency of Hawking radiation in AG N.
Sl. No. Mass of BHs (M)
30 1.99 10 M kg = × ⊙ Frequency of Hawking radiation
(Hz) Wavelength of Hawking radiation ( λ in
metre).
1 1×10 6 M⊙ 34.038 10 −× 10 7.428 10 ×
2 2×10 6 M⊙ 32.019 10 −× 10 14.858 10 ×
3 3×10 6 M⊙ 31.346 10 −× 10 22.288 10 ×
4 4×10 6 M⊙ 31.009 10 −× 10 29.732 10 ×
5 5×10 6 M⊙ 30.807 10 −× 10 37.174 10 ×
6 6×10 6 M⊙ 30.673 10 −× 10 44.576 10 ×
7 7×10 6 M⊙ 30.581 10 −× 10 51.635 10 ×
8 8×10 6 M⊙ 30.504 10 −× 10 59.523 10 ×
9 9×10 6 M⊙ 30.448 10 −× 10 66.964 10 ×
10 1×10 7 M⊙ 44.018 10 −× 11 7.428 10 ×
11 2×10 7 M⊙ 42.019 10 −× 11 14.858 10 ×
12 3×10 7 M⊙ 41.346 10 −× 11 22.288 10 ×
13 4×10 7 M⊙ 41.009 10 −× 11 29.732 10 ×
14 5×10 7 M⊙ 40.807 10 −× 11 37.174 10 ×
15 6×10 7 M⊙ 40.673 10 −× 11 44.576 10 ×
16 7×10 7 M⊙ 40.581 10 −× 11 51.635 10 ×

International Journal of Astrophysics and Space Sc ience 2013; 1(4): 45-51 49

Wavelength and frequency of Hawking radiation in AG N.
Sl. No. Mass of BHs (M)
30 1.99 10 M kg = × ⊙ Frequency of Hawking radiation
(Hz) Wavelength of Hawking radiation ( λ in
metre).
17 8×10 7 M⊙ 40.504 10 −× 11 59.523 10 ×
18 9×10 7 M⊙ 40.448 10 −× 11 66.964 10 ×
19 1×10 8 M⊙ 54.018 10 −× 12 7.466 10 ×
20 2×10 8 M⊙ 52.019 10 −× 12 14.858 10 ×
21 3×10 8 M⊙ 51.346 10 −× 12 22.288 10 ×
22 4×10 8 M⊙ 51.009 10 −× 12 29.732 10 ×
23 5×10 8 M⊙ 50.807 10 −× 12 37.174 10 ×
24 6×10 8 M⊙ 50.673 10 −× 12 44.576 10 ×
25 7×10 8 M⊙ 50.581 10 −× 12 51.635 10 ×
26 8×10 8 M⊙ 50.504 10 −× 12 59.523 10 ×
27 9×10 8 M⊙ 50.448 10 −× 12 66.964 10 ×
28 1×10 9 M⊙ 64.018 10 −× 13 7.466 10 ×
29 2×10 9 M⊙ 62.019 10 −× 13 14.858 10 ×
30 3×10 9 M⊙ 61.346 10 −× 13 22.288 10 ×
31 4×10 9 M⊙ 61.009 10 −× 13 29.732 10 ×
32 5×10 9 M⊙ 60.807 10 −× 13 37.174 10 ×
5. Figure

Figure 5.1: Electromagnetic spectrum from Electromagnetic radia tion, wikipedia, 2012
6. Electromagnetic and Gravitational
Waves
Electromagnetic radiation is classified according t o the
frequency of its wave. The electromagnetic spectrum , in
order of increasing frequency and decreasing wavele ngth,
consists of radio waves, microwaves, infrared radia tion,
visible light, ultraviolet radiation, X-rays and Y- rays. The
eyes of various organisms sense a small and somewha t
variable window of frequencies of EMR called the vi sible
spectrum. The electromagnetic spectrum is shown in the
figure 5.1(Electromagnetic radiation, wikipedia, 20 12).
According to Einstein’s general theory of relativit y, when the mass of a system accelerates, the space time su rrounding
the system undergoes a change in curvature and prop agates
outward gravitational waves at the speed of light a nd carries
the energy and angular momentum away from the syste m.
Detectable electromagnetic radiation usually origin ates from
the surface of objects, whereas the gravitational w aves
emanates inside an object. Thus the astronomers exp ect that
the universe as seen in the gravitational waves wil l look very
different from the universe seen in the electromagn etic
radiation (http://astro.berky.edu/imran/cosmology1. html).
On the basis of above data for the mass of black ho les, we
have calculated the frequency and wavelength of Haw king
radiation as tabulated in the table 1 for XRBs and in table 2
for AGN.

50 Dipo Mahto et al. : Frequency of Hawking radiation of black holes

7. Results and Discussion
While observing table 1, we see that the frequencie s /
wavelengths of Hawking radiations emitted by black holes
in XRBs are few x 10 2 Hz / few x 10 5 metre which are
within the range of long radio waves of electromagn etic
spectrum and hence these types of radiations may be
included in the series of electromagnetic spectrum.
While observing table 2, we see that the frequencie s /
wavelengths of Hawking radiations emitted by black holes
in AGN are lying between 4.018×10 -3 Hz to 4.018×10 -7 Hz /
7.428 x 10 10 m to 31.174 x 10 13 m which may be placed in the
order of long radio waves of electromagnetic spectr um and
hence these types of radiations may also be include d in the
series of electromagnetic spectrum of figure 5.1.
In the present work, the frequencies of Hawking
radiations have good agreement with gravitational w ave
detection efforts given in the reference of Curt & Thorne
(2002). Hence we can say that the Hawking radiation s may
be gravitational waves. We also see that electromag netic
waves and gravitational waves have quite separate
frequencies and wavelengths, but the gravitational waves
can be arranged in the order of electromagnetic spe ctrum.
In this work, we have converted the formula for the
frequency of Hawking radiation in terms of mass of the sun
(M⊙) also which closely agrees with reference to Denia l
Sigg (1998) and then in terms of Chandrasekhar limi t
(ch M ).
From tables 1 and 2, it is clear that the frequenci es of
Hawking radiations decrease with the increase of ma ss of
different test black holes, while with the increase of mass of
different black holes, the wavelength of the Hawkin g
radiation increases.
8. Conclusion
In the light of the present work, we can draw the f ollowing
conclusions:
(1) The Hawking radiations emitted by the black hol es
may be gravitational waves.
(2) The frequencies of the Hawking radiations emitt ed by
the black holes decrease with the increase of the m ass of
different test black holes, while the wavelengths o f the
Hawking radiations increase with the increase of th e mass of
different test black holes.
(3) The frequencies / wavelengths of Hawking radiat ions
emitted by black holes in XRBs are within the range of long
radio waves of electromagnetic spectrum and hence t hey
may be included in the series of electromagnetic sp ectrum.
(4) The Hawking radiations emitted by the black hol es in
XRBs are radio waves, while in AGN are very long ra dio
waves.
(5) The frequencies of the Hawking radiations emitt ed by
the black holes in AGN are below any other componen ts of
electromagnetic spectrum and hence they may be plac ed in
the order of long radio waves of electromagnetic sp ectrum.
(6) The wavelengths of the Hawking radiations emitt ed by
the black holes in AGN are above any other componen ts of electromagnetic spectrum and hence they may be plac ed in
the order of long radio waves of electromagnetic sp ectrum.
(7) The frequency ranges of the Hawking radiations
emitted by the black holes have better agreement wi th the
frequencies of gravitational waves detected by dete ctors like
LIGO, LISA, TAMA etc.
(8) The frequencies / wavelengths of the Hawking
radiations emitted by the black holes may be placed in the
series of electromagnetic spectrum without any dist urbance
to it.
(9) The frequency of the Hawking radiation emitted by
the black hole may be transformed in terms of the m ass of
sun ( M⊙) and then in terms of Chandrasekhar limit ( ch M ).
Acknowledgement
The authors are grateful to the referee for pointin g out the
errors in the original manuscript and making constr uctive
suggestions. The authors also acknowledge the help of Dr. D.
T. K. Dutta, Retired Professor, University Departme nt of
English, T. M. B. U., Bhagalpur and Prof. Vijoy Kan t Mishra,
H.O.D. Physics, Marwari College, Bhagalpur in proof
reading. The authors are obliged to Dr. Gopi Kant J ha,
Former Head & Prof. of Physics, L.N.M.U Darbhanga a nd
Dr. Neeraj Pant, Associate Professor, Dept. of Math ematics,
N.D.A. Khadakwasala, Pune for their inspiration and
motivation.

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