Open access peer-reviewed chapter

# The Black Hole Binary Gravitons and Related Problems

By Miroslav Pardy

Submitted: August 27th 2018Reviewed: November 22nd 2018Published: January 30th 2019

DOI: 10.5772/intechopen.82659

## Abstract

The energy spectrum of graviton emitted by the black hole binary is calculated in the first part of the chapter. Then, the total quantum loss of energy is calculated in the Schwinger theory of gravity. In the next part, we determine the electromagnetic shift of energy levels of H-atom electrons by calculating an electron coupling to the black hole thermal bath. The energy shift of electrons in H-atom is determined in the framework of nonrelativistic quantum mechanics. In the last section, we determine the velocity of sound in the black hole atmosphere, which is here considered as the black hole photon sea. Derivation is based on the thermodynamic theory of the black hole photon gas.

### Keywords

• graviton
• Schwinger source theory
• spectrum of H-atom
• Coulomb potential
• black hole spectrum
• energy shift
• sound

## 1. The graviton spectrum of the black hole binary

In 1916, Schwarzschild published the solution of the Einstein field equations [1] that were later understood to describe a black hole [2, 3], and in 1963, Kerr generalized the solution to rotating black holes [4]. The year 1970 was the starting point of the theoretical work leading to the understanding of black hole quasinormal modes [5, 6, 7], and in the 1990s, higher-order post-Newtonian calculations [8] were performed and later the extensive analytical studies of relativistic two-body dynamics were realized [9, 10]. These advances, together with numerical relativity breaks through in the past decade [11, 12, 13]. Numerous black hole candidates have now been identified through electromagnetic observations [14, 15, 16]. The black hole binary and their rotation and mergers are open problem of the astrophysics, and it is the integral part of the binary black hole physics.

The binary pulsar system PSR B1913+16 (also known as PSR J1915+1606) discovered by Hulse and Taylor [17] and subsequent observations of its energy loss by Taylor and Weisberg [18] demonstrated the existence of gravitational waves [19].

By the early 2000s, a set of initial detectors was completed, including TAMA 300 in Japan, GEO600 in Germany, the Laser Interferometer Gravitational-Wave Observatory (LIGO) in the United States, and Virgo in Italy. In 2015, Advanced LIGO became the first of a significantly more sensitive network of advanced detectors (a second-generation interferometric gravitational wave detector) to begin observations [20].

Taylor and Hulse, working at the Arecibo Radiotelescope, discovered the radio pulsar PSR B1913+16 in a binary, in 1974, and this is now considered as the best general relativistic laboratory [21].

Pulsar PSR B1913+16 is the massive body of the binary system where each of the rotating pairs is 1.4 times the mass of the Sun. These neutron stars rotate around each other in an orbit not much larger than the Sun’s diameter, with a period of 7.8 h. Every 59 ms, the pulsar emits a short signal that is so clear that the arrival time of a 5 min string of a set of such signals can be resolved within 15 μs.

A pulsar model based on strongly magnetized, rapidly spinning neutron stars was soon established as consistent with most of the known facts [22]; its electrodynamical properties were studied theoretically [23] and shown to be plausibly capable of generating broadband radio noise detectable over interstellar distances. The binary pulsar PSR B1913+16 is now recognized as the harbinger of a new class of unusually short-period pulsars, with numerous important applications.

Because the velocities and gravitational energies in a high-mass binary pulsar system can be significantly relativistic, strong-field and radiative effects come into play. The binary pulsar PSR B1913+16 provides significant tests of gravitation beyond the weak-field, slow-motion limit [24, 25].

We do not repeat here the derivation of the Einstein quadrupole formula in the Schwinger gravity theory [26]. We show that just in the framework of the Schwinger gravity theory, it is easy to determine the spectral formula for emitted gravitons and the quantum energy loss formula of the binary system. The energy loss formula is general, including black hole binary, and it involves arbitrarily strong gravity.

Since the measurement of the motion of the black hole binaries goes on, we hope that sooner or later the confirmation of our formula will be established.

### 1.1 The Schwinger approach for the problem

Source methods by Schwinger are adequate for the solution of the calculation of the spectral formula of gravitons and energy loss of binary. Source theory [27, 28] was initially constructed to describe the particle physics situations occurring in high-energy physics experiments. However, it was found that the original formulation simplifies the calculations in the electrodynamics and gravity, where the interactions are mediated by photon and graviton, respectively. The source theory of gravity forms the analogue of quantum electrodynamics because, while in QED the interaction is mediated by the photon, the gravitational interaction is mediated by the graviton [29]. The basic formula in the source theory is the vacuum-to-vacuum amplitude [30]:

0+0=eiWS,E1

where the minus and plus symbols refer to any time before and after the region of space–time with action of sources. The exponential form is postulated to express the physically independent experimental arrangements, with result that the associated probability amplitudes multiply and the corresponding Wexpressions add [27, 28].

In the flat space-time, the field of gravitons is described by the amplitude (1) with the action (c=1in the following text) [31]

WT=4πGdxdxTμνxD+xxTμνx12TxD+xxTx,E2

where the dimensionality of WThas the dimension of the Planck constant and Tμνis the momentum and energy tensor that, for a particle trajectory x=xt, is defined by the equation [32]

Tμνx=pμpνEδxxt,E3

where pμis the relativistic four-momentum of a particle with a rest mass mand

pμ=EpE4
pμpμ=m2,E5

and the relativistic energy is defined by the known relation

E=m1v2,E6

where v is the three-velocity of the moving particle.

Symbol Txin Eq. (2) is defined as T=gμνTμν, and D+xxis the graviton propagator whose explicit form will be determined later.

### 1.2 The power spectral formula in general

It may be easy to show that the probability of the persistence of vacuum is given by the following formula [27]:

0+02=exp2ImW=dexpdtdω1ωPωt,E7

where the so-called power spectral function Pωthas been introduced [27]. For the extraction of the spectral function from Im W, it is necessary to know the explicit form of the graviton propagator D+xx. This propagator involves the graviton property of spreading with velocity c. It means that its mathematical form is identical with the photon propagator form. With regard to Schwinger et al. [33], the x-representation of Dkin Eq. (2) is as follows:

D+xx=dk2π4eikxxDk,E8

where

Dk=1k2k02iϵ,E9

which gives

D+xx=i4π20sinωxx'xx'ett.E10

Now, using Eqs. (2), (7), and (10), we get the power spectral formula in the following form:

Pωt=4πGωdxdxdtsinωxxxx'cosωtt×TμνxtTμν(xt)12gμνTμν(xt)gαβTαβ(xt).E11

### 1.3 The power spectral formula for the binary system

In the case of the binary system with masses m1and m2, we suppose that they move in a uniform circular motion around their centre of gravity in the xyplane, with corresponding kinematical coordinates:

x1t=r1icosω0t+jsinω0tE12
x2t=r2icosω0t+π+jsinω0t+πE13

with

vit=dxi/dt,ω0=vi/ri,vi=vii=12.E14

For the tensor of energy and momentum of the binary, we have

Tμνx=p1μp1νE1δxx1t+p2μp2νE2δxx2t,E15

where we have omitted the tensor tμνG, which is associated with the massless, gravitational field distributed all over space and proportional to the gravitational constant G[32].

After the insertion of Eq. (15) into Eq. (11), we get [33]

Ptotalωt=P1ωt+P12ωt+P2ωt,E16

where (tt=τ)

P1ωt=r1πsin2ωr1sinω0τ/2sinω0τ/2cosωτ×E12ω02r12cosω0τ12m142E12,E17
P2ωt=r2πsin2ωr2sinω0τ/2sinω0τ/2cosωτ×E22ω02r22cosω0τ12m242E22,E18
P12ωt=4πsinωr12+r22+2r1r2cosω0τ1/2r12+r22+2r1r2cosω0τ1/2cosωτ×E1E2ω02r1r2cosω0τ+12m12m222E1E2.E19

### 1.4 The quantum energy loss of the binary

Using the following relations

ω0τ=φ+2πl,φππ,l=0,±1,±2,E20
l=l=cos2πlωω0=l=ω0δωω0l,E21

we get for Piωt, with ωbeing restricted to positive:

Piωt=l=1δωω0lPilωt.E22

Using the definition of the Bessel function J2lz

J2lz=12πππcoszsinφ2cos,E23

from which the derivatives and their integrals follow, we get for P1land P2lthe following formulae:

Pil=2ri(Ei2vi21mi42Ei202vildxJ2lx+ 4Ei2vi21vi2J2l2vil+4Ei2vi4J2l2vil),i=1,2.E24

Using r2=r1+ϵ, where ϵis supposed to be small in comparison with radii r1and r2, we obtain

r12+r22+2r1r2cosφ1/22acosφ2,E25

with

a=r11+ϵ2r1.E26

So, instead of Eq. (19), we get

P12ωt=2sin2ωacosω0τ/2cosω0τ/2]cosωτ×E1E2ω02r1r2cosω0τ+12m12m222E1E2.E27

Now, we can approach the evaluation of the energy loss formula for the binary from the power spectral of Eqs. (24) and (27). The energy loss is defined by the relation

dUdt=Pω=i,lδωω0lPil+P12ω=ddtU1+U2+U12.E28

From [34] we have Kapteyn’s formula:

l=1J2l2lvl2=v22.E29

After differentiating the last relation with respect to v, we have

l=1lJ2l2lv=0.E30

From [34] we learn other Kapteyn’s formulae:

l=12lJ2l2lv=v1v22,E31

and

l=1l02lvJ2lxdx=v331v23.E32

So, after the application of Eqs. (30), (31) and (32) to Eqs. (24) and (28), we get

dUidt=Gmi2vi3ω03ri1vi2313vi215.E33

Instead of using Kapteyn’s formulae for the interference term, we will perform a direct evaluation of the energy loss of the interference term by the ω-integration in (27) [35]. So, after some elementary modification in the ω-integral, we get

with

A=G,B=E1E2,C=v1v2,D=m12m222E1E2.E35

Using the definition of the δ-function and its derivative, we have, instead of Eq. (34), with v=aω0

dU12dt=Aω0πdxBCcosx+12Dcosx/2×δx2vcosx/2δx+2vcosx/2.E36

According to the Schwinger article [36], we express the delta function as follows:

δx±2vcosx/2=n=0±2vcosx/2nn!ddxnδx.E37

Then

δx±2vcosx/2=n=0±2vcosx/2nn!ddxn+1δx=E38

and it means that

δx+2vcosx/2δx2vcosx/2cosx/2=2n=12v2n1cosx/22n12n1!ddx2nδxE39

Now, we can write Eq. (36) in the following form after some elementary operations:

dU12dt=Aω0πdxBCcosx+12D×2n=12v2n1cosx/22n12n1!ddx2nδx,E40

where BCcosx+12Dcan be written as follows:

BCcosx+12D=4BC2(cos4x/2+4CB4BC2(cos2x/2+BC22CB+BD.E41

After the application of the per partes method, we get from Eq. (40) the following mathematical object:

dU12dt=2A4BC2ω0πdxδxn=1ddx2n2v2n1cosx/22n+22n1! 2A4CB4BC2ω0πdxδxn=1ddx2n2v2n1cosx/22n2n1! 2ABC22CB+BDω0πdxδxn=1ddx2n2v2n1cosx/22n12n1!.E42

We get after some elementary operations δfx=f0

J1=n=1ddx2n2v2n1cosx/22n+22n1!x=0=n=0fnv2n=Fv2,E43
J2=n=1ddx2n2v2n1cosx/22n2n1!x=0=n=0gnv2n=Gv2E44

and

J3=n=1ddx2n2v2n1cosx/22n12n1!x=0=n=0hnv2n=Hv2E45

where f,g,h,F,G,Hare functions which must be determined.

So we get instead of Eq. (41) the following final form:

dU12dt=2A4BC2ω0πGv22A4CB4BC2ω0πFv2 2A2CB+BC2+BDω0πHv2E46

Let us remark that we can use simple approximation in Eq. (41) as follows: cosx/22n+2cosx/22,cosx/22ncosx/22,cosx/22n1cosx/22. Then, after using the well-known formula

ddx2ncos2x/2=12cosx+πnE47

and

12cosx+πnx=0=121n.E48

So, instead of Eq. (46), we have

dU12dt=Aω0π2BC+BC2+BDn=12v2n11n2n1!.E49

## 2. Energy shift of H-atom electrons due to the black hole thermal bath

We here determine the electromagnetic shift of energy levels of H-atom electrons by calculating an electron coupling to the black hole thermal bath. The energy shift of electrons in H-atom is determined in the framework of nonrelativistic quantum mechanics.

The Gibbons-Hawking effect is the statement that a temperature can be associated to each solution of the Einstein field equations that contain a causal horizon.

Schwarzschild space-time involves an event horizon associated with temperature Tof a black hole of mass M. We consider here the influence of the heat bath of the Gibbons-Hawking photons on the energy shift of H-atom electrons.

The analogical problems are solved in the scientific respected journals. There is a general conviction of an analogy between the black hole and the hydrogen atom. Corda [37] used the model where Hawking radiation is a tunneling process. In his article the emission is expressed in terms of the black hole quantum levels. So, the Hawking radiation and black hole quasinormal modes by Corda [38] are analogical to hydrogen atom by Bohr.

In this model [39] the corresponding wave function is written in terms of a unitary evolution matrix. So, the final state is a pure quantum state with no information loss. Black hole is defined as the quantum systems, with discrete quantum spectra, with Hooft’s assumption that Schrödinger equations are universal for all universe dynamics.

Thermal photons by Gibbons and Hawking are blackbody photons, with the Planck photon distribution law [40, 41, 42], derived from the statistics of the oscillators inside of the blackbody. Later Einstein [43] derived the Planck formula from the Bohr model of atom where photons and electrons have the discrete energies related with the Bohr formula ω=EiEf, Ei,Efbeing the initial and final energies of electrons.

Now, we determine the modification of the Coulomb potential due to blackbody photons. At the start, the energy shift in the H-atom is the potential V0x, generated by nucleus of the H-atom. The potential at point V0x+δxis [44, 45]

V0x+δx=1+δx+12δx2+V0x.E50

The average of the last equation in space enables the elimination of the so-called the effective potential:

Vx=1+16δxT2Δ+V0x,E51

where δxT2is the average value of the square coordinate shift caused by the thermal photons. The potential shift follows from Eq. (51):

δVx=16δxT2ΔV0x.E52

The shift of the energy levels is given by the standard quantum formula [44]:

δEn=16δxT2ψnΔV0ψn.E53

In case of the Coulomb potential, which is the case of the H-atom, we have

V0=e24πx.E54

Then for the H-atom we can write

δEn=2π3δxT2e24πψn02,E55

where we used the following equation for the Coulomb potential

Δ1x=4πδx.E56

The motion of electron in the electric field is evidently described by elementary equation:

δx¨=emET,E57

which can be transformed by the Fourier transformation into the following equation

δx2=12e2m2ω4E2,E58

where the index ωconcerns the Fourier component of the above functions.

Using Bethe idea [46] of the influence of vacuum fluctuations on the energy shift of electron, the following elementary relations were applied by Welton [45], Akhiezer et al. [44] and Berestetzkii et al. [47]:

12Eω2=ω2,E59

and in case of the thermal bath of the blackbody, the last equation is of the following form [48]:

E2=ϱω=ω3π2c31eωkT1,E60

because the Planck law in (60) was written as

ϱω=Gω<Eω>=ω2π2c3ωeωkT1,E61

where the term

<Eω>=ωeωkT1E62

is the average energy of photons in the blackbody and

Gω=ω2π2c3E63

is the number of electromagnetic modes in the interval ω,ω+.

Then,

δx2=12e2m2ω4ω3π2c31eωkT1,E64

where δx2involves the number of frequencies in the interval ωω+.

So, after some integration, we get

δxT2=ω1ω212e2m2ω4ω3π2c3eωkT1=12e2m2π2c3Fω2ω1,E65

where Fωis the primitive function of the omega-integral with

1ω1eωkT1,E66

which is not elementary, and it is not in the tables of integrals.

Frequencies ω1and ω2can be determined from the field of thermal photons. It was performed for the Lamb shift [44, 47] caused by the interaction of the Coulombic atom with the field fluctuations. The Bethe-Welton method is valid here too and so we take Bethe-Welton frequencies. It means an electron does not respond to the fluctuating field if the frequency is much less than the atom binding energy given by the Rydberg constant [49] ERydberg=α2mc2/2. So, the lower frequency limit is

ω1=ERydberg/=α2mc22,E67

where α1/137is so-called the fine structure constant.

The second frequency follows from the cutoff, determined by the neglection of the relativistic effect in our theory. So, we write

ω2=mc2.E68

If we express the thermal function in the form of the geometric series

1eωkT1=q1+q2+q3+..;q=eωkT,E69
ω1ω2q1+q2+q3+..1ω=lnω+k=1ωkTkk!k+.;q=eωkTE70

and the first thermal contribution is

Thermalcontribution=lnω2ω1kTω2ω1,E71

then, with Eq. (55)

δEn2π3e2m2π2c3lnω2ω1kTω2ω1ψn02,E72

where according to Sokolov et al. [50]

ψn02=1πn2a02E73

with

a0=2me2.E74

Let us only remark that the numerical form of Eq. (72) has deep experimental astrophysical meaning.

Haroche [51] and his group performed experiments with the Rydberg atoms in a cavity. We used here Gibbons-Hawking black hole for the determination of the energy shift of H-atom electrons in the black hole gas.

## 3. Velocity of sound in the black hole photon gas

We have seen that the black hole can be modeled by the blackbody, and it means that there is the velocity of sound in the Gibbons-Hawking black hole thermal bath. So, let us derive the sound velocity from the thermodynamics of photon gas and energy mass relation.

In order to be pedagogically clear, we start with the derivation of the speed of sound in the real elastic rod.

Let Abe the cross-section of the element Adxof a rod on the axis x. Let φxtbe the deflection of Adxat point xat time t. The shift of the Adxat point x+dxis evidently

φ+φxdx.E75

Now, we suppose that the force tension Fxtacting on the Adxof the rod is given by Hooke’s law:

Fxt=EAφx,E76

where Eis Young’s modulus of elasticity. We easily derive that

Fx+dxFxEA2φx2dx.E77

The mass of Adxis ϱAdx, where ϱis the mass density of the rod and the dynamical equilibrium is expressed by Newton’s law of force:

or

φttv2φxx=0,E79

where

v=Eϱ1/2E80

is the velocity of sound in the rod.

The complete solution of Eq. (79) includes the initial and boundary conditions. We suppose that Eq. (80) is of the universal validity also for gas in the cylinder tube. If ΔL/Lis the relative prolongation of a rod, then an analogue for the tube of gas is ΔV/V, FΔp, where Vis the volume of a gas and pis gas pressure. Then, the modulus of elasticity as the analogue of Eq. (76) is

E=dpdVV.E81

The sound in ideal gas is the adiabatic thermodynamic process with no heat exchange. This is the model of the sound spreading in the gas of blackbody photons. Such process is described by the thermodynamic equation:

pVκ=const,E82

where κis the Poisson constant defined as κ=cp/cv, with cp,cvbeing the specific heat under constant pressure and under constant volume.

After differentiation of Eq. (82), we get the following equation:

dpVκ+κVκ1dV=0,E83

or

dpdV=κpV.E84

After inserting Eq. (84) into Eq. (81), we get from Eq. (80) the so-called Newton-Laplace formula:

v=κpϱ,E85

with ϱbeing the gas mass density.

The equilibrium radiation density has the Stefan-Boltzmann form:

u=aT4;a=7,5657.1016JK4m3.E86

Then, with regard to the thermodynamic definition of the specific heat,

cv=uTV=4aT3.E87

Similarly, with regard to the general thermodynamic theory,

cp=cv+uVT+pVTp=cv,E88

because VTT=0for photon gas, and in such a way, κ=1for photon gas. According to the theory of relativity, there is a relation for mass and energy, namely, m=E/c2. At the same time, the pressure and the internal energy of the blackbody gas are related as p=u/3. So, in our case

ϱ=u/c2=aT4c2;p=u3.E89

So, after the insertion of formulae in Eq. (88) into Eq. (85), the final formula for the sound velocity in photon blackbody sea is the following:

v=cκ3=c33,E90

which was derived by Partovi [52] using the QED theory of the photon gas. We correctly derived v/c<1.

So, we have performed the derivation of the velocity of sound in the relic photon sea. It is not excluded that the relic sound can be detected by the special microphones of Bell Laboratories. If we use van der Waals equation of state or the Kamerlingh Onnes virial equation, the obtained results will be modified with regard to the basic results.

Our derivation of the light velocity in the blackbody photon gas was based on the classical thermodynamic model with the adiabatic process (δQ=0), controlling the spreading of sound in the gas. Partovi [52] derived additional radiation corrections to the Planck distribution formula and the additional correction to the speed of sound in the relic photon sea. His formula is of the form

vsound=188π2α22025TTe4c3,E91

where αis the fine structure constant and Te=5.9G Kelvin. We see that our formula is the first approximation in the Partovi expression.

There is the Boltzmann statistical theory of transport of sound energy in a gas [53]. After the application of this theory to the photon gas or relic photon gas, we can obtain results involving the cross-section of the photon-photon interaction [47]:

σγγ=4,7α4cω2;ωmc2,E92

and

σγγ=97310125πα2re2ωmc26;ωmc2,E93

where re=e2/mc2=2,818×1013cm is the classical radius of electron and α=e2/cis the fine structure constant with numerical value 1/α=137,04.

## 4. Discussion and summary

We have derived the spectral density of gravitons and the total quantum loss of energy of the black hole binary. The energy loss is caused by the emission of gravitons during the motion of the two black hole binaries around each other under their gravitational interaction. The energy loss formulae of the production of gravitons are derived here by the Schwinger method. Because the general relativity and theory of gravity do not necessarily contain the last valid words to be written about the nature of gravity and it is not, of course, a quantum theory [21], they cannot give the answer on the production of gravitons and the quantum energy loss, respectively. So, this article is the original text that discusses the quantum energy loss caused by the production of gravitons by the black hole binary system. It is evident that the production of gravitons by the binary system forms a specific physical situation, where a general relativity can be seriously confronted with the source theory of gravity.

This article is an extended version of an older article by the present author [33], in which only the spectral formulae were derived. Here we have derived the quantum energy loss formulae, with no specific assumption concerning the strength of the gravitational field. We hope that future astrophysical observations will confirm the quantum version of the energy loss of the binary black hole.

In the next part of the chapter, the electromagnetic shift of energy levels of H-atom electrons was determined by calculating an electron coupling to the Gibbons-Hawking electromagnetic field thermal bath of the black hole. The energy shift of electrons in H-atom is determined in the framework of nonrelativistic quantum mechanics.

In the last section, we have determined the velocity of sound in the blackbody gas of photons inside of the black hole. Derivation was based on the thermodynamic theory of the photon gas and the Einstein relation between energy and mass. The spectral form for the n-dimensional blackbody was not here considered. The text is based mainly on the author articles published in the international journals of physics [33, 54, 55].

There is the fundamental problem concerning the maximal mass of the black hole. The theory of the space–time with maximal acceleration constant was derived by authors [56, 57]. In this theory the maximal acceleration constant is the analogue of the maximal velocity in special theory of relativity. Maximal acceleration determines the maximal black hole mass where the mass of the black hole is restricted by maximal acceleration of a body falling in the gravity field of the black hole.

Another question is what is the relation of our formulae to the results obtained by LIGO (Laser Interferometer Gravitational-Wave Observatory)? LIGO is the largest and most sensitive interferometer facility ever built. It has been periodically upgraded to increase its sensitivity. The most recent upgrade, Advanced LIGO (2015), detected for the first time the gravitational wave, with sensitivity far above the background noise. The event with number GW150914 was identified with the result of a merger of two black holes at a distance of about 400 Mpc from Earth [58]. Two additional significant detections, GW151226 and GW170104, were reported later. We can say that at this time it is not clear if the LIGO results involve information on the spectrum of gravitons calculated in this chapter.

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Miroslav Pardy (January 30th 2019). The Black Hole Binary Gravitons and Related Problems, New Ideas Concerning Black Holes and the Universe, Eugene Tatum, IntechOpen, DOI: 10.5772/intechopen.82659. Available from:

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