## 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 B**1913+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 B**1913+16** in a binary, in 1974, and this is now considered as the best general relativistic laboratory [21].

Pulsar PSR B**1913+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

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 B**1913+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 B**1913+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]:

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

In the flat space-time, the field of gravitons is described by the amplitude (1) with the action (

where the dimensionality of

where

and the relativistic energy is defined by the known relation

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

Symbol

### 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]:

where the so-called power spectral function

where

which gives

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

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

In the case of the binary system with masses

with

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

where we have omitted the tensor

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

where (

### 1.4 The quantum energy loss of the binary

Using the following relations

we get for

Using the definition of the Bessel function

from which the derivatives and their integrals follow, we get for

Using

with

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

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

From [34] we have Kapteyn’s formula:

After differentiating the last relation with respect to

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

and

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

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

with

Using the definition of the

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

Then

and it means that

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

where

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

We get after some elementary operations

and

where

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

Let us remark that we can use simple approximation in Eq. (41) as follows:

and

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

## 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

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

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

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

where

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

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

Then for the H-atom we can write

where we used the following equation for the Coulomb potential

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

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

where the index

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]:

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

because the Planck law in (60) was written as

where the term

is the average energy of photons in the blackbody and

is the number of electromagnetic modes in the interval

Then,

where

So, after some integration, we get

where

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

Frequencies

where

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

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

and the first thermal contribution is

then, with Eq. (55)

where according to Sokolov et al. [50]

with

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

Now, we suppose that the force tension

where

The mass of

or

where

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

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:

where

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

or

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

with

The equilibrium radiation density has the Stefan-Boltzmann form:

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

Similarly, with regard to the general thermodynamic theory,

because

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:

which was derived by Partovi [52] using the QED theory of the photon gas. We correctly derived

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 (

where

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]:

and

where

## 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.