Gravitational-magnetic-electric fields interaction

Gravitational-magnetic-electric fields interaction Yin Zhu (朱寅) Agriculture Department of Hubei Province, Wuhan, China [email protected] (April 24, 2017)

Abstract: From the observations that gravity has an action on light and strong magnetic field makes gravitational field varied, describing the gravitational redshift with graviton-photon interaction, relating the energy of the photons and gravitons to the energy density of the fields, we obtained equations, ∆𝑔 = √𝑓𝐺/𝜇0 ∆B and ∆𝑔 = √𝑓G𝜀0∆E, for the interaction between the gravitational and magnetic fields and between gravitational and electric ones, respectively. In the equations, the variation of gravitational acceleration by a magnetic or an electric field is determined respectively with the variation of magnetic flux density B or the electric field intensity E and the parameter ƒ which determines the variation of the frequency of photon by gravitational redshift. Treating the gravitational redshift on the surface as a constant, we obtained two Tables to show that as ∆B is varied from 1 T to 103 T, the weight of 1kg of a body in this field should be varied from 6.13×10-5g to 6.13×10-2g. Now, we can make a magnetic field with the strength of 2.8×103T. And, as ∆E is varied from 1kV/m to 104kV/m, the weight of 1kg of a body in this field should be varied from 6.53×10-11g to 6.53×10-7g. It indicates that the variation of gravitational field by a strong magnetic field could be easily measured.

Key words: Graviton-photon interaction; Gravitational-magnetic-electric fields interaction; Variation of weight by magnetic/electric field

1.

Introduction

The gravitational redshift, light bending by gravity and Shapiro time delay showed that gravity has an action on light. In 2016, it was observed that a strong magnetic field makes gravitational field varied from a neutron star surrounded by a very strong magnetic field (1013G).[1] Therefore, the observations indicate that, the gravitational and magnetic/electric fields can be interacted by each other. The Pound–Rebka experiment[2,3] showed that, in gravitational redshift, the frequency of a photon is varied by gravity for that this photon absorbed/emitted an energy of gravitational field. From Pound–Rebka experiment, it was presented that, according to the law of conservation of energy, the energy of the graviton is accordingly varied by the photon and the electromagnetic field is a good tool to vary the gravitational one.[4-6] And, because a magnetic field is stronger than the gravitational one, the variation of a gravitational field by a strong magnetic field can be measured.[7] Here, by describing the gravitational redshift with quantum field theory, according to the law of conservation of energy, we obtained equations for the interaction between gravitational and magnetic/electric fields.

2.

Graviton-photon interaction and gravity varied by magnetic/electric field

It is known that, in the gravitational redshift, the frequency 𝜔𝛾 of a photon is varied with

𝜔𝛾𝑟 = 𝜔𝛾𝑒 √1 − 𝑟𝑠 /𝑅𝑒

(1)

Where 𝜔𝛾𝑒 is the frequency of the light on the surface of the Earth, 𝜔𝛾𝑟 is the frequency at distance r > 𝑅𝑒 (i.e., it was shifted by the gravity), 𝑟𝑠 =

2GM 𝑐2

is the Schwarzschild radius, 𝑅𝑒 is the radius of the Earth.

The frequency of a photon varied by gravity is:

D𝜔𝛾 = 𝜔𝛾𝑟 − 𝜔𝛾𝑒 = 𝑓𝜔𝛾𝑒

(2)

Please note the symbol D. It is different from ∆. Here, D only denotes the variation by the gravitational redshift. And, 𝑓 = 1 − √1 − 𝑟𝑠 /𝑅𝑒

B2

The energy density of a magnetic field is U𝛾 =

2μ0

(3)

, where B and μ0 are the magnetic flux density and the

magnetic constant respectively. In a volume V, the energy of this field is

E𝛾 =

B2 2μ0

𝑉

(4)

In the same V, the total energy of the photons making up of the energy of this field is E𝛾 = nh𝜔𝛾 , where 𝜔𝛾 is the frequency of a photon, n is the numbers of the photons in the volume V. Therefore, there is,

nh𝜔𝛾 =

B2 2μ0

𝑉

(5)

Eq.(5) shows that, the energy of a magnetic field in a volume V is the sum of the energy of all the photons in this volume.

In Eq.(5), as taking V as a unit volume, n, h and 𝜇0 are constant. Thus, 𝜔𝛾 is only determined with 𝐵2 .Therefore, we have

𝜔𝛾 =

𝑉 2𝑛ℎ𝜇0

𝐵2

(6)

Or ∆𝜔𝛾 =

𝑉 2𝑛ℎ𝜇0

∆𝐵2

(7)

Eq.(7) means that the variation of ∆𝜔𝛾 is only determined with ∆𝐵2 .

As shown in Fig.1, in a given volume V, the original energies of the magnetic and gravitational field are respectively E𝛾 and E𝑔 . The total energy is 𝐸𝑇 = E𝛾 + 𝐸𝑔 . Because of the gravitational redshift, the energy of this magnetic field in the volume V is varied to E𝛾 + D𝐸𝛾 . The energy of the gravitational field in V is accordingly varied to E𝑔 − D𝐸𝑔 . The total energy is 𝐸𝑇 = (E𝛾 + D𝐸𝛾 ) + (𝐸𝑔 − 𝐷𝐸𝑔 ). For the law of conservation of

energy, there is D𝐸𝛾 = 𝐷𝐸𝑔 .

𝐸𝑇 = E𝛾 + 𝐸𝑔

𝐸𝑇 = (E𝛾 + D𝐸𝛾 )

+(𝐸𝑔 − 𝐷𝐸𝑔 )

A

B

Fig. 1. The variation of energy of magnetic and gravitational field in a given volume V. For the law of conservation of energy, although the energies of the gravitational and magnetic fields are varied by each other, in the same volume V, the total energy is still 𝐸𝑇 .

From Eqs.(5) and (7) we know, as E𝛾 is varied, 𝜔𝛾 also is varied. As pointed out in the above, ∆ is different from D. It means that, ∆𝜔𝛾 in Eq.(7) also can be varied by the gravitational redshift. And, from Eq. (2) and (3), there is

D𝜔𝛾 = 𝑓∆𝜔𝛾

(8)

And, D𝐸𝛾 = nhD𝜔𝛾 . Thus, there are two characteristics for the 𝐷𝐸𝑔 ：1) D𝐸𝛾 = 𝑛ℎ𝐷𝜔𝛾 = 𝐷𝐸𝑔 , and 2) the Pound–Rebka experiment[2,3] shows that a photon can absorb/emit an energy of ε𝛾 = hD𝜔𝛾 of the gravitational field. It means that the energy of the gravitational field absorbed/emitted by a photon is quantized. Therefore, as shown in Fig.2, it is clear, there is 𝐷𝐸𝑔 = 𝑛ℎ𝐷𝜔𝑔 , and

Dω𝑔 = D𝜔𝛾

(9)

It is noted that, 𝐷𝐸𝑔 = 𝑛ℎ𝐷𝜔𝑔 is accordant with the well-known theory in current physics that the Newtonian gravitational force is mediated by graviton.[8] In order to clarify this problem, we specially have the Supplementary Information 1.

𝐷𝐸𝑔

D𝐸𝛾

𝑛ℎ𝐷𝜔𝑔

𝑛ℎ𝐷𝜔𝛾

Fig.2. The redshift of a photon by graviton. The energy of gravitational field absorbed/emitted by a photon is quantized. Therefore, D𝐸𝛾 = 𝐷𝐸𝑔 and D𝐸𝛾 = 𝑛ℎ𝐷𝜔𝛾 clearly show that 𝐷𝐸𝑔 = 𝑛ℎ𝐷𝜔𝑔 and D𝜔𝑔 = D𝜔𝛾 .

Therefore, the total energy of gravitational field in V also can be expressed as E𝑔 = nh𝜔𝑔 .

We also can establish the energy density for a gravitational field in a given volume. Near the surface of the Earth, the gravitational field is analogous to a magnetic one. The gravitational lines are approximately parallelly directing outward the center of the Earth. Therefore, the energy density of a gravitational field in a given volume can be simply written as:

U𝑔 =

Where 𝑔 = G

𝑀 𝑅2

𝑔2

(10)

2G

, G is the Newtonian universe gravitational constant G.

The energy of the gravitational field in a given volume is made up of the sum of energy of all the gravitons nhω𝑔 . Just as the energy of a magnetic field in Eq.(5), we have

nh𝜔𝑔 =

𝑔2 2𝐺

𝑉

(11)

Therefore, when the frequency of this graviton is varied Dω𝑔 , the energy of the gravitational field in the

given volume V is varied with

nhDω𝑔 =

∆𝑔2

𝑉

(12)

∆𝑔2

(13)

∆B

(14)

2𝐺

From Eqs. (7), (8), (9) and (12), we have

𝑓 2𝜇0

∆𝐵2 =

1 2𝐺

Or

∆𝑔 = √𝑓

G 𝜇0

The deduction is also suitable for electric and magnetic fields. The energy density of an electric field is U𝑒 =

𝜀0 2

E 2 , where 𝜀0 is the permittivity of vacuum, E is the electric field intensity. Substituting the densities of

the electric and magnetic field into Eq.(13) we have

∆B = √𝜇0 𝜀0 ∆E

(15)

It is clear, we can obtain an equation for the gravitational and electric fields:

∆𝑔 = √𝑓G𝜀0∆E

(16)

In Eq.(14), in an given volume, because G and 𝜇0 are constant, √𝐺/𝜇0 = 2.28×10−2 kg −1 m𝐴, if ∆B is certain, ∆𝑔 is determined only with 𝑓 = 1 − √1 − 𝑟𝑠 /𝑅𝑒 . It means that, if we let ∆B = 1T, there is ∆𝑔 = 2.28×10−2 √𝑓ms −2 . And, according to the Pound–Rebka experiment,[8,9] we can select 𝑓 = 10−15 . In this case, as ∆B = 103 T, there is ∆𝑔 ≈ 7.2×10−7 ms −2 and

∆𝑔 𝑔

≈ 7.3×10−8 . It means that a weight of 1kg

of a body should be varied 7.3×10−5 g as this body is in a magnetic field with B = 103 𝑇. Now, we can make a stable magnetic field larger than 40 T[9] and the largest one with 2.8×103 T.[10] This variation can be easily measured.

It appears that 7.3×10−5 g is a very large variation. But, we can let the variation much larger for that we can

let 𝑓 = 1 − √1 − 𝑟𝑠 /𝑅𝑒 much larger.

∆𝑔 𝑔

≈ 7.3×10−8 is dependent on 𝑓 = 10−15 which was selected

simply according to the Pound–Rebka experiment.[2,3] It is a gravitational redshift by a height of 22.5m. But, Eq.(1) shows that, as a light with frequency of 𝜔𝛾0 is first produced on the surface of the Earth, it shall be redshifted with 𝜔𝛾𝑒 = 𝜔𝛾0 √1 − 𝑟𝑠 /𝑅𝑒 . i.e., 𝜔𝛾𝑒 can be determined with the radius of the Earth. Determined with the radius of the Earth, there is 𝑓 = 1 − √1 − 𝑟𝑠 /𝑅𝑒 ≈ 6.95×10−10 . In previous, 𝑓 is stressed. But, on the surface of the Earth, the gravitational redshift is almost same. Therefore, observed on the surface of the Earth, 𝑓 ≈ 6.95×10−10 can be treated as a constant. So, from Eq.(14), there is

∆𝑔 ≈ (6.01×10−7 kg −1 m𝐴)∆B

(17)

In Eq.(15), (6.01×10−7 kg −1 m𝐴) is constant, ∆g is only determined with ∆B. So, we have Table 1.

Table 1: The magnetic field strength and the variation of gravity ∆B (T)

∆𝑔 (ms-2)

∆𝑔/𝑔

Variation of 1kg (g)

1

6.01×10-7

6.13×10-8

6.13×10-5

10

6.01×10-6

6.13×10-7

6.13×10-4

102

6.01×10-5

6.13×10-6

6.13×10-3

103

6.01×10-4

6.13×10-5

6.13×10-2

However, the data in Table 1 need be confirmed with experiment. But, it indicates that, the variation of gravitational field by a strong magnetic field could be easily measured.

Analogy to Eq. (17), for Eq. (16), we have

∆𝑔 ≈ (6.4×10−16 kg −1 s𝐴)∆E

And, we have Table 2.

(18)

Table 2: The electric field intensity and the variation of gravity ∆E (kV/m)

∆𝑔 (ms-2)

1

6.40×10-13

6.53×10-14

6.53×10-11

10

6.40×10-12

6.53×10-13

6.53×10-10

102

6.40×10-11

6.53×10-12

6.53×10-9

103

6.40×10-10

6.53×10-11

6.53×10-8

∆𝑔/𝑔

Variation of 1kg (g)

It is noted that, as ∆g is varied, 𝑓 = 1 − √1 − 𝑟𝑠 /𝑅𝑒 also is varied. Therefore, as ∆𝑔/𝑔 is large, the variation of 𝑓 need be considered. From Eq.(1) we know, on the surface of the Earth, 𝑟𝑠 /𝑅𝑒 can be rewritten as 2𝑔 𝑐2

𝑅𝑒 . Therefore, Eq.(3) can be rewritten as

𝑓 = 1 − √1 −

2𝑔 𝑐2

𝑅𝑒

(19)

From Eq.(16), as shown in Table 3, we have the variations of 𝑓 by ∆𝑔.

Table 3. The relationship between ∆𝑔/𝑔 and 𝒇 ∆𝑔/𝑔

0.2

0.4

0.6

0.8

0.9

0.99

0.999

0.9999

𝑓(×10-10)

5.58

4.18

2.79

1.39

6.97×10-1

6.97×10-2

6.97×10-3

6.97×10-4

Table 3 shows that, in Eq.(14), as ∆𝑔 𝑔

∆𝑔 𝑔

< 0.9, the variation of 𝑓 by ∆𝑔 is not remarkable. While as

> 0.9, the variation of 𝑓 by ∆𝑔 is very significant. It indicates that: in Eq.(14),1) As

is a direct ratio of ∆B. 2) As

∆𝑔 𝑔

> 0.9, much larger ∆B is needed to vary 𝑔.

∆𝑔 𝑔

< 0.9, ∆𝑔 almost

3.

Discussions and conclusions

It is emphasized, Eqs.(14) and (16) are valid for that the factor √𝑓𝐺/𝜇0 and √𝑓G𝜀0 make the dimensionalities for them valid. If future experiment shows that the two equations need be revised, the factor √𝐺/𝜇0 and √G𝜀0 need not be revised. Only a much more accurate and precession ƒ is needed. And, it is assumed that in a given volume, the numbers of the graviton and photon are same. If the number of gravitons 𝑛𝑔 𝑛

𝑛

G

is different from that of photons nγ , Eq.(14) and (16) need be revised as ∆𝑔 = √𝑓 n𝑔 𝜇 ∆B and ∆𝑔 = √𝑓 𝑛𝑔 G𝜀0 . γ

0

𝛾

It is generally thought that nothing can escape from a black hole. Of course, a magnetic field also cannot escape from it. Intuitionally, it is easy to image that a very strong gravitational field can vary a magnetic field, as well as a very strong magnetic field can vary a gravitational one. In the Supplementary Information 2, we discussed 19 experiments testing the variation of gravity by superconductor and strong electric and magnetic field. These experiments may be possible evidence for Eqs.(14) and (16).

Eqs.(14), (15) and (16) show that: 1) The variation among electric, magnetic and gravitational fields by each other can be described with an analogous equation. 2) Maxwell equations show c =

1 √𝜇0 𝜀0

. Therefore, from

Eqs.(14), (15) and (16) we know, there are some of relationships among physical constant of 𝜀0 , 𝜇0 and G. 3) 𝜀0 and 𝜇0 were called the vacuum permittivity and vacuum permeability. It is thought that they reflect some features of the vacuum.[11-13] In quantum mechanics, a vacuum would not be “empty” but with vacuum fluctuations. Therefore, G also may reflect some of features of the vacuum. 4) In the microworld, the features of vacuum reflected by 𝜀0 and 𝜇0 may be same while in the macroworld they expressed as electric and magnetic fields. Because a graviton is different from a photon, G may be a different reflection of the features of the vacuum in the microworld.

From Eqs(14), (15) and (16) we have:

1 √𝜇0 𝜀0

√G𝜀0 =

= 3×108

m s

C 2.43×10−11 kg −2 C m

{√𝐺/𝜇0 = 2.28×10

(20)

kg s

Where C is the electric charge of Coulomb. Therefore, in Eq.(20), √G𝜀0 indicates the relationship Q

between the charge and mass, i.e., √G𝜀0 → , where Q and M are electric charge and mass respectively. Under M the condition that G and 𝜀0 reflect some of features of the vacuum, it means that the charge and mass reflect some Q

characteristic of the vacuum. While √𝐺/𝜇0 → v. It reflects the motion of M

Q

.

M

Now, we cannot know the exact meanings of Eq.(20). Now, it is generally thought that vacuum has some features in both gravity and quantum mechanics. There has been many exploring for it.[14,15] Fortunately, the constant 𝜀0 , 𝜇0 and G are that that have well shown some aspects of these features in experiment and theory. Therefore, Eq.(20) is a solid ground to further understand the vacuum. And, the relationship among charge, mass and vacuum may be known from Eq.(20).

Electromagnetic and gravitational fields are two very important fields. It is very significant to know the interaction between them. Eqs.(14) and (16) shows that, the gravitational field can be manipulated with electromagnetic field easily. Let Eq.(14) varied with time,

𝑔(t) = √𝑓

G 𝜇0

B(t)

(21)

The gravitational field could be manipulated as that an electromagnetic field is done with the Faraday’s law of induction. For example, gravitational communication[16] should be possible by a varying 𝑔(𝑡) in Eq.(21). And, a GemDrive[17] could be designed by varying the gravitational field with a strong magnetic/electric field in one part of a spacecraft to produce a gravitational potential difference between two parts of a spacecraft which propels this spacecraft to move. It shall lead to use the gravitational field. In astronomical observation, many gravitational and magnetic fields are very strong. Eq.(14) is useful for this observation.

Graviton-photon interaction was studied long time ago.[18,19] The graviton is usually detected with the gravitational radiation.[20,21] But, in recent, the graviton in the interactive field begun to study.[22,23] Our work could be a new way to study the graviton with the interactive force. (please see the Supplementary Information 1)

From Eq.(15), we can obtain the Faraday’s law of induction. From Eqs.(14) and (15), we know, the gravitational field is more analogous to the magnetic one. But, experiment is needed to exactly know the Faraday’s law of induction for Eq.(15). Because Eq.(16) is analogous to Eq.(15) , by analogous to the Maxwell equations, it could be concluded that there may be ∇×𝑔 = √𝑓G𝜀0

∂E ∂t

. But, we have not had the case to conclude the equation

about the 𝑔 and B in Eq.(14). So, observations and experiment are needed to exactly know how 𝑔 and B are varied by each other in Eq.(14).

For Eqs.(14) and (16), the two questions are very important: 1) Is the result of that the directions of the gravitational and magnetic/elecgric fields are parallel different from that the directions are vertical to each other? 2) Is the result of that the two fields are moving relatively different from that are static? These questions only can be known with experiment. Therefore, the two questions provide a new way to further know the gravitational field.

Acknowledgements: The author thanks Dr. Valery Timkov very much for his pointing out that in the previous version the calculation of the gravitational redshift on the surface of the Earth is wrong.

References [1] R. P. Mignani, V. Testa, D. Gonz´ alez Caniulef, R. Taverna, et al, Evidence for vacuum birefringence from the first optical polarimetry measurement of the isolated neutron star RX J1856.5−3754, Mon. Not. R. Astron. Soc. 465 (1), 492-500 (2017) [2] Pound, R. V., Rebka Jr. G. A., Gravitational Red-Shift in Nuclear Resonance, Phys. Rev. Lett. 3 (9), 439–441 (1959) [3] Pound, R. V.; Snider J. L., Effect of Gravity on Nuclear Resonance, Phys. Rev. Lett. 13 (18), 539–540(1964) [4]https://www.researchgate.net/publication/278111169_Graviton-photon_interaction_in_the_gravitational_redshift?ev=prf_pub

[5] https://www.researchgate.net/publication/301682815_Method_to_detect_gravitational_field_and_graviton?ev=prf_pub [6] https://www.researchgate.net/publication/304151459_Observation_of_Graviton_and_Ways_to_Manipulate_Gravitational_Field [7]https://www.researchgate.net/publication/311049619_An_outline_to_detect_graviton_and_attractive_force [8] Mistry N., A brief introduction to particle physics, Laboratory for Elementary Particle Physics, Cornell University [9] https://nationalmaglab.org/ [10] B.A. Boyko, A.I. Bykov, M.I. Dolotenko, N.P. Kolokolchikov, et al, With record magnetic fields to the 21st Century, IEEE Xplore (2002) [11] Leuchs G., Villar A.S. & Sánchez-Soto L. L., The quantum vacuum at the foundations of classical electrodynamics, Appl. Phys. B 100, 9–13 (2010) [12] Leuchs G. and Sa´ nchez-Soto L. L., A sum rule for charged elementary particles, Eur. Phys. J. D 67, 57 (2013) [13] Urban M., Couchot F ., Sarazin X., Djannati-Atai A., The quantum vacuum as the origin of the speed of light, Eur. Phys. J. D 67, 58 (2013) [14] Puthoff H. E., Advanced Space Propulsion Based on Vacuum (Spacetime Metric) Engineering, J. of the British Interplanetary Society, 63, 82-89 (2012) [15] Minami Y., Space Propulsion Physics toward Galaxy Exploration, J Aeronaut Aerospace Eng, 4, 2(2015) [16] Minami Y., The Latest Study of Gravitational Wave Communication System, J. of Earth Sci. and Engin. 6, 164-176 (2016) [17] https://www.researchgate.net/publication/313315115_A_Design_of_GemDrive?ev=prf_pub

[18] Weinberg S. Gravitation and cosmology. New York, Wiley, (1972) chapter 10.8 [19] Dyson F., Is a graviton detectable? I. J. Modern Phy., A 28, 1330041–1 (2013) [20] Abbott B. P. (LIGO Scientific Collaboration and Virgo Collaboration) et al, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett., 116 (6), 061102 (2016) [21] Taylor J. H., Fowler L.A., Weisberg J.M., Measurements of general relativistic effects in the binary pulsar PSR 1913+16, Nature 277, 437-440 (1979) [22] Bjerrumbohr N. E., Donoghue J. F., Holstein B. R., et al. Bending of light in quantum gravity, Phys. Rev. Lett., 114(6), 061301 (2015) [23] Ivanov M. A., A quantum gravitational model of redshifts, arXiv: 0409111

Supplementary Information 1

The analogy between gravitational field and electric field Yin Zhu (朱寅) Agriculture Department of Hubei Province, Wuhan, China [email protected] (March 18, 2017)

Abstract: The analogy between gravitational field and electric field in the current theory of gravity is simply reviewed. A well-known theory that the gravitational force is mediated by graviton is emphasized. It is shown that, the wavelength of graviton in the interactive (near) field is different from that in the radiative (far) field. The graviton in near field need be studied more deeply and could be detected. Studying and detecting the graviton in both interactive and radiative fields, the characteristics of graviton could be known well. Key words: Graviton—Interactive field—Radiative field

In current theory, gravity is usually described with the gravitational field. In the field modal, matter moves in certain ways in response to the curvature of spacetime which is described with Einstein’s general relativity. Usually, the Einstein’s field equations are studied and explained by analogous to the equations of electromagnetic field. Under the condition of weak field approximation, a gravitational field is studied by analogous to a moving charge. And, the retarded potential is used usually for the two fields.[1-3]

In the electrodynamics,[1] for a moving charge, the retarded potential can be expressed as the Líenard-Wiechert potential

𝑒

Φ(r, t) = |𝑟−𝑟(𝑡 ′)|−𝑣[𝑟−𝑟(𝑡 ′)]/𝑐

(1)

Where e is the charge, v is the velocity of the charge, c is the speed of light, t’ is the retarded time which is given by

t′ = t −

|𝑟−𝑟(𝑡 ′)|

(2)

𝑐

And

|𝑟 − 𝑟(𝑡 ′ )| = R = √[𝑥 − 𝑥(𝑡 ′ )]2 + [𝑦 − 𝑦(𝑡 ′ )]2 + [𝑧 − 𝑧(𝑡 ′ )]2

(3)

From Eqs.(1), (2) and (3), let β = v/c, γ2 = 1 − β2 , k = 1 − 𝐧 ∙ 𝛃 and n is a unit vector, we have

E(r, t) =

e γ2 k3 R2

(𝐧 − 𝛃) +

= interactive term +

e k3 Rc

̇ 𝐧×[(𝐧 − 𝛃)×𝛃]

(4)

radiative term

Eq.(4) shows that, the electric field of a moving charge is made up of two parts. The interactive part is corresponding to the Coulomb force. The radiative term is the electromagnetic radiation. It is corresponding to the electromagnetic wave. (It is noted that, a moving charge always produces a magnetic field. Here, the magnetic field is not considered.)

In electrodynamics, the Líenard-Wiechert potential is well-established. For example, it can be used to accurately describe a moving charge and to make the Maxwell equations linear in matter.

In the theory of gravity, in the weak field approximation, the retarded potential usually is written as[2]

ℎ𝜇𝜈 (𝑟, 𝑡) =

1 4𝜋

∙

16𝜋𝐺 𝑐2

1

∫ |𝑟−𝑟(𝑡 ′)| 𝑇𝜇𝜈 [𝑟(𝑡 ′ ), 𝑡 −

|𝑟−𝑟(𝑡 ′)| 𝑐

]𝑑 3 𝑥′

(5)

Where 𝑇𝜇𝜈 is the energy-momentum tensor.

From Eq.(5), we can obtain the Líenard-Wiechert potential field for a source m moving with velocity v. And an equation analogous to Eq.(4) for the moving mass can be arrived at:

g(r, t) = −Gm { =

α𝐧+[(2γ2 +1)k−4]γ𝛃 γ2 k3 R2

interactive term

+

̇ ̇) (𝐧∙𝛃̇)(αn−4γ𝛃)+k(α𝐧−4γ̇ 𝛃−4γ𝛃 ck3 R

+

}

(6)

radiative term

Where α ≡ 2γ − 1/γ.

It is clear, Eq.(6) is analogous to Eq.(4). The term of force and the term of

1 R

1 R2

is corresponding to the Newtonian gravitational

is corresponding to the gravitational radiation.

It is generally introduced that, from Eq.(5), the gravitational radiation of the binary star emitted in x direction is

ℎ𝜇𝜈 (𝑡, 𝑟) =

4𝑀𝐺𝑅 2 𝜔2 𝑐4𝑟

(

0 0 0 0

0 0 0 0

0 0 0 0 𝑐𝑜𝑠2𝜔𝑡𝑟 0 ) 0 −𝑐𝑜𝑠2𝜔𝑡𝑟

(7)

By researching the gravitational radiation of binary star PSR 1913+16, Hulse and Taylor[5] win the 1993 Nobel Prize in physics.

From Eqs.(1)-(6), it is clearly shown that the gravitational field of a moving mass is very analogous to the electric field of a moving charge. Both of them are made up of two terms: The interactive and radiative terms. It is well known that an electric field of a charge is made up of the near and far fields which are defined with the wavelengths. If the source dimensions are order of d and the wavelength is λ = 2πc/ω, there are three regions for this field:

Near (interactive) field:

d