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Yin Zhu

Agriculture Department of Hubei Province, Wuhan, China

Abstract From the Liénard-Wiechert potential in both the gravitational and the , it is shown that the speed of propagation of the () can be tested by comparing the measured speed of gravitational with the measured speed of Coulomb force. PACS: 04.20.Cv; 04.30.Nk; 04.80.Cc

Fomalont and Kopeikin [1] in 2002 claimed that to 20% accuracy they confirmed that the is equal to the speed of in . Their was immediately contradicted by Will [2] and other several physicists. [3-7] Fomalont and Kopeikin [1] accepted that their measurement is not sufficiently accurate to detect terms of order , which can experimentally distinguish Kopeikin interpretation from Will interpretation. Fomalont et al [8] reported their in 2009 and claimed that these measurements are more accurate than the 2002 VLBA experiment [1], but did not point out whether the terms of order have been detected. Within the post-Newtonian framework, several metric theories have studied the and propagation of gravitational waves. [9] For example, in the Rosen bi-metric theory, [10] the difference between the speed of gravity and the could be tested by comparing the arrival times of a gravitational and an electromagnetic wave from the same : a . Hulse and Taylor [11] showed the indirect evidence for gravitational radiation. However, the gravitational waves themselves have not yet been detected directly. [12] In electrodynamics the speed of electromagnetic waves appears in Maxwell equations as c = √휇0휀0, no such constant appears in any theory of gravity. Currently, our only experimental insights into the speed of the gravitational waves are derived from of the binary PSR 1913+16. Measurements of its agree at the 1% level with (GR), assuming that is emitted in the form of gravitational radiation. [13,14] At the low and in the weak-field limit, the Newtonian gravitational force can be the reflection of the curvature. [15] In this letter, it is suggested that, in principle, the speed of gravity can be studied or measured with the Newtonian gravitational force.

In the Newtonian theory of gravity, the gravitational interaction is instantaneous. Attempting to combine a finite gravitational speed with Newton's theory, Laplace [16] concluded that the speed of gravity is by analyzing the of the in 1805. Lightman [17] et al determined that with the purely central force, if the speed of gravity is finite, the in the two-body system no longer point toward the center of , and make

0 unstable. In 1998, Flandern [18] set a limit for the speed of gravity, based on a purely central

Email: [email protected]

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force of Newtonian theory from the observations of the solar system and binary . However, in Einstein’s general relativity the speed of gravity is postulated to be equal to the speed of light in vacuum. [19] Compared with the electromagnetic field, Carlip, [20] Marsh and Nissim-Sabat, [21] and Ibison et al [22] argued that in GR the gravitational interaction propagates at the speed of light, but the velocity-dependent terms in the interaction cancel the . Comparing the gravitational field with the electromagnetic field is a current way to understand and to study the speed of gravity [1, 10, 15, 20-22]. Here, I show that by comparing the measured speed of the Newtonian gravitational force with the measured speed of Coulomb force, some evidences for the speed of propagation of the gravitational field (waves) can be found. In GR, in the weak field limit, for the slowly moving mass, the gravitational field can be expressed as a perturbation of the -time metric. Adopted the units c=1, which can be written as

𝑔휇휈 = 휂휇휈 + ℎ휇휈, |ℎ휇휈| ≪ , (1)

where 휂휇푣 is the Minkowski metric, 휂휇푣 = 푑𝑖푎𝑔(− , + , + , + ), and ℎ휇푣 is the perturbation. Einstein’s field coupled to is ̅ ℎ휇푣 = − 6휋퐺푇휇푣, (2)

Where h̅ is the“trace-reversed” perturbation, ℎ̅ = ℎ − 휂 ℎ, and 푇 is the stress-energy tensor. μ푣 휇푣 휇휈 휇휈 휇휈 Solving Eq. (2) by defining the quadrupole moment tensor of the energy density of the source as

푖 푗 00 3 푞푖푗(푡) = ∫ 푥 푥 푇 (푡 , 푥)푑 x. In terms of the Fourier transform of the quadrupole moment, and transform back to t, the solution to Eq. (2) takes on the compact form

퐺 푑2푞 ℎ̅ (푡, 푥) = 푖푗 (푡 ), (3) 휇푣 푅 푑푡2 푚

where M is the mass of the moving source, R is the distance between the source and the observer, G is the ; and

푡푚 = 푡 − 푅, (4)

where 푡푚 is the time that the source mass is located at the retarded position, t is the time that the observer is affected by the corresponding potential. From Eq. (1), in a vacuum, we can recover the conventional relativistic wave equation

□ℎ휇푣 = , (5) where □ is the d’Alembertian operator. In the Newtonian limit, we have

ℎ푖푗 = − 훷훿푖푗, (6)

Where Φ is the conventional Newtonian potential, 훷 = −퐺푀 푅, 훿푖푗 is the Kronecker symbol. 휕2 The flat-space d’Alembertian operator has the form □ = − + 훻 , from Eqs. (5) and (6), we have 휕푡2 휕2훷 − ∇ 훷 = . (7) 휕푡2 Eq. (7) just is one of the d’Alembertian waves equation of the gravitational waves.

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Analogy with the of the electromagnetic field, letting the observer at the field point, and neglecting the speed of the source, from Eqs. (3), (4), (6) and (7), the retarded potential of the gravitational field can be written as [23] 푀(푡 훷(푥, 푡) = −퐺 푚), (8) 푅

푅 푡푚 = 푡 − . (9) 푐푔

where is the speed of gravity, = c. This deducing shows that, in the low speeds and weak-field limit, the curvature space-time [Eqs.(1-3)], gravitational waves [Eqs.(5-7)], the speed of gravity [Eqs.(8,9)], as well as the gravitational field, can be described and studied with the Newtonian . From Eqs. (8) and (9), treating the (or the Moon) as effectively point mass, its Liénard-Wiechert potential can be written as [2]

퐺푀 휙(r⃗, 푡) = , (10) |푟⃗−푟⃗푚(푡푚)|−푣⃗⃗·(푟⃗−푟⃗푚(푡푚)) 푐푔 where 푡푚 is “retarded” time given by the implicit equation

|푟⃗−푟⃗푚(푡푚)| 푡푚 = 푡 − , (11) 푐푔

휕푟⃗푚(푡푚) and |푟⃗ − 푟⃗푚(푡푚)| = 푅 = √[푥 − 푥푚(푡푚)] + [푦 − 푦푚(푡푚)] + [푧 − 푧푚(푡푚)] , ⃗ = . 휕푡푚

We can obtain Newtonian gravitational field strength from ℎ = 훻훷(푟⃗, 푡푚). The potential Φ is an explicit function of 푟⃗ and tm:

Φ = Φ(푟⃗, 푡푚). Thus, 휕Φ 훻훷(푟⃗, 푡푚) = 훻푟⃗훷(푟⃗, 푡푚) + 훻푡푚, 휕푡푚

휕푡 휕 |푟⃗−푟⃗ (푡 )| 푚 = (푡 − 푚 푚 ) 휕푡 휕푡 푐푔

휕푅 = − , 푐푔 휕푡

휕푅 휕푡 푅⃗⃗ where = − 푚 푛⃗⃗ ∙ ⃗, 푛⃗⃗ = , 휕푡 휕푡 푅

Substituting 휕푅 휕푡 into the 휕푡푚 휕푡 expression, we obtain 휕푡 푚 = = , 휕푡 ( −푛⃗⃗∙훽⃗⃗⃗) 푘

푣⃗⃗ where k = − 푛⃗⃗ ∙ 훽⃗ , 훽⃗ = . 푐푔

After a routine derivation, we can obtain the expression for the gravitational field: [20,24]

ℎ = 퐺푀 { (푛⃗⃗ − 훽⃗) + 푟푎푑𝑖푎푡𝑖푒 푡푒푟푚푠}, (12) 푘3훾2푅2 푣 where 훾 = − 훽 , β = . 푐푔

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The Newtonian gravitational field strength ℎ푠푡푟 is 푀 ℎ = 퐺 (푛⃗⃗ − 훽⃗) (13) 푠푡푟 푅2 푘3훾2 According to Carlip, [20] the terms (푛⃗⃗ − 훽⃗) could be interpreted, i.e., the retarded direction 푛⃗⃗ is linearly extrapolated toward the “instantaneous” direction; the gravitational directed toward the retarded position of the source is extrapolated toward its instantaneous position. In this case, “if a source abruptly stops moving at a point z(푠0), a test particle at position x will continue to accelerate toward the extrapolated position of the source until the time it takes for a to propagate from z(푠0) to x at light speed.”[20] In other words, during a solar/lunar when the Sun, Moon and are on the straight line EMS, observed at a point E on the Earth, the Sun is located at point S(r, t), but, the Newtonian force of the Sun on the Earth is from that the Sun was locating at point 푣 S′(푟푚, 푡푚), and 푡 − 푡푚 = 푅 ≈ 5 푠, 휃 = 푎푟 표푠 . where R is the distance between the Sun and the 푐푔 Earth, v is the speed of the Sun relative to the observer (which can be treated approximately as a uniform motion),

= c, θ is an angle between straight line EMS and ES’. Thus, Eqs. (11) and (13) mean that, during a solar/, the speed of the maximum Newtonian gravitational strength (force) along the line EMS is infinite. Assuming that P and E are two different points on the line EMS, from Eqs. (11) and (13) we know, the maximum Newtonian gravitational strength (force) of the Sun (or Moon) at points P(t) and E(t) emerge at the same time, t. However, the maximum strength (force) at point P(t) comes 푣 from the Sun (or Moon) located at point 푆푃(푟푃, 푡푃), 푡푃 = 푡 − 퐸푆푃 , and there is an angle 휃 = 푎푟 표푠 푐푔 between the lines EMS and 퐸푆푃, where 퐸푆푃 is a straight line connecting point E and 푆푃(푟푃, 푡푃). For the same reasons, we can obtain the same relationship between the points E(t) and 푆퐸(푟퐸, 푡퐸), as well as between the lines

EMS and 퐸푆퐸. Therefore, the speed of propagation of the gravitational field could be permitted to be = c. This conclusion is consistent with the observations of the absence of the gravity aberration in the solar system, which needs that the speed of the Newtonian gravitational force is infinite (or much fast than c under the assumption of pure central force [16-18]). This conclusion can be tested experimentally. If E and P are two points on the straight line EMS with a distance r between the two points, during a solar/lunar eclipse, the maximum Newtonian gravitational strength (force) from both the Sun and the Moon along the line EMS is unique with a unique direction along the line EMS. If the times of the maximum Newtonian gravitational strength (force) at points P and E have been measured, the speed of the Newtonian gravitational strength (force) along line EMS shall have been measured. In practical measurement, we can replace the Newtonian gravitational force by the . The tidal force 퐺푀푚 퐺푀푚 is the effect of the gravitational force. The Tidal force of the Sun on the Earth is 퐹 = − 푡푖푑푒 푅2 (푅+푟)2 (where M and m are respectively the mass of the Sun and Earth, R is the distance between the Sun and Earth, r is the radius of the Earth), the speed of the tidal force must be equal to the speed of the Newtonian gravitational force. The tidal force is very strong and easily detected (for example, it can make the sea level rise more than one meter). Observed from the Earth, the tidal force varies in a predictable manner, and maximizes when the tidal field is perpendicular to the surface of the Earth. Thus, the tidal force of the Moon or the Sun is a good target for identifying the direction of the Newtonian gravitational field.

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The measured speed of the Newtonian gravitational strength (force) can be used to study and to understand the speed of gravity. The Liénard-Wiechert potential has the same form in both the Newtonian theory of gravity and electrodynamics, [2, 24] Eq. (13) of the nonradiative gravitational field strength has the form as the same as that of the nonradiative Coulomb field strength. [20, 24] Therefore, by comparing the measured speed of the Newtonian gravitational force with that of Coulomb force, we can know whether or not the conclusions in Refs. [20-22] and the interpretation about Eq.(13) are right. If both the measured speed of the Newtonian gravitational force and that of the Coulomb force are (approximately) the same (infinite), we can have the conclusion that the speed of propagation of the gravitational field (waves) is equal to the speed of light in vacuum, and the conclusions in Refs. [20-22] and the interpretation of Eq.(13) are right. On the other hand, if the two measured speeds are very different (for example, the measured speed of Coulomb force is c), the conclusions in Refs. [20-22] and the interpretation of Eq.(13) are questioned and need to be reconsidered. Practically, for measuring the speed of the Coulomb force, we can let a source, which can act on two moving charges respectively at two different points with different distances between the charge and the source, suddenly produce and vanish. By observing the motion of the two charges, the speed of the Coulomb force shall be measured. The observations of the motion of the planets in the solar system show that there is no the gravity aberration, which has been described validly in Newtonian theory of gravity with the instantaneous interaction through a central force from the point mass between two bodies for more than 300 years. The absence of gravity aberration is difficult to be explained in Einstein’s general relativity, in which, the gravitational force is postulated to be propagated at the speed of light by the gravitational field, and the interaction, energy, and waves propagate at the same speed. Trying to explain why there is no the gravity aberration while the speed of propagation of the gravitational field is c, in Refs. [20-22] it is concluded that the gravitational field could be extrapolated from the retarded position of the source toward the present position of the moving body, and the gravity aberration is canceled. Two evidences imply the light to agree the conclusions in Refs. [20-22]: (i) In electrodynamics, the Liénard-Wiechert potential shows that the force on a charge at a given position and time depends on the position of the source at an earlier time due to the finite speed, c, at which electromagnetic information travels. (ii) The classical experiment which shows the electromagnetic lines of force by iron filings on paper can also show that the electromagnetic field can be extrapolated from the retarded position of the source to the present position of the moving body (charge or ). For example, the magnetic field lines of a magnetism can be extrapolated to the direction along the line connecting another moving magnetism. Therefore, the gravitational field could be analogously extrapolated from the and position of the source to the present time and position of the moving body. It is the problem that the conclusion of aberration cancel in the gravitational field has not been proven experimentally. Also, it has not been proven that, in the , the Coulomb force is directed not toward the retarded position but toward the instantaneous position of the moving charge although it was deduced early in the 1960s. [25]

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However, I showed an experimental method to test the two conclusions. It shall be an experimental evidence to understand and to study the speed of gravity. Before both the speed of the Newtonian gravitational force and the speed of Coulomb force have been measured, we cannot know how fast they are. Thus, the present result can be determined experimentally but cannot be derived theoretically. This result can only be obtained after the measured result has been obtained. Therefore, the problem is to have a measured result and to understand it. We cannot directly know the speed of propagation of the gravitational field (waves) from the measured result, but, the measured result is a direct evidence to test and to understand the speed of gravity and other aspects of the gravitational field for which the Newton’s gravitational (tidal) force, field strength and potential can be directly measured with high precision and we have not had other way to test and to understand them. 푣 Equation (13) implies that, for the moving body, there is an angle 휃 = 푎푟 표푠 between the direction of 푐푔 퐺푀푚 퐺푀푚 propagation of the gravitational field and the direction of the Newtonian gravitational force, 퐹 = . 퐹 = is 푅2 푅2 always directed along the straight line connecting point mass M and m at the present positions, but according to Eq.(13), it is not propagated along this line. It is produced at the same time by that the gravitational field propagates from different retarded times and retarded positions. Consequently, the observed speed of the Newtonian gravitational 퐺푀푚 force, 퐹 = , apparently is instantaneous, and the speed of propagation of the gravitational force (waves) could be 푅2 equal to the speed of light in vacuum. Equation (13) also implies that, under interaction, the gravitational fields of the point mass shall not be isotropic, they are directed toward each other as the electric fields of two charges under interaction.

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