_{1}

Subluminal values of the speed of gravitational waves (GW) were obtained that reproduce a hypothetical Hulse-Taylor binary pulsar undergoing circular orbit decay. Those values of speed were used to simulate, in the framework of gravitoelectromagnetism (GEM), the in-spiral process of 3 GW events. The calculated results show a significantly better agreement (with the results of the linear theory of relativity) than the ones obtained using the speed of light. A method was proposed to measure the speed of the GW. Constraints of 0.89c < cg < 1.11c and 0.98c < cg < 1.20c were obtained for GW170814 and GW170817 events respectively. Some assumptions for the extremely high constraints on the deviation of the GW speed from the light speed of the almost simultaneous GW170817/GRB170817A events were referenced and alternative scenarios presented. The need for model-independent GW arrival time delays at the detectors is remarked.

The Maxwell equations trigger the question on the potential existence of magnetic charges and currents to obtain the implied better symmetry of the partial differential equations representing the electric and magnetic field. Works in that direction were presented by, for example, Heaviside and Dirac. Ref. [

In Appendix of this paper it is proposed that the magnetic charges and their currents are of pure electrical origin and the potentials are accordingly given. The new gravitational vector potential was used (in addition to the standard one) as the base for determining the total power radiated during in-spiral of a binary system. Tentative calculations show that the impact on the orbital decay is a small increase (~ 0.5% relative deviation in the binaries’ separation at 0.5 s into the transient) using initial conditions and masses approximately matching the GW150914 event. During that study it was noticed that by changing the standard gravitational vector potential A e by a factor of about 2, the orbital decay significantly increased. The work presented here was triggered by that finding.

Researchers as Maxwell and Heaviside applied the electromagnetic equations to gravitation in the 19 century. This is currently known as gravitoelectromagnetism (GEM). Ref. [^{nd} law of Newton (ENET), to derive a relativistic Kepler’s 3^{rd} law (K3L), the results were equivalent to N ~ 8/5.

The finding about A e is interpreted as the possibility that the gravitational wave speed in vacuum (c_{g}) be smaller than the speed of light in vacuum (c) and therefore the gravitational permeability of free space, in analogy with electromagnetism, would be μ 0 g = 4 π G / c g 2 . The mentioned factor of 2 then would imply that c g = c / 2 ≈ 0.71 c . Note that the gravitational permittivity is kept as ε 0 g = 1 / ( 4 π G ) .

The detection of the almost simultaneous GW170817/GRB170817A [_{g} from c but, as noted in [

Even if c g = c is confirmed without doubts, the present work has teaching importance as a first introduction to GW without using the complicated physics and mathematics inherent in general relativity (GR) while at the same time recognizing the historical and remarkable scientific relevance of Einstein GR. Additionally the computationally efficiency of the numerical solution of many-body-problems can potentially be improved (with respect to numerical GR) by solving equations similar to the force balance equation presented in Subsection 2.1.

The main objective of this work is to assess the impact of the gravitational wave and force propagation speed on the orbital frequency of circular orbits under a GEM framework.

The rest of this work is structured as follows. In Section 2.1, K3L for circular orbits is extended considering GEM and ENET. In Section 2.2 the equations for the calculation of the power radiated during in-spirals of binaries are derived. In Section 2.3 the gravitational orbit decay calculation is described. In Section 3 a purely geometric method to calculate c_{g} is proposed. Section 4 presents a comparison of the results obtained here with the ones from LGR for 3 simulated GW events. In this section c_{g} is also calculated along with its deterministic constraints for 2 GW events. Section 5 presents a summary and some concluding remarks.

The balance between the Lorentz force extended to gravitation and the inertial force can be written as [

F b a = − m a ( E b a + v a × B b a ) = m a a a ( 1 − β a 2 ) 3 (1)

where

F b a : Gravito-electromagnetic (GEM) force acting on a point mass m a due to another point mass m b ;

E b a : Gravito-electric (GE) field acting on body a;

v a : Velocity of body a in the center of mass reference frame;

B b a : Gravito-magnetic (GM) field acting on body a;

a a : Newtonian acceleration, in polar coordinates: a a = ( r ¨ a − r a θ ˙ a 2 ) e r _ a + ( r a θ ¨ a + 2 r ˙ a θ ˙ a ) e θ _ a

β a = v a / c ; v a 2 = r ˙ a 2 + r a 2 θ ˙ a 2

r a , θ a : Radial and azimuthal coordinates respectively.

e r _ a , e θ _ a : Radial (pointing from the origin of the laboratory reference frame to a) and azimuthal (counter clock wise from x to y) unit vectors respectively which can be written as

e r _ a = x ˙ ^ cos ( θ a ( t ) ) + y ^ sin ( θ a ( t ) ) ,

e θ _ a = − x ˙ ^ sin ( θ a ( t ) ) + y ^ cos ( θ a ( t ) )

The number of dots on top of a variable represents the order of the time derivative.

Note that the Newtonian acceleration is corrected by ( 1 − β a 2 ) 3 = γ a − 6 where γ a = ( 1 − β a 2 ) − 1 2 is the ubiquitous Lorentz factor. This correction yields the correct values of the intrinsic (two-body problem) perihelion precession of the planets of the solar system [

The Lienard-Wiechert potentials extended to gravitation (ELW) is written as

V = G m c d r c d − r • v (2)

A = G k c m v r c d − r • v (3)

k = k w / k d , c d = c / k d

where

k d : Coefficient to consider the impact of the speed (delay) of the GEM force on the orbit decay.

k w : Coefficient to consider the impact of the GM permeability on the orbit decay ⇒ c w = c / k w .

c w : Gravitational wave speed.

r : Distance from the position of the point mass to the observation point (separation).

To obtain ELW potentials the following conversions are made: 1 4 π ε 0 → G and μ 0 4 π → k w G c 2 .

Following similar approach as the ones described in [

E = G c d 2 c m b 1 3 [ b 1 ( v ( k − c c d ) − k ⋅ r a c d ) + b 2 ( c c d r − k r c d v ) ] (4)

B = G k w c d c 2 m b 1 3 ( b 1 a × r + b 2 v × r ) (5)

b 1 = r c d − r • v , b 2 = c d 2 − v 2 + r • a

where, r , v , a : Separation vector, velocity and acceleration of the source of the fields at the retarded time.

The acceleration is to be calculated as a = a N ( 1 − β 2 ) 3 a N : Newtonian acceleration.

Assuming circular orbits with piece-wise constant angular speed (ω) the fields acting on mass a are written as:

E b a = G c d c m b ( r ⋅ c d ) 2 [ ( ω 2 r r b ( k − c d c ) + ( r b ω ) 2 − c d 2 ) e r _ b + ( k ⋅ f / c d − c d 2 c ω ⋅ r 2 ) e θ _ b ] (6)

B b a = G m b ( r ⋅ c d ) 2 k w c ⋅ ( − f / c + c d 2 c r b ω ) z (7)

f = ω 3 ( r b 3 − r r b 2 g b ) , g b = ( 1 − ( ω ⋅ r b ) 2 c 2 ) 3

where,

e r _ b , e θ _ b : Radial unit vector (pointing from the origin of the laboratory reference frame to b) and azimuthal (counter clock wise, from x to y) unit vector respectively which are determined as

e r _ b ( t b a ) = − x ˙ ^ cos ( ω t b a ) − y ^ sin ( ω t b a ) , e θ _ b ( t b a ) = x ˙ ^ sin ( ω t b a ) − y ^ cos ( ω t b a )

Equation (1) for circular orbit with a constant angular speed can be written as:

− ( E b a + v a × B b a ) = − r a ω 2 g a e r _ a = − v a 2 r a g a e r _ a (8)

ω = θ ˙ , g a = ( 1 − ( ω ⋅ r a ) 2 c 2 ) 3

Substituting Equation (6) and (7) into Equation (8), considering small r ( e r _ b ( t b a ) ≈ e r _ b ( t ) ), a CM reference frame ( e r _ a ( t ) = − e r _ b ( t ) ) and ignoring in the left hand side the dependency on e θ _ b , leads to the following extended Kepler’s 3^{rd} law for circular orbits:

− k w k d 3 η b ( η a 3 − η a 2 g b ) r 4 c 4 ω 4 + ( k d 2 ( η b η a + ( k − c d c ) η a + η a 2 ) r 2 c 2 + r 3 G M g a ) ⋅ ω 2 − 1 = 0 (9)

where, η a = m a / M , η b = m b / M , M = m a + m b .

The CM relations r a = m b m a + m b r and r b = − m a m a + m b r were assumed to hold when deriving Equation (9).

Note that making c = ∞ in Equation (9), Kepler’s 3^{rd} law for circular orbit is recovered.

For a binary system the superposition of the ELW potentials at the observation point R are written as

Φ ( R , t ) = G m a R a − R a • v a / c d + G m b R b − R b • v b / c d (10)

A = k w G c 2 m a v a R a − R a • v a / c d + k w G c 2 m b v b R b − R b • v b / c d (11)

where, R a , R b are the distances from mass a and b respectively, to the observation point. The positions and velocities are evaluated at their respective retarded time: t R a = t − R a / c and t R b = t − R b / c .

The positions and velocities (in the CM) of the components of the binary system moving with piece-wise constant angular speed are:

r a = r a [ x ^ cos ( ω ⋅ t R a ) + y ^ sin ( ω ⋅ t R a ) ] ,

r b = − r b [ x ^ cos ( ω ⋅ t R b ) + y ^ sin ( ω ⋅ t R b ) ]

v a = ω ⋅ r a [ − x ^ sin ( ω ⋅ t R a ) + y ^ cos ( ω ⋅ t R a ) ] ,

v b = ω ⋅ r b [ x ^ sin ( ω ⋅ t R b ) − y ^ cos ( ω ⋅ t R b ) ]

and

r = r [ x ^ cos ( ω ⋅ t R ) + y ^ sin ( ω ⋅ t R ) ] , t R = t − R / c

Expanding in Taylor series the velocities around t_{R}, using the binomial expansion in the law of cosines relating R a , R b with r a , r b , considering R ≫ r a , r b and assuming that R ≈ R a ≈ R b the following is obtained:

A ( R , t ) = k w G c 2 R m a [ v a ( t R ) + d v a d t | t R Δ t a ] d e n a + k w G c 2 R m b [ v b ( t R ) + d v b d t | t R Δ t b ] d e n b (12)

d e n a = 1 − 1 c d R ^ a • [ v a ( t R ) + d v a d t | t R Δ t a ] ,

d e n b = 1 − 1 c d R ^ b • [ v b ( t R ) + d v b d t | t R Δ t b ]

Δ t a = 1 c d [ R ^ • r a ] , Δ t b = 1 c d [ R ^ • r b ]

Rearranging Equation (12) as A T ( R , t ) = p T + F T , and expressing the Cartesian unit vectors in spherical coordinates with the choice of ϕ = 0 as in Ref. [

p T = k w G ω ⋅ r c 2 R η M ( 1 d e n a − 1 d e n b ) ( − cos θ sin ( ω ⋅ t R ) θ ^ + cos ( ω ⋅ t R ) ϕ ^ ) = c p p 1 ( t ) p 2 ( t ) (13)

F T = − k w k d G ω 2 r 2 2 c 3 R η ( m b g a d e n a + m a g b d e n b ) sin θ [ cos θ ( 1 + cos ( 2 ω ⋅ t R ) ) θ ^ + sin ( 2 ω ⋅ t R ) ϕ ^ ] = − c F ⋅ F 1 ( t ) F 2 ( t ) (14)

where,

η = m a m b ( m a + m b ) 2

d e n a = 1 + k d ω ⋅ r a c sin θ sin ( ω ⋅ t R ) + k d 2 g a ω 2 r a 2 2 c 2 sin 2 θ ( 1 + cos ( 2 ω ⋅ t R ) ) (15)

d e n b = 1 − k d ω ⋅ r b c sin θ sin ( ω ⋅ t R ) + k d 2 g b ω 2 r b 2 2 c 2 sin 2 θ ( 1 + cos ( 2 ω ⋅ t R ) ) (16)

The assumption R ^ a ≈ R ^ b ≈ R ^ was made to obtain Equations (15), (16). The inertial accelerations were corrected with the relativistic factors g a , g b .

The GE radiation field is determined as g r a d = − ∂ A T ∂ t which depends only on the components of the vector potential A T that are transverse to the observation direction:

g r a d = − c p ( ∂ p 1 ∂ t p 2 + p 1 ∂ p 2 ∂ t ) + c F ⋅ ( ∂ F 1 ∂ t F 2 + F 1 ∂ F 2 ∂ t ) (17)

∂ p 1 ∂ t = ω ( n a d e n a 2 + n b d e n b 2 ) , ∂ p 2 ∂ t = ω ( − cos θ ⋅ cos ( ω ⋅ t R ) θ ^ − sin ( ω ⋅ t R ) ϕ ^ )

n a = − k d ω ⋅ r a c sin θ cos ( ω ⋅ t R ) + k d 2 g a ω 2 r a 2 c 2 sin 2 θ sin ( 2 ω ⋅ t R )

n b = − k d ω ⋅ r b c sin θ cos ( ω ⋅ t R ) − k d 2 g b ω 2 r b 2 c 2 sin 2 θ sin ( 2 ω ⋅ t R )

∂ F 1 ∂ t = ω ( m b g a n a d e n a 2 − m a g b n b d e n b 2 ) ,

∂ F 2 ∂ t = 2 ω ( sin θ [ − cos θ sin ( 2 ω ⋅ t R ) θ ^ + cos ( 2 ω ⋅ t R ) ϕ ^ ] )

Note that ω , r a , r b , r , were considered constants for the calculation of the derivatives in Equation (17).

The power radiated per unit area in the direction of a monochromatic wave propagation is determined as S = c / k w 4 π G g r a d 2 ( t ) S ^ where the GM radiation field is included.

Note that the average value, which is optional in EM and required in GR [

The total power radiated is calculated as

d E d t = ∫ S • d A , d A = R 2 sin θ d θ d ϕ S ^ , d E d t = d E d t | θ ^ + d E d t | ϕ ^ ,

d E d t | θ ^ = c / k w R 2 2 G ∫ 0 π g → r a d _ θ ^ 2 sin θ d θ , d E d t | ϕ ^ = c / k w R 2 2 G ∫ 0 π g → r a d _ ϕ ^ 2 sin θ d θ .

Note that, d E d t does not depend on R.

The total classical mechanical energy of the binary system can be written as U T o t = 1 2 η M ω 2 r 2 − G m a m b r , therefore

d U t o t d t = η M ( ω ω ˙ ⋅ r 2 + ω 2 r r ˙ ) + G m a m b r 2 r ˙ (18)

From Equation (9),

ω 2 = 1 d + n d ω 4 = f 0 + f 4 ω 4 ,

d = k d 2 ( η b η a + ( k − c d c ) η a + η a 2 ) r 2 c 2 ,

n = k w k d 3 η b ( η a 3 − η a 2 g b ) r 4 c 4

Taking the time derivative in both sides:

ω ˙ = f ˙ 0 + f ˙ 4 ω 4 2 ω − 4 f 4 ω 3 (19)

where,

f ˙ 0 = − d ˙ d 2 = q 0 r ˙ + q g a ω ˙ , q 0 = − d ˙ r / d 2 ,

d ˙ r = 2 k d 2 ( η b η a + ( k − c d c ) η a + η a 2 ) r c 2 + 3 r 2 G M g a + r 3 G M p a ω ,

p a = − 6 η b 2 c 2 ( 1 − ( ω ⋅ η b ⋅ r ) 2 c 2 ) 2 ⋅ ω ⋅ r , q g a = − d ˙ ω / d 2 , d ˙ ω = r 4 G M p a ,

f ˙ 4 = n ˙ ⋅ d − n ⋅ d ˙ d 2 = q 4 r ˙ + q g b ω ˙ , q 4 = n ˙ r ⋅ d − n ⋅ d ˙ r d 2 ,

n ˙ r = k w k d 3 η b c 4 ( − η a 2 p b ω ⋅ r 4 + 4 ( η a 3 − η a 2 g b ) r 3 ) ,

p b = − 6 η a 2 c 2 ( 1 − ( ω ⋅ η a ⋅ r ) 2 c 2 ) 2 ω ⋅ r , q g b = n ˙ ω ⋅ d − n ⋅ d ˙ ω d 2 , n ˙ ω = − k w k d 3 η b c 4 η a 2 p b r 5

Substituting into Equation (19):

ω ˙ = q 0 + q 4 ω 4 2 ⋅ ω − 4 f 4 ( r ) ⋅ ω 3 − q g a − q g b ω 4 r ˙

Plug it into Equation (18):

d U t o t d t = q r ˙ = − d E d t (20)

q = η M ( ω q 0 + q 4 ω 4 2 ⋅ ω − 4 f 4 ( r ) ⋅ ω 3 − q g a − q g b ω 4 r 2 + ω 2 r ) + G m a m b r 2

As pointed out before, under some assumptions relative to the emission time of the sources and to the intergalactic medium the coincident events GW170817/GRB170817A provided a very tight constrain on c g − c , the bounds of which can be circumvented if the GW-EW delay is allowed to be greater than 10 s or if the EW can precede the GW [_{g} based on the detection of continuous monochromatic signals emitted from, for example, rapidly rotating neutron stars.

In this section a method to measure c_{g} is proposed that is also not limited by the same systematics without having the expectation to compete with simultaneous GW-EW event-based measurements. Assuming that accurate and model-independent GW arrival time delays among the detectors can be obtained, c_{g} could be determined as follows:

The arrival time delays between the 3 detectors are related to the GW speed through the following equations:

Δ t V − L = Δ R V − L / c g , Δ t L − H = Δ R L − H / c g

where,

Δ t : Arrival time delay between indicated detector sites.

Δ R : Distance travelled by the wave front (arriving first at the first indicated site).

Designating

α : Angle between the wave propagation direction and the line joining Virgo and Livingston sites.

β : Angle between the wave propagation direction and the line joining Livingston and Hanford sites.

Then

Δ R V − L = D V − L cos α , Δ R L − H = D L − H cos β

D: Straight distance (through earth) between indicated detector sites.

Note that the wave direction is perpendicular to a vector originating at the second indicated site and ending at the wave vector. Note also that the projection of the wave vector on the detectors’ plane does not change those angles and that the angles are the same along their corresponding joining lines as long as the wave source is far enough from the detectors.

Therefore

Δ t V − L = D V − L cos ( α ) / c g (21)

Δ t L − H = D L − H cos ( β ) / c g (22)

The angles are related through

α = 180 − ( β + γ L ) (23)

γ L : Angle between the line joining Virgo and Livingston and the line joining Livingston and Hanford.

From the law of cosines (generalized Pythagoras theorem)

γ L = arccos ( D L − H 2 + D V − L 2 − D V − H 2 2 D L − H D V − L ) (24)

Solving for c g in Equation (22) and substituting it (along with Equation (23)) into Equation (21) the following is obtained:

β = arccos [ D V − L Δ t L − H D L − H Δ t V − L cos ( 180 − γ L − β ) ] (25)

c g is then obtained from Equation (22) or (21).

Note that if the angles could be measured (or accurately inferred from measurements), detectors’ signals, not complying with the equality of c g in Equation (21) and in Equation (22): D V − L cos ( α ) Δ t V − L ≈ D L − H cos ( β ) Δ t L − H , could be rejected.

The values of k d and k w will be determined in such a way that the period decay, d T d t , of the Hulse-Taylor (H&T) binary pulsar (PSR 1913+16) assuming a circular orbit, is reproduced (in another work the elliptical orbit case will be addressed). The value to be reproduced is the one from GTR [

T ˙ = d T d t = 192 π m a m b 5 c 5 ( m a + m b ) 1 / 3 ( 2 π G T ) 5 / 3 f ( e )

assuming f ( e ) = 1 + ( 73 / 24 ) e 2 + ( 37 / 96 ) e 4 ( 1 − e 2 ) 7 / 2 = 1 . The GTR result (for elliptical orbit) is in very good agreement with experiment. The hypothetical H&T result for circular orbit is used as a target because the H&T value for elliptical orbit is very robust: direct measurements of orbital parameters are made and it has been systematically reproduced using experimental results from more than 40 years (measurements can still be made).

Relatively recent detected gravitational waves (e.g. GW150914 and GW170817) are not used as a target considering that they are not as robust as the H&T experimental results: the GW measured data are not direct measurements of orbital parameters, and they are not reproducible (the experiment cannot be repeated on demand).

Additionally in Ref. [

The GW170817/GRB170817A coincidental events (which apparently implies that the speed of the gravitational wave is the same as the speed of light: k w = 1 ), have non trivial measurement/interpretation uncertainties due to the uniqueness of the events. For example Ref. [^{−10}) of a coincidental origin of the events for the model of LIGO-Virgo which reports a probability of 5 × 10^{−8} that both events happened by chance.

^{nd} set ( k w = 2.53 , k d = 1 ) implies c g = 0.63 c while the gravitational force news is transmitted at the speed c. The last set ( k w = 1.75 , k d = k w = 1.32 ⇒ c g = 0.76 c ) represents a case for which both the GEM force news and the gravitational wave propagate at the same speed as it happens in classical EM. The T ˙ shown in

The distance to the GRB170817A event that results in a time coincidence with the GW170817 event at the GW detectors for c g = 0.76 c , was calculated (assuming no common distance to the source and assuming equal time of emission) For arrival time coincidence of the signals (moving in vacuum):

Δ R = R g ( c c g − 1 ) (26)

So for R g = 40 Mpc and c g = 0.76 c ⇒ Δ R ≈ 12.6 Mpc . Therefore, in this example, an isotropic (or a disk/beam reaching the earth) GRB source located at about 12.6 Mpc behind the GW source will be detected in time-coincidence with a GW moving at about 0.76c. Note that time coincidences could also happen for astronomical events resulting in emission of EM waves at any distance if no

K_{w} | K_{d} | dT/dt (s/s) | Cw | Cd |
---|---|---|---|---|

1.00 | 1.00 | −5.01E−14 | 1.00 | 1.00 |

2.53 | 1.00 | −2.02E−13 | 0.63 | 1.00 |

1.75 | 1.32 | −2.03E−13 | 0.76 | 0.76 |

emission time coincidence is assumed. Note also that if a large difference of c g − c really exists no real coincidence will be detected on earth for simultaneous time emissions and common spatial origin of the events since as pointed out in [

Figures 1-4 show in-spiral simulations of three GW events. The integration step for all transients was Δ t = 10 − 5 s .

K3L (Kepler’s 3^{rd} law): Make c = ∞ in Equation (9) and in q (Equation (20)).

EM (Electro-Magnetism): Make g a = g b = 1 and p a = p b = 0 .

EM-ENET: Use Equation (9) and Equation (20) without approximation.

LGR (Linear General Relativity): r 4 = r 0 4 − N η ⋅ c r s 3 ( t − t 0 ) [

The input data for the transient of

For k w = 2.53 and k d = 1 ( c g = 0.63 c ) it is expected to yield similar results since calculations for the GW150914 event indicated that. The following values of N were calculated (using Equation (4)) that approximately reproduced the results for the GW150914 event: N = 28.8 5 , 17.7 5 , 22.5 5 for K3L, EM and EM-ENET respectively taking c g = 0.76 c . Note that for LGR N = 32 5 , K3L is used and c g = c .

x = ( R n + h ) cos ϕ ⋅ cos λ , y = ( R n + h ) cos ϕ ⋅ sin λ ,

z = ( b 2 a 2 R n + h ) sin ϕ ,

R n = a 2 a 2 cos 2 ϕ + b 2 sin 2 ϕ .

where f: Latitude, λ: Longitude, h: Ellipsoidal height, a: Ellipsoid semi-major axis: 6378137.0 m, b: Ellipsoid semi-minor axis: 6356752.0 m. The axis values correspond to the World Geodetic System 1984. In this application h does not make significant impact on the results. The distance between the site i and j was calculated as

D i − j = ( x i − x j ) 2 + ( y i − y j ) 2 + ( z i − z j ) 2 (27)

Equation (24) yields γ L = 84.02 deg .

_{g} is about 0.99c and 1.08c for GW1708 (14 and 17) respectively.

Site | Longitude (rad) | Latitude (rad) | D (km) | |
---|---|---|---|---|

H | −2.08405676917 | 0.81079526383 | V − H | 8180.731 |

L | −1.58430937078 | 0.53342313506 | V − L | 7929.013 |

V | 0.18333805213 | 0.76151183984 | L − H | 3001.776 |

Event | V − L (msec) | L − H (msec) | Alpha (deg) | Beta (deg) | C_{g} (km/s) |
---|---|---|---|---|---|

GW170814 | 14.00 | 8.00 | 58.4 | 37.6 | 297183.2 |

GW170817 | 21.87 | 3.33 | 27.0 | 69.0 | 323096.5 |

The same results are obtained if the Cartesian coordinates of Ref. [_{g} deviates in −0.1 km/s). If the approximation D V − L = D V − H = 27 msec (light travel time) is made and D L − H = 10 msec as noted in [

The uncertainty of the GW speed can be determined making further calculations based on the uncertainties of the arrival time delays. Uncertainty of ±1 msec. in the arrival time delays were estimated in [_{g} < 1.11c. For GW170817 assuming 10% uncertainty [_{g} < 1.20c. These results put significantly more constraint than the bounds of 0.55c < c_{g} < 1.42c reported in [_{g}.

Note that the calculated values of c_{g} are model dependent because the determination of the time delays assumes c g = c in some part of the experimental procedure to detect and localize the GW [

It is noted that Ref. [_{g} < 1.22c based on the Jovian deflection experiment (measurements of the propagation of quasar’s radio signal past Jupiter), in that reference a distinction is made on the meaning of c in different parts of the GR equations.

Note that if c_{g} is given, the expected Δ t V − L could be inferred for the GW150914 event (VIRGO was not operational during that time) using Equations (22)-(25). Perhaps the angular sky location could be improved as well. For example assuming c g = c and using Δ t L − H = 6 . 9 msec [

Speeds of GW and force propagation were obtained that account for the expected period decay of a hypothetical Hulse-Taylor binary pulsar moving in circular orbits. The so-obtained speeds were used to simulate the in-spiral process of 3 GW events. The results are in significantly better agreement (with the results of LGR) than the results obtained using the speed of light for both the GW and the gravitational force. A method to calculate the speed of GW was proposed and applied to 2 GW events. Uncertainties of the arrival time delays were used to deterministically constrain the GW speed. It is remarked the need for measuring and/or calculating model-independent delays to determine and constrain the GW speed.

It is noted that the accuracy level of the results obtained here (concerning the 3 GW events) is not determined necessarily by the agreement with the LGR results but by its agreement with, for example, model-independent measured profiles of the detectors’ strain (h) and/or with orbital parameters as, for example, the orbital frequency, calculated using experimental raw data from the LIGO-VIRGO detectors.

It is envisioned to extent the GEM model developed here to consider elliptical orbits.

It could be worthy to assess the impact of the implied kinetic energy of ENET on the orbital decay and also, if needed, the impact of the magnetic charges and currents using the potentials obtained in Appendix.

The author declares no conflicts of interest regarding the publication of this paper.

Quintero-Leyva, B. (2020) On the Gravitational Wave and Force Propagation Speed Impact on the GEM Decay of Circular Orbits. Open Access Library Journal, 7: e6683. https://doi.org/10.4236/oalib.1106683

Ref. [

∇ • E = ρ e ε 0 , ∇ • B = μ 0 ρ m , ∇ × E = − μ 0 J m − ∂ B ∂ t , ∇ × B = μ 0 J e + μ 0 ε 0 ∂ E ∂ t

The conservation equations were given by ∇ • J e = − ∂ ρ e ∂ t , ∇ • J m = − ∂ ρ m ∂ t . The electric and magnetic field were written as E = − ∇ V e − ∂ A e ∂ t − ∇ × A e m , B = − ∇ V m − μ 0 ε 0 ∂ A m ∂ t + ∇ × A e .

Note that the variables with subscript “m” represent the new magnetic sources.

Assuming Lienard-Wiechert potentials for point magnetic charges also:

V e = 1 4 π ε 0 q e r − r • v c , V m = μ 0 4 π q m r − r • v c , A e = μ 0 4 π q e v r − r • v c , A m = μ 0 4 π q m v r − r • v c

Assuming that all magnetic charges are of electrical origins, the new potentials are expressed as V m = μ 0 4 π q e v r − r • v c , A m = μ 0 4 π q e v r − r • v c v . Note the quadratic dependence of A m on the speed.

The vector potentials for gravitation are then written as A e = G c 2 m v r − r • v c , A m = G c 2 m ⋅ v v r − r • v c .

A e is calculated in the main text ( k = k w = k d = 1 ). A m is similarly determined as A m = A 1 + A 2 + A 3 + A 4 , where,

A 1 = G ω 2 r 2 c 2 R ( m a η b 2 d e n a − m b η a 2 d e n b ) ( − cos θ ⋅ sin ( ω ⋅ t R ) θ ^ + cos ( ω ⋅ t R ) ϕ ^ ) = c p . m p m .1 ( t ) p m .2 ( t ) = p m

A 2 + A 3 = − c K . m K m .1 ( t ) K m .2 ( t ) = K m ,

c K . m = G ω 3 r 3 2 c 3 R , K m .1 ( t ) = m a η b 3 d e n a + m b η a 3 d e n b

K m .2 ( t ) = sin θ cos θ ⋅ ( 1 + cos ( 2 ω ⋅ t R ) + sin ( 2 ω ⋅ t R ) ) θ ^ + sin θ ( sin ( 2 ω ⋅ t R ) − 1 − cos ( 2 ω ⋅ t R ) ) ϕ ^

A 4 = − G ω 4 r 4 2 c 4 R ( m a η b 4 d e n a − m b η a 4 d e n b ) ( k 4. θ ^ θ ^ + k 4. ϕ ^ ϕ ^ ) = − c F . m F m .1 ( t ) F m .2 ( t ) = F m

k 4. θ ^ = cos θ ⋅ sin 2 θ cos ( ω ⋅ t R ) ( 1 + cos ( 2 ω ⋅ t R ) ) ,

k 4. ϕ ^ = sin 2 θ ⋅ cos ( ω ⋅ t R ) sin ( 2 ω ⋅ t R )

Then

B m − r a d = − 1 c 2 ∂ A T ∂ t , E m . r a d = − ∇ × A T = c [ B m − r a d . φ θ ^ − B m − r a d . θ ϕ ^ ]

The total radiation field (Coulomb gauge) is then written as

E r a d = − ∂ A e ∂ t − ∇ × A m = E e . r a d + E m . r a d ,

B r a d = − 1 c 2 ∂ A m ∂ t + ∇ × A e = B m . r a d + B e . r a d

The Poynting vector is calculated as S = c 8 π G ( E r a d 2 + c 2 B r a d 2 ) ⋅ s ^ .

where,

E r a d 2 = E r a d _ θ ^ 2 + E r a d _ ϕ ^ 2 , B r a d 2 = B r a d _ θ ^ 2 + B r a d _ ϕ ^ 2

E r a d _ θ ^ 2 = [ ( d p e + d F e ) | ∂ t − ( d p m + d K m + d F m ) | ∇ × ] θ ^ 2

B r a d _ θ ^ 2 = [ ( d p m + d K m + d F m ) | ∂ t + ( d p e + d F e ) | ∇ × ] θ ^ 2

Similarly for ϕ ^ .

where,

( d p e + d F e ) | ∂ t = − ∂ A e ∂ t = [ ∂ p e . θ ∂ t θ ^ + ∂ p e . ϕ ∂ t ϕ ^ ] + [ ∂ F e . θ ∂ t θ ^ + ∂ F e . ϕ ∂ t ϕ ^ ]

− ( d p m + d K m + d F m ) | ∇ × = − ∇ × A m = − 1 c [ − ∂ p m . ϕ ∂ t θ ^ + ∂ p m . θ ∂ t ϕ ^ ] − 1 c [ − ∂ K m . ϕ ∂ t θ ^ + ∂ K m . θ ∂ t ϕ ^ ] − 1 c [ − ∂ F m . ϕ ∂ t θ ^ + ∂ F m . θ ∂ t ϕ ^ ]

( d p m + d K m + d F m ) | ∂ t = − 1 c 2 ∂ A m ∂ t = 1 c 2 [ ∂ p m . θ ∂ t θ ^ + ∂ p m . ϕ ∂ t ϕ ^ ] + 1 c 2 [ ∂ K m . θ ∂ t θ ^ + ∂ K m . ϕ ∂ t ϕ ^ ] + 1 c 2 [ ∂ F m . θ ∂ t θ ^ + ∂ F m . ϕ ∂ t ϕ ^ ]

( d p e + d F e ) | ∇ × = ∇ × A e = 1 c [ − ∂ p e . ϕ ∂ t θ ^ + ∂ p e . θ ∂ t ϕ ^ ] + 1 c [ − ∂ F e . ϕ ∂ t θ ^ + ∂ F e . θ ∂ t ϕ ^ ]

The total power radiated is then determined as d E d t = ∫ S • d A .