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The Analogy Between the Kepler Problem and Retarded Electrodynamic Potentials

In my book, Seeing Gravity, I commented on the analogy between the eccentricity and a velocity. It is more than a mere analogy: The conic section have refraction properties. In particular, if an ellipse of eccentricity e is rotated about its major axis, it produces an ellipsoid of refractive index 1/e. That is, rays of light parallel to the major axis will be refracted so as to pass through the further of the two foci. Snell's law is obeyed: sin i/sin r=1/e, where i and r are the incident and refractive angles.


The equation for elliptic motion is:


s=s'(1-e^2)/[1+e cos v]

where s' is the semi-major axis and the angle v is the true anomaly. To make the connection with retarded potentials and the (lack) of radiation, I identify the radial coordinate s with the present position of a charge and s' as its past position. Both, are functions of time, the present time t and the past time t': s'=t-t' in natural units. Using e to represent the relative velocity, the electric field.

V[E}=q{V[s']-V[e}s']/G^3{s'-V[e].V[s]}*3,

where V[] denotes vector. E is the electric field, q the charge, G the Lorentz factor, and the present position is

V[s]=V[s']-V[e]s'.

It is no big deal (and certainly not worth writing a paper about it, viz., Ibison et al. "The speed of gravity revisited") to conclude that the electric field always points in the direction of the present position of the charge--provided that the charge does not accelerate! In order to transform the electric field into

V[E]=q V[n]{(1-e^2)/s^2 Q^{3},

where V[n] is the unit normal pointing in the direction of V[s] and Q={1-e^2 sin^2v}^{1/2}, the true anomaly is the angle between V[e] and V[s].


In order to arrive at the above formula, the relation,

s Q=s'(1-e cos E)

had to be used. The angle E, not to be confused with the electric field, is the eccentric anomaly. The two angles are related by

cos v= Q{[cos E-e]/[1-e cos E]},

making it look like an aberration formula for Q=1. Analogously.

sin v = Q sin E/[1-e cos E].

Thus, the Keplerian orbit is transformed into:

s=s'(1-e^2)/[Q+e cos v].

Although it appears as a minor addition, it has the following effect upon the equation of the trajectory.


Letting u=1/s, the equation of the trajectory of a Keplerian orbit its

u"+u=h^2/m,

where the primes mean differentiation with respect to the true anomaly, h is the angular momentum and m is the gravitational constant. Whereas, in the analogous case of a retarded potential, the equation of the trajectory is

u"+u=(1-e^2)/Q^3,

and the right-hand side is no longer a constant. But, what does it represent?


Transforming back to s as the dependent variable and integrating gives the energy equation

(ds/dt)^2+h^2/s^2= -2 int V[E].dV[s]

where int stand for the integral, and V[E] is none other than the retarded electric potential. The total energy is no longer a constant of the motion!


Even more can be said when we introduce the radiation component of the electric field:

V[E]_rad=q{V[s']x(V[s]xdV[e]/dt)/s^3(1-e cos E)^3,

where the time derivative of the eccentricity plays the role of an acceleration. In a resistive medium, the eccentricity decreases in time. It is said [P.C. Peter's thesis] that "stars will have zero eccentricity by the time gravitational radiation becomes an important energy loss mechanism.


According to Leopold Infeld, gravitational radiation can be made whatever we wish by choosing the coordinate system appropriately. However, his calculation identifies the bilinear products of the third order derivatives with the decrease in energy. Apart from the fact that energy is not localized in general relativity, the introduction of an acceleration is incompatible with the geodesic equations of general relativity!


The radiative term in the electric field has two components: one along the direction of the motion and the other in the direction of the acceleration vector. Considering the component in the direction of the motion, the numerator of the field will be

1-e^2+s' cos A de/dt,

where A is the angle between V[s'] and dV[e]/dt. This is analogous to numerator in Weber's force:

F_W= (q/r^2) {1-v^2/2+ r dv/dt}.

The lack of induction is given by the condition

v^2/r=dv/dt,

where v is the velocity, since it renders the vector potential, v/r, independent of time. Thus, we can conclude that a varying eccentricity can be related to some type of gravitational induction, when we replace the charge q by the gravitational parameter, m.


What type of radiation would it be? Dipole radiation is excluded because the center of mass remains constant if the charge to mass ratio is constant. But here, the acceleration is in the rate of change of the eccentricity. If it so happens that the angle A is equal to the angle between past and present positions, viz., the induction angle, then the power loss will be proportional to

(de/dt)^2 sin^2 A,

which is Larmor's formula.


Finally, for geodesic motion, Weber's force must vanish giving the geodesic equation:

dv/dt= (1/2r)v^2-1.

It can't be clearer than this that geodesic equations do not "radiate", and this is the extent to which general relativity arrives at. Any fudging of the equations to make gravitational radiation pop out is clearly in contraction with the geodesic equations. And recall, that the equation of motion is not independent of the field equations of general relativity like the Lorentz equation is independent of Maxwell's equations in electromagnetism.






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