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Can a Velocity for Gravity be Defined?

Updated: Mar 5, 2023

The first step between Newton's and Coulomb's laws--both inversed square laws--was taken by Ampere who considered the motion of charges by taking into consideration their angle dependencies. A full, dynamical law of force was written down by Weber which consisted of a combination of Coulomb and Ampere laws.

It wasn't long after Weber formulated his law for the interaction of a pair of charges in motion that the astronomers made an attempt to apply his law to gravitation, and in particular, to the anomalous precession of Mercury. The idea was that the gravitational interaction of heavenly bodies would be modified by their motion.

Yet, Weber's law and its Newtonian analogue are quite distinct: the former restrains the particles to constant velocity, while the latter to the conservation of angular momentum. The latter prohibits a definition of the maximum speed of gravity, contrary to the ad hoc hypothesis made by general relativity.

To Newton,

d(theta)/ds= r=1/r (1+z^2)^(1/2)= sin(alpha) (1)

where (alpha) is the complementary angle of the angle between the force directed to the source and the force directed to the instantaneous center of curvature. Newton's slope, z=dr/d(theta)/r, and s is the arc length. Since the velocity v=ds/dt, we can rearrange (1) to read

r d(theta)/dt = v sin(alpha). (2)

To Ampere, s represented an element of element of circuit, and the angle (theta), the angle it makes with a straight line connecting the two elements of the circuit.

Newton, in his second edition of Principia considered the triangle YPS where S is the source, P, the point that the tangent to the curve intersects the trajectory, and Y, the normal drawn from S to the tangent. Thus,

YS/PS= sin(alpha) =r d(theta)/dt / v.

Since PS=r, YS is the ratio of the angular momentum (always per unit mass) to the velocity v

YS= r^2 d(theta)/dt / v.

Which of the two is held constant will determine the nature of the force law.

The radius of curvature is

(rho) = r /sin^3(alpha) (1+z^2-z'),

where the sine term is essentially a correction for the curve not being unit speed. The prime indicates differentiation with respect to (theta) so that

z' = d^2r/dt^2[r d(theta)/dt] - (dr/dt)^2/[r^2(d(theta)/dt)^2]-

dr/dt d^2 d^2(theta)dt^2/r (d(theta)/dt)^3.

The last term is crucial to Newton, and zero for Ampere.

The force at the instantaneous center of curvature is related to the force at the source by

F_0= F_s sin(alpha)= v^2/(rho)=v^2 sin(alpha)/r X [1+z^2-z'] (3)

In Newton's case it becomes

F_s= [r d(theta)/dt]^2/r {1-d^2r/dt^2/r (d(theta)/dt)^2} (4)

Multiplying numerator and denominator by r^2 gives

F_s= h^2/r^3 -d^2r/dt^2, (5)

which is Newton's law. Notice that the numerator cancels the denominator of the second term in (4) giving Newton's law (5), and the fact that square terms in (4) have cancelled out to the conservation of angular momentum, h=r^2 d(theta)/dt= constant.

The expression for the Newtonian force, (5), can be converted into the orbital equation by writing u=1/r, and t->(theta), i.e., dr/dt / d(theta)/dt =dr/d(theta), by using the conservation of angular momentum. The orbital equation is

u" + u =F_s/u^2 h^2 = GM/h^2, (5a)

where the prime denotes differentiation with respect to (theta), the second equality holds if F_s is Newton's inverse square law. Now, if we make the substitution e^2-> GM, the gravitational fine-structure constant is (the peripherical mass, m, is unity here)

(alpha) = GM/h c.

It is clear that if

r_cl =GM/c^2

multiplication by 1/(alpha) gives the gravitational Compton wavelength

r_comp =h/c,

and a furth multiplication by the same factor gives

r_b= h^2/GM.

The three characteristic lengths delineate the domains of validity. In the non-relativistic limit, the Bohr radius, r_b, commands, and the second equality in (5a) is its inverse. Hence, Kepler ellipses care entirely non-relativistic. If we multiply it be the square of the gravitational fine-structure constant we get the gravitational analogue of the classical radius of the electron and this is precisely what appears in Einstein's modification of (5a), viz.,

u"+u = 1/r_b + 3 r_cl u^2. (5b)

This shows that the correction term is ultra-relativistic; it cannot be gravitational itself but the effect that gravitation has on light. In fact, the magnitude of the correction to the perihelion shift of Mercury is proportional to the square of the gravitational fine-structure constant. The gravitational Compton radius is independent of gravitation, since GM cancels when r_c is multiplied by the inverse of the fine-structure constant. The interactions at the Compton level are not gravitational, but still relativistic since it contains, c. Gravitational interactions are from the Compton radius down, while those smaller than it are those of gravitation upon light.

The Compton radius could not appear in the second term in (5b) because it is gravitationally independent. If this interpretation is correct, gravity has no speed.

Rather had we considered constant angular velocity w= r d(theta)/dt, we would have obtained

F_s =v^2 sin^2(alpha)/r {1+ (1/w^2)[2(dr/dt)^2-r d^2 r/dt^2]}. (6)

Now, there is a limiting speed, w, appearing in the denominator of the motional terms in (6). Introducing w/v=sin(alpha), gives

F_s = w^2/r X {1+ (1/w^2)[2(dr/dt)^2-r d^2 r/dt^2]}. (7)

This is still not Weber's law since the signs in the second term have to be switched and the numerical value of the coefficients are different, yet, it indicates that w, the constant speed is a limiting speed particle speed. It plays the role of the c, which, according to Weber, is "that velocity which electrical masses must have and must retain, if they are not to act on each other at all.

The appearance of the squared term in Weber's law,

F_w= e^2/r x{1+(1/c^2)[2r d^2r/dt^2- (dr/dt)^2]} (8)

allows for a distinction of the forces between longitudinal and transverse motions of the electrical particles according to Weber's empirical determination. For neutral particles we would expect only attraction (change in sign of the acceleration) and the inverse of the force ratio found by Ampere for electrically charged particles.

It is therefore quite amazing that no such determination is possible for Newton's law (5), which gives the correct orbital equation, and that no limiting speed for gravitational interaction exists!

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