Dual Laws and Dual Observations

Updated: Oct 26, 2019

The solar system formed in such a way that plaentary and lunar orbits are almost circular. For this reason Newton was infatuated with circular, or nearly, circular orbits and sought conditions for their existence through central power laws. Even when they were not entirely circular, Newton gave an expression for the apsidal angle in terms of the characteristic exponent of the power law (cf. Seeing Gravity p. 98)/

Newton singled out two laws of force, and noted their similarity in that only for these laws were the orbits conic sections. He then gave a brilliant argument that connected the inverse square law for a force directed to the sun at the focus and a line force law that would result if the elliptical orbit were instead a circular orbit by a fictitious body at the center. The inverse square law would thus be replaced by a linear, harmonic, oscillator one.

I recently noted a wonderful analogy between constant sectional curvature (the harmonic oscillator), and non-constant sectional curvature given by a tidal force (inverse square). That is, if the constant density were replaced in the harmonic oscillator equation by the ratio of the mass to the volume, one would go from a linear force law to an inverse square one. This would show, then, their apparent equivalence.

This is analogous to the distinction between Laplace's and Poisson's equation. Because of the inverse square law, the force acting on a particle inside a spherical shell is zero. Hence, the gravitational potential inside the shell must be constant, for, otherwise, the force would not be zero. We can calculare it anywhere inside the shell, say at the center:

dV=-Gdm/r=4 pi G rho rdr,

where rho is the constant density. So inside a shell of radius C, the potential is

V=-2 pi G rho(C^2-r^2).

If we evaluate it from the center, we have

V-v_0=-(4/3)pi G rho r^2-2 pi G rho(C2-r^2)=-(2/3)pi G rho(3C^2-r^2),

which just changes the former by a mere numerical factor.

The force inside the shell is

f=(4/3)pi G rho r,

and this gives Newton's law as

r**=-(4/3)pi G rho r,

although we previously agreed upon the fact that the force was zero inside the shell. The sectional curvature, r**/r is -(4/3)pi G rho, a constant.

Now, exterior to the shell, Newton's law reads

V=-(4/3)pi G rho C^3/r

and the external force is: f=(4/3)pi G rho C^3/r^2,

where we could, if we wanted to, replace (4/3)pi rho C^3 by a constant mass M, that could be concentrated at the center of the shell of radius C. Again, invoking Newton's law

r** = GM/r^2

we have a sectional curvature equal to GM/r^3. This reduces to a constant sectional curvature by multiply through by the volume, proportional to r^3. The linear and inverse square laws are said to be duals of one another. Notice, however, the different signs in the two curvatures.

Over two centuries were to pass before the mathematicians got a hold of the dualism. The realm of real numbers was extended to complex numbers. When a circle associated with a complex number w=z is deformed into an ellipse by z=z+1/z, and then squared, w^2=z^2+1/z^2+z, the origin is shifted to a focus of the ellipse. This was termed a "transmutation" of the force law--a big name that massed the physical essence of what is going on. It was attributed to Bohlin by Arnold, and to Krasner by Neeham, both working at the beginning of the twentieth century.

As mathematicians will, they attempted to generalize the dual law into law(s) by the following theorem:

"The trajectories of the motion of the point w in the complex plane in a central gravitational field whose strength is proportional to the distance from the center raised to the power m goes under the transformation Z-w^m into trajectories of the motion in a central field whose strength is proportional to the distance from the center raised to the power A if

(a+3)(A+3)=4, m=(a+3)/2" (Arnold). (I)

"a" is the dual exponent of A. The conclusion is therefore that to every power law there is a unique dual law. In the case (-2,1), Arnold demonstrates the dualism Z=w^2 explicitly. Whereas w satisfies Hooke's law, w**=-w, Z satisfies Z''=-Z/|Z|^3 where the second time derivative ** refers to time t, while '' refers to time T. The two are related by Kepler's law of areas in equal times, dT/dt=|Z|^2/|w|^2. Whereas the former has sectional curvature, -1, the latter has sectional curvature, -1/|Z|^3, just like before, except that the signs are now the same.

So anything that obeys Eq. (I) should be result in a central force law, and, consequently, a periodic orbit. Right? Wrong! I was even unsure of it when I wrote @Seeing Gravity", but the proof seem legit. Let's look at the dual power laws that satisfy the equation:

(-2,1) (-4,-7) and (-5,-5).

Although the first is beyond discussion it provides an inkling of what is behind the equation. The m=-3 exponent is missing because that would violate equation (I). And in fact, this corresponds to Cotes' spirals which are unstable. Skipping for the moment the second pair, we see that the last pair is a self-dual. That is what happens in one space happens also in the image space.

It is not difficult to see that the self-pair describes the circular orbit

r=2 a cos theta,

where a is the radius of the circle. Danby wrties u=1/r and proceeds to differentiate so as to derive the orbit equation

u'' + u=(1/h^2 u*2)f(1/u)=1/a cos^3 theta,

where h is the conserved angular momentum. Eliminating cos theta, the central force law is found to be


which is a candidate for the (-5,-5) self-dual. Using Kepler's third, h v=GM, where v=rw, the rotational velocity, the sectional curvature,

r**/r=2(2GM/w r^3)^2.

At constant density, the sectional curvature is

r**/r=2(2 G rho/w)^2,

is constant, but this time proportional to the square of the density. This is indicate of two particle interactions.

It is well-known that a field whose central attraction decreases as the fifth power of the distance has circular trajectories that pass through the origin. Since the sun is placed at the center of the coordinate system, this represents a collision singularity---and this is the reason for the product of densities in the sectional curvature!

So the actual dualism is between sectional curvatures of constant and non-constant curvatures that can be interconverted by multiplying by the volume r^3, or its square. That, I believe, is the origin of the 3's in equation (I).

In general, the potential can be expanded in terms of Legendre polynomials,

V=-(GM/r){1-J_2(a/r)^2 P_2(cos theta)+J_3(a/r)^3 P_3(cos theta)+...},

where a is usually the unperturbed radius of a planet, and the J_i are dimensionless constants that reflected the mass distribution within the planet or body under consideration. For instance, J_2 is interpreted in terms of the 3 moments of inertia.

The squared term is missing in the expansion. It would correspond to a field of force, 1/r^3. Newton found that when this added attraction was added on to the equation of motion, the new orbit resembles the old one except that the latter is continuously rotating. This is because the added term affect the magnitude of the anglular momentum. If h is the old value of the angular momentum, and h' is the new value, the apsidal angles are increased by the factor h'/h. This is Prop XLV in Newton's Principia, and Arnold hints that it may have some baring on Eq (I) above. No, it does not since it is the result of a 1/r^3 force and it is not listed in the duals.

But, what listed is the (-4,-7) pair. this corresponds to the P_2 term in the potential expansion and gives rise to a 1/r^4 force. By expanding the (1-2GM/rc^2) term in the dr^2 of the coefficient of the Schwarzschild metric, Einstein obtain the modified orbit equation,

u''+u=GM/h^2+(3GM/c^2) u^2.

Comparing this to the standard form of the orbit equation, given above, it is easy to dientify the force

f=GM/r^2 +3 GM h^2/c^2/^4..

Who says there are no forces in general relativity?! It arises as a tidal force say through the action of a satellite on a planet. Kepler's third law tells us the there is a balance between gravitational and centrifugal forces.

However, due to the presence of a satellite, like the moon, there will be an excess of either gravity or centrifugal forces at the equation due to tidal shear. The force is

f=GM/r^2+3GM a^2H_2 P_2(cos theta)/r^4

Equating this with Einstein's expression yields

h^2=c^2a^2 J_2(3 cos 2 theta-1).

But, in the presence of an external torque, the angular momentum, h, is not constant, but, rather the speed v=aw. We thus obtain, the second-order correction

v^2/c^2=J_2(3 cos 2theta+1).

Both the deflection of light and the advance of the perihelion of Mercury are expressed in terms of the square of the ratio. Introducing the Kepler's third law into it gives the ratio of the Schwarzschild radius to the radial distance R_s/r. The advance of the perihelion is 1.5 R_s/r, while the deflection of light is 2R_s/r, roughly the same order of magnitude, and both proportional to 10^(-6). Oddly enough this is the same order as the oblateness of the sun!

Excluding rotation, the flattening, or obtaleness of a star or planet is (3/2)J_2. The sun is almost spherical, and perfectly aligned to the eclptic so that its oblateness is three orders of magnitude smaller than the earth's which is 3.35x10^(-3) Now the deflection of light is 1.75 arc seconds or 8.57x10^(-6), which is the same order of magnitude as the sun's oblateness.

The same applies to the perihelion of Mercury, and both involve the sun!

Writing in the 29/10/1920 issue of Science, the author of duality of power laws, Edward Kasner had this say about these two phenomena:

"It follows that the gravitational field of the sun can be exploited either by observations on the orbit of the planets or by observations on the deflection of light rays. It is not necessary to use both observations. There is, in particular a connection between the deflection of light (1.75" at the sun's limb) and the motion of the perihelion of Mercury (43" per century); either could have been theoretically predicted from the other--but in fairyland who can lay down the boundary between theory and practice?"

If there was a choice between remembering Kasner for his duality laws or dual observations, it would certainly be the latter. The pair (-4,-7) in Eq (I) has no role in the duality of central forces. The actual duality is between constant sectional curvature and tidal forces that are expressed in terms of GM/r^3, and can be transformed into constant curvature by replacing the expression with a uniform density. This is the same as the transition between Laplace's and Poisson's equations!

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