Time Dilation and Space Contraction from Keplerian Orbits

We have discussed repulsive gravity within the context of the conformal equivalence between the squares of Hookean ellipses and the orbits in a gravitational field. In particular attractive oscillators corresponds to attractive gravity while a negative Hookean law is equivalent to repulsive gravity. Here, we show that oscillators and gravitational forces lead to non-Euclidean geometries. This has been discussed before by Milnor, "On the geometry of the Kepler problem" in Not. Am Math Soc90 (1983) 353-365. However, Milnor had to resort to inverse velocities in order to derive the Poincare' disc metric for hyperbolic geometry. Unlike space, velocities have a limit, and the use of inverse velocities would tend to eradicate such a limit.

We will also see that time dilation and space contraction are not, at all, limited to relativistic phenomena. They also appear in conformal mappings. Consider the conformal mapping

$$w=z^2.$$ 

We would like to think that Kepler's laws are invariant under such mappings. In particular Kepler's second law, which states that equal areas are swept out in equal times, is invariant:

$$\frac{|z|^2\dot{\varphi}}{dt}=\frac{|w|^2\dot{\varphi}}{d\tau}=L=\mbox{const.},$$

which implies the conservation of angular momentum, $L$, per unit mass. Thus, the time $\tau$ is related to $t$ via

$$d\tau=|z|^2dt.$$

If $z$ is the solution to the harmonic oscillator equation,

$$\frac{d^2z}{dt^2}=-z,$$

its total energy,

$$|\dot{z}|^2+|z|^2=2E,$$

is an integral of the motion. Solving for $|z|$, and introducing it into the expression for $d\tau$,

$$d\tau=\left(2E-|\dot{z}|^2\right)dt,$$

shows that $\tau$ undergo time dilation. Space contraction follows from taking the derivative of the conformal mapping and then absolute values

$$\left|\frac{dw}{dz}\right|=2|z|=2\sqrt{2E-|\dot{z}|^2}.$$

We can write down the metric from the equation of motion; taking absolute values of both sides gives

$$\left|\frac{d\dot{z}}{dt}\right|=|z|=\sqrt{2E-|\dot{z}|^2}.$$

Rearranging and squaring give

$$dt^2=\frac{|d\dot{z}|^2}{2E-|\dot{z}|^2},$$

which does not look like the hyperbolic metric of Poincare' because the denominator should be squared. Yet, it should be recalled that the velocity metric is

$$dt^2=\frac{2E|d\dot{z}|^2-|\dot{z}\wedge d\dot{z}|^2}{(2E-|\dot{z}|^2)^2}.$$

Writing the second term in the numerator as

$$|\dot{z}\wedge d\dot{z}|^2=|\dot{z}|^2|d\dot{z}|^2-|\dot{z}\cdot d\dot{z}|^2,$$

we observe that for uniform acceleration, the increment in the velocity $d\dot{z}$ is always perpendicular to the tangential velocity, $\dot{z}$ so that the last term vanishes. In other words, centripetal acceleration invariably points inward to the center of the circular orbit. Parenthetically, if $v$ is the velocity, the radius of curvature is

$$R=\frac{|v|^3}{|v\wedge\dot{v}|}.$$

For uniform acceleration, the denominator reduces to the product $|v| |\dot{v}|$, so that

$$R=\frac{|v|^2|}{|\dot{v}|}$$

which upon rearranging gives the centripetal acceleration.

$$|\dot{v}|=\frac{|v|^2}{R},$$

Hence, we come out with the above line element. 

Thus, it is as Newton claimed: Given the curvature of the force, the path is uniquely determined. All inverse square laws lead to conic sections, ellipses and hyperbolas---and to their respective geometries, elliptic and hyperbolic, respectively.

What can we say about the line element for $w$? To this end we make use of the Maupertuis-Jacobi principle

$$E-U(z)=\left|\frac{dw}{dz}\right|^2\left(E^{\prime}-U^{\prime}(w)\right),$$

where the total energies, $E$ and $E^{\prime}$, and potential energies, $U(z)$ and $U(w)$, satisfy

$$EE^{\prime}=U(z)U^{\prime}(w)=\mbox{a negative const}.,$$

Consider first the case of attraction. The oscillator energy is $U(z)=|z|^2$. This requires a negative total energy on the right-hand side, $E^{\prime}=-\frac{1}{4}$. Now, a constant total oscillator energy, $E=1$, requires an attractive gravitational energy of

$$U^{\prime}(w)=-\frac{\kappa}{4|w|},$$

where $\kappa$ is a constant. (This numerical values are of no concern.) 

Applying the conservation of energy to both the oscillator and gravitational fields leads to

$$\frac{|\dot{z}|^2}{2}=E-|z|^2,$$

and

$$\frac{|\dot{w}|^2}{2}=\frac{\kappa}{|w|}-|E^{\prime}|,$$

respectively, where the absolute sign is necessary because $E^{\prime}<0$.

Newton's law of gravitation is

$$\frac{d^2w}{d\tau^2}=-\frac{\kappa w}{|w|^3}.$$

Note that Arnol'd, in Huygens and Barrow, Newton and Hooke, and other authors introduce a constant $C$ on the right-hand side which can be both positive (elliptic orbits), or negative (hyperbolic orbits). The constant $C$ is related to the total energy, $E^{\prime}$, but, according to these authors, it is not the energy which changes sign, but rather the coefficient. This camouflages the actual physics of what is going on.

Taking the absolute value of both sides of Newton's law,

$$\left|\frac{d\dot{w}}{d\tau}\right|=\frac{\kappa}{|w|^2},$$

and changing times,

$$|d\dot{w}|=dt\frac{\kappa}{|w|}=dt\left(|E^{\prime}|+|\dot{w}|^2/2\right),$$

gives the line element

$$dt^2=\frac{4|d\dot{w}|^2}{(2|E^{\prime}|+|\dot{w}|^2)^2},$$

which is the stereographic line element of elliptic geometry.  Milnor resorts to the introduction of an inverted velocity coordinate $v=\dot{w}/|w|^2$, by claiming that he wants to "describe what happens in a neighborhood of infinity." However, a "neighborhood of infinity" for velocities does not exist. In other words, it is not true that $v\cdot v>2E^{\prime}$ "must always be satisfied."

Just consider repulsive gravity for which the line element is

$$dt^2=\frac{4|d\dot{w}|^2}{(2E^{\prime}-|\dot{w}|^2)^2}.$$

This is the Poincare' disc metric of hyperbolic where $2E^{\prime}>|\dot{w}|^2$. Likewise, the repulsive oscillator line element is

$$dt^2=\frac{4|d\dot{z}|^2}{2|E|+|\dot{z}|^2}.$$

It is quite surprising that these velocity metrics were derived in a relativistic framework, from the relativistic addition law of velocities [cf. Fock, The Theory of Space, Time and Gravitation,  in particular Sec. 17], yet the can be derived from Keplerian orbits. Consequently, the Riemannian metric, 

$$dt^2=\frac{4|dv|^2}{(1-\alpha|v|^2)^2},$$ 

of constant curvature $\alpha$, whether it be positive (hyperbolic) or negative (elliptic), is not restricted solely to relativistic phenomena. This includes the metric in the penultimate expression, which is not quite of the Riemannian form, and which applies to uniform acceleration.