Evolution of Revolution

How fast would you have to fire a cannonball for it to never hit the ground?

Newton's very first ideas about gravitational orbits are said to have come about from a thought experiment. A cannonball was known to lose 5 metres of altitude a second after being fired horizontally, but the Earth - being round - curves away from the cannonball as it flies forward. So it occurred to Newton to ask: How fast would the cannonball have to be fired for the curvature to completely compensate for the vertical loss? If a cannonball was fired at this speed it would never lose any altitude, and end up orbiting the Earth.

The diagram above shows that the answer can be found using simply trigonometry and comes to
$$
\begin{align}
v &= \sqrt{gr} \\
&= \sqrt{9.81 ms^{-2} \times 6.371\times 10^6 m} \\
&= 7868\space ms^{-1} \\
&= 17603\space mph
\end{align}
$$

In general, (non-relativistic) orbits are elliptical

The next stage was to look at more general orbits.  Newton wasn't in fact the person who discovered that planetary orbits were ellipses rather than circles.  It was Kepler who made detailed observations and calculations and came up with the following two rules

1. The area swept out by a line between the Sun and a planet per unit of time is constant
2. The shape of the orbit is elliptical with the Sun at one of the foci.

Newton is usually credited with the discovery of the inverse r squared force rule, but he didn't do this either.  In fact it was already widely believed that the force pulling a body towards the Sun was proportional to one over the distance squared.  (It makes intuitive sense that the strength of the pull would be diluted in proportion to the area it passes through.)

What Newton did do - and this was an amazing achievement - was to show that the inverse square rule predicted the orbits described by Kepler.  Kepler's first rule was easy to show - it's just a restatement that angular momentum is conserved.  Kepler's second rule was much much harder, and Newton proved it in Principia Mathematica using a complicated geometric argument that is nicely summarized in the book Feynman's Lost Lecture.

A much simpler proof - illustrated above - was provided some time later by Laplace.  The key idea is to use a slightly unusual parameter to describe the trajectory, namely the heading $\phi$ relative to the radial direction.  If you know this as a function of the distance $r$ then the whole orbit is completely determined.

In the illustration, the LHS shows how starting with Newton's inverse square law we can determine how $\phi$ varies with $r$  $^\dagger$.  The RHS shows that if we start with the assumption that the orbit is elliptical, with the Sun at a focal point, then we end up with exactly the same equation!

The Schwarzschild metric

The final stage in the evolution of ideas about orbits came with relativity.  Einstein took Newton's first law and generalized it to arbitrary coordinate systems.  The derivative operator in $\frac{d^2x^m}{dt^2} = 0$ was replaced with the covariant derivative and the new rule became $\frac{D^2x^\mu}{d\tau^2} = 0$.  As a tensor equation this held in every coordinate system not just "inertial" ones.

Applying the definition of the covariant derivative we obtain
$$
\frac{d^2x^\mu}{d\tau^2} = - \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}
$$

Which boils down to an equation involving the metric $g$ because
$$
\Gamma^\mu_{\alpha\beta} = \frac{1}{2}g^{\mu\nu}(\partial_\alpha g_{\beta\nu} + \partial_\beta g_{\alpha\nu} - \partial_\nu g_{\alpha\beta})
$$

We can do a number of things from this point.  One is to choose some coordinate names $t, r, \theta, \phi$, then make the assumption that at large $r$ the metric becomes Minkowskian and low speed orbits become Newtonian, and then calculate what the metric must look like.  If you do that you end up with the Schwarzschild metric illustrated above$^{\dagger_2}$.  Another thing we can do is start off with the Schwarzschild metric and show that it results in Newtonian orbits in the limiting case.  Let's do that.

First note that we can make the low speed approximation
$$
\frac{d^2x^m}{d\tau^2} = - \Gamma^m_{tt}\frac{dt}{d\tau}\frac{dt}{d\tau}
$$
or better still
$$
\frac{d^2x^m}{dt^2} = - \Gamma^m_{tt}
$$
Next, note that we can ignore $m = \theta, \phi$ as we know the acceleration is radial, and we can ignore non-diagonal values in the metric, and we can use units in which $c=1$ to get
$$
\begin{align}
\frac{d^2r}{dt^2} &= - \Gamma^r_{tt} \\
&=- \frac{1}{2}g^{rr}(\partial_t g_{tr} + \partial_t g_{tr} - \partial_r g_{tt}) \\
&\approx \frac{1}{2} \partial_r g_{tt} \\
&= - \frac{GM}{r^2}
\end{align}
$$

which is Newton's law of gravitation!

The interesting results, however, come when you don't apply the non-relativistic assumptions.  We made two in the above calculations.  The first was that the object passing by the sun is moving slowly.  If we do not make this assumption then the trajectory predicted is very different to the one predicted by Newton.  For example, light (which is moving as fast as it is possible to move) is deflected by the sun through an angle twice as large as that predicted by the Newtonian model.  This prediction was confirmed by Eddington during the solar eclipse of 1919 that made Einstein a household name$^{\dagger_3}$.  The second assumption we made was that the orbit was a large distance from the Sun.  Again, if we ditch this assumption we end up with a different orbit, specifically we end up with an advancing perihelion.  Mercury's advancing perihelion had already been observed by astronomers but until the General Theory of Relativity there had been no explanation for it!


FOOTNOTES

• $\dagger$: It may not be entirely obvious that the first two statements are equivalent to the inverse square law.   The first says total energy is conserved, where PE is minus the integral of force.  The second says angular momentum is conserved, which is true because the force is radial.
• $\dagger_2$: Only two of the coordinates $r$, and $\phi$ are shown with $\theta$ taken to be $\frac{\pi}{2}$, and $t$ a "don't care".  The curvature in the picture illustrates that a unit of the $r$ coordinate nearer the Sun contributes slightly more to proper distance than a unit of the $r$ coordinate further away. 
• $\dagger_3$: Although in The Golem, Collins and Pinch argue convincingly that in fact the results from the expedition to Brazil and the island of Principe were inconclusive, but Eddington and his team were so convinced of the theory that they only reported results that supported Einstein and ignored those that supported Newton.  The moral of the story is that you should never trust tidy narratives in science history.