4.3  Free-Fall Equations

 

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody?

                                                                                                Galileo Galilei, 1638

 

The equation of two-body non-rotating radial free-fall in Newtonian theory is formally identical to the one-body radial free-fall solution in Einstein's theory (as is Kepler's third law), provided we identify Newton's radial distance with the Schwarzschild parameter r, and Newton's time with the proper time of the falling particle. Therefore, it's worthwhile to explicitly derive the cycloidal form of this solution. From the Newtonian point of view we can begin with the inverse-square law of gravitation for the radial separation s(t) between two identical non-rotating particles of mass m

 

 

where dots signify derivatives with respect to time. Integrating this over ds from an arbitrary initial separation s(0) to the separation s(t) at some other time t gives

 

 

Notice that the left hand integral can be rewritten

 

 

Therefore, the previous equation can easily be integrated to give

 

 

which shows that the quantity

 

is invariant for all t. Solving the equation for , we have

 

 

Rearranging, this gives

 

 

To simplify the expressions, we put s0 = s(0),  v0 =  and r = s(t)/s0. In these terms, the preceding expression can be written

 

 

There are two cases to consider. If K is positive, then the trajectory is bounded, and there is some point on the trajectory (the apogee) at which v = 0. Choosing this point as our time origin t = 0, we have K=1, and the standard integral gives

 

 

This equation describes a (scaled) cycloidal relation between t and r, which can be expressed parametrically in terms of a fictitious angle q as follows

 

 

To verify that these two equations are equivalent to the preceding equation, we can solve the second for q and substitute into the first to give

 

 

Using the trigonometric identity    we see that the first term on the right side is

 

 

Also, letting f = invcos(2r-1), we can use the trigonometric identity

 

 

to show that this angle is

 

 

so the second term on the right side of (2) is

 

 

which completes the demonstration that the cycloid relation given by (2) is equivalent to the free-fall relation (1).

 

The second case is when K is negative. For this case we can conveniently express the equations in terms of the positive parameter k = -K. The standard integral

 

 

tells us that, for any two points s0 and s1 on the trajectory, the time interval is related to the separations according to

 

 

where

 

Notice that if we define  S0 = s0 / k  and  R  =  kr,  then this becomes

 

 

Thus, if we define the normalized time parameter

 

 

then the normalized equation of motion is

 

 

This represents the shape of every non-rotating separation between identical particles of mass m for which k is positive, which means that the absolute value of  v0 exceeds 2.  These are the unbound radial orbits for which R goes to infinity, as opposed to the case when the absolute value of v0 is less than this threshold, which gives bound radial orbits in the shape of a cycloid in accord with equation (1).

 

It's interesting to note the "removable singularity" of (3) at R = 0. Physically the parameter R is always non-negative by definition, so it abruptly reverses slope at the origin, even though the position may vary monotonically with respect to an external coordinate system.

 

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