I just read Middleton's captivating article about trajectories on 2D surfaces1 with large amounts of awe, copious quantities of appreciation, and a wee bit of envy. Connecting the fascinating marble orbits in gravity wells to the actual trajectories possible in general relativity in a very tangible way, it legitimately fleshes out further the oft-made, but heretofore largely unspecified and certainly undelivered, promise of the rubber sheet model of general relativity to which many texts and practitioners allude. It is to be celebrated.

I have one concern to report, though. Professor Middleton's analysis assumes that the rotational kinetic energy of the rolling sphere of moment of inertia I, radius a, and mass m is negligible, and includes the assertion that the results so obtained are unaltered from the results one would obtain by assuming that the “scalar” model of rolling without slipping holds.2 However, I believe there is at least one significant difference between these two cases, which can be seen most readily by comparing terms in Middleton's Eq. (11) to Eq. (8) of English and Mareno,3 the latter of which has a factor of B = 1 + I/ma2 = 7/5 (for uniform solid spheres) in the denominator of the uniform gravitational field term compared to the former. Middleton's analysis does not include this factor, which, from the perspective of this reader, does not appear to be a big deal since its effect can be readily scaled away by pretending that the gravitational field is weaker by a factor of B than previously supposed. Thus, for example, the center-of-mass motion of a uniform solid sphere that rolls without slipping in the “scalar” approximation on such a surface in a uniform gravitational field of strength g has the same motion as that of a point particle moving without friction on the same surface in a uniform gravitational field given by field strength g/B = 5 g/7 (as one can readily check in the simpler case of one dimensional motion down an incline).

For a concrete example of how this adjustment plays out in orbital phenomena, consider the case of nearly circular orbits of objects moving on the surface of a cone with surface z = kr. For a puck sliding without friction to produce an orbit that mimics a planetary ellipse, one needs k=2, as noted in Ref. 1 just after Eq. (21); however, in order for a small sphere rolling with its axis of rotation parallel to the cone's surface to produce that same planet-like path, one needs a cone with k=14/5 (see endnote 13 in Ref. 2 for details). Simply setting the rotational kinetic energy to zero misses this subtlety. Thus, I am claiming that care must be used in extending the results of the paper from the case of orbits of frictionless pucks to the case of orbits of rollers in the scalar approximation, particularly Eq. (23). To accomplish this extension, one must first correctly specify the orientation of the spin axis of the roller, then one replaces g wherever it appears by g/B, where B = 7/5 for marbles, B = 5/3 for hollow rolling spheres, etc.

1.
Chad A.
Middleton
, “
The 2D surfaces that generate Newtonian and general relativistic orbits with small eccentricities
,”
Am. J. Phys
.
83
(
7
),
608
615
(
2015
).
2.
Gary D.
White
, “
On trajectories of rolling marbles in cones and other funnels
,”
Am. J. Phys.
81
(
12
),
890
898
(
2013
) provides more information on the scalar case and how it differs from the full-blown vector version of the no-slip condition.
3.
L. Q.
English
and
A.
Mareno
, “
Trajectories of rolling marbles on various funnels
,”
Am. J. Phys.
80
(
11
),
996
1000
(
2012
). One might also compare to Eq. (24) of Ref. 2, which agrees with the English and Mareno result in the case of the scalar approximation.