HPS 0410 Einstein for Everyone

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Appendix to Gravity Near a Massive Body
Geodesics of Space Near the Sun

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

Why does the disturbance to the geometry of three-dimensional space due to the presence of the sun lead the geodesics of space to curve away from the sun?

To see why, recall that a geodesic is a line of shortest spatial distance between two points in space. Take the Euclidean geometry of space as in Newtonian theory:

What makes this curve a geodesic is that it is the shortest of all curves connecting points A and B. The figure below shows nearby curves that connect A and B. The total length of each curve is indicated. The red curve is the shortest. Neighboring curves on either side are longer.

The presence of the sun disturbs the geometry of space in its vicinity. With that slight disturbance, the lengths of these curves connecting A and B will be slightly altered, so that another of these curves becomes the shortest:

To see why it is this curve that bows away from the sun, recall what happens as we approach the sun. Circles surrounding the sun do not lose circumferential length quite as quickly as they would if the geometry were exactly Euclidean. The curves connecting A and B lie close to these circles when in the vicinity of the sun and their lengths will be affected similarly. Hence, as we consider those curves closer to the sun, they do not lose length quite as quickly as they would in the Euclidean case.

This effect is indicated in the revision below to the above figure. As the curves come closer to the sun, we must add successively larger corrections to their lengths to conform with the new geometry. The curves are slightly longer than the corresponding curves in the Euclidean geometry; and this effect increases the closer we come to the sun. These additions are indicated as +2, +4, +6, etc.

The overall effect is that the shortest of the curves connecting A and B has moved one curve up in the figure and is indicated in red. That curve is bowed away from the sun, if we gauge it by the original Euclidean geometry. (It is, of course, "straight" according to the new geometry.)

Note for experts: The numbers inserted are adapted to the geometry of geodesics. The curves considered are very close to each other, which means that linear and quadratic approximations are quite accurate.

Consider what happens as we approach the sun in equal distance steps. The curves we intersect that connect A and B become shorter until their lengths reach a minimum at the geodesic. Then they increase again. That is, their lengths have a turning point at the geodesic, which means that their derivative with respect to the distance from the sun is zero. Thus, to good approximation, these lengths vary quadratically with distance from the sun as the geodesic is approached. Hence the lengths vary as ..., 9, 4, 1, 0, 1, 4, 9, ...

It is otherwise with the corrections to the lengths of these curves due to the new geometry. There is no turning point. They can be well approximated in this tiny neighborhood as a linear function of distance from the sun. Hence they grow linearly as 0, 2, 4, 6, ...

Copyright John D. Norton.May 30, 2021