Geometrizing Gravity

Einstein's Idea:

A Path to the Geometrization of Gravity--'Suggested' by Experiments

...This is a 'new' organization of ideas...somewhat different from what is in the book...

We take the point of view that whatever theoretical foundation we put down for gravitation and other phenomena, we base it as closely as possible to experimental results.

We take the following experimental underpinnings as given [i.e. well tested and verified]:

The Radar-ranging experiment measured the time it takes for the radar to go from the earth to Mercury and back.

Among the various possibilities of things to change, we take the Einstein point of view that it is the geometric structure which is to encode the information related to gravitation. This point of view suggests that we consider warping space in addition to time.

This means that we consider the following form for the interval between events:

ds2 = - c2 ( 1 -2GM/(c2 r))dt2 + gxxdx2 + gyydy2 + gzzdz2

where the gxx, gyy, and gzz are functions to be determined so that the resulting spacetime structure is consistent both with the Pound-Rebka experiment and the Shapiro radar-ranging experiments.

The important point to make is: The 1915 geometric theory of gravitation of A. Einstein does this without any freely adjustable parameters!

Four Important Ideas in Geometrizing Gravity

Geodesics of Spacetime

In Euclidean space, the shortest distance between two points is a 'straight' line...

However, on the surface of a sphere the shortest distance between two points on the sphere is not the Euclidean distance between those must stay on the surface.

On the surface of the sphere, there are paths which connect pairs of points in the shortest way : the great circles do this:

The equator-circling curve is closed, of finite length, and the is an ideal curve in the sense that it is the shortest one which goes around the equator and returns to its original point.

The geodesic curves on a two-dimensional surface are defined to be those which have the shortest possible lenght as measured by observers who live and move in the surface without knowledge of any higher-dimensional space outside.

For the General Theory of Relativity, Einstein chose to identify these shortest possible curves in SOME four-dimentional spacetime with the paths along which test particles move.


It was not realized until almost 1918 that another important element which is needed is this concept of Hermann Weyl and Tuillio Levi-Civita...

The basic idea:

In order to compare quantities at different spatial points one needs to have a method whereby the objects which live at those points can be copied and brought to the same point for the comparison.

Consider the problem of comparing two vectors at different points A and B. In Euclidean space we do this as follows.