Geometrizing Gravity

Einstein's Idea:

• pick up on Minkowski's spacetime model, encoding the idea that the spacetime interval between events is fundamental and elevating it to an assumption about the very nature of the geometry of four dimensional spacetime

• incorporate the principle of equivalence by encoding the gravitational redshift/blueshift as some appropriate structures imposed on the set of spatio-temporal events
• remove the restriction of prior spacetime models which prevented the geometric structures from incluence by material content...i.e. removed assumption of absolute structure.

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]:

• All prior experiments testing the postulates of the special theory, which we now interpret as providing local information about spacetime structure.
• The organization of ideas of Minkowski as an integrative scheme for accounting for the results of the tests of the postulates of the special theory.
• Thus the local spacetime structure is given by the interval between events in a local region:

ds2 = - c2dt2 + dx2 + dy2 + dz2

Note that Minkowski spacetime possesses the causal structure imposed by the lightcone.

• The Pound-Rebka/Pound-Snider experiments imply that clocks run at different rates depending on their position in relation to sources of gravitation [heavy masses...]. This is taken to imply that the Minkowski spacetime interval needs revision for non-local experiments:
• Using g = G M/ R2e we could show that the form is

ds2 = - c2 ( 1 -2GM/(c2 r))dt2 + dx2 + dy2 + dz2

where the combination rs = 2GM/c2 is called the Schwarzschild radius and will play an important role later. Note that as we get far from the object whose mass is M we approach the Minkowksi spacetime model.

• Radar-Ranging Experiments: This is a set of experiments performed by I. Shapiro and collaborators beginning in the mid 1960's in which they sent a radar pulse from the earth to Mercury, having it graze the outer limb of the Sun. The figure below shows the idea: The Radar-ranging experiment measured the time it takes for the radar to go from the earth to Mercury and back.

• Using the time-only warped spacetime, one can calculate the time it would take for the radar to go from the earth and return...the result is in disagreement with the experimental findings of Shapiro.
• One concludes that the time-only warped spacetime can not be valid in general
• Thus, there is a need to change something more about the geometry....

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!