Physics of Irving Ezra Segal
Mark Loewe, Libertarian, for Congress, United States Representative, District 35, Texas
This list of publications of Irving Ezra Segal does not include unpublished manuscripts, such as "The Nature of Gravity", I. E. Segal (1998).

In 1989, Segal estimated the radius of the universe to be R = 160 ± 40 Mpc.  In 2014, my attempt to use the same observed data to confirm Segal's estimate found R = 504.6 million lightyears = 154.7 million parsecs.  The difference of 160 Mpc from 154.7 Mpc is well within Segal's estimated uncertainty of (at most) 40 Mpc.

For an Einstein universe, three dimensional space is approximated as a 3-sphere of radius R and Segal proposes that an Einstein energy operator, which governs time evolution, is the sum of a Minkowski energy operator and a gravitational energy operator.  This note uses stereographic projection for the simpler case of a 1-sphere (which may be viewed as any great circle of a 3-sphere) to derive an analogous relation.

Segal's chronometric redshift is given by z = tangent2(θ/2) = tangent2[ct/(2R)], where R is the radius of the universe, c is the speed of light, t is the Einstein time for light to travel from a source to an observer on a great circle of radius R, ct is the great circle distance from the source to the observer, and θ is the great circle angular separation of the source from the observer.  Chronometric redshift does not require any assumption that the source moves with respect to the observer; if the source has some peculiar motion toward or away from the observer, then chronometric redshift is modified by some Doppler blueshift or Doppler redshift.
Light from galaxy GN-z11 is received with a redshift of z = 11.09.  If this redshift is purely chronometric (negligible peculiar motion), then the chronometric estimate of the fraction of our universe that is farther than GN-z11 from us is less than 4 percent.  If the radius of the universe is roughly R = 504.6 million lightyears, then the chronometric estimate of the great circle distance of GN-z11 from us is roughly 1.29 billion lightyears.

This note confirms statements in "The Nature of Gravity", I. E. Segal (1998) (unpublished), that "The mean and variance of m would then be expected to be of the same order of magnitude.  With the estimate of R in (54), me = 1037 in chronounits; this implies that the dispersion in m would be less than 1 part in 1017, consistently with the observed sharp value."

This note is an attempt to confirm a statement in "The Nature of Gravity", I. E. Segal (1998) (unpublished), that "The two theoretical predictions are thus equally consistent with the presently accepted empirical value."  One prediction is based on Schwarzschild's solution of general relativity and the other prediction seems to be based on Einstein's equivalence principle.  General relativity and Segal's chronometric theory seem to be different extensions of Einstein's equivalence principle.

More to come ...
Mark Loewe, Libertarian, for Congress, United States Representative, District 35, Texas